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2631 y Fu(\()p Fr(H)2111 2643 y Fq(\024)2178 2631 y Fp(\000)23 b Fr(E)2327 2643 y Fq(\024)2370 2631 y Fu(\))p Fr(U)2468 2601 y Fo(\003)2459 2651 y Fq(\024)2541 2631 y Fu(con)n(v)n(erges)33 b(in)i(norm)f(resolv)n(en)n(t)g(sense)g(to)0 2739 y(a)h(Hamiltonian)g @beginspecial @setspecial @endspecial 584 2718 a(^)562 2739 y Fr(H)631 2751 y Fo(1)737 2739 y Fu(when)g Fr(\024)h Fp(!)g(1)f Fu(while)h Fr(U)1564 2751 y Fq(\024)1642 2739 y Fu(con)n(v)n(erges)d(strongly)h(to)h(a)g(unitary)g(transformation)e @beginspecial @setspecial @endspecial Fr(U)3468 2751 y Fo(1)3574 2739 y Fu(\(in)j(other)0 2846 y(w)n(ords)e(no)g(c)n(hange)g (of)h(represen)n(tation)e(is)i(necessary\).)57 b(The)35 b(Hamiltonian)g @beginspecial @setspecial @endspecial Fr(H)42 b Fu(:=)35 b Fr(U)2810 2816 y Fo(\003)2801 2867 y(1)2893 2825 y Fu(^)2871 2846 y Fr(H)2940 2858 y Fo(1)3010 2846 y Fr(U)3067 2858 y Fo(1)3172 2846 y Fu(is)g(called)g(the)g Fn(Nelson)0 2946 y(Hamiltonian)p Fu(.)j(After)28 b(Nelson's)f(pap)r(er) 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y(the)i(fact)g(that)g(the)g(Nelson)f(Hamiltonian)g(is)h (only)f(de\014ned)h(as)f(the)h(resolv)n(en)n(t)e(limit)i(of)g(the)g (cuto\013)g(Hamiltonians.)100 3544 y(Let)34 b(us)g(no)n(w)g(describ)r (e)g(the)h(con)n(ten)n(t)f(of)g(the)h(pap)r(er.)57 b(In)34 b(Sect.)58 b @beginspecial @setspecial @endspecial(2)34 b(w)n(e)g(recall)f(classical)g(notations)h(related)f(to)h(F)-7 b(o)r(c)n(k)0 3643 y(spaces,)27 b(in)n(tro)r(duce)g(some)g (de\014nitions)h(and)f(pro)n(v)n(e)f(some)h(extensions)g(of)g (Glimm-Ja\013e's)h @beginspecial @setspecial @endspecial Fr(N)2942 3655 y Fq(\034)3011 3643 y Fu(estimates.)100 3743 y(In)j(Sect.)47 b @beginspecial @setspecial @endspecial(3)31 b(w)n(e)f(recall)g(the)i(construction)e(of)h(the)g(Nelson)g (Hamiltonian)g(follo)n(wing)e @beginspecial @setspecial @endspecial([21].)47 b(In)31 b(Sect.)47 b(4)31 b(w)n(e)f(pro)n(v)n(e)g (the)0 3843 y(so-called)25 b Fn(higher)31 b(or)l(der)e(estimates)p Fu(,)f(follo)n(wing)d(an)i(idea)f(of)h(Rosen)f @beginspecial @setspecial @endspecial([22)o(].)37 b(In)26 b(Sect.)37 b(5)26 b(w)n(e)h(study)g(the)g(sp)r(ectral)f(theory)g(for)0 3942 y(the)j(con\014ned)f(Nelson)g(Hamiltonian.)39 b(W)-7 b(e)29 b(pro)n(v)n(e)e(an)h(HVZ)g(theorem)g(and)h(a)f(p)r(ositiv)n(e)g (comm)n(utator)f(estimate.)39 b(Sect.)g @beginspecial @setspecial @endspecial(6)0 4042 y(is)32 b(dev)n(oted)g(to)g(the)g (scattering)f(theory)h(for)f(the)i(Nelson)f(mo)r(del.)50 b(W)-7 b(e)33 b(sho)n(w)e(the)h(existence)g(of)g(asymptotic)g(\014elds) g(and)0 4141 y(that)d(the)g(CCR)f(represen)n(tation)f(they)i(de\014ne)f (is)h(of)f(F)-7 b(o)r(c)n(k)28 b(t)n(yp)r(e,)h(using)f(an)h(argumen)n (t)e(from)h @beginspecial @setspecial @endspecial([12)o(].)40 b(In)29 b(Sect.)40 b(7)28 b(w)n(e)g(pro)n(v)n(e)0 4241 y(v)-5 b(arious)24 b(propagation)g(estimates)h(for)g(the)h(Nelson)f (Hamiltonian.)36 b(Finally)25 b(the)h(asymptotic)f(completeness)g(of)h (the)g(w)n(a)n(v)n(e)0 4341 y(op)r(erators)g(is)h(sho)n(wn)g(in)h (Sect.)37 b @beginspecial @setspecial @endspecial(8.)100 4540 y Fz(Ac)m(kno)m(wledgemen)m(t)0 4640 y Fu(The)27 b(author)g(w)n(ould)f(lik)n(e)h(to)g(think)h(Pr.)36 b(C.)27 b(G)n(\023)-39 b(erard)25 b(for)i(helpful)h(discussions)e(and)h(for)g (suggestions)e(during)i(the)g(writing)0 4739 y(of)h(this)f(pap)r(er.) 1318 4958 y @beginspecial @setspecial @endspecial(2.)46 b Fv(Present)-6 b(a)g(tion)32 b(of)g(the)f(model)100 5107 y Fu(In)24 b(this)g(section)g(w)n(e)g(de\014ne)g(the)h(Nelson)f (mo)r(del.)35 b(W)-7 b(e)25 b(start)e(with)i(a)f(review)f(of)h(the)h (basic)e(construction)h(and)g(the)g(main)0 5207 y(notations)32 b(related)g(to)h(b)r(osonic)f(F)-7 b(o)r(c)n(k)32 b(spaces.)52 b(F)-7 b(or)32 b(a)g(more)g(detailed)h(exp)r(osition)f(w)n(e)h(refer)f (the)h(reader)e(to)i @beginspecial @setspecial @endspecial([3)o (],[6],[7].)0 5307 y(In)e(Subsect.)47 b @beginspecial @setspecial @endspecial(2.2)30 b(w)n(e)g(giv)n(e)g(some)g(tec)n(hnical) h(estimates)f(obtained)h(b)n(y)f(adaptation)g(of)h @beginspecial @setspecial @endspecial Fr(N)2872 5319 y Fq(\034)2913 5307 y Fp(\000)p Fu(estimates)f([16)o(].)47 b(Finally)31 b(in)0 5406 y(Subsect.)55 b @beginspecial @setspecial @endspecial(2.3)33 b(w)n(e)g(in)n(tro)r(duce)h(the)g(formal)e (Hamiltonian)i(of)f(the)h(in)n(teracting)f(system)g(of)h @beginspecial @setspecial @endspecial Fr(P)46 b Fu(con\014ned)33 b(nonrelativistic)0 5506 y(particles)21 b(\(n)n(ucleons\))h(with)h(a)f (relativistic)f(scalar)g(\014eld)h(\(mesons\).)35 b(In)22 b(order)f(to)h(giv)n(e)g(sense)f(to)h(the)h(formal)e(Hamiltonian,)p eop %%Page: 3 3 3 2 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)958 b(3)0 489 y Fu(w)n(e)24 b(put)h(a)f(high-momen)n (tum)g(cuto\013)h(in)g(the)g(in)n(teraction)e(and)i(w)n(e)f(sho)n(w)f (that)i(the)g(cuto\013)g(Hamiltonian)f(is)g(a)g(w)n(ell)h(de\014ned)0 589 y(selfadjoin)n(t)i(op)r(erator.)0 657 y @beginspecial @setspecial @endspecial 84 x @beginspecial @setspecial @endspecial(2.1.)47 b Fz(Basic)30 b(de\014nitions)f(and)i(notations.)41 b Fu(Let)26 b Fm(h)h Fu(b)r(e)g(a)f(complex)g(Hilb)r(ert)h(space.)36 b(Let)27 b Fp(\012)3001 711 y Fq(n)3001 761 y(s)3045 741 y Fm(h)g Fu(denote)g(the)f(symmetric)0 840 y(n-fold)i(tensor)e(p)r (o)n(w)n(er)h(of)g @beginspecial @setspecial @endspecial Fm(h)p Fu(.)37 b(W)-7 b(e)28 b(in)n(tro)r(duce)g(the)g(b)r(osonic)f(F) -7 b(o)r(c)n(k)27 b(space)g(b)n(y)1645 988 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\000\()p Fm(h)p Fu(\))d(:=)1939 909 y Fl(M)1939 1088 y Fq(n)p Fo(\025)p Fs(0)2079 988 y Fp(\012)2144 954 y Fq(n)2144 1009 y(s)2188 988 y Fm(h)p Fr(:)p @beginspecial @setspecial @endspecial 0 1219 a @beginspecial @setspecial @endspecial Fp(\012)65 1189 y Fs(0)65 1239 y Fq(s)102 1219 y Fm(h)29 b Fu(:=)f Fk(C)15 b Fu(,)38 b(iden)n(ti\014ed)31 b(as)f(subspace)g(of)h @beginspecial @setspecial @endspecial(\000\()p Fm(h)p Fu(\),)i(represen)n(ts)c(the)i(space)f(of)h(zero-particle)e(states.)46 b(W)-7 b(e)31 b(denote)g(b)n(y)g @beginspecial @setspecial @endspecial(\012)g(the)0 1319 y(v)n(ector)23 b(\(1,0,...\))36 b(usually)24 b(called)g Fn(vacuum)i(ve)l(ctor)f Fu(and)f(b)n(y)g @beginspecial @setspecial @endspecial(\000)1922 1331 y Fs(\014n)2001 1319 y Fu(\()p Fm(h)p Fu(\))h(the)g(subspace)e(of)i (\014nite)g(particle)e(states,)i(whic)n(h)f(is)h(the)0 1418 y(subspace)j(of)g(\014nite)h(sum)f(of)h(v)n(ectors)d(in)j @beginspecial @setspecial @endspecial Fp(\012)1373 1388 y Fq(n)1373 1439 y(s)1418 1418 y Fm(h)p Fu(.)39 b(Among)28 b(the)h(main)f(op)r(erators)e(acting)i(on)g @beginspecial @setspecial @endspecial(\000\()p Fm(h)p Fu(\),)i(w)n(e)e(will)g (\014rst)g(recall)g(the)0 1518 y(de\014nitions)g(of)f(the)h(most)g (familiar)f(as)g @beginspecial @setspecial @endspecial Fr(number)d(oper)r(ator)31 b @beginspecial @setspecial @endspecial(N)36 b Fu(giv)n(en)27 b(in)h(its)g(sp)r(ectral)f(decomp)r (osition)1709 1654 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)1776 1682 y Fo(j\012)1848 1666 y Fj(n)1848 1699 y(s)1888 1682 y Fm(h)1958 1654 y Fu(:=)c Fr(n)p Fu(1)-15 b(l)o Fr(:)p @beginspecial @setspecial @endspecial 0 1823 a Fu(Creation)27 b(op)r(erators,)e(whic)n(h)j(are)e(un)n(b)r(ounded)i(op)r(erators)e (densely)h(de\014ned)h(on)g @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)2784 1770 y Fi(1)p 2784 1779 29 3 v 2784 1813 a(2)2826 1823 y Fu(\),)g(are)f(giv)n(en)g(b)n(y) 1265 1970 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)1309 1936 y Fo(\003)1347 1970 y Fu(\()p Fr(h)p Fu(\))1459 1999 y Fo(j\012)1531 1982 y Fj(n)1531 2015 y(s)1572 1999 y Fm(h)1642 1970 y Fu(:=)1753 1895 y Fl(p)p 1836 1895 258 4 v 75 x Fu(\()p Fr(n)18 b Fu(+)g(1\))p Fr(S)2144 1982 y Fq(n)p Fs(+1)2273 1970 y Fr(h)h Fp(\012)f Fu(1)-15 b(l)2472 1999 y Fo(\012)2524 1982 y Fj(n)2524 2015 y(s)2565 1999 y Fm(h)2612 1970 y Fr(;)p @beginspecial @setspecial @endspecial 0 2122 a Fu(where)37 b @beginspecial @setspecial @endspecial Fr(S)301 2134 y Fq(n)384 2122 y 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@beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(W)g Fu(\()p Fr(h)492 3202 y Fs(1)530 3190 y Fu(\))18 b Fp(\000)g Fr(W)12 b Fu(\()p Fr(h)833 3202 y Fs(2)871 3190 y Fu(\)\))p Fr(u)p Fp(jj)23 b(\024)g Fr(C)1199 3202 y Fq(\017)1245 3190 y Fp(jj)p Fr(h)1339 3202 y Fs(1)1394 3190 y Fp(\000)18 b Fr(h)1525 3202 y Fs(2)1562 3190 y Fp(jj)1608 3155 y Fq(\017)1640 3190 y Fu(\(\()p Fp(jj)p Fr(h)1798 3202 y Fs(1)1836 3190 y Fp(jj)1882 3155 y Fs(2)1938 3190 y Fu(+)g Fp(jj)p Fr(h)2115 3202 y Fs(2)2152 3190 y Fp(jj)2198 3155 y Fs(2)2235 3190 y Fu(\))2278 3133 y Fj(\017)p 2278 3142 29 3 v 2278 3175 a Fi(2)2320 3190 y Fp(jj)p Fr(u)p Fp(jj)g Fu(+)g Fp(jj)p Fu(\()p Fr(N)28 b Fu(+)18 b(1\))2902 3133 y Fj(\017)p 2901 3142 V 2901 3175 a Fi(2)2943 3190 y Fr(u)p Fp(jj)p Fu(\))p Fr(;)37 b Fu(0)23 b Fp(\024)f Fr(\017)h Fp(\024)g Fu(1)p Fr(;)82 b @beginspecial @setspecial @endspecial -3573 w Fu(\(2.1\))762 3375 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 8 w(lim)762 3426 y Fq(s)p Fo(!)p Fs(0)944 3375 y Fu(sup)907 3448 y Fo(jj)p Fq(h)p Fo(jj\024)p Fq(c)1120 3375 y Fr(s)1159 3340 y Fo(\000)p Fs(1)1248 3375 y Fp(jj)p Fu(\()p Fr(W)12 b Fu(\()p Fr(sh)p Fu(\))19 b Fp(\000)f Fu(1)-15 b(l)18 b Fp(\000)g Fu(i)p Fr(\036)p Fu(\()p Fr(h)p Fu(\)\)\()p Fr(N)29 b Fu(+)18 b(1\))2321 3340 y Fo(\000)2382 3318 y Fi(1)p 2382 3327 V 2382 3360 a(2)2421 3340 y Fo(\000)p Fq(\017)2504 3375 y Fp(jj)23 b Fu(=)g(0)p Fr(;)37 b Fu(0)22 b Fr(<)h(\017:)83 b @beginspecial @setspecial @endspecial -3055 w Fu(\(2.2\))100 3578 y(Let)23 b @beginspecial @setspecial @endspecial Fm(h)g Fu(b)r(e)g(a)f(Hilb)r(ert)i(space.)34 b(Let)23 b @beginspecial @setspecial @endspecial Fm(f)g Fu(:)g Fm(h)h Fp(!)f Fm(h)g Fu(b)r(e)g(an)f(\(un)n(b)r(ounded\))i(op)r (erator.)34 b(W)-7 b(e)23 b(denote)f(b)n(y)h @beginspecial @setspecial @endspecial(d\000\()p Fm(f)p Fu(\))g(the)h(ampli\014cation) 0 3677 y(of)k @beginspecial @setspecial @endspecial Fm(f)f Fu(to)h(the)g(whole)f(space)g @beginspecial @setspecial @endspecial(\000\()p Fm(h)p Fu(\))1231 3880 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(d\000\()p Fm(f)p Fu(\))1420 3909 y Fo(j\012)1492 3892 y Fj(n)1492 3925 y(s)1533 3909 y Fm(h)1603 3880 y Fu(:=)1754 3776 y Fq(n)1714 3801 y Fl(X)1717 3978 y Fq(j)s Fs(=1)1862 3880 y Fu(1)-15 b(l)1911 3843 y Fo(\012)p Fs(\()p Fq(j)s Fo(\000)p Fs(1\))2153 3880 y Fp(\012)18 b Fm(f)h Fp(\012)f Fu(1)-15 b(l)2414 3843 y Fo(\012)p Fs(\()p Fq(n)p Fo(\000)p Fq(j)s Fs(\))2646 3880 y Fr(:)p @beginspecial @setspecial @endspecial 0 4104 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fm(h)192 4116 y Fq(i)220 4104 y Fr(;)g(i)23 b Fu(=)h(1)p Fr(;)14 b Fu(2)27 b(b)r(e)h(t)n(w)n(o)g(Hilb)r(ert)g (spaces.)38 b(Let)28 b @beginspecial @setspecial @endspecial Fm(g)c Fu(:)g Fm(h)1711 4116 y Fs(1)1772 4104 y Fp(!)g Fm(h)1922 4116 y Fs(2)1988 4104 y Fu(b)r(e)k(a)g(b)r(ounded)g(op)r (erator.)37 b(W)-7 b(e)29 b(de\014ne)f(the)g(op)r(erator)f @beginspecial @setspecial @endspecial(\000\()p Fm(g)p Fu(\))0 4204 y(b)n(y)1680 4340 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\000\()p Fm(g)p Fu(\))c(:)g Fm(h)1950 4352 y Fs(1)2011 4340 y Fp(!)g Fm(h)2160 4352 y Fs(2)2197 4340 y Fr(;)p @beginspecial @setspecial @endspecial 1601 4519 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\000\()p Fm(g)p Fu(\))1759 4548 y Fo(j\012)1831 4531 y Fj(n)1831 4565 y(s)1872 4548 y Fm(h)1915 4556 y Fi(1)1974 4519 y Fu(:=)g Fm(g)2127 4485 y Fo(\012)p Fs(\()p Fq(n)p Fs(\))2276 4519 y Fr(:)p @beginspecial @setspecial @endspecial 0 4671 a Fu(A)28 b(less)f(familiar)g(op)r(erator)f(is)h @beginspecial @setspecial @endspecial(d\000\()p Fm(f)p Fr(;)14 b Fm(g)p Fu(\),)28 b(where)g @beginspecial @setspecial @endspecial Fm(f)p Fr(;)14 b Fm(g)27 b Fu(are)g(t)n(w)n(o)f(op)r (erators)g(on)h @beginspecial @setspecial @endspecial Fm(h)2482 4683 y Fs(1)2548 4671 y Fu(in)n(to)g Fm(h)2759 4683 y Fs(2)2796 4671 y Fu(.)37 b(It)28 b(is)g(de\014ned)g(as)f(in)h @beginspecial @setspecial @endspecial([11)o(]:)1508 4807 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(d\000\()p Fm(f)p Fr(;)14 b Fm(g)p Fu(\))24 b(:)f(\000\()p Fm(h)1973 4819 y Fs(1)2010 4807 y Fu(\))h Fp(!)f Fu(\000\()p Fm(h)2299 4819 y Fs(2)2337 4807 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1183 5046 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(d\000\()p Fm(f)p Fr(;)14 b Fm(g)p Fu(\))1451 5075 y Fo(j\012)1523 5058 y Fj(n)1523 5091 y(s)1564 5075 y Fm(h)1607 5083 y Fi(1)1667 5046 y Fu(:=)1817 4942 y Fq(n)1778 4967 y Fl(X)1780 5144 y Fq(j)s Fs(=1)1925 5046 y Fm(f)1952 5012 y Fo(\012)p Fs(\()p Fq(j)s Fo(\000)p Fs(1\))2195 5046 y Fp(\012)k Fm(g)g Fp(\012)g Fm(f)2448 5012 y Fo(\012)p Fs(\()p Fq(n)p Fo(\000)p Fq(j)s Fs(\))2694 5046 y Fr(:)p @beginspecial @setspecial @endspecial 0 5270 a Fu(W)-7 b(e)22 b(notice)g(that)g @beginspecial @setspecial @endspecial(d\000\()p Fm(f)p Fr(;)14 b Fm(f)p Fu(\))24 b(=)e Fr(N)9 b Fu(\000\()p Fm(f)p Fu(\))23 b(and)e(if)i @beginspecial @setspecial @endspecial Fm(h)1425 5282 y Fs(1)1485 5270 y Fu(=)g Fm(h)1616 5282 y Fs(2)1653 5270 y Fu(,)h(w)n(e)d(ha)n(v)n(e)g @beginspecial @setspecial @endspecial(d\000\(1)-15 b(l)o Fr(;)14 b Fm(g)p Fu(\))23 b(=)g(d\000\()p Fm(g)p Fu(\).)36 b(If)22 b @beginspecial @setspecial @endspecial Fp(jj)p Fm(f)p Fp(jj)h(\024)g Fu(1,)g(the)f(follo)n(wing)f(inequalit)n(y)0 5370 y(holds)1261 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1383 5472 y Fo(\000)1445 5449 y Fi(1)p 1445 5458 V 1445 5492 a(2)1501 5506 y Fu(d\000\()p Fm(f)p Fr(;)14 b Fm(g)p Fu(\))p Fr(u)p Fp(jj)23 b(\024)g(jj)p Fu(d\000\()p Fm(g)2192 5472 y Fo(\003)2230 5506 y Fm(g)p Fu(\))2314 5449 y Fi(1)p 2314 5458 V 2314 5492 a(2)2356 5506 y Fr(u)p Fp(jj)p Fr(:)83 b @beginspecial @setspecial @endspecial -2556 w Fu(\(2.3\))p eop %%Page: 4 4 4 3 bop 0 226 a Fs(4)1728 b(Z.)29 b(AMMARI)0 489 y Fu(Let)40 b @beginspecial @setspecial @endspecial Fr(i)190 501 y Fs(1)241 489 y Fu(\()p Fr(r)r(esp:)14 b(i)498 501 y Fs(2)535 489 y Fu(\))40 b(b)r(e)g(the)g(injection)g(of)g Fm(h)1388 501 y Fs(1)1439 489 y Fu(\()p Fr(r)r(esp:)14 b Fm(h)1710 501 y Fs(2)1748 489 y Fu(\))40 b(in)n(to)g @beginspecial @setspecial @endspecial Fm(h)2044 501 y Fs(1)2108 489 y Fp(\010)26 b Fm(h)2242 501 y Fs(2)2280 489 y Fu(.)73 b(There)39 b(exists)h(a)f(unitary)g(transformation)f @beginspecial @setspecial @endspecial Fr(U)0 589 y Fu(iden)n(tifying)28 b(\000\()p Fm(h)540 601 y Fs(1)578 589 y Fu(\))18 b Fp(\012)g Fu(\000\()p Fm(h)838 601 y Fs(2)876 589 y Fu(\))28 b(with)g @beginspecial @setspecial @endspecial(\000\()p Fm(h)1252 601 y Fs(1)1308 589 y Fp(\010)18 b Fm(h)1434 601 y Fs(2)1472 589 y Fu(\),)28 b(de\014ned)g(as)f(follo)n(ws)783 815 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(U)9 b(u)18 b Fp(\012)g Fr(v)26 b Fu(:=)1175 664 y Fl(s)p 1258 664 291 4 v 1268 759 a Fu(\()p Fr(p)18 b Fu(+)g Fr(q)s Fu(\)!)p 1268 796 271 4 v 1339 872 a Fr(p)p Fu(!)p Fr(q)s Fu(!)1549 815 y Fr(S)1600 827 y Fq(p)p Fs(+)p Fq(q)1721 815 y Fu(\000\()p Fr(i)1834 827 y Fs(1)1871 815 y Fu(\))p Fr(u)h Fp(\012)f Fu(\000\()p Fr(i)2166 827 y Fs(2)2203 815 y Fu(\))p Fr(v)s(;)83 b(u)23 b Fp(2)g(\012)2598 781 y Fq(p)2598 836 y(s)2636 815 y Fm(h)2679 827 y Fs(1)2717 815 y Fr(;)k(v)g Fp(2)c(\012)2977 781 y Fq(q)2977 836 y(s)3013 815 y Fm(h)3056 827 y Fs(2)3094 815 y Fr(:)p @beginspecial @setspecial @endspecial 0 1014 a Fu(This)28 b(transformation)e(has)h(the)h(follo)n(wing)e(prop)r (erties:)0 1114 y @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))1710 1253 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(U)9 b Fu(\012)18 b Fp(\012)g Fu(\012)23 b(=)f(\012)p Fr(:)p @beginspecial @setspecial @endspecial 0 1392 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(Let)f Fr(h)310 1404 y Fs(1)370 1392 y Fp(2)d Fm(h)492 1404 y(1)538 1392 y Fr(;)14 b(h)623 1404 y Fs(2)683 1392 y Fp(2)23 b Fm(h)804 1404 y(2)1153 1535 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)1197 1501 y Fq(])1228 1535 y Fu(\()p Fr(h)1308 1547 y Fs(1)1364 1535 y Fp(\010)18 b Fr(h)1495 1547 y Fs(2)1532 1535 y Fu(\))p Fr(U)32 b Fu(=)23 b Fr(U)9 b Fu(\()p Fr(a)1883 1501 y Fq(])1914 1535 y Fu(\()p Fr(h)1994 1547 y Fs(1)2031 1535 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fr(a)2511 1501 y Fq(])2542 1535 y Fu(\()p Fr(h)2622 1547 y Fs(2)2659 1535 y Fu(\)\))p Fr(;)p @beginspecial @setspecial @endspecial 1192 1713 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\036)p Fu(\()p Fr(h)1321 1725 y Fs(1)1377 1713 y Fp(\010)g Fr(h)1508 1725 y Fs(2)1545 1713 y Fu(\))p Fr(U)32 b Fu(=)23 b Fr(U)9 b Fu(\()p Fr(\036)p Fu(\()p Fr(h)1981 1725 y Fs(1)2019 1713 y Fu(\))18 b Fp(\012)g Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fr(\036)p Fu(\()p Fr(h)2583 1725 y Fs(2)2621 1713 y Fu(\)\))p Fr(:)p @beginspecial @setspecial @endspecial 0 1852 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))28 b(Let)g Fm(f)316 1864 y Fq(i)367 1852 y Fu(:)23 b Fm(h)456 1864 y Fq(i)507 1852 y Fp(!)g Fm(h)656 1864 y Fq(i)707 1852 y Fr(;)14 b(i)23 b Fu(=)f(1)p Fr(;)14 b Fu(2)27 b(b)r(e)h(t)n(w)n(o)f(op)r(erators)1160 1991 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(d\000\()p Fm(f)1317 2003 y Fs(1)1373 1991 y Fp(\010)18 b Fm(f)1483 2003 y Fs(2)1521 1991 y Fu(\))p Fr(U)32 b Fu(=)22 b Fr(U)9 b Fu(\(d\000\()p Fm(f)1984 2003 y Fs(1)2022 1991 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)17 b Fp(\012)h Fu(d\000\()p Fm(f)2614 2003 y Fs(2)2652 1991 y Fu(\)\))p Fr(;)p @beginspecial @setspecial @endspecial 1439 2169 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(U)k Fu(\000\()p Fm(f)1629 2181 y Fs(1)1667 2169 y Fu(\))c Fp(\012)g Fu(\000\()p Fm(f)1911 2181 y Fs(2)1949 2169 y Fu(\))23 b(=)g(\000\()p Fm(f)2203 2181 y Fs(1)2259 2169 y Fp(\010)18 b Fm(f)2369 2181 y Fs(2)2406 2169 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 2308 a Fu(W)-7 b(e)28 b(de\014ne)g(the)g Fn(sc)l(attering)i(identi\014c)l(ation)g(op)l(er)l(ator)f @beginspecial @setspecial @endspecial Fr(I)1410 2447 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(I)h Fu(:)23 b(\000)1574 2459 y Fs(\014n)1652 2447 y Fu(\()p Fm(h)p Fu(\))d Fp(\012)e Fu(\000)1914 2459 y Fs(\014n)1992 2447 y Fu(\()p Fm(h)p Fu(\))24 b Fp(!)f Fu(\000)2281 2459 y Fs(\014n)2359 2447 y Fu(\()p Fm(h)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1007 2712 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(I)7 b(u)18 b Fp(\012)g Fr(v)26 b Fu(:=)1376 2561 y Fl(s)p 1459 2561 291 4 v 1469 2656 a Fu(\()p Fr(p)18 b Fu(+)h Fr(q)s Fu(\)!)p 1469 2693 271 4 v 1540 2769 a Fr(p)p Fu(!)p Fr(q)s Fu(!)1773 2712 y Fr(S)1824 2724 y Fq(p)p Fs(+)p Fq(q)1960 2712 y Fr(u)f Fp(\012)g Fr(v)s(;)60 b(u)22 b Fp(2)i(\012)2449 2678 y Fq(p)2449 2733 y(s)2487 2712 y Fm(h)p Fr(;)k(v)e Fp(2)d(\012)2790 2678 y Fq(q)2790 2733 y(s)2827 2712 y Fm(h)p Fr(:)p @beginspecial @setspecial @endspecial 0 2912 a Fu(W)-7 b(e)28 b(can)f(also)g(de\014ne)h @beginspecial @setspecial @endspecial Fr(I)34 b Fu(b)n(y)28 b(the)g(follo)n(wing)e (form)n(ula:)802 3116 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(I)895 3009 y Fq(p)859 3038 y Fl(Y)858 3214 y Fq(i)p Fs(=1)980 3116 y Fr(a)1024 3082 y Fo(\003)1062 3116 y Fu(\()p Fr(h)1142 3128 y Fq(i)1170 3116 y Fu(\)\012)18 b Fp(\012)1401 3009 y Fq(q)1364 3038 y Fl(Y)1363 3214 y Fq(i)p Fs(=1)1485 3116 y Fr(a)1529 3082 y Fo(\003)1567 3116 y Fu(\()p Fr(g)1639 3128 y Fq(i)1666 3116 y Fu(\)\012)24 b(:=)1930 3009 y Fq(q)1893 3038 y Fl(Y)1892 3214 y Fq(i)p Fs(=1)2014 3116 y Fr(a)2058 3082 y Fo(\003)2096 3116 y Fu(\()p Fr(g)2168 3128 y Fq(i)2195 3116 y Fu(\))2278 3009 y Fq(p)2242 3038 y Fl(Y)2241 3214 y Fq(i)p Fs(=1)2363 3116 y Fr(a)2407 3082 y Fo(\003)2445 3116 y Fu(\()p Fr(h)2525 3128 y Fq(i)2553 3116 y Fu(\)\012)p Fr(;)83 b(h)2799 3128 y Fq(i)2826 3116 y Fr(;)14 b(g)2903 3128 y Fq(i)2954 3116 y Fp(2)23 b Fm(h)p Fr(:)p @beginspecial @setspecial @endspecial 0 3328 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(\031)j Fu(b)r(e)d(the)g(follo)n(wing)e(map)1528 3529 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1569 3461 a @beginspecial @setspecial @endspecial Fr(\031)h Fu(:)c Fm(h)18 b Fp(\010)g Fm(h)24 b Fp(!)f Fm(h)p Fr(;)p @beginspecial @setspecial @endspecial 1569 3596 a Fu(\()p Fr(h)1649 3608 y Fs(0)1687 3596 y Fr(;)14 b(h)1772 3608 y Fo(1)1842 3596 y Fu(\))23 b Fp(!)g Fr(h)2051 3608 y Fs(0)2107 3596 y Fu(+)18 b Fr(h)2238 3608 y Fo(1)2308 3596 y Fr(:)2372 3529 y @beginspecial @setspecial @endspecial 0 3731 a Fu(Then)28 b(w)n(e)f(can)g(express)g @beginspecial @setspecial @endspecial Fr(I)34 b Fu(as)27 b(follo)n(wing)1750 3870 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(I)j Fu(=)23 b(\000\()p Fr(\031)s Fu(\))p Fr(U:)p @beginspecial @setspecial @endspecial 0 4013 a Fu(W)-7 b(e)28 b(notice)f(that)h @beginspecial @setspecial @endspecial Fr(I)35 b Fu(is)27 b(un)n(b)r(ounded)h(since)g Fp(jj)p Fr(\031)s Fp(jj)23 b Fu(=)1607 3945 y Fp(p)p 1676 3945 42 4 v 68 x Fu(2.)0 4113 y(Let)28 b @beginspecial @setspecial @endspecial Fr(i)22 b Fu(=)h(\()p Fr(i)349 4125 y Fs(0)386 4113 y Fr(;)14 b(i)452 4125 y Fo(1)522 4113 y Fu(\))28 b(b)r(e)g(a)f(pair)g (of)h(maps)f(from)g Fm(h)h Fu(to)g @beginspecial @setspecial @endspecial Fm(h)p Fu(.)37 b(W)-7 b(e)28 b(de\014ne)1363 4252 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(I)7 b Fu(\()p Fr(i)p Fu(\))24 b(:)f(\000)1621 4264 y Fs(\014n)1699 4252 y Fu(\()p Fm(h)p Fu(\))c Fp(\012)f Fu(\000)1960 4264 y Fs(\014n)2039 4252 y Fu(\()p Fm(h)p Fu(\))23 b Fp(!)h Fu(\000)2328 4264 y Fs(\014n)2406 4252 y Fu(\()p Fm(h)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1525 4430 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(I)7 b Fu(\()p Fr(i)p Fu(\))24 b(:=)e Fr(I)f Fu(\000\()p Fr(i)1965 4442 y Fs(0)2002 4430 y Fu(\))e Fp(\012)f Fu(\000\()p Fr(i)2249 4442 y Fo(1)2319 4430 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4569 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(i)22 b Fu(=)h(\()p Fr(i)349 4581 y Fs(0)386 4569 y Fr(;)14 b(i)452 4581 y Fo(1)522 4569 y Fu(\))p Fr(;)g(j)28 b Fu(=)23 b(\()p Fr(j)807 4581 y Fs(0)845 4569 y Fr(;)14 b(j)916 4581 y Fo(1)986 4569 y Fu(\))28 b(b)r(e)g(t)n(w)n(o)f(pairs)f(of)i(maps)f(from)g @beginspecial @setspecial @endspecial Fm(h)h Fu(to)g Fm(h)p Fu(.)37 b(W)-7 b(e)28 b(de\014ne)1302 4708 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(d)p Fr(I)7 b Fu(\()p Fr(i;)14 b(j)5 b Fu(\))23 b(:)h(\000)1682 4720 y Fs(\014n)1760 4708 y Fu(\()p Fm(h)p Fu(\))19 b Fp(\012)f Fu(\000)2021 4720 y Fs(\014n)2100 4708 y Fu(\()p Fm(h)p Fu(\))23 b Fp(!)h Fu(\000)2389 4720 y Fs(\014n)2467 4708 y Fu(\()p Fm(h)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 977 4886 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(d)p Fr(I)7 b Fu(\()p Fr(i;)14 b(j)5 b Fu(\))24 b(:=)e Fr(I)f Fu(\(d\000\()p Fr(i)1617 4898 y Fs(0)1655 4886 y Fr(;)14 b(j)1726 4898 y Fs(0)1763 4886 y Fu(\))19 b Fp(\012)f Fu(\000\()p Fr(i)2010 4898 y Fo(1)2080 4886 y Fu(\))h(+)f(\000\()p Fr(i)2327 4898 y Fs(0)2364 4886 y Fu(\))g Fp(\012)g Fu(d\000\()p Fr(i)2656 4898 y Fs(0)2694 4886 y Fr(;)c(j)2765 4898 y Fo(1)2835 4886 y Fu(\)\))p Fr(:)p @beginspecial @setspecial @endspecial 0 5025 a Fu(If)28 b @beginspecial @setspecial @endspecial Fr(i)112 5037 y Fs(0)149 5025 y Fr(i)178 4995 y Fo(\003)178 5045 y Fs(0)234 5025 y Fu(+)18 b Fr(i)346 5037 y Fo(1)416 5025 y Fr(i)445 4995 y Fo(\003)445 5045 y(1)538 5025 y Fp(\024)23 b Fu(1)k(w)n(e)g(ha)n(v)n(e)g(the)g (estimates)920 5176 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(N)1065 5188 y Fs(0)1120 5176 y Fu(+)18 b Fr(N)1270 5188 y Fo(1)1340 5176 y Fu(\))1372 5142 y Fo(\000)1435 5120 y Fi(1)p 1435 5129 29 3 v 1435 5162 a(2)1477 5176 y Fu(d)p Fr(I)1566 5142 y Fo(\003)1604 5176 y Fu(\()p Fr(i;)c(j)5 b Fu(\))p Fr(u)p Fp(jj)23 b(\024)g(jj)p Fu(d\000\()p Fr(j)2188 5188 y Fs(0)2225 5176 y Fr(j)2264 5142 y Fo(\003)2259 5197 y Fs(0)2321 5176 y Fu(+)18 b Fr(j)2438 5188 y Fo(1)2509 5176 y Fr(j)2548 5142 y Fo(\003)2543 5197 y(1)2613 5176 y Fu(\))2655 5120 y Fi(1)p 2655 5129 V 2655 5162 a(2)2698 5176 y Fr(u)p Fp(jj)p Fr(;)82 b @beginspecial @setspecial @endspecial -2897 w Fu(\(2.4\))217 5367 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(j)p Fu(\()p Fr(u)320 5379 y Fs(2)357 5367 y Fp(j)p Fu(d)p Fr(I)469 5333 y Fo(\003)507 5367 y Fu(\()p Fr(i;)14 b(j)5 b Fu(\))p Fr(u)724 5379 y Fs(1)761 5367 y Fu(\))p Fp(j)23 b(\024)g(jj)p Fu(d\000\()p Fp(j)p Fr(j)1160 5379 y Fs(0)1198 5367 y Fp(j)p Fu(\))1263 5310 y Fi(1)p 1263 5319 V 1263 5353 a(2)1324 5367 y Fp(\012)18 b Fu(1)-15 b(l)13 b Fr(u)1518 5379 y Fs(2)1555 5367 y Fp(jj)23 b(jj)p Fu(d\000\()p Fp(j)p Fr(j)1857 5379 y Fs(0)1895 5367 y Fp(j)p Fu(\))1960 5310 y Fi(1)p 1960 5319 V 1960 5353 a(2)2003 5367 y Fr(u)2051 5379 y Fs(1)2088 5367 y Fp(jj)18 b Fu(+)g Fp(jj)p Fu(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fp(j)p Fr(j)2619 5379 y Fo(1)2690 5367 y Fp(j)p Fu(\))2755 5310 y Fi(1)p 2755 5319 V 2755 5353 a(2)2798 5367 y Fr(u)2846 5379 y Fs(2)2882 5367 y Fp(jj)24 b(jj)p Fu(d\000\()p Fp(j)p Fr(j)3185 5379 y Fo(1)3256 5367 y Fp(j)p Fu(\))3321 5310 y Fi(1)p 3321 5319 V 3321 5353 a(2)3363 5367 y Fr(u)3411 5379 y Fs(1)3448 5367 y Fp(jj)p Fr(:)83 b @beginspecial @setspecial @endspecial -3600 w Fu(\(2.5\))0 5506 y(F)-7 b(or)27 b(other)g(prop)r(erties)g(and)g(equiv)-5 b(alen)n(t)27 b(de\014nitions)h(of)g(these)f(op)r(erators,)f(w)n(e)h(refer)g(the)h (reader)e(to)i @beginspecial @setspecial @endspecial([12)o(].)p eop %%Page: 5 5 5 4 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)958 b(5)100 489 y Fu(Let)27 b @beginspecial @setspecial @endspecial Fp(K)j Fu(b)r(e)e(an)f(auxiliary)f(Hilb)r(ert)i (space.)36 b(Let)28 b Fr(v)e Fp(2)e(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)19 b(\012)f Fm(h)p Fu(\).)38 b(W)-7 b(e)28 b(de\014ne)g(an)f(extended)h(creation)e(op)r(erator:)1340 632 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)1384 598 y Fo(\003)1422 632 y Fu(\()p Fr(v)s Fu(\))e(:)f Fp(K)d(\012)e Fu(\000)1817 644 y Fs(\014n)1896 632 y Fu(\()p Fm(h)p Fu(\))23 b Fp(!)h(K)19 b(\012)f Fu(\000)2350 644 y Fs(\014n)2429 632 y Fu(\()p Fm(h)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1085 830 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(a)1129 796 y Fo(\003)1167 830 y Fu(\()p Fr(v)s Fu(\))1274 859 y Fo(jK\012\012)1449 842 y Fj(n)1449 875 y(s)1490 859 y Fm(h)1561 830 y Fu(:=)1671 755 y Fl(p)p 1754 755 258 4 v 75 x Fu(\()p Fr(n)h Fu(+)f(1\))23 b(1)-15 b(l)2084 842 y Fo(K)2157 830 y Fp(\012)18 b Fr(S)2291 842 y Fq(n)p Fs(+1)2458 830 y Fr(v)j Fp(\012)d Fu(1)-15 b(l)2652 859 y Fo(\012)2704 842 y Fj(n)2704 875 y(s)2744 859 y Fm(h)2792 830 y Fr(:)p @beginspecial @setspecial @endspecial 0 989 a @beginspecial @setspecial @endspecial(a)44 959 y Fo(\003)82 989 y Fu(\()p Fr(v)s Fu(\))41 b(is)f(closable)f (densely)g(de\014ned)i(op)r(erator)d(since)i(its)g(adjoin)n(t)f @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(v)s Fu(\))i(is)f(densely)g(de\014ned.)74 b(W)-7 b(e)40 b(de\014ne)g(the)h(\014eld)0 1089 y(op)r(erator)25 b @beginspecial @setspecial @endspecial Fr(\036)p Fu(\()p Fr(v)s Fu(\))j(as)e(in)h(the)g(scalar)f(case.)35 b(When)28 b Fm(h)23 b Fu(=)g Fr(L)1753 1059 y Fs(2)1789 1089 y Fu(\()p Fk(R)1876 1059 y Fq(d)1920 1089 y Fr(;)14 b(dk)s Fu(\),)28 b @beginspecial @setspecial @endspecial Fr(v)e Fp(2)d(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)k(\012)f Fm(h)p Fu(\))27 b(can)f(b)r(e)h(represen)n(ted)f(as)g(a)h(function)0 1189 y @beginspecial @setspecial @endspecial Fr(k)f Fp(!)d Fr(v)s Fu(\()p Fr(k)s Fu(\))h Fp(2)37 b(B)s Fu(\()p Fp(K)q Fu(\),)28 b(suc)n(h)f(that)h(for)f @beginspecial @setspecial @endspecial Fr(x)c Fp(2)h(K)q Fr(;)k(v)s Fu(\()p Fr(k)s Fu(\))p Fr(x)c Fu(:=)f Fr(v)s(x)p Fu(\()p Fr(k)s Fu(\))p Fr(;)29 b(k)d(a:e:)h Fu(and)959 1374 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(K)20 b(\002)e(K)25 b(3)e Fu(\()p Fr(x;)14 b(y)s Fu(\))24 b Fp(!)1613 1261 y Fl(Z)1696 1374 y Fu(\()p Fr(v)s Fu(\()p Fr(k)s Fu(\))1881 1340 y Fo(\003)1934 1374 y Fr(v)s Fu(\()p Fr(k)s Fu(\))p Fr(x)p Fp(j)14 b Fr(y)s Fu(\))2247 1386 y Fo(K)2326 1374 y Fr(dk)26 b Fu(=)c(\()p Fr(v)s(x)p Fp(j)p Fr(v)s(y)s Fu(\))2789 1403 y Fo(K\012)p Fm(h)2941 1374 y @beginspecial @setspecial @endspecial 0 1568 a Fu(is)27 b(a)h(con)n(tin)n(uous)e(quadratic)h(form.)36 b(A)28 b(stronger)e(condition)h(is)h @beginspecial @setspecial @endspecial Fr(v)e Fp(2)e Fr(L)2232 1538 y Fs(2)2268 1568 y Fu(\()p Fk(R)2355 1538 y Fq(d)2399 1568 y Fr(;)14 b Fp(B)s Fu(\()p Fp(K)q Fu(\)\),)28 b(i.e.:)1529 1753 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)1626 1753 y Fp(jj)p Fr(v)s Fu(\()p Fr(k)s Fu(\))p Fp(jj)1871 1719 y Fs(2)1871 1776 y Fo(B)q Fs(\()p Fo(K)p Fs(\))2038 1753 y Fr(dk)39 b(<)e Fp(1)p Fr(:)p @beginspecial @setspecial @endspecial 0 1939 a Fu(Assume)28 b(that)g @beginspecial @setspecial @endspecial 23 w([)p Fr(v)577 1909 y Fo(\003)574 1960 y Fs(1)615 1939 y Fu(\()p Fr(k)s Fu(\))p Fr(;)37 b(v)825 1951 y Fs(2)863 1939 y Fu(\()p Fr(k)941 1909 y Fo(0)964 1939 y Fu(\)])24 b(=)e(0)p Fr(;)28 b Fp(8)p Fr(k)s(;)14 b(k)1399 1909 y Fo(0)1421 1939 y Fu(.)37 b(Then:)1246 2231 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1288 2081 a @beginspecial @setspecial @endspecial([)p Fr(a)p Fu(\()p Fr(v)1427 2093 y Fs(1)1465 2081 y Fu(\))p Fr(;)14 b(a)1578 2047 y Fo(\003)1616 2081 y Fu(\()p Fr(v)1688 2093 y Fs(2)1726 2081 y Fu(\)])23 b(=)g Fr(v)1935 2047 y Fo(\003)1932 2102 y Fs(1)1987 2081 y Fr(v)2027 2093 y Fs(2)2083 2081 y Fp(\012)32 b Fu(1)-15 b(l)2229 2110 y Fs(\000\()p Fm(h)p Fs(\))2369 2081 y Fr(;)p @beginspecial @setspecial @endspecial 1288 2231 a Fu([)p Fr(\036)p Fu(\()p Fr(v)1432 2243 y Fs(1)1470 2231 y Fu(\))p Fr(;)14 b(\036)p Fu(\()p Fr(v)1660 2243 y Fs(2)1699 2231 y Fu(\)])23 b(=)g(iIm\()p Fr(v)2062 2196 y Fo(\003)2059 2251 y Fs(1)2114 2231 y Fr(v)2154 2243 y Fs(2)2192 2231 y Fu(\))33 b Fp(\012)18 b Fu(1)-15 b(l)2389 2259 y Fs(\000\()p Fm(h)p Fs(\))2529 2231 y Fr(;)p @beginspecial @setspecial @endspecial 1288 2380 a Fu([)p Fr(\036)p Fu(\()p Fr(v)1432 2392 y Fs(1)1470 2380 y Fu(\))p Fr(;)14 b(W)e Fu(\()p Fr(v)1701 2392 y Fs(2)1739 2380 y Fu(\)])24 b(=)e(Im\()p Fr(v)2079 2345 y Fo(\003)2076 2400 y Fs(1)2132 2380 y Fr(v)2172 2392 y Fs(2)2209 2380 y Fu(\))d Fp(\012)32 b Fr(W)12 b Fu(\()p Fr(v)2519 2392 y Fs(2)2557 2380 y Fu(\))p Fr(:)2654 2231 y @beginspecial @setspecial @endspecial 0 2477 a @beginspecial @setspecial @endspecial 66 x @beginspecial @setspecial @endspecial Fu(2.2.)47 b Fz(T)-8 b(ec)m(hnical)26 b(Estimates.)39 b Fu(In)22 b(this)g(subsection)f(w)n(e)h(will)g(collect)f(some)g(tec)n (hnical)h(estimates)f(whic)n(h)h(are)f(adaptation)0 2642 y(of)28 b(Glimm-Ja\013e's)f @beginspecial @setspecial @endspecial Fr(N)692 2654 y Fq(\034)733 2642 y Fp(\000)g Fu(estimates.)0 2742 y(W)-7 b(e)28 b(recall)f(the)h(sym)n(b)r(olic)f (annihilation)g(and)g(creation)g(op)r(erators)e(in)j(the)g(case)f(of)h (a)f(F)-7 b(o)r(c)n(k)27 b(space)g(constructed)g(o)n(v)n(er)f(the)0 2842 y(space)h(of)g(square)g(in)n(tegrable)f(functions)i @beginspecial @setspecial @endspecial Fm(h)23 b Fu(:=)g Fr(L)1555 2811 y Fs(2)1592 2842 y Fu(\()p Fk(R)1678 2811 y Fq(d)1723 2842 y Fr(;)14 b(dk)s Fu(\).)37 b(Let)28 b(\011)22 b Fp(2)i Fu(\000)2308 2854 y Fs(\014n)2386 2842 y Fu(\()p Fm(h)p Fu(\))g(:)979 2997 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\()p Fr(a)p Fu(\()p Fr(k)s Fu(\)\011\))1262 2963 y Fs(\()p Fq(n)p Fs(\))1359 2997 y Fu(\()p Fr(k)1434 3009 y Fs(1)1472 2997 y Fr(;)14 b(:::;)g(k)1658 3009 y Fq(n)1704 2997 y Fu(\))23 b(:=)g(\()p Fr(n)18 b Fu(+)g(1\))2137 2941 y Fi(1)p 2137 2950 29 3 v 2137 2983 a(2)2193 2997 y Fu(\011)2258 2963 y Fs(\()p Fq(n)p Fs(+1\))2439 2997 y Fu(\()p Fr(k)s(;)c(k)2597 3009 y Fs(1)2634 2997 y Fr(;)g(:::;)g(k)2820 3009 y Fq(n)2866 2997 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 701 3251 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(a)777 3216 y Fo(\003)816 3251 y Fu(\()p Fr(k)s Fu(\))g(\011\))1037 3216 y Fs(\()p Fq(n)p Fs(\))1134 3251 y Fu(\()p Fr(k)1209 3263 y Fs(1)1247 3251 y Fr(;)g(:::;)g(k)1433 3263 y Fq(n)1478 3251 y Fu(\))23 b(:=)37 b Fr(n)1708 3216 y Fo(\000)1770 3194 y Fi(1)p 1770 3203 V 1770 3236 a(2)1879 3147 y Fq(n)1840 3172 y Fl(X)1842 3349 y Fq(j)s Fs(=1)1987 3251 y Fr(\016)s Fu(\()p Fr(k)22 b Fp(\000)c Fr(k)2250 3263 y Fq(j)2285 3251 y Fu(\))c(\011)2396 3216 y Fs(\()p Fq(n)p Fo(\000)p Fs(1\))2578 3251 y Fu(\()p Fr(k)2653 3263 y Fs(1)2691 3251 y Fr(;)g(:::;)2836 3229 y Fu(^)2834 3251 y Fr(k)2877 3263 y Fq(j)2912 3251 y Fr(;)g(:::;)g(k)3098 3263 y Fq(n)3143 3251 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 0 3499 a Fu(where)41 b @beginspecial @setspecial @endspecial 256 3478 a(^)254 3499 y Fr(k)297 3511 y Fq(j)374 3499 y Fu(means)h(that)g Fr(k)880 3511 y Fq(j)957 3499 y Fu(is)f(omitted.)80 b(Let)42 b @beginspecial @setspecial @endspecial Fr(S)5 b Fu(\()p Fk(R)1743 3469 y Fq(d)1788 3499 y Fu(\))42 b(b)r(e)g(the)g(Sc)n(h)n(w)n(artz)e(space.)79 b(W)-7 b(e)42 b(can)g(de\014ne)g(the)g(monomial)0 3602 y @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)119 3614 y Fs(1)157 3602 y Fu(\))p Fr(:::a)p Fu(\()p Fr(k)377 3614 y Fq(s)413 3602 y Fu(\))28 b(as)f(an)g(op)r(erator)f(from)h(\000) 1273 3614 y Fs(\014n)1352 3602 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)1526 3572 y Fq(d)1571 3602 y Fu(\)\))28 b(in)n(to)f @beginspecial @setspecial @endspecial Fr(S)5 b Fu(\()p Fk(R)1973 3572 y Fq(ds)2049 3602 y Fu(\))19 b Fp(\012)f Fu(\000)2235 3614 y Fs(\014n)2313 3602 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)2487 3572 y Fq(d)2532 3602 y Fu(\)\).)0 3702 y(Let)22 b @beginspecial @setspecial @endspecial Fp(K)i Fu(b)r(e)f(an)f(auxiliary)f(Hilb)r(ert)i(space.)34 b(Let)22 b Fr(w)j Fu(b)r(e)e(un)n(b)r(ounded)g(op)r(erator)d(from)i @beginspecial @setspecial @endspecial Fp(K)9 b(\012)f Fm(h)2789 3672 y Fo(\012)p Fq(s)2899 3702 y Fu(in)n(to)22 b Fp(K)9 b(\012)f Fm(h)3250 3672 y Fo(\012)p Fq(r)3338 3702 y Fu(,)24 b(with)f(a)e(domain)0 3801 y(con)n(taining)g @beginspecial @setspecial @endspecial Fp(K)8 b(\012)e Fr(S)f Fu(\()p Fk(R)682 3771 y Fq(ds)757 3801 y Fu(\).)35 b(A)22 b Fn(Wick)j(monomial)e Fu(with)f Fn(symb)l(ol)h @beginspecial @setspecial @endspecial Fr(w)h Fu(is)d(the)h(follo)n (wing)f(sesquilinear)f(form)h(on)g @beginspecial @setspecial @endspecial Fp(K)8 b(\012)e Fu(\000)3538 3813 y Fs(\014n)3616 3801 y Fu(\()p Fr(S)f Fu(\()p Fk(R)3791 3771 y Fq(d)3835 3801 y Fu(\)\))1255 3944 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1333 3956 y Fq(r)n(;s)1440 3944 y Fu(:=)37 b Fr(a)1609 3910 y Fo(\003)1647 3944 y Fu(\()p Fr(k)1725 3910 y Fo(0)1722 3965 y Fs(1)1760 3944 y Fu(\))p Fr(:::a)1905 3910 y Fo(\003)1943 3944 y Fu(\()p Fr(k)2021 3910 y Fo(0)2018 3965 y Fq(r)2055 3944 y Fu(\))14 b Fr(w)j(a)p Fu(\()p Fr(k)2296 3956 y Fs(1)2334 3944 y Fu(\))p Fr(:::a)p Fu(\()p Fr(k)2554 3956 y Fq(s)2590 3944 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4088 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(!)i Fu(b)r(e)e(a)f(p)r(ositiv)n(e)g(regular) f(function)i(satisfying:)1192 4356 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -196 x Fl(8)1192 4235 y(>)1192 4260 y(<)1192 4410 y(>)1192 4434 y(:)1266 4239 y @beginspecial @setspecial @endspecial Fr(!)d Fp(2)f Fr(C)1487 4209 y Fo(1)1558 4239 y Fu(\()p Fk(R)1644 4209 y Fq(d)1688 4239 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1266 4359 a Fp(j)p Fr(@)1338 4329 y Fq(\013)1333 4382 y(k)1385 4359 y Fr(!)s Fu(\()p Fr(k)s Fu(\))p Fp(j)f(\024)g Fr(c)1720 4371 y Fq(\013)1767 4359 y Fu(\(1)18 b(+)g Fp(j)p Fr(k)s Fp(j)p Fu(\))2066 4329 y Fq(N)2119 4337 y Fj(\013)2211 4359 y @beginspecial @setspecial @endspecial Fu(for)27 b Fr(\013)d Fp(2)f Fk(N)2547 4329 y Fq(d)2592 4359 y Fr(;)p @beginspecial @setspecial @endspecial 1266 4478 a(!)s Fu(\()p Fr(k)s Fu(\))g Fp(\025)g Fr(m)f(>)h Fu(0)p Fr(:)825 b @beginspecial @setspecial @endspecial 2708 4356 a @beginspecial @setspecial @endspecial 100 4623 a Fu(W)-7 b(e)28 b(set:)1688 4766 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)1755 4778 y Fq(\034)1819 4766 y Fu(:=)23 b(d\000\()p Fr(!)2115 4732 y Fq(\034)2157 4766 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4909 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 200 x Fz(Lemma)30 b(2.1.)40 b Fn(L)l(et)26 b Fr(r)j Fn(b)l(e)d(a)h(p)l(ositive)h(inte)l(ger)e(and)h Fr(\034)33 b Fu(:=)23 b(\()p Fr(\034)1875 5121 y Fq(i)1903 5109 y Fu(\))1935 5121 y Fs(1)p Fq(::r)2071 5109 y Fn(b)l(e)j(a)h(se)l (quenc)l(e)f(of)h(r)l(e)l(al)g(numb)l(ers.)37 b(F)-6 b(or)27 b @beginspecial @setspecial @endspecial Fr( )f Fp(2)d(D)r Fu(\()3630 5005 y Fq(r)3593 5030 y Fl(Y)3592 5207 y Fq(i)p Fs(=1)3714 5109 y Fr(N)3800 5038 y Fi(1)p 3800 5047 V 3800 5080 a(2)3781 5119 y Fq(\034)3812 5127 y Fj(i)3842 5109 y Fu(\))p Fn(:)787 5422 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)884 5318 y Fq(r)847 5343 y Fl(Y)847 5520 y Fq(i)p Fs(=1)968 5422 y Fr(N)1054 5351 y Fi(1)p 1054 5360 V 1054 5393 a(2)1035 5431 y Fq(\034)1066 5439 y Fj(i)1096 5422 y Fr( )s Fp(jj)1199 5387 y Fs(2)1260 5422 y Fu(=)1391 5318 y Fq(r)1347 5343 y Fl(X)1350 5520 y Fq(j)s Fs(=1)1495 5309 y Fl(Z)1592 5422 y Fr(P)1657 5387 y Fq(\034)1645 5442 y(r)n(;j)1729 5422 y Fu(\()p Fr(k)1804 5434 y Fs(1)1842 5422 y Fr(;)14 b(:::;)g(k)2028 5434 y Fq(j)2063 5422 y Fu(\))g Fp(jj)2207 5314 y Fq(j)2170 5343 y Fl(Y)2169 5520 y Fq(i)p Fs(=1)2290 5422 y Fr(a)p Fu(\()p Fr(k)2409 5434 y Fq(i)2437 5422 y Fu(\))p Fr( )s Fp(jj)2572 5387 y Fs(2)2610 5422 y Fr(dk)2696 5434 y Fs(1)2734 5422 y Fr(:::dk)2889 5434 y Fq(j)2924 5422 y Fr(;)83 b @beginspecial @setspecial @endspecial -3030 w Fu(\(2.6\))p eop %%Page: 6 6 6 5 bop 0 226 a Fs(6)1728 b(Z.)29 b(AMMARI)0 543 y Fn(wher)l(e)h @beginspecial @setspecial @endspecial Fr(P)299 513 y Fq(\034)287 565 y(r)n(;j)401 543 y Fn(is)g(a)g(sum)g(of)g(homo)l(gene)l (ous)h(functions)e(in)h(the)g(variables)i @beginspecial @setspecial @endspecial Fr(!)s Fu(\()p Fr(k)2410 555 y Fq(i)2438 543 y Fu(\))e Fn(of)g(de)l(gr)l(e)l(e)2888 439 y Fq(r)2844 464 y Fl(X)2851 641 y Fq(i)p Fs(=1)2978 543 y Fr(\034)3014 555 y Fq(i)3072 543 y Fn(and)g(satisfying)1018 823 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(P)1083 789 y Fq(\034)1071 844 y(r)n(;j)1155 823 y Fu(\()p Fr(k)1230 835 y Fs(1)1267 823 y Fr(;)14 b(:::;)g(k)1453 835 y Fq(j)1489 823 y Fu(\))23 b(=)1770 744 y Fl(X)1632 940 y Fs(\()p Fq(i)1681 948 y Fj(s)1713 940 y Fs(\))1739 948 y Fi(1)p Fj(::r)1839 940 y Fo(2)p Fm(S)1953 948 y Fj(r)o(;j)2043 823 y Fr(!)s Fu(\()p Fr(k)2173 835 y Fq(i)2196 843 y Fi(1)2233 823 y Fu(\))2265 789 y Fq(\034)2296 797 y Fi(1)2332 823 y Fr(:::!)s Fu(\()p Fr(k)2531 835 y Fq(i)2554 843 y Fj(r)2593 823 y Fu(\))2625 789 y Fq(\034)2656 797 y Fj(r)2693 823 y Fr(;)83 b @beginspecial @setspecial @endspecial -2799 w Fu(\(2.7\))0 1114 y Fn(wher)l(e)35 b @beginspecial @setspecial @endspecial Fm(S)308 1126 y Fq(r)n(;j)427 1114 y Fn(is)g(the)f(set,)i(c)l(onstructe)l(d)e(by)h(induction,)i(of)f (surje)l(ctive)f(maps)g Fp(f)p Fu(1)p Fr(;)14 b(:::;)g(r)r Fp(g)31 b(!)h(f)p Fu(1)p Fr(;)14 b(:::;)g(j)5 b Fp(g)p Fn(,)35 b(such)g(that)g @beginspecial @setspecial @endspecial Fr(i)3698 1126 y Fq(s)3765 1114 y Fp(\024)c Fr(s)0 1214 y Fn(and)f Fu(\()p Fr(i)222 1226 y Fq(s)258 1214 y Fu(\))290 1226 y Fs(1)p Fq(::r)r Fo(\000)p Fs(1)514 1214 y Fn(is)g(in)g @beginspecial @setspecial @endspecial Fm(S)774 1226 y Fq(r)r Fo(\000)p Fs(1)p Fq(;j)976 1214 y Fn(or)g(in)g Fm(S)1254 1226 y Fq(r)r Fo(\000)p Fs(1)p Fq(;j)s Fo(\000)p Fs(1)1540 1214 y Fn(and)h Fm(S)1771 1226 y Fq(j;j)s Fs(+1)1959 1214 y Fu(=)23 b Fm(S)2116 1226 y Fs(1)p Fq(;)p Fs(0)2229 1214 y Fu(=)g Fp(;)p Fn(.)0 1237 y @beginspecial @setspecial @endspecial 100 1475 a Fz(Pro)s(of.)38 b @beginspecial @setspecial @endspecial 23 w @beginspecial @setspecial @endspecial 459 1367 a Fq(j)421 1396 y Fl(Y)420 1573 y Fq(i)p Fs(=1)542 1475 y Fr(a)p Fu(\()p Fr(k)661 1487 y Fq(i)689 1475 y Fu(\))19 b(can)g(b)r(e)g(de\014ned)g(as)f(op)r (erator)g(on)g(\000)1843 1487 y Fs(\014n)1922 1475 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)2096 1445 y Fq(d)2141 1475 y Fu(\)\))19 b(in)n(to)g @beginspecial @setspecial @endspecial Fr(S)5 b Fu(\()p Fk(R)2526 1445 y Fq(d)-7 b(j)2594 1475 y Fu(\))q Fp(\012)q Fu(\000)2745 1487 y Fs(\014n)2824 1475 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)2998 1445 y Fq(d)3043 1475 y Fu(\)\).)34 b(F)-7 b(or)19 b @beginspecial @setspecial @endspecial Fr( )26 b Fp(2)d Fu(\000)3515 1487 y Fs(\014n)3593 1475 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)3768 1445 y Fq(d)3812 1475 y Fu(\)\),)0 1744 y(\()p Fr(k)75 1756 y Fs(1)113 1744 y Fr(;)14 b(:::;)g(k)299 1756 y Fq(j)334 1744 y Fu(\))30 b Fp(!)g(jj)607 1636 y Fq(j)569 1665 y Fl(Y)569 1842 y Fq(i)p Fs(=1)690 1744 y Fr(a)p Fu(\()p Fr(k)809 1756 y Fq(i)837 1744 y Fu(\))p Fr( )s Fp(jj)i Fu(is)g(a)f(function)h (in)g @beginspecial @setspecial @endspecial Fr(S)5 b Fu(\()p Fk(R)1737 1714 y Fq(d)-7 b(j)1806 1744 y Fu(\).)49 b(This)31 b(implies)h(that)g(the)g(r.h.s.)49 b(of)31 b(\()p @beginspecial @setspecial @endspecial(2.6\))h(is)f(w)n(ell)h (de\014ned)g(for)0 1995 y Fr( )26 b Fp(2)e Fu(\000)211 2007 y Fs(\014n)289 1995 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)463 1965 y Fq(d)508 1995 y Fu(\)\))24 b(.)36 b(The)24 b(h)n(yp)r(othesis)f (on)h @beginspecial @setspecial @endspecial Fr(!)i Fu(imply)e(that)h @beginspecial @setspecial @endspecial(\000)1875 2007 y Fs(\014n)1953 1995 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)2127 1965 y Fq(d)2172 1995 y Fu(\)\))24 b(is)g(a)f(core)g(for)h Fr(N)2786 1925 y Fi(1)p 2786 1934 29 3 v 2786 1967 a(2)2768 2005 y Fq(\034)2853 1995 y Fu(and)f(for)g @beginspecial @setspecial @endspecial 3171 1892 a Fq(r)3134 1917 y Fl(Y)3133 2093 y Fq(i)p Fs(=1)3255 1995 y Fr(N)3341 1925 y Fi(1)p 3341 1934 V 3341 1967 a(2)3322 2005 y Fq(\034)3353 2013 y Fj(i)3383 1995 y Fu(.)36 b(If)24 b(the)g(lemma)0 2247 y(holds)k(for)g @beginspecial @setspecial @endspecial Fr( )g Fp(2)d Fu(\000)560 2259 y Fs(\014n)638 2247 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)813 2217 y Fq(d)857 2247 y Fu(\)\))30 b(then)f(it)g(can)f(b)r(e)h(extended)g(to)f @beginspecial @setspecial @endspecial Fr( )g Fp(2)d(D)r Fu(\()2246 2143 y Fq(r)2209 2168 y Fl(Y)2208 2345 y Fq(i)p Fs(=1)2330 2247 y Fr(N)2416 2176 y Fi(1)p 2416 2185 V 2416 2218 a(2)2397 2257 y Fq(\034)2428 2265 y Fj(i)2458 2247 y Fu(\).)40 b(In)29 b(fact)g(since)f @beginspecial @setspecial @endspecial(\000)3079 2259 y Fs(\014n)3157 2247 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)3332 2217 y Fq(d)3376 2247 y Fu(\)\))29 b(is)g(a)f(core)g(for)37 2412 y Fq(r)1 2437 y Fl(Y)0 2614 y Fq(i)p Fs(=1)121 2516 y Fr(N)207 2445 y Fi(1)p 207 2454 V 207 2488 a(2)188 2526 y Fq(\034)219 2534 y Fj(i)249 2516 y Fu(,)k(then)g(w)n(e)e(can)h(extend)g @beginspecial @setspecial @endspecial 1089 2408 a Fq(j)1052 2437 y Fl(Y)1051 2614 y Fq(i)p Fs(=1)1172 2516 y Fr(a)p Fu(\()p Fr(k)1291 2528 y Fq(i)1319 2516 y Fu(\))h(to)f(b)r(ounded)g(op) r(erator)e(from)i Fp(D)r Fu(\()2504 2412 y Fq(r)2467 2437 y Fl(Y)2466 2614 y Fq(i)p Fs(=1)2588 2516 y Fr(N)2673 2445 y Fi(1)p 2673 2454 V 2673 2488 a(2)2655 2526 y Fq(\034)2686 2534 y Fj(i)2716 2516 y Fu(\))g(in)n(to)g @beginspecial @setspecial @endspecial Fr(L)3008 2486 y Fs(2)3044 2516 y Fu(\()p Fk(R)3131 2486 y Fq(d)-7 b(j)3199 2516 y Fr(;)14 b Fu(\()p Fr(P)3333 2486 y Fq(\034)3321 2538 y(r)n(;j)3405 2516 y Fu(\))3447 2464 y Fi(1)p 3448 2473 V 3448 2506 a(2)3490 2516 y Fr(dk)s Fu(\))21 b Fp(\012)f Fu(\000\()p Fm(h)p Fu(\).)0 2698 y(Let)28 b(us)f(pro)n(v)n(e)f(the)i(lemma)g(for)f @beginspecial @setspecial @endspecial Fr( )f Fp(2)d Fu(\000)1227 2710 y Fs(\014n)1306 2698 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)1480 2668 y Fs(3)1523 2698 y Fu(\)\))28 b(b)n(y)g(induction)g(in)f Fr(r)r Fu(.)0 2798 y(F)-7 b(or)27 b @beginspecial @setspecial @endspecial Fr(r)f Fu(=)c(1)1242 3079 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1283 2982 a @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1415 2911 y Fi(1)p 1415 2920 V 1415 2954 a(2)1396 2992 y Fq(\034)1427 3000 y Fi(1)1463 2982 y Fr( )s Fp(jj)1566 2952 y Fs(2)1687 2982 y Fu(=)82 b(\()p Fr( )s Fp(j)p Fr(N)2013 2994 y Fq(\034)2044 3002 y Fi(1)2081 2982 y Fr( )s Fu(\))p @beginspecial @setspecial @endspecial 1687 3160 a(=)1834 3047 y Fl(Z)1931 3160 y Fr(!)s Fu(\()p Fr(k)s Fu(\))2096 3126 y Fq(\034)2127 3134 y Fi(1)2164 3160 y Fp(jj)p Fr(a)p Fu(\()p Fr(k)s Fu(\))p Fr( )s Fp(jj)2467 3126 y Fs(2)2504 3160 y Fr(dk)s(:)2658 3079 y @beginspecial @setspecial @endspecial 100 3381 a Fu(W)-7 b(e)33 b(see)g(that)h @beginspecial @setspecial @endspecial Fr(P)639 3344 y Fq(\034)670 3352 y Fi(1)627 3403 y Fs(1)p Fq(;)p Fs(1)717 3381 y Fu(\()p Fr(k)s Fu(\))f(=)f Fr(!)s Fu(\()p Fr(k)s Fu(\))1122 3350 y Fq(\034)1153 3358 y Fi(1)1190 3381 y Fu(.)54 b(\(2.6\),)13 b(\(2.7\))33 b(are)g(satis\014ed)g(for)f @beginspecial @setspecial @endspecial Fr(r)k Fu(=)c(1.)54 b(Assume)33 b(that)h(\(2.6\),)48 b(\(2.7\))33 b(hold)g(for)g Fr(r)r Fu(.)0 3493 y(Using)27 b(the)h(fact)g(that)g @beginspecial @setspecial @endspecial Fr(N)787 3505 y Fq(\034)856 3493 y Fu(preserv)n(es)d(\000)1264 3505 y Fs(\014n)1343 3493 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)1517 3463 y Fq(d)1562 3493 y Fu(\)\))28 b(and)f(the)h(induction)g(h)n(yp)r(othesis,)f(w)n(e)h (ha)n(v)n(e)428 3755 y @beginspecial @setspecial @endspecial Fp(jj)488 3651 y Fq(r)r Fs(+1)494 3676 y Fl(Y)493 3853 y Fq(i)p Fs(=1)619 3755 y Fr(N)705 3684 y Fi(1)p 705 3693 V 705 3726 a(2)686 3765 y Fq(\034)717 3773 y Fj(i)747 3755 y Fr( )s Fp(jj)850 3721 y Fs(2)970 3755 y @beginspecial @setspecial @endspecial Fu(=)1162 3651 y Fq(r)1118 3676 y Fl(X)1120 3853 y Fq(j)s Fs(=1)1266 3642 y Fl(Z)1362 3755 y Fr(P)1427 3721 y Fq(\034)1415 3775 y(r)n(;j)1499 3755 y Fu(\()p Fr(k)1574 3767 y Fs(1)1612 3755 y Fr(;)14 b(:::;)g(k)1798 3767 y Fq(j)1833 3755 y Fu(\))g Fp(jj)1978 3647 y Fq(j)1940 3676 y Fl(Y)1939 3853 y Fq(i)p Fs(=1)2061 3755 y Fr(a)p Fu(\()p Fr(k)2180 3767 y Fq(i)2208 3755 y Fu(\))p Fr(N)2326 3684 y Fi(1)p 2326 3693 V 2326 3726 a(2)2307 3765 y Fq(\034)2338 3773 y Fj(r)q Fi(+1)2446 3755 y Fr( )s Fp(jj)2549 3721 y Fs(2)2586 3755 y Fr(dk)0 3897 y @beginspecial @setspecial @endspecial 887 4139 a @beginspecial @setspecial @endspecial 83 w Fu(=)1162 4035 y Fq(r)1118 4060 y Fl(X)1120 4236 y Fq(j)s Fs(=1)1266 4026 y Fl(Z)1362 4139 y Fr(P)1427 4104 y Fq(\034)1415 4159 y(r)n(;j)1513 4139 y Fp(jj)p Fu(\()p Fr(N)1658 4151 y Fq(\034)1689 4159 y Fj(r)q Fi(+1)1816 4139 y Fu(+)1944 4031 y Fq(j)1899 4060 y Fl(X)1905 4236 y Fq(i)p Fs(=1)2033 4139 y Fr(!)s Fu(\()p Fr(k)2163 4151 y Fq(i)2191 4139 y Fu(\))2223 4104 y Fq(\034)2254 4112 y Fj(r)q Fi(+1)2362 4139 y Fu(\))2404 4082 y Fi(1)p 2404 4091 V 2404 4124 a(2)2499 4031 y Fq(j)2461 4060 y Fl(Y)2460 4236 y Fq(i)p Fs(=1)2582 4139 y Fr(a)p Fu(\()p Fr(k)2701 4151 y Fq(i)2729 4139 y Fu(\))p Fr( )s Fp(jj)2864 4104 y Fs(2)2902 4139 y Fr(dk)0 4281 y @beginspecial @setspecial @endspecial 970 4522 a @beginspecial @setspecial @endspecial Fu(=)1162 4418 y Fq(r)1118 4443 y Fl(X)1120 4620 y Fq(j)s Fs(=1)1266 4409 y Fl(Z)1362 4522 y Fr(P)1427 4488 y Fq(\034)1415 4543 y(r)n(;j)1513 4522 y Fp(f)1599 4415 y Fq(j)1555 4443 y Fl(X)1561 4620 y Fq(i)p Fs(=1)1689 4522 y Fr(!)s Fu(\()p Fr(k)1819 4534 y Fq(i)1846 4522 y Fu(\))1878 4488 y Fq(\034)1909 4496 y Fj(r)q Fi(+1)2032 4522 y Fp(jj)2130 4415 y Fq(j)2092 4443 y Fl(Y)2092 4620 y Fq(i)p Fs(=1)2213 4522 y Fr(a)p Fu(\()p Fr(k)2332 4534 y Fq(i)2360 4522 y Fu(\))p Fr( )s Fp(jj)2495 4488 y Fs(2)2551 4522 y Fu(+)k Fp(jj)p Fr(N)2766 4451 y Fi(1)p 2766 4460 V 2766 4494 a(2)2747 4532 y Fq(\034)2778 4540 y Fj(r)q Fi(+1)2938 4415 y Fq(j)2901 4443 y Fl(Y)2900 4620 y Fq(i)p Fs(=1)3021 4522 y Fr(a)p Fu(\()p Fr(k)3140 4534 y Fq(i)3168 4522 y Fu(\))p Fr( )s Fp(jj)3303 4488 y Fs(2)3341 4522 y Fp(g)p Fr(dk)0 4665 y @beginspecial @setspecial @endspecial 970 4906 a @beginspecial @setspecial @endspecial Fu(=)1120 4802 y Fq(r)r Fs(+1)1118 4827 y Fl(X)1120 5004 y Fq(j)s Fs(=1)1266 4793 y Fl(Z)1349 4906 y Fu([)p Fr(P)1437 4872 y Fq(\034)1425 4926 y(r)n(;j)1567 4798 y(j)1522 4827 y Fl(X)1529 5004 y Fq(i)p Fs(=1)1656 4906 y Fr(!)s Fu(\()p Fr(k)1786 4918 y Fq(i)1814 4906 y Fu(\))1846 4872 y Fq(\034)1877 4880 y Fj(r)q Fi(+1)2004 4906 y Fu(+)g Fr(P)2152 4872 y Fq(\034)2140 4926 y(r)n(;j)s Fo(\000)p Fs(1)2309 4906 y Fr(!)s Fu(\()p Fr(k)2439 4918 y Fq(j)2474 4906 y Fu(\))2506 4872 y Fq(\034)2537 4880 y Fj(r)q Fi(+1)2646 4906 y Fu(])23 b Fp(jj)2790 4798 y Fq(j)2752 4827 y Fl(Y)2752 5004 y Fq(i)p Fs(=1)2873 4906 y Fr(a)p Fu(\()p Fr(k)2992 4918 y Fq(i)3020 4906 y Fu(\))p Fr( )s Fp(jj)3155 4872 y Fs(2)3193 4906 y Fr(dk)s(;)100 5168 y Fu(where)k @beginspecial @setspecial @endspecial Fr(P)393 5180 y Fq(r)n(;r)r Fs(+1)586 5168 y Fu(=)22 b Fr(P)726 5180 y Fq(r)n(;)p Fs(0)836 5168 y Fu(=)g(0.)37 b(Then)28 b(w)n(e)f(obtain)g(the)h(follo)n(wing)f (iterated)g(relation)452 5433 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(P)517 5399 y Fq(\034)505 5453 y(r)r Fs(+1)p Fq(;j)677 5433 y Fu(\()p Fr(k)752 5445 y Fs(1)789 5433 y Fr(;)14 b(:::;)g(k)975 5445 y Fq(j)1011 5433 y Fu(\))23 b(=)g Fr(P)1219 5399 y Fq(\034)1207 5453 y(r)n(;j)1291 5433 y Fu(\()p Fr(k)1366 5445 y Fs(1)1403 5433 y Fr(;)14 b(:::;)g(k)1589 5445 y Fq(j)1625 5433 y Fu(\))1729 5325 y Fq(j)1685 5354 y Fl(X)1691 5531 y Fq(i)p Fs(=1)1818 5433 y Fr(!)s Fu(\()p Fr(k)1948 5445 y Fq(i)1976 5433 y Fu(\))2008 5399 y Fq(\034)2039 5407 y Fj(r)q Fi(+1)2166 5433 y Fu(+)k Fr(P)2314 5399 y Fq(\034)2302 5453 y(r)n(;j)s Fo(\000)p Fs(1)2471 5433 y Fu(\()p Fr(k)2546 5445 y Fs(1)2584 5433 y Fr(;)c(:::;)g(k)2770 5445 y Fq(j)s Fo(\000)p Fs(1)2890 5433 y Fu(\))p Fr(!)s Fu(\()p Fr(k)3052 5445 y Fq(j)3087 5433 y Fu(\))3119 5399 y Fq(\034)3150 5407 y Fj(r)q Fi(+1)3259 5433 y Fr(:)83 b @beginspecial @setspecial @endspecial -3365 w Fu(\(2.8\))p eop %%Page: 7 7 7 6 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)958 b(7)0 543 y Fu(W)-7 b(e)27 b(note)f(that)h Fr(P)569 509 y Fq(\034)557 564 y(r)n(;r)643 543 y Fu(\()p Fr(k)718 555 y Fs(1)756 543 y Fr(;)14 b(:::;)g(k)942 555 y Fq(r)979 543 y Fu(\))23 b(=)1159 439 y Fq(r)1123 464 y Fl(Y)1122 641 y Fq(i)p Fs(=1)1243 543 y Fr(!)s Fu(\()p Fr(k)1373 555 y Fq(i)1401 543 y Fu(\))1433 509 y Fq(\034)1464 517 y Fj(i)1494 543 y Fu(.)37 b(It)27 b(is)f(easy)g(to)g(see,)h(using)f(induction)h(h)n(yp)r(othesis)f(\()p @beginspecial @setspecial @endspecial(2.7\))g(for)g Fr(r)k Fu(and)c(\()p @beginspecial @setspecial @endspecial(2.8\),)0 745 y(that)i Fr(P)245 711 y Fq(\034)233 765 y(r)r Fs(+1)p Fq(;j)405 745 y Fu(\()p Fr(k)480 757 y Fs(1)517 745 y Fr(;)14 b(:::;)g(k)703 757 y Fq(r)r Fs(+1)824 745 y Fu(\))24 b(=)1178 666 y Fl(X)967 862 y Fs(\()p Fq(i)1016 870 y Fj(s)1049 862 y Fs(\))1075 870 y Fi(1)p Fj(::r)q Fi(+1)1245 862 y Fo(2)p Fm(S)1359 870 y Fj(r)q Fi(+1)p Fj(;j)1523 745 y Fr(!)s Fu(\()p Fr(k)1653 757 y Fq(i)1676 765 y Fi(1)1713 745 y Fu(\))1745 711 y Fq(\034)1776 719 y Fi(1)1813 745 y Fr(:::!)s Fu(\()p Fr(k)2012 757 y Fq(i)2035 765 y Fj(r)q Fi(+1)2144 745 y Fu(\))2176 711 y Fq(\034)2207 719 y Fj(r)q Fi(+1)2315 745 y Fu(.)p @beginspecial @setspecial @endspecial 1497 w Fh(\004)0 886 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 198 x Fz(Corollary)33 b(2.2.)40 b Fn(L)l(et)29 b Fr(\013;)14 b(\027)36 b Fn(,)p Fr(\034)d Fu(:=)23 b(\()p Fr(\034)1199 1096 y Fq(i)1227 1084 y Fu(\))1259 1096 y Fs(1)p Fq(::r)1398 1084 y Fn(b)l(e)30 b(a)g(se)l(quenc)l(e)f(of)i(r)l(e)l(al)f(numb)l (ers.)38 b(F)-6 b(or)30 b @beginspecial @setspecial @endspecial Fr( )c Fp(2)d(D)r Fu(\()p Fr(N)3016 1050 y Fq(\013)3007 1105 y(\027)3129 981 y(r)3093 1005 y Fl(Y)3092 1182 y Fq(i)p Fs(=1)3213 1084 y Fr(N)3299 1013 y Fi(1)p 3299 1022 29 3 v 3299 1056 a(2)3280 1094 y Fq(\034)3311 1102 y Fj(i)3341 1084 y Fu(\))377 1393 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)499 1359 y Fq(\013)490 1413 y(\027)612 1289 y(r)575 1314 y Fl(Y)574 1491 y Fq(i)p Fs(=1)696 1393 y Fr(N)781 1322 y Fi(1)p 781 1331 V 781 1364 a(2)763 1403 y Fq(\034)794 1411 y Fj(i)824 1393 y Fr( )s Fp(jj)927 1359 y Fs(2)987 1393 y Fu(=)1118 1289 y Fq(r)1075 1314 y Fl(X)1077 1491 y Fq(j)s Fs(=1)1209 1280 y Fl(Z)1305 1393 y Fr(P)1370 1359 y Fq(\034)1358 1413 y(r)n(;j)1442 1393 y Fu(\()p Fr(k)1517 1405 y Fs(1)1555 1393 y Fr(;)14 b(:::;)g(k)1741 1405 y Fq(j)1776 1393 y Fu(\))24 b Fp(jj)p Fu(\()p Fr(N)1977 1405 y Fq(\027)2037 1393 y Fu(+)2164 1285 y Fq(j)2120 1314 y Fl(X)2126 1491 y Fq(i)p Fs(=1)2253 1393 y Fr(!)s Fu(\()p Fr(k)2383 1405 y Fq(i)2411 1393 y Fu(\))2443 1359 y Fq(\027)2485 1393 y Fu(\))2517 1359 y Fq(\013)2617 1285 y(j)2579 1314 y Fl(Y)2578 1491 y Fq(i)p Fs(=1)2700 1393 y Fr(a)p Fu(\()p Fr(k)2819 1405 y Fq(i)2847 1393 y Fu(\))p Fr( )s Fp(jj)2982 1359 y Fs(2)3019 1393 y Fr(dk)3105 1405 y Fs(1)3143 1393 y Fr(:::dk)3298 1405 y Fq(j)3334 1393 y Fr(;)83 b @beginspecial @setspecial @endspecial -3440 w Fu(\(2.9\))0 1621 y Fn(wher)l(e)30 b Fr(P)299 1591 y Fq(\034)287 1642 y(r)n(;j)401 1621 y Fn(is)g(the)g(function)g (de\014ne)l(d)g(by)g(\()p @beginspecial @setspecial @endspecial(2.7\).) 0 1654 y @beginspecial @setspecial @endspecial 123 x @beginspecial @setspecial @endspecial Fz(Lemma)g(2.3.)40 b Fn(L)l(et)32 b Fr(p;)14 b(q)32 b Fp(2)d Fk(N)43 b Fn(and)48 b Fr(\034)1224 1747 y Fo(0)1276 1777 y Fu(:=)29 b(\()p Fr(\034)1470 1747 y Fo(0)1461 1798 y Fq(i)1494 1777 y Fu(\))1526 1789 y Fs(1)p Fq(::p)1638 1777 y Fr(;)14 b(\034)38 b Fu(:=)29 b(\()p Fr(\034)1934 1789 y Fq(i)1962 1777 y Fu(\))1994 1789 y Fs(1)p Fq(::q)2136 1777 y Fn(b)l(e)34 b(two)f(se)l(quenc)l(es)f(of)i(r)l(e)l(al)g(numb)l(ers.)47 b(L)l(et)33 b @beginspecial @setspecial @endspecial Fr(W)3635 1789 y Fq(r)n(;s)3752 1777 y Fn(b)l(e)h(a)0 1876 y(Wick)c(monomial)i (such)d(that)h @beginspecial @setspecial @endspecial Fr(r)c Fp(\024)d Fr(p;)14 b(s)22 b Fp(\024)h Fr(q)s Fn(.)39 b(Then)470 2082 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()585 1974 y Fq(p)549 2003 y Fl(Y)548 2180 y Fq(i)p Fs(=1)656 2082 y Fu(\()p Fr(N)755 2096 y Fq(\034)793 2076 y Fg(0)786 2115 y Fj(i)838 2082 y Fu(+)18 b(1\))995 2047 y Fo(\000)1056 2025 y Fi(1)p 1056 2034 V 1056 2067 a(2)1099 2082 y Fu(\))23 b Fr(W)1232 2094 y Fq(r)n(;s)1368 1974 y(q)1331 2003 y Fl(Y)1330 2180 y Fq(i)p Fs(=1)1438 2082 y Fu(\()p Fr(N)1537 2094 y Fq(\034)1568 2102 y Fj(i)1617 2082 y Fu(+)18 b(1\))1774 2047 y Fo(\000)1835 2025 y Fi(1)p 1835 2034 V 1835 2067 a(2)1878 2082 y Fp(jj)23 b(\024)f(jj)p Fu(\()p Fr(P)2177 2047 y Fq(\034)2215 2022 y Fg(0)2165 2102 y Fq(p;r)2257 2082 y Fu(\))2289 2047 y Fo(\000)2351 2025 y Fi(1)p 2351 2034 V 2351 2067 a(2)2407 2082 y Fr(w)17 b Fu(\()p Fr(P)2580 2047 y Fq(\034)2568 2102 y(q)r(;s)2656 2082 y Fu(\))2688 2047 y Fo(\000)2750 2025 y Fi(1)p 2750 2034 V 2750 2067 a(2)2792 2082 y Fp(jj)2838 2110 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)3055 2094 y Fg(\012)p Fj(s)3133 2110 y Fq(;)p Fo(K\012)p Fm(h)3299 2094 y Fg(\012)p Fj(r)3377 2110 y Fs(\))3407 2082 y Fr(:)p @beginspecial @setspecial @endspecial 100 2319 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Let)27 b @beginspecial @setspecial @endspecial Fr( )s(;)14 b(\036)24 b Fp(2)f(K)d(\012)e Fu(\000)1035 2331 y Fs(\014n)1113 2319 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)1288 2289 y Fq(d)1332 2319 y Fu(\)\).)438 2525 y @beginspecial @setspecial @endspecial Fp(j)p Fu(\()p Fr(W)571 2537 y Fq(r)n(;s)670 2525 y Fr( )s Fp(j)14 b Fr(\036)p Fu(\))p Fp(j)83 b(\024)g(j)p Fu(\()p Fr(w)1282 2421 y Fq(s)1244 2446 y Fl(Y)1244 2623 y Fq(i)p Fs(=1)1379 2525 y Fr(a)p Fu(\()p Fr(k)1498 2537 y Fq(i)1526 2525 y Fu(\))p Fr( )s Fp(j)1713 2421 y Fq(r)1676 2446 y Fl(Y)1675 2623 y Fq(i)p Fs(=1)1810 2525 y Fr(a)p Fu(\()p Fr(k)1932 2491 y Fo(0)1929 2545 y Fq(i)1958 2525 y Fu(\))p Fr(\036)p Fu(\))2071 2554 y Fo(K\012)p Fm(h)2217 2537 y Fg(\012)p Fj(r)2296 2554 y Fo(\012)p Fs(\000\()p Fm(h)p Fs(\))2511 2525 y Fp(j)0 2656 y @beginspecial @setspecial @endspecial 951 2841 a(\024)g(j)p Fu(\(\()p Fr(P)1251 2807 y Fq(\034)1289 2782 y Fg(0)1239 2862 y Fq(p;r)1331 2841 y Fu(\))1363 2807 y Fo(\000)1425 2785 y Fi(1)p 1425 2794 V 1425 2827 a(2)1481 2841 y Fr(w)17 b Fu(\()p Fr(P)1654 2807 y Fq(\034)1642 2862 y(q)r(;s)1730 2841 y Fu(\))1762 2807 y Fo(\000)1824 2785 y Fi(1)p 1824 2794 V 1824 2827 a(2)1880 2841 y Fu(\()p Fr(P)1977 2807 y Fq(\034)1965 2862 y(s;q)2053 2841 y Fu(\))2095 2785 y Fi(1)p 2096 2794 V 2096 2827 a(2)2190 2737 y Fq(s)2153 2762 y Fl(Y)2152 2939 y Fq(i)p Fs(=1)2287 2841 y Fr(a)p Fu(\()p Fr(k)2406 2853 y Fq(i)2434 2841 y Fu(\))p Fr( )s Fp(j)d Fu(\()p Fr(P)2657 2807 y Fq(\034)2695 2782 y Fg(0)2645 2862 y Fq(p;r)2737 2841 y Fu(\))2779 2785 y Fi(1)p 2779 2794 V 2779 2827 a(2)2873 2737 y Fq(r)2836 2762 y Fl(Y)2835 2939 y Fq(i)p Fs(=1)2970 2841 y Fr(a)p Fu(\()p Fr(k)3092 2807 y Fo(0)3089 2862 y Fq(i)3117 2841 y Fu(\))p Fr(\036)p Fu(\))p Fp(j)0 2972 y @beginspecial @setspecial @endspecial 951 3157 a(\024)83 b(jj)p Fu(\()p Fr(P)1242 3123 y Fq(\034)1280 3098 y Fg(0)1230 3178 y Fq(p;r)1321 3157 y Fu(\))1353 3123 y Fo(\000)1416 3101 y Fi(1)p 1416 3110 V 1416 3143 a(2)1472 3157 y Fr(w)16 b Fu(\()p Fr(P)1644 3123 y Fq(\034)1632 3178 y(q)r(;s)1721 3157 y Fu(\))1753 3123 y Fo(\000)1815 3101 y Fi(1)p 1815 3110 V 1815 3143 a(2)1857 3157 y Fp(jj)1903 3186 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)2120 3170 y Fg(\012)p Fj(s)2197 3186 y Fq(;)p Fo(K\012)p Fm(h)2363 3170 y Fg(\012)p Fj(r)2441 3186 y Fs(\))2494 3157 y Fu(\()2526 3044 y Fl(Z)2624 3157 y Fr(P)2689 3123 y Fq(\034)2677 3178 y(q)r(;s)2765 3157 y Fp(jj)2863 3054 y Fq(s)2825 3078 y Fl(Y)2825 3255 y Fq(i)p Fs(=1)2960 3157 y Fr(a)p Fu(\()p Fr(k)3079 3169 y Fq(i)3107 3157 y Fu(\))p Fr( )s Fp(jj)3242 3123 y Fs(2)3280 3157 y Fr(dk)s Fu(\))3411 3101 y Fi(1)p 3411 3110 V 3411 3143 a(2)0 3289 y @beginspecial @setspecial @endspecial 1099 3474 a Fp(\002)p Fu(\()1196 3361 y Fl(Z)1293 3474 y Fr(P)1358 3439 y Fq(\034)1396 3414 y Fg(0)1346 3494 y Fq(p;r)1437 3474 y Fp(jj)1534 3370 y Fq(r)1497 3395 y Fl(Y)1497 3572 y Fq(i)p Fs(=1)1632 3474 y Fr(a)p Fu(\()p Fr(k)1754 3439 y Fo(0)1751 3494 y Fq(i)1779 3474 y Fu(\))p Fr(\036)p Fp(jj)1906 3439 y Fs(2)1944 3474 y Fr(dk)2033 3439 y Fo(0)2056 3474 y Fu(\))2098 3417 y Fi(1)p 2099 3426 V 2099 3459 a(2)0 3605 y @beginspecial @setspecial @endspecial 951 3794 a @beginspecial @setspecial @endspecial Fp(\024)83 b(jj)p Fu(\()p Fr(P)1242 3759 y Fq(\034)1280 3734 y Fg(0)1230 3814 y Fq(p;r)1321 3794 y Fu(\))1353 3759 y Fo(\000)1416 3737 y Fi(1)p 1416 3746 V 1416 3779 a(2)1472 3794 y Fr(w)16 b Fu(\()p Fr(P)1644 3759 y Fq(\034)1632 3814 y(q)r(;s)1721 3794 y Fu(\))1753 3759 y Fo(\000)1815 3737 y Fi(1)p 1815 3746 V 1815 3779 a(2)1857 3794 y Fp(jj)1903 3822 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)2120 3806 y Fg(\012)p Fj(s)2197 3822 y Fq(;)p Fo(K\012)p Fm(h)2363 3806 y Fg(\012)p Fj(r)2441 3822 y Fs(\))2494 3794 y Fp(jj)2592 3686 y Fq(q)2555 3715 y Fl(Y)2554 3892 y Fq(i)p Fs(=1)2676 3794 y Fr(N)2761 3723 y Fi(1)p 2761 3732 V 2761 3765 a(2)2743 3803 y Fq(\034)2774 3811 y Fj(i)2804 3794 y Fr( )s Fp(jj)42 b(\002)18 b(jj)3128 3686 y Fq(p)3092 3715 y Fl(Y)3092 3892 y Fq(i)p Fs(=1)3213 3794 y Fr(N)3299 3723 y Fi(1)p 3299 3732 V 3299 3765 a(2)3280 3821 y Fq(\034)3318 3801 y Fg(0)3311 3840 y Fj(i)3344 3794 y Fr(\036)p Fp(jj)p Fr(:)0 4089 y Fu(This)37 b(inequalit)n(y)f(sho)n(ws)g(that)h(the)g (quadratic)e(form)i @beginspecial @setspecial @endspecial 1804 3981 a Fq(p)1769 4010 y Fl(Y)1768 4187 y Fq(i)p Fs(=1)1875 4089 y Fu(\()p Fr(N)1974 4103 y Fq(\034)2012 4083 y Fg(0)2005 4122 y Fj(i)2063 4089 y Fu(+)24 b(1\))2226 4055 y Fo(\000)2288 4032 y Fi(1)p 2288 4041 V 2288 4075 a(2)2330 4089 y Fr(W)2408 4101 y Fq(r)n(;s)2544 3981 y(q)2507 4010 y Fl(Y)2506 4187 y Fq(i)p Fs(=1)2614 4089 y Fu(\()p Fr(N)2713 4101 y Fq(\034)2744 4109 y Fj(i)2799 4089 y Fu(+)g(1\))2962 4055 y Fo(\000)3023 4032 y Fi(1)p 3023 4041 V 3023 4075 a(2)3102 4089 y Fu(can)37 b(b)r(e)g(extended)g (to)f(a)0 4283 y(b)r(ounded)28 b(op)r(erator)e(with)i(norm)f(less)g (than)h @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(P)1570 4253 y Fq(\034)1608 4227 y Fg(0)1558 4303 y Fq(p;r)1649 4283 y Fu(\))1681 4253 y Fo(\000)1743 4230 y Fi(1)p 1743 4239 V 1743 4273 a(2)1786 4283 y Fr(w)16 b Fu(\()p Fr(P)1958 4253 y Fq(\034)1946 4303 y(q)r(;s)2035 4283 y Fu(\))2067 4253 y Fo(\000)2129 4230 y Fi(1)p 2129 4239 V 2129 4273 a(2)2171 4283 y Fp(jj)2217 4311 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)2434 4295 y Fg(\012)p Fj(s)2511 4311 y Fq(;)p Fo(K\012)p Fm(h)2677 4295 y Fg(\012)p Fj(r)2755 4311 y Fs(\))2785 4283 y Fu(.)p @beginspecial @setspecial @endspecial 1027 w Fh(\004)0 4327 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 123 x Fz(Corollary)33 b(2.4.)40 b Fn(L)l(et)29 b Fr(v)799 4462 y Fq(i)850 4450 y Fp(2)24 b(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)19 b(\012)f Fm(h)p Fu(\);)29 b Fr(i)22 b Fu(=)h(1)p Fr(;)14 b(:::;)g(n)p Fn(.)38 b(Then)30 b(ther)l(e)g(exists)g @beginspecial @setspecial @endspecial Fr(c)23 b(>)f Fu(0)29 b Fn(such)h(that)700 4868 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 742 4656 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))14 b Fp(jj)p Fu(\()p Fr(N)27 b Fu(+)18 b(1\))1144 4622 y Fq(p)1244 4552 y(n)1211 4577 y Fl(Y)1210 4754 y Fq(i)p Fs(=1)1332 4656 y Fr(a)1376 4622 y Fq(])1407 4656 y Fu(\()p Fr(v)1479 4668 y Fq(i)1507 4656 y Fu(\)\()p Fr(N)28 b Fu(+)18 b(1\))1823 4622 y Fo(\000)p Fq(p)p Fo(\000)1970 4599 y Fj(n)p 1970 4608 37 3 v 1974 4642 a Fi(2)2021 4656 y Fp(jj)37 b(\024)f Fr(c)2302 4552 y Fq(n)2270 4577 y Fl(Y)2269 4754 y Fq(i)p Fs(=1)2390 4656 y Fp(jj)p Fr(v)2476 4668 y Fq(i)2504 4656 y Fp(jj)2550 4685 y Fo(B)q Fs(\()p Fo(K)p Fq(;)p Fo(K\012)p Fm(h)p Fs(\))2868 4656 y Fr(:)p @beginspecial @setspecial @endspecial 746 4934 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))24 b Fp(jj)p Fu(\()p Fr(N)j Fu(+)18 b(1\))1180 4900 y Fq(p)1280 4830 y(n)1247 4855 y Fl(Y)1246 5032 y Fq(i)p Fs(=1)1368 4934 y Fr(\036)p Fu(\()p Fr(v)1489 4946 y Fq(i)1517 4934 y Fu(\)\()p Fr(N)28 b Fu(+)18 b(1\))1833 4900 y Fo(\000)p Fq(p)p Fo(\000)1981 4878 y Fj(n)p 1981 4887 V 1985 4920 a Fi(2)2032 4934 y Fp(jj)37 b(\024)f Fr(c)2313 4830 y Fq(n)2280 4855 y Fl(Y)2280 5032 y Fq(i)p Fs(=1)2401 4934 y Fp(jj)p Fr(v)2487 4946 y Fq(i)2515 4934 y Fp(jj)2561 4963 y Fo(B)q Fs(\()p Fo(K)p Fq(;)p Fo(K\012)p Fm(h)p Fs(\))2879 4934 y Fr(:)p @beginspecial @setspecial @endspecial 746 5145 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)f Fr({)p Fu(\))14 b Fn(L)l(et)43 b Fr(r)n(;)14 b(s)24 b Fp(2)f Fk(N)t Fr(;)14 b Fu(\()p Fr(\034)1407 5111 y Fo(0)1397 5166 y Fq(i)1436 5145 y Fu(\))1468 5157 y Fs(1)p Fq(::r)1578 5145 y Fr(;)g Fu(\()p Fr(\034)1683 5157 y Fq(i)1711 5145 y Fu(\))1743 5157 y Fs(1)p Fq(::s)1879 5145 y Fn(b)l(e)30 b(se)l(quenc)l(es)f(of)h(r)l(e)l(al)h(numb)l(ers.)37 b(Then)3200 4868 y @beginspecial @setspecial @endspecial 330 5400 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()446 5296 y Fq(r)409 5321 y Fl(Y)408 5498 y Fq(i)p Fs(=1)516 5400 y Fu(\()p Fr(N)615 5414 y Fq(\034)653 5394 y Fg(0)646 5433 y Fj(i)697 5400 y Fu(+)18 b(1\))854 5365 y Fo(\000)916 5343 y Fi(1)p 916 5352 29 3 v 916 5385 a(2)958 5400 y Fu(\))c Fr(W)1082 5412 y Fq(r)n(;s)1219 5296 y(s)1182 5321 y Fl(Y)1181 5498 y Fq(i)p Fs(=1)1288 5400 y Fu(\()p Fr(N)1387 5412 y Fq(\034)1418 5420 y Fj(i)1467 5400 y Fu(+)k(1\))1624 5365 y Fo(\000)1686 5343 y Fi(1)p 1686 5352 V 1686 5385 a(2)1728 5400 y Fp(jj)23 b(\024)g(jj)1982 5296 y Fq(r)1945 5321 y Fl(Y)1945 5498 y Fq(i)p Fs(=1)2066 5400 y Fr(!)s Fu(\()p Fr(k)2199 5365 y Fo(0)2196 5420 y Fq(i)2224 5400 y Fu(\))2256 5365 y Fo(\000)2318 5328 y Fj(\034)2351 5311 y Fg(0)2346 5347 y Fj(i)p 2318 5352 56 3 v 2332 5385 a Fi(2)2401 5400 y Fr(w)2529 5296 y Fq(s)2492 5321 y Fl(Y)2491 5498 y Fq(i)p Fs(=1)2612 5400 y Fr(!)s Fu(\()p Fr(k)2742 5412 y Fq(i)2770 5400 y Fu(\))2802 5365 y Fo(\000)2864 5333 y Fj(\034)2892 5346 y(i)p 2864 5352 55 3 v 2877 5385 a Fi(2)2933 5400 y Fp(jj)2979 5428 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)3196 5412 y Fg(\012)p Fj(s)3273 5428 y Fq(;)p Fo(K\012)p Fm(h)3439 5412 y Fg(\012)p Fj(r)3517 5428 y Fs(\))3547 5400 y Fr(:)p @beginspecial @setspecial @endspecial eop %%Page: 8 8 8 7 bop 0 226 a Fs(8)1728 b(Z.)29 b(AMMARI)100 489 y Fz(Pro)s(of.)51 b @beginspecial @setspecial @endspecial 28 w Fu(Clearly)33 b(_)-27 b Fr({)p Fu(\))30 b(giv)n(es)k(_)-28 b Fr({)t Fu(_)h Fr({)p Fu(\).)45 b(F)-7 b(or)30 b Fr(p)d Fu(=)h(0)i(and)g Fr(n)d Fu(=)g(1)p Fr(;)46 b Fu(_)-28 b Fr({)p Fu(\))31 b(follo)n(ws)e(from)h(the)h(last)f(lemma)g(b)n(y)g (taking)g @beginspecial @setspecial @endspecial Fr(!)g Fu(=)d(1)j(and)0 589 y Fr(r)f Fu(=)d Fr(p)g Fu(=)g(1)p Fr(;)14 b(s)26 b Fu(=)g Fr(q)j Fu(=)d(0)k(or)e @beginspecial @setspecial @endspecial Fr(r)h Fu(=)d Fr(p)h Fu(=)f(0)p Fr(;)14 b(s)25 b Fu(=)h Fr(q)k Fu(=)c(1.)42 b(F)-7 b(or)29 b Fr(p;)14 b(n)26 b Fp(2)h Fk(N)t Fu(,)36 b(comm)n(utation)29 b(prop)r(erties)g(reduces)g(the)h(inequalit)n(y)0 688 y(to)d(the)h(case)f @beginspecial @setspecial @endspecial Fr(p)c Fu(=)g(0)p Fr(;)14 b(n)22 b Fu(=)h(1.)36 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))28 b(is)g(a)f(direct)g(application)g(of)h(Lemma)f @beginspecial @setspecial @endspecial(2.3.)1444 b Fh(\004)100 788 y Fu(W)-7 b(e)28 b(recall)e(no)n(w)h(w)n(ell)h(kno)n(wn)e (estimates,)i(see)f @beginspecial @setspecial @endspecial([22)o(].)0 809 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 129 x Fz(Lemma)j(2.5.)40 b Fn(L)l(et)29 b @beginspecial @setspecial @endspecial Fr(b)g Fn(b)l(e)h(a)g(p)l (ositive)h(op)l(er)l(ator.)40 b(Then)1158 1135 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1200 1067 a @beginspecial @setspecial @endspecial Fu(d\000\()p Fr(b)1366 1037 y Fq(\013)1409 1045 y Fi(1)1445 1067 y Fu(\))24 b Fp(\024)e Fr(N)1664 1037 y Fs(1)p Fo(\000)p Fq(\013)1792 1045 y Fi(1)1843 1067 y Fu(d\000\()p Fr(b)p Fu(\))2041 1037 y Fq(\013)2084 1045 y Fi(1)2121 1067 y Fr(;)36 b(w)r(her)r(e)29 b(\013)2488 1079 y Fs(1)2548 1067 y Fp(\024)23 b Fu(1)p Fr(:)p @beginspecial @setspecial @endspecial 1200 1202 a Fu(d\000\()p Fr(b)1366 1167 y Fq(\034)1397 1175 y Fi(1)1433 1202 y Fu(\))1465 1167 y Fq(\013)1508 1175 y Fi(1)1568 1202 y Fp(\024)f Fu(d\000\()p Fr(b)1821 1167 y Fq(\034)1852 1175 y Fi(2)1889 1202 y Fu(\))1921 1167 y Fq(\013)1964 1175 y Fi(2)2024 1202 y Fu(d\000\()p Fr(b)2190 1167 y Fq(\034)2221 1175 y Fi(3)2257 1202 y Fu(\))2289 1167 y Fq(\013)2332 1175 y Fi(3)2369 1202 y Fr(;)2742 1135 y @beginspecial @setspecial @endspecial 0 1333 a Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial Fr(\013)287 1345 y Fs(1)348 1333 y Fu(=)22 b Fr(\013)488 1345 y Fs(2)544 1333 y Fu(+)c Fr(\013)680 1345 y Fs(3)718 1333 y Fr(;)29 b Fn(and)45 b Fr(\013)999 1345 y Fs(1)1036 1333 y Fr(\034)1072 1345 y Fs(1)1132 1333 y Fu(=)23 b Fr(\013)1273 1345 y Fs(2)1310 1333 y Fr(\034)1346 1345 y Fs(2)1402 1333 y Fu(+)18 b Fr(\013)1538 1345 y Fs(3)1576 1333 y Fr(\034)1612 1345 y Fs(3)1649 1333 y Fn(.)0 1349 y @beginspecial @setspecial @endspecial 100 1483 a Fu(Com)n(bining)27 b(Lemma)g @beginspecial @setspecial @endspecial(2.5)g(and)g(Lemma)h (2.3)e(w)n(e)i(obtain)f(a)g(sligh)n(tly)g(more)g(general)f(estimate.)0 1499 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 139 x Fz(Lemma)k(2.6.)40 b Fn(L)l(et)33 b Fr(r)n(;)14 b(s;)g(p;)g(q)34 b Fp(2)d Fk(N)t Fn(.)58 b(L)l(et)64 b Fr(\034)1420 1608 y Fo(0)1444 1650 y Fq(j)1510 1638 y Fu(:=)30 b(\()p Fr(\034)1705 1608 y Fo(0)1730 1596 y Fq(i)1730 1658 y(j)1765 1638 y Fu(\))1797 1650 y Fs(1)p Fq(::p)1908 1638 y Fr(;)76 b(\034)2043 1650 y Fq(j)2109 1638 y Fu(:=)30 b(\()p Fr(\034)2304 1608 y Fq(i)2295 1659 y(j)2333 1638 y Fu(\))2365 1650 y Fs(1)p Fq(::q)2474 1638 y Fr(;)e(j)36 b Fu(=)30 b(1)p Fr(::)p Fu(3)k Fn(b)l(e)g(se)l(quenc)l(es)f(of)i(r)l(e)l(al)g(numb)l(ers)0 1759 y(such)30 b(that)g @beginspecial @setspecial @endspecial Fr(\034)404 1729 y Fo(0)428 1717 y Fq(i)428 1780 y Fs(1)488 1759 y Fu(=)22 b Fr(\034)620 1729 y Fo(0)644 1717 y Fq(i)644 1780 y Fs(2)700 1759 y Fu(+)c Fr(\034)828 1729 y Fo(0)852 1717 y Fq(i)852 1780 y Fs(3)889 1759 y Fr(;)60 b(\034)1017 1729 y Fq(i)1008 1780 y Fs(1)1069 1759 y Fu(=)23 b Fr(\034)1202 1729 y Fq(i)1193 1780 y Fs(2)1249 1759 y Fu(+)18 b Fr(\034)1377 1729 y Fq(i)1368 1780 y Fs(3)1435 1759 y Fn(.)39 b(Then)303 1965 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)399 1857 y Fq(p)363 1886 y Fl(Y)363 2063 y Fq(i)p Fs(=1)470 1965 y Fu(\()p Fr(N)569 1984 y Fq(\034)607 1967 y Fg(0)629 1964 y Fj(i)629 2002 y Fi(2)684 1965 y Fu(+)18 b(1\))841 1930 y Fo(\000)902 1908 y Fi(1)p 902 1917 29 3 v 902 1950 a(2)945 1965 y Fu(\()p Fr(N)1044 1984 y Fq(\034)1082 1967 y Fg(0)1104 1964 y Fj(i)1104 2002 y Fi(3)1159 1965 y Fu(+)g(1\))1316 1930 y Fo(\000)1377 1908 y Fi(1)p 1377 1917 V 1377 1950 a(2)1420 1965 y Fr(W)1498 1977 y Fq(r)n(;s)1634 1857 y(q)1597 1886 y Fl(Y)1596 2063 y Fq(i)p Fs(=1)1704 1965 y Fu(\()p Fr(N)1803 1984 y Fq(\034)1841 1964 y Fj(i)1834 2002 y Fi(2)1889 1965 y Fu(+)g(1\))2046 1930 y Fo(\000)2108 1908 y Fi(1)p 2108 1917 V 2108 1950 a(2)2150 1965 y Fu(\()p Fr(N)2249 1984 y Fq(\034)2287 1964 y Fj(i)2280 2002 y Fi(3)2335 1965 y Fu(+)g(1\))2492 1930 y Fo(\000)2554 1908 y Fi(1)p 2554 1917 V 2554 1950 a(2)2596 1965 y Fp(jj)23 b(\024)g(jj)p Fu(\()p Fr(P)2896 1930 y Fq(\034)2934 1905 y Fg(0)2956 1938 y Fi(1)2884 1985 y Fq(p;r)2993 1965 y Fu(\))3025 1930 y Fo(\000)3087 1908 y Fi(1)p 3087 1917 V 3087 1950 a(2)3143 1965 y Fr(w)16 b Fu(\()p Fr(P)3315 1930 y Fq(\034)3346 1938 y Fi(1)3303 1985 y Fq(q)r(;s)3392 1965 y Fu(\))3424 1930 y Fo(\000)3486 1908 y Fi(1)p 3486 1917 V 3486 1950 a(2)3528 1965 y Fp(jj)p Fr(:)p @beginspecial @setspecial @endspecial 100 2187 a Fu(The)27 b(follo)n(wing)g(estimate) g(is)h(an)f(immediate)h(application)f(of)h(Lemma)f @beginspecial @setspecial @endspecial(2.6.)0 2203 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 134 x Fz(Corollary)33 b(2.7.)40 b Fn(L)l(et)29 b @beginspecial @setspecial @endspecial Fr(r)n(;)14 b(s;)g(\013;)g(\014)35 b Fn(b)l(e)29 b(p)l(ositive)j(inte)l(gers)d(such)h(that)g Fr(\013)23 b Fp(\024)g Fr(r)n(;)14 b(\014)27 b Fp(\024)c Fr(s)p Fn(.)39 b(Then)30 b(the)g(fol)t(lowing)i(assertion)f(holds)710 2478 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 752 2487 a @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(N)c Fu(+)18 b(1\))1081 2452 y Fo(\000)1143 2427 y Fj(r)q Fg(\000)p Fj(\013)p 1143 2440 113 3 v 1185 2473 a Fi(2)1283 2487 y Fu(\(d\000\()p Fr(!)s Fu(\))h(+)f(1\))1708 2452 y Fo(\000)1770 2430 y Fj(\013)p 1770 2439 38 3 v 1775 2473 a Fi(2)1835 2487 y Fr(W)1913 2499 y Fq(r)n(;s)2012 2487 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))2295 2452 y Fo(\000)2357 2426 y Fj(s)p Fg(\000)p Fj(\014)p 2357 2440 108 3 v 2397 2473 a Fi(2)2479 2487 y Fu(\(d\000\()p Fr(!)s Fu(\))h(+)f(1\))2904 2452 y Fo(\000)2966 2426 y Fj(\014)p 2966 2440 35 3 v 2969 2473 a Fi(2)3015 2487 y Fp(jj)23 b(\024)3190 2478 y @beginspecial @setspecial @endspecial 803 2733 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 215 w Fu(inf)905 2793 y Fg(f)p Fj(\034)969 2779 y Fg(0)964 2814 y Fj(i)991 2793 y(;\034)1038 2806 y(i)1064 2793 y Fg(2)p Fi([0)p Fj(;)p Fi(1])p Fg(j)803 2820 y Ff(P)884 2864 y Fq(\034)922 2844 y Fg(0)915 2883 y Fj(i)944 2864 y Fs(=)p Fq(\013;)1069 2820 y Ff(P)1150 2864 y Fq(\034)1181 2872 y Fj(i)1207 2864 y Fs(=)p Fq(\014)s Fo(g)1370 2733 y Fp(jj)1467 2629 y Fq(r)1430 2654 y Fl(Y)1430 2831 y Fq(i)p Fs(=1)1565 2733 y Fr(!)s Fu(\()p Fr(k)1698 2699 y Fo(0)1695 2754 y Fq(i)1723 2733 y Fu(\))1755 2699 y Fo(\000)1817 2661 y Fj(\034)1850 2644 y Fg(0)1845 2680 y Fj(i)p 1817 2686 56 3 v 1831 2719 a Fi(2)1914 2733 y Fr(w)2042 2629 y Fq(s)2004 2654 y Fl(Y)2004 2831 y Fq(i)p Fs(=1)2139 2733 y Fr(!)s Fu(\()p Fr(k)2269 2745 y Fq(i)2297 2733 y Fu(\))2329 2699 y Fo(\000)2391 2667 y Fj(\034)2419 2680 y(i)p 2391 2686 55 3 v 2404 2719 a Fi(2)2459 2733 y Fp(jj)2505 2762 y Fo(B)q Fs(\()p Fo(K\012)p Fm(h)2722 2745 y Fg(\012)p Fj(s)2800 2762 y Fq(;)p Fo(K\012)p Fm(h)2966 2745 y Fg(\012)p Fj(r)3044 2762 y Fs(\))3074 2733 y Fr(:)p @beginspecial @setspecial @endspecial 0 2941 a @beginspecial @setspecial @endspecial 71 x Fu(2.3.)47 b Fz(The)33 b(Nelson)e(mo)s(del.)38 b Fu(The)28 b(Nelson)g(mo)r(del)h ([21)o(])f(describ)r(es)g(a)g(system)g(of)g @beginspecial @setspecial @endspecial Fr(P)40 b Fu(nonrelativistic)28 b(particles)f(coupled)0 3112 y(to)c(a)f(scalar)f(relativistic)h (\014eld)h(of)g(b)r(osons)f(b)n(y)g(a)h(lo)r(cal,)g(translation)f(in)n (v)-5 b(arian)n(t)21 b(in)n(teraction.)35 b(It)23 b(exhibits)g(a)f (relativ)n(ely)g(mild)0 3211 y(ultra)n(violet)29 b(div)n(ergence,)g (and)h(w)n(as)f(the)i(\014rst)f(QFT)g(mo)r(del,)h(on)f(whic)n(h)g(a)f (renormalization)f(pro)r(cedure)i(w)n(as)f(rigorously)0 3311 y(carried)d(on.)0 3410 y(W)-7 b(e)28 b(consider)f(the)h(atomic)f (Hamiltonian)g(of)h(the)g(system)f(of)g @beginspecial @setspecial @endspecial Fr(P)40 b Fu(nonrelativistic)27 b(con\014ned)g(particles)g(as)g(follo)n(ws)1338 3622 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(K)i Fu(:=)1603 3566 y(1)p 1558 3603 132 4 v 1558 3679 a(2)p Fr(M)1747 3518 y Fq(P)1713 3543 y Fl(X)1716 3720 y Fq(j)s Fs(=1)1861 3622 y Fr(D)1932 3588 y Fs(2)1930 3643 y Fq(j)2001 3622 y Fu(+)18 b Fr(V)h Fu(\(x)2227 3634 y Fs(1)2264 3622 y Fr(;)14 b(:::;)g Fu(x)2451 3634 y Fq(P)2507 3622 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3853 a Fu(It)24 b(acts)e(on)h(the)h(Hilb)r (ert)g(space)f @beginspecial @setspecial @endspecial Fr(L)1059 3823 y Fs(2)1095 3853 y Fu(\()p Fk(R)1182 3823 y Fs(3)p Fq(P)1276 3853 y Fr(;)14 b(d)p Fu(x\))24 b(whic)n(h)f(w)n(e)g (denote)g(in)g(the)h(sequel)f(b)n(y)g @beginspecial @setspecial @endspecial Fp(K)q Fu(.)36 b(W)-7 b(e)24 b(assume)e(that)i Fr(V)42 b Fp(2)23 b Fr(L)3600 3823 y Fs(2)3600 3877 y(lo)r(c)3687 3853 y Fu(\()p Fk(R)3773 3823 y Fs(3)p Fq(P)3868 3853 y Fu(\))0 3955 y(and)29 b @beginspecial @setspecial @endspecial Fr(V)44 b Fp(\025)25 b Fu(0.)41 b(Kato's)27 b(inequalit)n(y)i(giv)n(es) f(that)h @beginspecial @setspecial @endspecial Fr(K)34 b Fu(is)29 b(essen)n(tially)f(selfadjoin)n(t)h(on)g Fr(C)2660 3925 y Fo(1)2654 3976 y Fs(0)2730 3955 y Fu(\()p Fk(R)2817 3925 y Fs(3)p Fq(P)2911 3955 y Fu(\),)h(see)e @beginspecial @setspecial @endspecial([23)o(,)i(Thm.)41 b(X.28].)g(W)-7 b(e)0 4063 y(set)28 b Fp(h)p Fu(x)p Fp(i)23 b Fu(:=)g(\()p Fp(j)p Fu(x)p Fp(j)494 4033 y Fs(2)550 4063 y Fu(+)18 b(1\))717 4011 y Fi(1)p 716 4020 29 3 v 716 4053 a(2)759 4063 y Fu(.)37 b(W)-7 b(e)28 b(will)g(also)e(assume)1529 4215 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(V)42 b Fp(\025)22 b Fr(c)1756 4136 y Fl(X)1804 4313 y Fq(i)1876 4215 y Fp(h)p Fu(x)1952 4227 y Fq(i)1980 4215 y Fp(i)2012 4181 y Fq(\013)2106 4215 y Fr(;)14 b(\013)23 b(>)g Fu(2)p Fr(:)p @beginspecial @setspecial @endspecial 0 4451 a Fu(W)-7 b(e)28 b(notice)f(that)h(for)f @beginspecial @setspecial @endspecial(0)c Fp(\024)g Fr(\014)k Fp(\024)c Fu(1,)k Fp(h)p Fr(D)r Fp(i)1236 4421 y Fq(\014)1281 4451 y Fu(\()p Fr(K)d Fu(+)18 b(1\))1565 4421 y Fo(\000)1627 4394 y Fj(\014)p 1627 4408 35 3 v 1630 4441 a Fi(2)1676 4451 y Fr(;)28 b Fp(h)p Fu(x)p Fp(i)1835 4421 y Fq(\014)1880 4451 y Fu(\()p Fr(K)c Fu(+)18 b(1\))2164 4421 y Fo(\000)2226 4394 y Fj(\014)p 2226 4408 V 2229 4441 a Fi(2)2303 4451 y Fu(are)26 b(b)r(ounded)i(op)r(erators.)100 4550 y(The)d(b)r(oson)g (one)h(particle)f(space)g(is)g(the)h(Hilb)r(ert)g(space)f @beginspecial @setspecial @endspecial Fm(h)f Fu(:=)e Fr(L)2133 4520 y Fs(2)2170 4550 y Fu(\()p Fk(R)2256 4520 y Fs(3)2300 4550 y Fr(;)14 b(dk)s Fu(\))p Fr(;)26 b Fu(where)f Fr(k)j Fu(denotes)e(the)g(b)r(oson)f(momen)n(tum)0 4654 y(observ)-5 b(able.)47 b(The)31 b(b)r(oson)g(p)r(osition)g(observ)-5 b(able)30 b @beginspecial @setspecial @endspecial Fp(\000)1667 4619 y Fo(r)1722 4628 y Fj(k)p 1667 4635 92 4 v 1703 4682 a Fs(i)1800 4654 y Fu(will)i(b)r(e)f(denotes)g(b)n(y)h(the)f (italic)h(letter)f @beginspecial @setspecial @endspecial Fr(x)p Fu(.)49 b(This)31 b(should)g(not)h(b)r(e)0 4753 y(confused)26 b(with)g(the)g(n)n(ucleon)f(p)r(osition)g(observ)-5 b(able)25 b(denoted)g(b)n(y)h(the)g(roman)e(letter)i @beginspecial @setspecial @endspecial(x.)36 b(The)26 b(free)f(b)r(osonic)g(Hamiltonian)0 4853 y(is)i(de\014ned)h(b)n(y)g (the)g(second)f(quan)n(tization)f(of)i(a)f(single)g(b)r(oson)g(energy) -7 b(.)36 b(It)28 b(acts)f(on)g(the)h(b)r(osonic)f(F)-7 b(o)r(c)n(k)28 b(space)e @beginspecial @setspecial @endspecial(\000\()p Fm(h)p Fu(\).)1365 5062 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1406 4987 a @beginspecial @setspecial @endspecial Fr(H)1475 4999 y Fq(b)1532 4987 y Fu(:=)c(d\000\()p Fr(!)s Fu(\))p Fr(;)1406 5135 y(!)s Fu(\()p Fr(k)s Fu(\))h(:=)g(\()p Fp(j)p Fr(k)s Fp(j)1829 5101 y Fs(2)1885 5135 y Fu(+)18 b Fr(m)2041 5101 y Fs(2)2078 5135 y Fu(\))2120 5079 y Fi(1)p 2120 5088 29 3 v 2120 5121 a(2)2163 5135 y Fr(;)60 b(m)23 b(>)f Fu(0)p Fr(:)2535 5062 y @beginspecial @setspecial @endspecial 0 5271 a Fu(It)28 b(is)f(essen)n(tially)g(selfadjoin)n(t)g(on)h @beginspecial @setspecial @endspecial(\000)1138 5283 y Fs(\014n)1216 5271 y Fu(\()p Fp(D)r Fu(\()p Fr(!)s Fu(\)\).)0 5371 y(The)g(Hilb)r(ert)g(space)f(of)g(the)h(join)n(t)g (system)f(is)h @beginspecial @setspecial @endspecial Fp(H)c Fu(:=)e Fp(K)e(\012)e Fu(\000\()p Fm(h)p Fu(\).)38 b(The)28 b(Hamiltonian)f(without)h(in)n(teraction)f(is)g(giv)n(en)g(b)n (y)1527 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(H)1596 5518 y Fs(0)1657 5506 y Fu(:=)c Fr(K)g Fp(\012)18 b Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fr(H)2316 5518 y Fq(b)2349 5506 y Fr(:)p @beginspecial @setspecial @endspecial eop %%Page: 9 9 9 8 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)958 b(9)0 489 y Fu(This)38 b(is)h(a)e(selfadjoin)n(t)i (op)r(erator)d(since)i @beginspecial @setspecial @endspecial Fr(K)31 b Fp(\012)26 b Fu(1)-15 b(l)37 b(and)i @beginspecial @setspecial @endspecial(1)-15 b(l)25 b Fp(\012)g Fr(H)2031 501 y Fq(b)2103 489 y Fu(comm)n(ute)38 b(on)g Fp(H)q Fu(.)69 b(The)38 b(lo)r(cal)g(translation)f(in)n(v)-5 b(arian)n(t)0 589 y(in)n(teraction)27 b(b)r(et)n(w)n(een)g(n)n(ucleons) g(and)g(b)r(osons)g(is)h(giv)n(en)f(b)n(y)g(the)h(formal)f(expression) 668 804 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 702 700 a Fq(P)668 725 y Fl(X)671 902 y Fq(j)s Fs(=1)801 804 y Fr(')p Fu(\(x)931 816 y Fq(j)967 804 y Fu(\))p Fr(;)74 b Fu(where)13 b Fr(')p Fu(\(x\))24 b(:=)f([2\(2)p Fr(\031)s Fu(\))1840 769 y Fs(3)1877 804 y Fu(])1900 769 y Fo(\000)1962 747 y Fi(1)p 1962 756 29 3 v 1962 789 a(2)2041 691 y Fl(Z)2138 804 y Fu(e)2175 769 y Fo(\000)p Fs(i)p Fo(h)p Fq(k)q(;)p Fs(x)p Fo(i)2395 804 y Fu(\()p Fr(a)2471 769 y Fo(\003)2509 804 y Fu(\()p Fr(k)s Fu(\))c(+)f Fr(a)p Fu(\()p Fp(\000)p Fr(k)s Fu(\)\))3046 748 y Fr(dk)p 2982 785 218 4 v 2982 873 a(!)s Fu(\()p Fr(k)s Fu(\))3157 821 y Fi(1)p 3157 830 29 3 v 3157 864 a(2)3209 804 y Fr(:)p @beginspecial @setspecial @endspecial 0 1026 a Fu(The)34 b(in)n(teraction)e(term)i (cannot)f(b)r(e)g(de\014ned)h(as)f(an)g(op)r(erator)f(on)h @beginspecial @setspecial @endspecial Fp(H)i Fu(with)f(a)f(dense)g (domain.)54 b(This)34 b(comes)f(from)g(the)0 1130 y(fact)h(that)h @beginspecial @setspecial @endspecial Fr(!)412 1100 y Fo(\000)473 1077 y Fi(1)p 473 1086 V 473 1120 a(2)558 1130 y Fr(=)-51 b Fp(2)48 b Fr(L)709 1100 y Fs(2)746 1130 y Fu(\()p Fk(R)832 1100 y Fs(3)875 1130 y Fr(;)14 b(dk)s Fu(\),)36 b(b)r(ecause)e(the)h(in)n(tegral)d(div)n(erges)h(for)g (large)g @beginspecial @setspecial @endspecial Fr(k)s Fu(.)56 b(This)34 b(phenomenon)g(is)g(kno)n(wn)f(as)g(an)0 1229 y @beginspecial @setspecial @endspecial Fr(ul)r(tr)r(av)s(iol)r (et)22 b(pr)r(obl)r(em)p Fu(.)36 b(In)28 b(order)e(to)i(ha)n(v)n(e)e(a) h(w)n(ell)g(de\014ned)h(op)r(erator,)e(one)i(in)n(tro)r(duces)f (cuto\013)g(in)n(teractions:)922 1411 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(')976 1423 y Fq(\024)1019 1411 y Fu(\(x\))d(:=)e([2\(2)p Fr(\031)s Fu(\))1482 1377 y Fs(3)1520 1411 y Fu(])1543 1377 y Fo(\000)1605 1355 y Fi(1)p 1605 1364 V 1605 1397 a(2)1684 1298 y Fl(Z)1781 1411 y Fu(e)1818 1377 y Fo(\000)p Fs(i)p Fo(h)p Fq(k)q(;)p Fs(x)p Fo(i)2037 1411 y Fu(\()p Fr(a)2113 1377 y Fo(\003)2152 1411 y Fu(\()p Fr(k)s Fu(\))d(+)f Fr(a)p Fu(\()p Fp(\000)p Fr(k)s Fu(\)\))2631 1355 y Fr(\037)2683 1367 y Fq(\024)2726 1355 y Fu(\()p Fr(k)s Fu(\))p 2625 1392 218 4 v 2625 1480 a Fr(!)s Fu(\()p Fr(k)s Fu(\))2800 1429 y Fi(1)p 2800 1438 29 3 v 2800 1471 a(2)2866 1411 y Fr(dk)s(;)p @beginspecial @setspecial @endspecial 1073 1739 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(I)1109 1751 y Fq(\024)1176 1739 y Fu(:=)1321 1635 y Fq(P)1287 1660 y Fl(X)1289 1837 y Fq(j)s Fs(=1)1434 1739 y Fr(')1488 1751 y Fq(\024)1531 1739 y Fu(\(x)1607 1751 y Fq(j)1643 1739 y Fu(\))p Fr(;)37 b(v)1775 1751 y Fq(\024)1841 1739 y Fu(:=)23 b([\(2)p Fr(\031)s Fu(\))2131 1704 y Fs(3)2168 1739 y Fu(])2191 1704 y Fo(\000)2253 1682 y Fi(1)p 2253 1691 V 2253 1724 a(2)2312 1682 y Fr(\037)2364 1694 y Fq(\024)2407 1682 y Fu(\()p Fr(k)s Fu(\))p 2306 1720 218 4 v 2306 1808 a Fr(!)s Fu(\()p Fr(k)s Fu(\))2481 1756 y Fi(1)p 2481 1765 29 3 v 2481 1799 a(2)2547 1739 y Fu(e)2584 1704 y Fo(\000)p Fs(i)p Fo(h)p Fq(k)q(;)p Fs(x)p Fo(i)2804 1739 y Fr(:)p @beginspecial @setspecial @endspecial 0 1971 a Fu(Here)33 b @beginspecial @setspecial @endspecial Fr(\037)g Fu(a)f(is)h(p)r(ositiv)n(e)g(function)g(in)g Fr(C)1260 1941 y Fo(1)1331 1971 y Fu(\()p Fk(R)1417 1941 y Fs(3)1460 1971 y Fu(\))h(suc)n(h)e(that)i @beginspecial @setspecial @endspecial(0)d Fp(\024)h Fr(\037)p Fu(\()p Fr(k)s Fu(\))g Fp(\024)g Fu(1)p Fr(;)45 b(\037)p Fu(\()p Fr(k)s Fu(\))33 b(=)e(1)i(for)f Fp(j)p Fr(k)s Fp(j)g(\024)g Fu(1)g(and)h @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()p Fr(k)s Fu(\))f(=)g(0)h(for)0 2073 y Fp(j)p Fr(k)s Fp(j)23 b(\025)g Fu(2.)36 b(W)-7 b(e)28 b(set)g @beginspecial @setspecial @endspecial Fr(\037)629 2085 y Fq(\024)672 2073 y Fu(\()p Fr(k)s Fu(\))23 b(:=)g Fr(\037)p Fu(\()1011 2040 y Fq(k)p 1010 2054 39 4 v 1010 2102 a(\024)1059 2073 y Fu(\).)0 2102 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 123 x Fz(Lemma)30 b(2.8.)40 b Fn(One)29 b(has)h(for)h @beginspecial @setspecial @endspecial Fr(\013)23 b Fp(\025)g Fu(0)1611 2426 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1653 2358 a @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))14 b(d\000\()p Fr(!)s Fu(\))1943 2324 y Fq(\013)2013 2358 y Fp(\024)23 b Fr(H)2177 2324 y Fq(\013)2170 2378 y Fs(0)2224 2358 y Fr(;)p @beginspecial @setspecial @endspecial 1657 2492 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))14 b Fr(K)1829 2458 y Fq(\013)1899 2492 y Fp(\024)23 b Fr(H)2063 2458 y Fq(\013)2056 2513 y Fs(0)2110 2492 y Fr(:)2289 2426 y @beginspecial @setspecial @endspecial 100 2650 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(F)-7 b(or)27 b Fr( )f Fp(2)d Fr(C)796 2620 y Fo(1)790 2671 y Fs(0)867 2650 y Fu(\()p Fk(R)953 2620 y Fs(3)996 2650 y Fu(\))c Fp(\012)f Fu(\000)1182 2662 y Fs(\014n)1260 2650 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)1434 2620 y Fs(3)1478 2650 y Fu(\)\),)28 b(whic)n(h)g(is)f(a)g (core)g(for)g @beginspecial @setspecial @endspecial Fr(H)2355 2662 y Fs(0)2392 2650 y Fu(,)h(one)f(has)1464 2856 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1506 2788 a @beginspecial @setspecial @endspecial (\(d\000\()p Fr(!)s Fu(\))p Fr( )s(;)14 b( )s Fu(\))24 b Fp(\024)e Fu(\()p Fr(H)2150 2800 y Fs(0)2188 2788 y Fr( )s(;)14 b( )s Fu(\))p Fr(;)1506 2922 y Fu(\()p Fr(K)6 b( )s(;)14 b( )s Fu(\))23 b Fp(\024)g Fu(\()p Fr(H)2010 2934 y Fs(0)2047 2922 y Fr( )s(;)14 b( )s Fu(\))p Fr(:)2436 2856 y @beginspecial @setspecial @endspecial 0 3062 a Fu(This)32 b(means)f(that)h @beginspecial @setspecial @endspecial(d\000\()p Fr(!)s Fu(\))e Fp(\024)g Fr(H)1047 3074 y Fs(0)1116 3062 y Fu(and)h Fr(K)36 b Fp(\024)29 b Fr(H)1551 3074 y Fs(0)1589 3062 y Fu(.)49 b(Since)32 b @beginspecial @setspecial @endspecial Fr(H)1951 3074 y Fs(0)1988 3062 y Fr(;)14 b(K)37 b Fu(and)32 b @beginspecial @setspecial @endspecial(d\000\()p Fr(!)s Fu(\))g(comm)n(ute,)h(the)f (sp)r(ectral)f(theorem)g(giv)n(es)0 3161 y(the)d(inequalities)f (announced)g(in)h(the)g(lemma.)p @beginspecial @setspecial @endspecial 2338 w Fh(\004)0 3265 y @beginspecial @setspecial @endspecial Fr(I)36 3277 y Fq(\024)106 3265 y Fu(are)d(w)n (ell-de\014ned)i(op)r(erators)d(on)i Fp(D)r Fu(\(\()p Fr(N)g Fu(+)16 b(1\))1565 3213 y Fi(1)p 1564 3222 29 3 v 1564 3255 a(2)1607 3265 y Fu(\))26 b(as)g(long)g(as)g @beginspecial @setspecial @endspecial Fr(\024)d(<)f Fp(1)27 b Fu(and)f(they)h(are)e Fr(H)2866 3277 y Fs(0)2903 3265 y Fu(-b)r(ounded)i(with)g(in\014nitesimal)0 3365 y(b)r(ound.)37 b(W)-7 b(e)28 b(set)g @beginspecial @setspecial @endspecial Fr(H)630 3377 y Fq(\024)696 3365 y Fu(:=)23 b Fr(H)876 3377 y Fs(0)931 3365 y Fu(+)18 b Fr(I)1050 3377 y Fq(\024)1094 3365 y Fu(.)0 3377 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 139 x Fz(Theorem)30 b(2.9.)40 b Fn(F)-6 b(or)30 b @beginspecial @setspecial @endspecial Fr(\024)23 b(<)g Fp(1)p Fn(,)53 b Fr(H)1139 3528 y Fq(\024)1212 3516 y Fn(is)30 b(a)g(selfadjoint)i(op)l(er)l(ator) f(on)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)2387 3528 y Fs(0)2425 3516 y Fu(\))p Fn(.)0 3537 y @beginspecial @setspecial @endspecial 100 3668 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Using)27 b(Corollary)e(2.4)i(and)g(Lemma)h(2.8,)e(w)n(e)i(pro)n(v)n(e)e(for)h Fr(c)g Fu(indep)r(enden)n(t)i(from)e @beginspecial @setspecial @endspecial Fr(\025)1390 3815 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(I)1472 3827 y Fq(\024)1515 3815 y Fu(\()p Fr(H)1616 3827 y Fs(0)1672 3815 y Fu(+)18 b Fr(\025)p Fu(\))1835 3781 y Fo(\000)p Fs(1)1925 3815 y Fp(jj)23 b(\024)37 b Fr(c)14 b(\025)2194 3781 y Fo(\000)2256 3758 y Fi(1)p 2256 3767 V 2256 3801 a(2)2312 3815 y Fp(jj)p Fr(v)2398 3827 y Fq(\024)2441 3815 y Fp(jj)p Fr(:)p @beginspecial @setspecial @endspecial 0 3953 a Fu(Then)28 b(b)n(y)f(the)h (Kato-Rellic)n(h)e(theorem,)h(one)h(sees)f(that)g @beginspecial @setspecial @endspecial Fr(H)1872 3965 y Fq(\024)1943 3953 y Fu(is)h(a)f(selfadjoin)n(t)g(op)r(erator)f(on)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)3112 3965 y Fs(0)3150 3953 y Fu(\).)630 b Fh(\004)1009 4134 y @beginspecial @setspecial @endspecial Fu(3.)45 b Fv(Constr)n(uction)32 b(of)f(the)h(Nelson)f(Hamil)-6 b(tonian)100 4284 y Fu(In)26 b(this)g(section)g(w)n(e)g(recall)f(the)i (construction)e(in)i @beginspecial @setspecial @endspecial([21)o(])f (of)g(the)h(Nelson)f(Hamiltonian.)36 b(It)26 b(consists)g(in)g (applying)g(to)g(the)0 4383 y(cuto\013)i(Hamiltonians)f @beginspecial @setspecial @endspecial Fr(H)812 4395 y Fq(\024)883 4383 y Fu(a)g(cuto\013)h(dep)r(enden)n(t)g(unitary)f (transformation)f @beginspecial @setspecial @endspecial Fr(U)2502 4395 y Fq(\024)2545 4383 y Fu(,)i(letting)g(then)g Fr(\024)f Fu(to)h Fp(1)p Fu(.)0 4541 y @beginspecial @setspecial @endspecial(3.1.)47 b Fz(Dressing)31 b(T)-8 b(ransformation.)40 b Fu(F)-7 b(or)27 b(a)g(\014xed)h Fr(\024)1744 4553 y Fs(0)1809 4541 y Fu(and)f Fr(\024)c(<)g Fp(1)p Fu(,)k(w)n(e)h(de\014ne)312 4739 y @beginspecial @setspecial @endspecial Fr(g)352 4751 y Fq(\024)394 4739 y Fu(\()p Fr(k)s Fu(\))84 b(:=)e Fp(\000)p Fu(i)856 4683 y(\(2)p Fr(\031)s Fu(\))1012 4653 y Fo(\000)1074 4631 y Fi(3)p 1074 4640 V 1074 4673 a(2)p 856 4720 261 4 v 878 4808 a Fr(!)943 4757 y Fi(1)p 942 4766 29 3 v 942 4799 a(2)985 4808 y Fu(\()p Fr(k)s Fu(\))1150 4683 y Fr(\037)1202 4695 y Fq(\024)1245 4683 y Fu(\()p Fr(k)s Fu(\))19 b Fp(\000)f Fr(\037)1509 4695 y Fq(\024)1548 4703 y Fi(0)1585 4683 y Fu(\()p Fr(k)s Fu(\))p 1150 4720 545 4 v 1228 4806 a Fr(!)s Fu(\()p Fr(k)s Fu(\))h(+)1521 4774 y Fq(k)1557 4757 y Fi(2)p 1505 4788 103 4 v 1505 4835 a Fs(2)p Fq(M)1742 4739 y Fp(2)k Fr(C)1885 4705 y Fo(1)1879 4760 y Fs(0)1956 4739 y Fu(\()p Fk(R)2042 4705 y Fs(3)2085 4739 y Fu(\))p Fr(;)-2140 b Fu(\(3.1\))0 4860 y @beginspecial @setspecial @endspecial 396 5059 a @beginspecial @setspecial @endspecial Fr(G)461 5071 y Fq(\024)588 5059 y @beginspecial @setspecial @endspecial Fu(:=)793 4955 y Fq(P)758 4980 y Fl(X)761 5157 y Fq(j)s Fs(=1)892 5059 y Fu(e)929 5025 y Fo(\000)p Fs(i)p Fo(h)p Fq(k)q(;)p Fs(x)1118 5033 y Fj(j)1149 5025 y Fo(i)1194 5059 y Fr(g)1234 5071 y Fq(\024)1276 5059 y Fu(\()p Fr(k)s Fu(\))38 b Fp(2)23 b(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)20 b(\012)e Fm(h)p Fu(\))p Fr(;)-1957 b Fu(\(3.2\))0 5201 y @beginspecial @setspecial @endspecial 402 5362 a @beginspecial @setspecial @endspecial Fr(C)461 5374 y Fq(\024)588 5362 y @beginspecial @setspecial @endspecial Fu(:=)768 5305 y(1)p 768 5342 42 4 v 768 5419 a(2)905 5283 y Fl(X)834 5461 y Fs(1)p Fo(\024)p Fq(j;`)p Fo(\024)p Fq(P)1124 5249 y Fl(Z)1221 5362 y Fp(j)p Fr(g)1284 5374 y Fq(\024)1327 5362 y Fu(\()p Fr(k)s Fu(\))p Fp(j)1460 5327 y Fs(2)1525 5362 y Fu(sin)p Fp(h)p Fr(k)s(;)14 b Fu(\(x)1818 5374 y Fq(j)1872 5362 y Fp(\000)k Fu(x)1999 5374 y Fq(`)2030 5362 y Fu(\))p Fp(i)c Fr(dk)41 b Fp(2)23 b Fr(S)5 b Fu(\()p Fk(R)2455 5327 y Fs(3)p Fq(P)2550 5362 y Fu(\))p Fr(;)-2605 b Fu(\(3.3\))0 5506 y @beginspecial @setspecial @endspecial eop %%Page: 10 10 10 9 bop 0 226 a Fs(10)1695 b(Z.)29 b(AMMARI)400 530 y Fr(E)461 542 y Fq(\024)588 530 y Fu(:=)82 b @beginspecial @setspecial @endspecial 879 474 a Fr(P)p 768 511 288 4 v 768 587 a Fu(2\(2)p Fr(\031)s Fu(\))966 563 y Fo(\000)p Fs(3)1093 417 y Fl(Z)1262 474 y Fu(1)p 1200 511 166 4 v 1200 587 a Fr(!)s Fu(\()p Fr(k)s Fu(\))1399 474 y(\()p Fr(\037)1483 486 y Fq(\024)1526 474 y Fu(\()p Fr(k)s Fu(\))19 b Fp(\000)f Fr(\037)1790 486 y Fq(\024)1829 494 y Fi(0)1865 474 y Fu(\()p Fr(k)s Fu(\)\))2007 443 y Fs(2)p 1399 511 647 4 v 1495 597 a Fu(\()p Fr(!)s Fu(\()p Fr(k)s Fu(\))h(+)1821 564 y Fq(k)1857 547 y Fi(2)p 1804 578 103 4 v 1804 625 a Fs(2)p Fq(M)1916 597 y Fu(\))2069 530 y Fr(dk)44 b Fp(\000)2383 474 y Fr(P)p 2292 511 246 4 v 2292 587 a Fu(\(2)p Fr(\031)s Fu(\))2448 563 y Fo(\000)p Fs(3)2562 417 y Fl(Z)2669 474 y Fr(\037)2721 486 y Fq(\024)2764 474 y Fu(\()p Fr(k)s Fu(\))p 2669 511 206 4 v 2689 587 a Fr(!)s Fu(\()p Fr(k)s Fu(\))2894 474 y Fr(\037)2946 486 y Fq(\024)2989 474 y Fu(\()p Fr(k)s Fu(\))19 b Fp(\000)f Fr(\037)3253 486 y Fq(\024)3292 494 y Fi(0)3328 474 y Fu(\()p Fr(k)s Fu(\))p 2894 511 545 4 v 2939 597 a(\()p Fr(!)s Fu(\()p Fr(k)s Fu(\))h(+)3265 564 y Fq(k)3301 547 y Fi(2)p 3248 578 103 4 v 3248 625 a Fs(2)p Fq(M)3361 597 y Fu(\))3462 530 y Fr(dk)s(;)-3574 b Fu(\(3.4\))0 650 y @beginspecial @setspecial @endspecial 316 806 a Fr(r)353 818 y Fq(\024)396 806 y Fu(\(x\))84 b(:=)e Fp(\000)p Fu(i)p Fr(k)s Fu(e)929 772 y Fo(\000)p Fs(i)p Fo(h)p Fq(k)q(;)p Fs(x)p Fo(i)1149 806 y Fr(g)1189 818 y Fq(\024)1231 806 y Fu(\()p Fr(k)s Fu(\))p Fr(;)-1364 b Fu(\(3.5\))0 852 y @beginspecial @setspecial @endspecial 405 1008 a @beginspecial @setspecial @endspecial Fr(U)462 1020 y Fq(\024)588 1008 y @beginspecial @setspecial @endspecial Fu(:=)82 b(e)795 974 y Fs(i)p Fq(\036)p Fs(\(G)931 982 y Fj(\024)970 974 y Fs(\)+iC)1113 982 y Fj(\024)1156 1008 y Fr(;)-1179 b Fu(\(3.6\))0 1054 y @beginspecial @setspecial @endspecial 414 1190 a(^)393 1211 y Fr(H)462 1223 y Fq(\024)588 1211 y @beginspecial @setspecial @endspecial Fu(:=)82 b Fr(U)815 1223 y Fq(\024)858 1211 y Fu(\()p Fr(H)959 1223 y Fq(\024)1021 1211 y Fp(\000)18 b Fr(E)1165 1223 y Fq(\024)1208 1211 y Fu(\))p Fr(U)1306 1177 y Fo(\003)1297 1232 y Fq(\024)1344 1211 y Fr(:)-1367 b Fu(\(3.7\))0 1381 y(In)28 b(order)e(to)h(simplify)i(the)f(writing)f(of)g(some)g (form)n(ulas,)g(w)n(e)g(will)h(replace)e(often)i @beginspecial @setspecial @endspecial Fr(r)2636 1393 y Fq(\024)2680 1381 y Fu(\(x)2756 1393 y Fq(j)2791 1381 y Fu(\))p Fr(;)14 b(v)2900 1393 y Fq(\024)2943 1381 y Fu(\(x)3019 1393 y Fq(j)3054 1381 y Fu(\))28 b(b)n(y)g Fr(r)3269 1351 y Fq(j)3267 1402 y(\024)3310 1381 y Fr(;)14 b(v)3390 1351 y Fq(j)3387 1402 y(\024)3430 1381 y Fu(.)0 1405 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 157 x Fz(Lemma)30 b(3.1.)40 b Fn(F)-6 b(or)32 b(a)i(\014xe)l(d)e @beginspecial @setspecial @endspecial Fr(\024)1004 1574 y Fs(0)1073 1562 y Fn(and)i(for)f Fr(\024)1421 1574 y Fs(0)1487 1562 y Fr(<)28 b(\024)g(<)g Fp(1)p Fn(,)34 b @beginspecial @setspecial @endspecial 1912 1541 a Fu(^)1891 1562 y Fr(H)1960 1574 y Fq(\024)2035 1562 y Fn(is)f(a)g(selfadjoint)i(op)l(er)l(ator)f(on)f(the)g(domain)h Fp(D)r Fu(\()p Fr(H)3666 1574 y Fs(0)3704 1562 y Fu(\))e Fn(and)0 1661 y(e)l(qual)e(to)1085 1843 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1153 1822 a Fu(^)1132 1843 y Fr(H)1201 1855 y Fq(\024)1267 1843 y Fu(=)22 b Fr(H)1423 1855 y Fs(0)1479 1843 y Fu(+)1649 1764 y Fl(X)1562 1942 y Fs(1)p Fo(\024)p Fq(i)f Fu(2)p Fr(\024)3840 3089 y Fs(0)3877 3077 y Fu(.)0 3192 y(The)31 b(in)n(tegrand)f(of)h @beginspecial @setspecial @endspecial Fr(V)714 3149 y Fs(\(1\))695 3202 y Fq(\024)803 3192 y Fr(;)14 b(V)907 3149 y Fs(\(2\))888 3202 y Fq(\024)1027 3192 y Fu(con)n(tain)30 b(the)i(term)f @beginspecial @setspecial @endspecial(1)20 b Fp(\000)g Fr(\037)1871 3204 y Fq(\024)1910 3212 y Fi(0)1947 3192 y Fu(,)32 b(then)f(there)g (exist)g Fr(\024)2658 3204 y Fs(0)2726 3192 y Fu(suc)n(h)g(that)g @beginspecial @setspecial @endspecial Fr(V)3167 3149 y Fs(\(1\))3148 3202 y Fq(\024)3270 3192 y Fu(\(resp)p Fr(:)14 b(V)3554 3149 y Fs(\(2\))3536 3202 y Fq(\024)3644 3192 y Fu(\))31 b(has)f(a)0 3292 y(small)d @beginspecial @setspecial @endspecial Fr(L)274 3262 y Fo(1)344 3292 y Fu(\()p Fk(R)430 3262 y Fs(3)474 3292 y Fu(\))c(\(resp)p Fr(:)14 b(L)804 3262 y Fq(s)839 3292 y Fu(\()p Fk(R)925 3262 y Fs(3)968 3292 y Fu(\))p Fr(;)g(s)23 b(>)g Fu(2\))28 b(norm)f(for)g @beginspecial @setspecial @endspecial Fr(\024)c(>)g Fu(2)p Fr(\024)1882 3304 y Fs(0)1919 3292 y Fu(.)37 b(After)28 b(a)f(c)n(hange)g(of)g(v)-5 b(ariables)27 b(to)g(separate)g(the)h(motion)f(of)0 3408 y(the)32 b(cen)n(ter)g(of)g (mass,)h(the)f(term)g @beginspecial @setspecial @endspecial Fr(V)1148 3365 y Fs(\(2\))1129 3417 y Fq(\024)1269 3408 y Fu(b)r(ecomes)g @beginspecial @setspecial @endspecial Fr(V)1671 3365 y Fs(\(3\))1652 3417 y Fq(\024)1781 3408 y Fu(+)21 b Fr(V)1934 3365 y Fs(\(4\))1915 3417 y Fq(\024)2054 3408 y Fp(2)31 b Fr(L)2197 3378 y Fo(1)2288 3408 y Fu(+)21 b Fr(L)2431 3378 y Fq(q)2499 3408 y Fu(where)32 b(2)e Fp(\024)g Fr(q)k(<)c Fp(1)p Fu(.)50 b(Using)32 b(the)h(Sob)r(olev)0 3519 y(injection)24 b @beginspecial @setspecial @endspecial Fr(H)411 3489 y Fs(1)448 3519 y Fu(\()p Fk(R)534 3489 y Fs(3)577 3519 y Fu(\))g Fp(!)f Fr(L)796 3489 y Fq(q)828 3464 y Fg(0)854 3519 y Fu(\()p Fk(R)941 3489 y Fs(3)984 3519 y Fu(\))p Fr(;)28 b Fu(2)23 b Fp(\024)f Fr(q)1259 3489 y Fo(0)1306 3519 y Fp(\024)g Fu(6,)i(w)n(e)f(obtain,)h(b)n(y)g(a)f (con)n(v)n(enien)n(t)f(c)n(hoice)h(of)g @beginspecial @setspecial @endspecial Fr(q)k Fu(and)c Fr(q)3055 3489 y Fo(0)3078 3519 y Fu(,)i(that)f Fr(V)3369 3476 y Fs(\(2\))3350 3529 y Fq(\024)3481 3519 y Fu(is)g Fr(H)3630 3531 y Fs(0)3667 3519 y Fp(\000)p Fu(form)0 3619 y(b)r(ounded)k @beginspecial @setspecial @endspecial([21)o(].)3344 b Fh(\004)0 3639 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 135 x Fz(Theorem)30 b(3.6.)40 b Fn(Ther)l(e)31 b(is)f(a)g(unique)g(selfadjoint)i(op)l(er)l(ator)2012 3753 y Fu(^)1990 3774 y Fr(H)2059 3786 y Fo(1)2159 3774 y Fn(acting)f(on)f @beginspecial @setspecial @endspecial Fp(H)q Fn(,)g(satisfying)1209 4014 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1250 3912 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))65 b(lim)1346 3962 y Fq(\024)p Fo(!1)1518 3912 y Fu(\()1572 3891 y(^)1550 3912 y Fr(H)1619 3924 y Fq(\024)1680 3912 y Fp(\000)18 b Fr(z)t Fu(\))1838 3878 y Fo(\000)p Fs(1)1950 3912 y Fu(=)23 b(\()2092 3891 y(^)2070 3912 y Fr(H)2139 3924 y Fo(1)2228 3912 y Fp(\000)18 b Fr(z)t Fu(\))2386 3878 y Fo(\000)p Fs(1)2474 3912 y Fr(;)p @beginspecial @setspecial @endspecial 1255 4095 a Fu(_)-28 b Fr({)t Fu(_)h Fr({)p Fu(\))23 b Fr(s)c Fp(\000)46 b Fu(lim)1500 4144 y Fq(\024)p Fo(!1)1685 4095 y Fu(e)1722 4060 y Fo(\000)p Fs(i)p Fq(t)1834 4045 y Fs(^)1818 4060 y Fq(H)1872 4068 y Fj(\024)1951 4095 y Fu(=)23 b(e)2076 4060 y Fo(\000)p Fs(i)p Fq(t)2188 4045 y Fs(^)2172 4060 y Fq(H)2226 4068 y Fg(1)2305 4095 y Fn(for)28 b Fr(t)23 b Fp(2)h Fk(R)p Fr(:)2691 4014 y @beginspecial @setspecial @endspecial 0 4294 a Fn(The)31 b(domain)g(of)f @beginspecial @setspecial @endspecial 586 4273 a Fu(^)564 4294 y Fr(H)633 4306 y Fo(1)733 4294 y Fn(satis\014es)g Fp(D)r Fu(\()1166 4273 y(^)1143 4294 y Fr(H)1212 4306 y Fo(1)1283 4294 y Fu(\))24 b Fp(\032)e(D)r Fu(\()1547 4273 y(^)1524 4294 y Fr(H)1611 4223 y Fi(1)p 1611 4232 V 1611 4265 a(2)1593 4303 y Fo(1)1664 4294 y Fu(\))i(=)e Fp(D)r Fu(\()p Fr(H)1992 4223 y Fi(1)p 1992 4232 V 1992 4265 a(2)1974 4316 y Fs(0)2034 4294 y Fu(\))p Fn(.)0 4316 y @beginspecial @setspecial @endspecial 100 4441 a Fz(Pro)s(of.)67 b @beginspecial @setspecial @endspecial 35 w Fu(The)35 b(pro)r(of)f(is)h(based)g(on)f (the)h(Thm.)60 b(A.1)35 b(in)g(the)g(app)r(endix.)59 b(Let)35 b(us)g(apply)g(no)n(w)f(this)h(theorem)g(with)0 4563 y @beginspecial @setspecial @endspecial Fr(B)63 4575 y Fq(\024)106 4563 y Fu(\()p Fr(\036)p Fu(\))24 b(=)f(\()450 4484 y Fl(X)363 4663 y Fs(1)p Fo(\024)p Fq(i)g Fu(0)p Fr(;)g Fu(is)h(norm)f(con)n(tin)n (uous,)h(w)n(e)f(obtain)1063 1878 y @beginspecial @setspecial @endspecial 166 w Fp(jj)p Fr(U)1341 1844 y Fo(\003)1332 1899 y Fq(\024)1379 1878 y Fu(\()1433 1857 y(^)1411 1878 y Fr(H)1480 1890 y Fq(\024)1542 1878 y Fp(\000)18 b Fr(z)t Fu(\))1700 1844 y Fo(\000)p Fs(1)1788 1878 y Fr(U)1845 1890 y Fq(\024)1907 1878 y Fp(\000)g Fr(U)2056 1844 y Fo(\003)2047 1899 y(1)2117 1878 y Fu(\()2171 1857 y(^)2149 1878 y Fr(H)2218 1890 y Fo(1)2307 1878 y Fp(\000)g Fr(z)t Fu(\))2465 1844 y Fo(\000)p Fs(1)2553 1878 y Fr(U)2610 1890 y Fo(1)2680 1878 y Fp(jj)24 b(\024)0 1924 y @beginspecial @setspecial @endspecial 1229 2041 a Fr(c)p Fu(\()p Fp(jj)p Fu(\()p Fr(W)12 b Fu(\()p Fr(G)1562 2053 y Fq(\024)1606 2041 y Fu(\))19 b Fp(\000)f Fr(W)12 b Fu(\()p Fr(G)1927 2053 y Fo(1)1998 2041 y Fu(\)\)\()p Fr(N)28 b Fu(+)18 b(1\))2346 2007 y Fo(\000)p Fq(\017)2429 2041 y Fp(jj)0 2087 y @beginspecial @setspecial @endspecial 1229 2212 a Fu(+)p Fp(jj)p Fu(\()1394 2191 y(^)1372 2212 y Fr(H)1441 2224 y Fq(\024)1502 2212 y Fp(\000)g Fr(z)t Fu(\))1660 2177 y Fo(\000)p Fs(1)1767 2212 y Fp(\000)g Fu(\()1905 2191 y(^)1882 2212 y Fr(H)1951 2224 y Fo(1)2041 2212 y Fp(\000)g Fr(z)t Fu(\))2199 2177 y Fo(\000)p Fs(1)2287 2212 y Fp(jj)1229 2378 y Fu(+)p Fp(jj)p Fu(e)1377 2343 y Fs(i)p Fq(C)1444 2351 y Fj(\024)1504 2378 y Fp(\000)g Fu(e)1624 2343 y Fs(i)p Fq(C)1691 2351 y Fg(1)1755 2378 y Fp(jj)p Fu(\))p Fr(:)0 2515 y Fu(The)j(application)g(of)g(Lemma)g @beginspecial @setspecial @endspecial(3.2)f(completes)h(the)h(pro)r(of) f(of)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)o Fu(\).)40 b(_)-27 b Fr({)t Fu(_)f Fr({)p Fu(\))22 b(follo)n(ws)e(from)h(the)g(equiv)-5 b(alence)21 b(of)g(the)h(con)n(v)n (ergence)d(in)0 2615 y(the)26 b(strong)f(resolv)n(en)n(t)f(sense)h(and) h(the)g(strong)f(con)n(v)n(ergence)e(of)j(unitary)f(groups)f(\(T)-7 b(rotter)26 b(theorem\).)36 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)f Fr({)t Fu(_)h Fr({)p Fu(\))26 b(is)g(ob)n(vious.)p @beginspecial @setspecial @endspecial Fh(\004)100 2715 y Fu(Let)e @beginspecial @setspecial @endspecial Fr(\037)297 2684 y Fo(0)297 2735 y Fq(\024)365 2715 y Fu(b)r(e)g(another)g(cuto\013)h(function)g(and)f (de\014ne)g @beginspecial @setspecial @endspecial Fr(H)1798 2684 y Fo(0)1846 2715 y Fu(to)g(b)r(e)h(the)g(Nelson)f(Hamiltonian)g (constructed)g(using)g(the)h(later)0 2814 y(cuto\013.)-228 b @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 151 x Fz(Prop)s(osition)30 b(3.9.)40 b Fn(Ther)l(e)31 b(exists)e(a)h(\014nite)g(c)l(onstant)f Fr(E)5 b Fn(,)30 b(such)g(that)1629 3103 y @beginspecial @setspecial @endspecial Fr(H)1705 3069 y Fo(0)1751 3103 y Fu(=)23 b Fr(H)i Fu(+)18 b Fr(E)5 b(:)100 3254 y Fz(Pro)s(of.)48 b @beginspecial @setspecial @endspecial 27 w Fu(W)-7 b(e)29 b(de\014ne)h @beginspecial @setspecial @endspecial Fr(H)896 3224 y Fo(00)952 3254 y Fu(\(resp)p Fr(:)1178 3233 y Fu(^)1156 3254 y Fr(H)1232 3224 y Fo(00)1225 3275 y Fq(\024)1274 3254 y Fu(\))g(to)f(b)r(e)h(the)g(Nelson)g(Hamiltonian)f (\(resp.)42 b(the)30 b(cuto\013)g(mo)r(di\014ed)g(Hamiltonian\))0 3395 y(obtained)37 b(using)f(the)i(dressing)e(transformation)f(giv)n (en)h(b)n(y)h @beginspecial @setspecial @endspecial Fr(g)2028 3365 y Fo(00)2025 3415 y Fq(\024)2109 3395 y Fu(:=)h Fp(\000)p Fu(i)2333 3355 y Fs(\(2)p Fq(\031)r Fs(\))2459 3326 y Fg(\000)2513 3310 y Fi(3)p 2513 3317 V 2513 3348 a(2)p 2332 3376 224 4 v 2352 3450 a Fq(!)2406 3406 y Fi(1)p 2406 3413 29 3 v 2406 3444 a(2)2448 3450 y Fs(\()p Fq(k)q Fs(\))2589 3348 y Fq(\037)2630 3323 y Fg(0)2630 3365 y Fj(\024)2669 3348 y Fs(\()p Fq(k)q Fs(\))p Fo(\000)p Fq(\037)2850 3356 y Fj(\024)2884 3368 y Fi(0)2922 3348 y Fs(\()p Fq(k)q Fs(\))p 2589 3376 422 4 v 2655 3440 a Fq(!)r Fs(\()p Fq(k)q Fs(\)+)2851 3418 y Fj(K)2901 3406 y Fi(2)p 2849 3427 87 3 v 2849 3460 a(2)p Fj(M)3021 3395 y Fu(.)65 b(It)37 b(is)g(easy)f(to)h(see)g(that)0 3552 y @beginspecial @setspecial @endspecial Fr(H)76 3522 y Fo(0)125 3552 y Fu(=)25 b Fr(H)291 3522 y Fo(00)353 3552 y Fu(+)19 b Fr(E)5 b Fu(,)30 b(where)f Fr(E)34 b Fu(is)29 b(a)g(\014nite)h(constan)n(t.)41 b(Using)29 b(similar)g(calculus)g(with)h(\()p @beginspecial @setspecial @endspecial(3.12\),)f(w)n(e)f(obtain)h(that)3504 3531 y(^)3482 3552 y Fr(H)3551 3564 y Fo(1)3647 3552 y Fu(=)3759 3531 y(^)3738 3552 y Fr(H)3814 3522 y Fo(0)o(0)3807 3572 y(1)3877 3552 y Fu(.)0 3651 y(Since)f @beginspecial @setspecial @endspecial Fr(U)274 3663 y Fo(1)367 3651 y Fu(=)22 b Fr(U)520 3621 y Fo(00)511 3672 y(1)609 3651 y Fu(w)n(e)27 b(ha)n(v)n(e)g @beginspecial @setspecial @endspecial Fr(H)j Fu(=)22 b Fr(H)1185 3621 y Fo(00)1227 3651 y Fu(.)37 b(Then)28 b @beginspecial @setspecial @endspecial Fr(H)1580 3621 y Fo(0)1626 3651 y Fu(=)23 b Fr(H)i Fu(+)18 b Fr(E)5 b Fu(.)1855 b Fh(\004)1392 3830 y @beginspecial @setspecial @endspecial Fu(4.)45 b Fv(Higher)32 b(order)f(estima)-6 b(tes)100 3979 y Fu(In)25 b(this)g(section)g(w)n(e)g(pro)n(v)n(e)e (some)i(estimates)g(whic)n(h)g(allo)n(w)f(to)h(b)r(ound)g(p)r(o)n(w)n (ers)f(of)h @beginspecial @setspecial @endspecial Fr(N)34 b Fu(and)25 b Fr(H)3038 3991 y Fs(0)3101 3979 y Fu(b)n(y)g(p)r(o)n(w)n (ers)e(of)i Fr(H)7 b Fu(.)36 b(They)0 4079 y(pla)n(y)20 b(an)h(imp)r(ortan)n(t)g(role.)34 b(F)-7 b(r\177)-42 b(ohlic)n(h)20 b(has)g(pro)n(v)n(ed)g(a)h(higher)f(estimates)h(in)g (the)g(massless)f(case)g @beginspecial @setspecial @endspecial([14)o (],)j(but)f(they)f(are)f(di\013eren)n(t)0 4178 y(from)27 b(what)f(w)n(e)h(in)n(tend)g(to)g(pro)n(v)n(e.)35 b(Their)27 b(pro)r(ofs)f(are)g(based)g(in)h(the)g(follo)n(wing)f(principle)h(of)g (cuto\013)g(indep)r(endence)h @beginspecial @setspecial @endspecial([22)o(].)0 4199 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 131 x Fz(Lemma)i(4.1.)40 b Fn(L)l(et)29 b Fp(f)p Fr(N)781 4342 y Fq(j)815 4330 y Fp(g)g Fn(and)i Fp(f)p Fr(H)1159 4342 y Fq(j)1193 4330 y Fp(g)e Fn(b)l(e)h(se)l(quenc)l(es)f(of)i(op)l (er)l(ators)g(such)e(that,)i(for)f(c)g(indep)l(endent)h(of)f(j,)1223 4467 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1336 4479 y Fq(j)1371 4467 y Fr( )s Fp(jj)23 b(\024)g Fr(c)14 b Fp(jj)p Fr(H)1750 4479 y Fq(j)1785 4467 y Fr( )s Fp(jj)p Fr(;)37 b Fn(for)44 b Fr( )26 b Fp(2)e(D)r Fu(\()p Fr(H)2420 4479 y Fq(j)2455 4467 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 4605 a Fn(Supp)l(ose)33 b(that)h Fr(N)560 4617 y Fq(j)627 4605 y Fn(is)g(selfadjoint)h(and)f(that)f Fr(N)1529 4617 y Fq(j)1592 4605 y Fp(!)d Fr(N)41 b Fn(in)34 b(the)f(str)l(ong)f(r)l(esolvent)i(sense,)g(wher)l(e)g(N)e(is)i (selfadjoint,)i(and)0 4705 y @beginspecial @setspecial @endspecial Fr(H)69 4717 y Fq(j)127 4705 y Fp(!)23 b Fr(H)37 b Fn(in)29 b(the)h(str)l(ong)g(gr)l(aph)h(limit.)39 b(Then)1264 4842 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)9 b( )s Fp(jj)23 b(\024)g Fr(c)14 b Fp(jj)p Fr(H)7 b( )s Fp(jj)p Fr(;)37 b Fn(for)44 b Fr( )26 b Fp(2)e(D)r Fu(\()p Fr(H)7 b Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 100 4994 a Fu(Let)30 b @beginspecial @setspecial @endspecial(\000)14 b(\()p Fr(r)r(esp:)g Fu(\000)597 5006 y Fq(j)632 4994 y Fu(\))30 b(denotes)g(the)g(graph)f (of)h Fr(H)20 b Fu(\()p Fr(r)r(esp:)14 b(H)1863 5006 y Fq(j)1899 4994 y Fu(\).)44 b(W)-7 b(e)31 b(recall)e(that)h @beginspecial @setspecial @endspecial Fr(H)2619 5006 y Fq(j)2681 4994 y Fp(!)d Fr(H)37 b Fu(in)30 b(the)g(strong)f(graph)g (sense)h(if)0 5093 y(for)d(all)g @beginspecial @setspecial @endspecial(\()p Fr( )s(;)14 b(')p Fu(\))24 b Fp(2)g Fu(\000,)j(there)h(exist)f(a)g(sequence)g @beginspecial @setspecial @endspecial(\()p Fr( )1567 5105 y Fq(j)1603 5093 y Fr(;)14 b(')1694 5105 y Fq(j)1729 5093 y Fu(\))23 b Fp(2)h Fu(\000)1915 5105 y Fq(j)1977 5093 y Fu(whic)n(h)k(con)n(v)n (erges)d(to)i @beginspecial @setspecial @endspecial(\()p Fr( )s(;)14 b(')p Fu(\))29 b(in)f Fp(H)19 b(\002)f(H)q Fu(.)100 5193 y(The)27 b(follo)n(wing)g(easy)f(\014rst)i(order)e (estimate)i(follo)n(ws)e(from)h(the)h(pro)r(of)f(of)h(Thm.)37 b @beginspecial @setspecial @endspecial(3.6.)0 5209 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 135 x Fz(Lemma)30 b(4.2.)40 b Fn(Ther)l(e)30 b(exists)g Fr(c)23 b(>)f Fu(0)29 b Fn(indep)l(endent)i(for)f Fr(\024)g Fn(such)g(that)g(for)g @beginspecial @setspecial @endspecial Fr(\024)2422 5356 y Fs(0)2482 5344 y Fp(\024)23 b Fr(\024)g Fp(\024)f(1)1673 5485 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(H)1742 5497 y Fs(0)1802 5485 y Fp(\024)h Fu(\()1944 5464 y(^)1922 5485 y Fr(H)1991 5497 y Fq(\024)2053 5485 y Fu(+)18 b Fr(c)p Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial eop %%Page: 14 14 14 13 bop 0 226 a Fs(14)1695 b(Z.)29 b(AMMARI)0 489 y @beginspecial @setspecial @endspecial Fu(4.1.)47 b Fz(Rosen)27 b(estimates.)39 b Fu(In)25 b(this)g(subsection)g(w)n(e)f(pro)n(v)n(e)f (higher)i(order)e(estimates)i(using)f(a)h(tec)n(hnique)g(due)g(to)f (Glimm)0 589 y(and)k(Ja\013e)f(to)h(pro)n(v)n(e)e(similar)h(estimates)h (for)f(the)h @beginspecial @setspecial @endspecial Fr(Y)1642 601 y Fs(2)1708 589 y Fu(and)f(\()p Fr(')1955 558 y Fs(4)1993 589 y Fu(\))2025 601 y Fs(2)2091 589 y Fu(mo)r(dels.)37 b(It)28 b(has)g(b)r(een)g(tak)n(en)f(up)h(b)n(y)g(Rosen)f(in)i @beginspecial @setspecial @endspecial([22)o(])f(for)0 688 y(the)33 b(general)f(\()p Fr(')526 658 y Fs(2)p Fq(n)605 688 y Fu(\))637 700 y Fs(2)707 688 y Fu(mo)r(del)h(and)g(F)-7 b(r\177)-42 b(ohlic)n(h)32 b @beginspecial @setspecial @endspecial([14)o(].)53 b(This)33 b(tec)n(hnique)g(is)f(based)h(in)g (the)g(so)f(called)h Fn(pul)t(l-thr)l(ough)i(formula)0 788 y Fu(whic)n(h)f(is)g(the)h(iden)n(tit)n(y)f(that)g(w)n(e)g(obtain,) i(in)e(a)g(formal)f(w)n(a)n(y)-7 b(,)35 b(when)f(w)n(e)g(mo)n(v)n(e)f (the)h(resolv)n(en)n(t)f(through)g(a)h(pro)r(duct)g(of)0 887 y(annihilation)21 b(op)r(erators)d @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)934 899 y Fq(i)962 887 y Fu(\).)35 b(But)22 b(some)e(care)g(m)n(ust)h(b)r(e)g (tak)n(en)f(when)h(w)n(e)g(w)n(an)n(t)f(to)h(rigorously)d(pro)n(v)n(e)h (the)i(pull)h(through)0 987 y(form)n(ula)29 b(,)j(since)e(w)n(e)g(need) g(a)g(dense)h(subspace)e @beginspecial @setspecial @endspecial Fp(H)1610 999 y Fs(0)1675 987 y Fp(\032)f(D)r Fu(\()p Fr(H)1935 999 y Fs(0)1972 987 y Fu(\))j(on)f(whic)n(h)h @beginspecial @setspecial @endspecial -62 x Fl(Q)2472 1012 y Fq(i)2514 987 y Fr(a)p Fu(\()p Fr(k)2633 999 y Fq(i)2661 987 y Fu(\))f(acts)g(as)g(an)g(op)r(erator)f(and)h (satis\014es)0 1024 y Fl(Q)78 1112 y Fq(i)120 1087 y Fr(a)p Fu(\()p Fr(k)239 1099 y Fq(i)267 1087 y Fu(\))p Fp(H)369 1099 y Fs(0)430 1087 y Fp(\032)22 b(D)r Fu(\()p Fr(H)684 1099 y Fs(0)722 1087 y Fu(\))27 b(and)g @beginspecial @setspecial @endspecial -63 x Fl(Q)1020 1112 y Fq(i)1062 1087 y Fr(a)p Fu(\()p Fr(k)1181 1099 y Fq(i)1209 1087 y Fu(\))g(can)f(b)r(e)h(de\014ned)g(as)f(op)r(erator)f(on)h Fr(H)2434 1099 y Fs(0)2471 1087 y Fp(H)2541 1099 y Fs(0)2579 1087 y Fu(.)36 b(W)-7 b(e)27 b(need)g(also)f(a)g(resolv)n(en)n(t)f(con) n(trol)g(of)0 1186 y(the)k(comm)n(utator)e(of)h(the)h(mo)r(di\014ed)f (cuto\013)h(in)n(teraction)e(with)i @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)2152 1198 y Fq(i)2180 1186 y Fu(\),)g(whic)n(h)f(allo)n(ws)f(to)h(de\014ne)g(it)h(as)f(lo)r (cally)f(in)n(tegrable)0 1286 y(function)h(with)g(v)-5 b(alues)27 b(in)h(b)r(ounded)g(op)r(erators)e(on)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)1847 1298 y Fs(0)1885 1286 y Fu(\).)37 b(This)28 b(requires)e Fr(H)2548 1298 y Fs(0)2585 1286 y Fp(H)2655 1298 y Fs(0)2721 1286 y Fu(to)h(b)r(e)h(dense.)100 1386 y(Let)c @beginspecial @setspecial @endspecial Fp(f)p Fr(J)333 1398 y Fq(i)361 1386 y Fp(g)403 1398 y Fs(1)p Fq(::j)s Fs(+1)618 1386 y Fu(b)r(e)i(a)e(set)h(of)f(disjoin)n(t)h(subsets)f(of)h @beginspecial @setspecial @endspecial Fp(f)p Fu(1)p Fr(;)14 b(:::;)g(n)p Fp(g)23 b Fu(so)h(that)h(within)g(eac)n(h)f(subset)h Fr(J)3037 1398 y Fq(i)3089 1386 y Fu(the)g(elemen)n(ts)g(are)f(tak)n (en)0 1485 y(in)k(their)f(natural)g(order.)36 b(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(notation:)908 1800 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 949 1618 a @beginspecial @setspecial @endspecial 14 w Fr(H)1039 1583 y Fq(J)1032 1638 y(\024)1108 1618 y Fu(:=)23 b([)p Fr(a)p Fu(\()p Fr(k)1361 1630 y Fq(i)1384 1638 y Fi(1)1421 1618 y Fu(\))p Fr(;)14 b(:::;)g Fu([)p Fr(a)p Fu(\()p Fr(k)1738 1630 y Fq(i)1761 1638 y Fj(j)1798 1618 y Fu(\))p Fr(;)1877 1597 y Fu(^)1867 1618 y Fr(I)1903 1630 y Fq(\024)1946 1618 y Fu(])p Fr(:::)p Fu(])p Fr(;)74 b Fu(where)37 b Fr(J)31 b Fu(=)22 b Fp(f)p Fr(i)2643 1630 y Fs(1)2680 1618 y Fr(;)14 b(:::;)g(i)2852 1630 y Fq(j)2886 1618 y Fp(g)p Fr(;)p @beginspecial @setspecial @endspecial 949 1761 a(R)q Fu(\()p Fr(z)t Fu(\))23 b(:=)f(\()p Fr(z)g Fp(\000)1451 1740 y Fu(^)1429 1761 y Fr(H)1498 1773 y Fq(\024)1541 1761 y Fu(\))1573 1727 y Fo(\000)p Fs(1)1663 1761 y Fr(;)949 1913 y(R)1012 1925 y Fq(`)1044 1913 y Fu(\()p Fr(z)t Fu(\))h(:=)g(\()p Fr(z)f Fp(\000)1461 1834 y Fl(X)1509 2011 y Fq(i)1594 1913 y Fr(!)s Fu(\()p Fr(k)1724 1925 y Fq(i)1752 1913 y Fu(\))d Fp(\000)1908 1892 y Fu(^)1886 1913 y Fr(H)1955 1925 y Fq(\024)1998 1913 y Fu(\))2030 1879 y Fo(\000)p Fs(1)2119 1913 y Fr(;)2992 1800 y @beginspecial @setspecial @endspecial 0 2131 a Fu(where)27 b(the)h(sum)g(runs)f(o)n(v)n(er)f @beginspecial @setspecial @endspecial Fr(i)c Fp(2)i Fr(J)1098 2143 y Fq(`)1148 2131 y Fp([)19 b Fr(J)1268 2143 y Fq(`)p Fs(+1)1384 2131 y Fr(:::)g Fp([)g Fr(J)1592 2143 y Fq(j)s Fs(+1)1711 2131 y Fu(.)0 2154 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 202 x Fz(Lemma)30 b(4.3.)40 b Fn(L)l(et)29 b Fr(\013)h Fn(,)p Fr(\034)j Fu(:=)23 b(\()p Fr(\034)1028 2368 y Fq(i)1056 2356 y Fu(\))1088 2368 y Fs(1)p Fq(::r)1227 2356 y Fn(b)l(e)30 b(a)g(se)l(quenc)l(e)g(of)g(r)l(e)l(al)g(numb)l(ers.)38 b(F)-6 b(or)30 b @beginspecial @setspecial @endspecial Fr( )c Fp(2)e(D)r Fu(\(\()p Fr(H)2871 2368 y Fs(0)2927 2356 y Fu(+)18 b(1\))3084 2321 y Fq(\013)3197 2252 y(r)3160 2277 y Fl(Y)3159 2454 y Fq(i)p Fs(=1)3281 2356 y Fr(N)3366 2285 y Fi(1)p 3366 2294 29 3 v 3366 2327 a(2)3348 2365 y Fq(\034)3379 2373 y Fj(i)3409 2356 y Fu(\))208 2670 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(H)355 2682 y Fs(0)411 2670 y Fu(+)g(1\))568 2636 y Fq(\013)681 2566 y(r)644 2591 y Fl(Y)643 2768 y Fq(i)p Fs(=1)765 2670 y Fr(N)850 2599 y Fi(1)p 850 2608 V 850 2641 a(2)832 2680 y Fq(\034)863 2688 y Fj(i)893 2670 y Fr( )s Fp(jj)996 2636 y Fs(2)1056 2670 y Fu(=)1187 2566 y Fq(r)1144 2591 y Fl(X)1146 2768 y Fq(j)s Fs(=1)1278 2557 y Fl(Z)1374 2670 y Fr(P)1439 2636 y Fq(\034)1427 2691 y(r)n(;j)1511 2670 y Fu(\()p Fr(k)1586 2682 y Fs(1)1624 2670 y Fr(;)c(:::;)g(k)1810 2682 y Fq(j)1845 2670 y Fu(\))24 b Fp(jj)p Fu(\()p Fr(H)2048 2682 y Fs(0)2104 2670 y Fu(+)2231 2562 y Fq(j)2187 2591 y Fl(X)2193 2768 y Fq(i)p Fs(=1)2321 2670 y Fr(!)s Fu(\()p Fr(k)2451 2682 y Fq(i)2478 2670 y Fu(\))19 b(+)f(1\))2686 2636 y Fq(\013)2786 2562 y(j)2748 2591 y Fl(Y)2747 2768 y Fq(i)p Fs(=1)2869 2670 y Fr(a)p Fu(\()p Fr(k)2988 2682 y Fq(i)3016 2670 y Fu(\))p Fr( )s Fp(jj)3151 2636 y Fs(2)3188 2670 y Fr(dk)3274 2682 y Fs(1)3312 2670 y Fr(:::dk)3467 2682 y Fq(j)3503 2670 y Fr(:)83 b @beginspecial @setspecial @endspecial -3609 w Fu(\(4.1\))100 2919 y Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 23 w Fu(This)27 b(lemma)f(is)h(similar)f(to)g(Corollary)f(2.2.)35 b(W)-7 b(e)27 b(pro)n(v)n(e)e(\()p @beginspecial @setspecial @endspecial(4.1\))i(for)f Fr( )g Fp(2)e(D)r Fu(\()p Fr(K)6 b Fu(\))16 b Fp(\012)h Fu(\000)3070 2931 y Fs(\014n)3148 2919 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)3322 2889 y Fs(3)3365 2919 y Fu(\)\))28 b(and)e(then)h(w)n(e)0 3103 y(extend)22 b(it)h(to)e @beginspecial @setspecial @endspecial Fr( )27 b Fp(2)c(D)r Fu(\(\()p Fr(H)795 3115 y Fs(0)840 3103 y Fu(+)7 b(1\))986 3069 y Fq(\013)1098 2999 y(r)1062 3024 y Fl(Y)1061 3201 y Fq(i)p Fs(=1)1182 3103 y Fr(N)1268 3032 y Fi(1)p 1268 3041 V 1268 3074 a(2)1249 3113 y Fq(\034)1280 3121 y Fj(i)1310 3103 y Fu(\).)36 b(This)22 b(extends)g @beginspecial @setspecial @endspecial 1920 2995 a Fq(j)1883 3024 y Fl(Y)1882 3201 y Fq(i)p Fs(=1)2003 3103 y Fr(a)p Fu(\()p Fr(k)2122 3115 y Fq(i)2150 3103 y Fu(\))g(to)g(b)r(ounded)h(op) r(erator)d(from)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\(\()p Fr(H)3352 3115 y Fs(0)3398 3103 y Fu(+)7 b(1\))3544 3069 y Fq(\013)3656 2999 y(r)3619 3024 y Fl(Y)3618 3201 y Fq(i)p Fs(=1)3740 3103 y Fr(N)3825 3032 y Fi(1)p 3825 3041 V 3825 3074 a(2)3807 3113 y Fq(\034)3838 3121 y Fj(i)3868 3103 y Fu(\))0 3372 y(in)n(to)27 b @beginspecial @setspecial @endspecial(\()p Fr(H)269 3384 y Fs(0)325 3372 y Fu(+)453 3264 y Fq(j)408 3293 y Fl(X)415 3470 y Fq(i)p Fs(=1)542 3372 y Fr(!)s Fu(\()p Fr(k)672 3384 y Fq(i)700 3372 y Fu(\))19 b(+)f(1\))908 3338 y Fo(\000)p Fq(\013)1007 3372 y Fr(L)1064 3338 y Fs(2)1100 3372 y Fu(\()p Fk(R)1187 3338 y Fs(3)p Fq(j)1261 3372 y Fr(;)c Fu(\()p Fr(P)1395 3338 y Fq(\034)1383 3392 y(r)n(;j)1467 3372 y Fu(\))1509 3315 y Fi(1)p 1509 3324 V 1509 3358 a(2)1552 3372 y Fr(dk)s Fu(\))k Fp(\012)g(H)q Fr(:)28 b @beginspecial @setspecial @endspecial 1939 w Fh(\004)100 3567 y Fu(Using)e(the)h(fact)f(that)h @beginspecial @setspecial @endspecial(\()p Fr(H)916 3579 y Fs(0)969 3567 y Fu(+)1050 3504 y Fl(P)1138 3525 y Fq(j)1138 3591 y(i)p Fs(=1)1263 3567 y Fr(!)s Fu(\()p Fr(k)1393 3579 y Fq(i)1421 3567 y Fu(\))16 b(+)g(1\))1624 3536 y Fo(\000)p Fs(1)1713 3567 y Fr(H)1782 3579 y Fs(0)1845 3567 y Fu(is)27 b(b)r(ounded)g(and)f (\()p @beginspecial @setspecial @endspecial(4.1\),)g(w)n(e)g(ha)n(v)n (e)f(for)h Fr( )g Fp(2)e Fu(\()p Fr(H)3343 3579 y Fs(0)3396 3567 y Fu(+)16 b(1\))3551 3536 y Fo(\000)p Fs(1)3640 3567 y Fp(D)r Fu(\()p Fr(N)3824 3514 y Fj(r)p 3824 3523 30 3 v 3825 3556 a Fi(2)3868 3567 y Fu(\))495 3782 y @beginspecial @setspecial @endspecial -113 x Fl(Z)592 3782 y Fp(jj)p Fr(N)724 3721 y Fj(p)p Fg(\000)p Fj(r)p 724 3735 105 3 v 762 3768 a Fi(2)843 3782 y Fr(H)912 3794 y Fs(0)1001 3679 y Fq(r)964 3703 y Fl(Y)963 3880 y Fq(i)p Fs(=1)1085 3782 y Fr(a)p Fu(\()p Fr(k)1204 3794 y Fq(i)1232 3782 y Fu(\))p Fr( )s Fp(jj)1367 3748 y Fs(2)1404 3782 y Fr(dk)86 b @beginspecial @setspecial @endspecial Fp(\024)d Fr(c)1774 3669 y Fl(Z)1871 3782 y Fp(jj)p Fu(\()p Fr(H)2018 3794 y Fs(0)2074 3782 y Fu(+)2200 3679 y Fq(r)2157 3703 y Fl(X)2205 3880 y Fq(i)2291 3782 y Fr(!)s Fu(\()p Fr(k)2421 3794 y Fq(i)2448 3782 y Fu(\))19 b(+)f(1\))2707 3679 y Fq(r)2670 3703 y Fl(Y)2670 3880 y Fq(i)p Fs(=1)2791 3782 y Fr(a)p Fu(\()p Fr(k)2910 3794 y Fq(i)2938 3782 y Fu(\))p Fr(N)3056 3721 y Fj(p)p Fg(\000)p Fj(r)p 3056 3735 V 3094 3768 a Fi(2)3175 3782 y Fr( )s Fp(jj)3278 3748 y Fs(2)3316 3782 y Fr(dk)0 3913 y @beginspecial @setspecial @endspecial 1576 4104 a @beginspecial @setspecial @endspecial Fp(\024)83 b Fr(c)1774 3991 y Fl(Z)1871 4104 y Fp(jj)1968 4001 y Fq(r)1931 4025 y Fl(Y)1931 4202 y Fq(i)p Fs(=1)2052 4104 y Fr(a)p Fu(\()p Fr(k)2171 4116 y Fq(j)2206 4104 y Fu(\)\()p Fr(H)2339 4116 y Fs(0)2396 4104 y Fu(+)18 b(1\))p Fr(N)2638 4043 y Fj(p)p Fg(\000)p Fj(r)p 2638 4057 V 2677 4090 a Fi(2)2757 4104 y Fr( )s Fp(jj)2860 4070 y Fs(2)2898 4104 y Fr(dk)0 4235 y @beginspecial @setspecial @endspecial 1576 4368 a Fp(\024)83 b Fr(c)23 b Fp(jj)p Fr(N)1915 4307 y Fj(p)p 1915 4321 31 3 v 1916 4354 a Fi(2)1960 4368 y Fu(\()p Fr(H)2061 4380 y Fs(0)2117 4368 y Fu(+)18 b(1\))p Fr( )s Fp(jj)2377 4334 y Fs(2)2414 4368 y Fr(:)0 4517 y Fu(Hence)36 b @beginspecial @setspecial @endspecial -62 x Fl(Q)334 4476 y Fq(r)334 4542 y(i)p Fs(=1)459 4517 y Fr(a)p Fu(\()p Fr(k)578 4529 y Fq(i)606 4517 y Fu(\))h(can)e(b)r(e)i(de\014ned)f(as)g(b)r(ounded)g(op)r(erator) e(from)i @beginspecial @setspecial @endspecial(\()p Fr(H)2358 4529 y Fs(0)2420 4517 y Fu(+)24 b(1\))2583 4487 y Fo(\000)p Fs(1)2671 4517 y Fp(D)r Fu(\()p Fr(N)2856 4461 y Fj(p)p 2856 4475 V 2857 4507 a Fi(2)2901 4517 y Fu(\))36 b(in)n(to)g @beginspecial @setspecial @endspecial Fr(L)3203 4487 y Fs(2)3240 4517 y Fu(\()p Fk(R)3326 4487 y Fs(3)p Fq(r)3402 4517 y Fr(;)14 b(dk)s Fu(\))24 b Fp(\012)g Fu(\()p Fr(H)3774 4529 y Fs(0)3835 4517 y Fu(+)0 4635 y(1\))74 4605 y Fo(\000)p Fs(1)163 4635 y Fp(D)r Fu(\()p Fr(N)347 4578 y Fj(p)p Fg(\000)p Fj(r)p 347 4592 105 3 v 386 4625 a Fi(2)466 4635 y Fu(\).)35 b(It)22 b(is)f(easy)f(to)h(see,)h(using)f(comm)n (utation)f(relations)g(and)h(Lemma)g @beginspecial @setspecial @endspecial(3.1,)g(that)g(\()p Fr(N)15 b Fu(+)5 b(1\))3207 4605 y Fo(\000)3268 4578 y Fj(p)p 3268 4592 31 3 v 3269 4625 a Fi(2)3323 4614 y Fu(^)3313 4635 y Fr(I)3349 4647 y Fq(\024)3393 4635 y Fr(N)3478 4578 y Fj(p)p 3478 4592 V 3479 4625 a Fi(2)3523 4635 y Fu(\()p Fr(H)3624 4647 y Fs(0)3667 4635 y Fu(+)g(1\))3811 4605 y Fo(\000)p Fs(1)0 4743 y Fu(is)24 b(b)r(ounded)g(for)f @beginspecial @setspecial @endspecial Fr(\024)g(<)g Fp(1)p Fu(,)h(hence)g @beginspecial @setspecial @endspecial 1064 4722 a(^)1054 4743 y Fr(I)1090 4755 y Fq(\024)1158 4743 y Fu(can)f(b)r(e)h(de\014ned)h(as)e(b)r (ounded)h(op)r(erator)e(from)h @beginspecial @setspecial @endspecial(\()p Fr(H)2755 4755 y Fs(0)2804 4743 y Fu(+)11 b(1\))2954 4713 y Fo(\000)p Fs(1)3042 4743 y Fp(D)r Fu(\()p Fr(N)3227 4686 y Fj(p)p 3227 4700 V 3228 4733 a Fi(2)3271 4743 y Fu(\))25 b(in)n(to)e @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)3677 4686 y Fj(p)p 3677 4700 V 3678 4733 a Fi(2)3721 4743 y Fu(\).)36 b(So)0 4859 y(the)28 b(comm)n(utator)f Fr(H)688 4816 y Fo(f)p Fs(1)p Fo(g)681 4869 y Fq(\024)821 4859 y Fu(acts)g(as)g(b)r(ounded)h (op)r(erator)e(from)i @beginspecial @setspecial @endspecial(\()p Fr(H)2066 4871 y Fs(0)2122 4859 y Fu(+)18 b(1\))2279 4829 y Fo(\000)p Fs(1)2368 4859 y Fp(D)r Fu(\()p Fr(N)2552 4802 y Fj(p)p 2552 4816 V 2553 4849 a Fi(2)2597 4859 y Fu(\))28 b(in)n(to)g Fr(L)2883 4829 y Fs(2)2919 4859 y Fu(\()p Fk(R)3006 4829 y Fs(3)3049 4859 y Fr(;)14 b(dk)3172 4871 y Fs(1)3209 4859 y Fu(\))19 b Fp(\012)f(D)r Fu(\()p Fr(N)3528 4802 y Fj(p)p Fg(\000)p Fi(1)p 3528 4816 104 3 v 3566 4849 a(2)3646 4859 y Fu(\).)38 b(No)n(w)0 4958 y(for)27 b @beginspecial @setspecial @endspecial Fr(J)k Fu(:=)23 b Fp(f)p Fu(1)p Fr(;)14 b(:::;)g(r)r Fp(g)27 b Fu(w)n(e)g(can)g(pro)n(v)n(e)f(b)n(y)h(induction)h(on)g @beginspecial @setspecial @endspecial Fr(r)i Fu(that)996 5170 y Fr(H)1072 5136 y Fo(f)p Fq(J)5 b Fo(g)1065 5191 y Fq(\024)1209 5170 y Fu(:)24 b(\()p Fr(H)1357 5182 y Fs(0)1413 5170 y Fu(+)18 b(1\))1570 5136 y Fo(\000)p Fs(1)1658 5170 y Fp(D)r Fu(\()p Fr(N)1843 5109 y Fj(p)p 1843 5123 31 3 v 1844 5156 a Fi(2)1888 5170 y Fu(\))23 b Fp(!)g Fr(L)2106 5136 y Fs(2)2143 5170 y Fu(\()p Fk(R)2229 5136 y Fs(3)p Fq(r)2305 5170 y Fr(;)2379 5066 y Fq(r)2343 5091 y Fl(Y)2342 5268 y Fq(i)p Fs(=1)2463 5170 y Fr(dk)2549 5182 y Fq(i)2577 5170 y Fu(\))c Fp(\012)f(D)r Fu(\()p Fr(N)2895 5109 y Fj(p)p Fg(\000)p Fj(r)p 2895 5123 105 3 v 2934 5156 a Fi(2)3015 5170 y Fu(\))p Fr(:)-3070 b Fu(\(4.2\))0 5392 y(F)-7 b(or)29 b @beginspecial @setspecial @endspecial(#)p Fr(J)36 b Fu(=)27 b(1)i(\(4.2\))h(is)g(already)f(done.) 44 b(Assume)30 b(that)g(\()p @beginspecial @setspecial @endspecial(4.2\))g(holds)g(for)f @beginspecial @setspecial @endspecial(#)p Fr(J)36 b Fu(=)27 b Fr(r)r Fu(.)45 b(Using)30 b(the)g(fact)g(that)h Fr(a)p Fu(\()p Fr(k)3527 5404 y Fq(r)r Fs(+1)3648 5392 y Fu(\))f(maps)0 5506 y Fr(L)57 5476 y Fs(2)94 5506 y Fu(\()p Fk(R)180 5476 y Fs(3)p Fq(r)256 5506 y Fr(;)293 5444 y Fl(Q)371 5464 y Fq(r)371 5531 y(i)p Fs(=1)497 5506 y Fr(dk)583 5518 y Fq(i)611 5506 y Fu(\))24 b Fp(\012)g(D)r Fu(\()p Fr(N)940 5449 y Fj(p)p Fg(\000)p Fj(r)p 940 5463 V 979 5496 a Fi(2)1059 5506 y Fu(\))37 b(in)n(to)f @beginspecial @setspecial @endspecial Fr(L)1362 5476 y Fs(2)1399 5506 y Fu(\()p Fk(R)1485 5476 y Fs(3\()p Fq(r)r Fs(+1\))1697 5506 y Fr(;)1734 5444 y Fl(Q)1812 5464 y Fq(r)r Fs(+1)1812 5531 y Fq(i)p Fs(=1)1947 5506 y Fr(dk)2033 5518 y Fq(i)2061 5506 y Fu(\))24 b Fp(\012)g(D)r Fu(\()p Fr(N)2390 5449 y Fj(p)p Fg(\000)p Fj(r)q Fg(\000)p Fi(1)p 2390 5463 179 3 v 2465 5496 a(2)2583 5506 y Fu(\))36 b(and)g(maps)g @beginspecial @setspecial @endspecial(\()p Fr(H)3148 5518 y Fs(0)3210 5506 y Fu(+)24 b(1\))3373 5476 y Fo(\000)p Fs(1)3462 5506 y Fp(D)r Fu(\()p Fr(N)3646 5449 y Fj(p)p 3646 5463 31 3 v 3647 5496 a Fi(2)3691 5506 y Fu(\))36 b(in)n(to)p eop %%Page: 15 15 15 14 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(15)0 491 y @beginspecial @setspecial @endspecial Fr(L)57 460 y Fs(2)94 491 y Fu(\()p Fk(R)180 460 y Fs(3)223 491 y Fr(;)14 b(dk)346 503 y Fq(r)r Fs(+1)467 491 y Fu(\))19 b Fp(\012)f Fu(\()p Fr(H)702 503 y Fs(0)758 491 y Fu(+)g(1\))915 460 y Fo(\000)p Fs(1)1004 491 y Fp(D)r Fu(\()p Fr(N)1188 434 y Fj(p)p Fg(\000)p Fi(1)p 1188 448 104 3 v 1226 480 a(2)1306 491 y Fu(\),)29 b(w)n(e)e(pro)n(v)n (e)f(\()p @beginspecial @setspecial @endspecial(4.2\).)0 590 y(A)i(simple)g(computation)f(giv)n(es:)110 1038 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 152 813 a @beginspecial @setspecial @endspecial Fr(H)228 779 y Fo(f)p Fs(1)p Fo(g)221 833 y Fq(\024)356 813 y Fu(=)488 757 y(1)p 454 794 111 4 v 454 810 a Fp(p)p 523 810 42 4 v 69 x Fu(2)622 709 y Fq(P)588 734 y Fl(X)591 911 y Fq(j)s Fs(=1)722 813 y Fr(v)765 779 y Fq(j)762 833 y(\024)801 841 y Fi(0)837 813 y Fu(\()p Fr(k)912 825 y Fs(1)950 813 y Fu(\))19 b(+)1139 757 y(1)p 1094 794 132 4 v 1094 870 a(2)p Fr(M)1283 709 y Fq(P)1249 734 y Fl(X)1251 911 y Fq(j)s Fs(=1)1383 813 y Fr(r)1422 779 y Fq(j)1420 833 y(\024)1463 813 y Fu(\()p Fr(k)1538 825 y Fs(1)1576 813 y Fu(\))p Fr(a)1652 779 y Fo(\003)1690 813 y Fu(\()p Fr(r)1761 779 y Fq(j)1759 833 y(\024)1803 813 y Fu(\))g(+)f Fr(r)1976 779 y Fq(j)1974 833 y(\024)2017 813 y Fu(\()p Fr(k)2092 825 y Fs(1)2130 813 y Fu(\))p Fr(a)p Fu(\()p Fr(r)2277 779 y Fq(j)2275 833 y(\024)2319 813 y Fu(\))h Fp(\000)2453 740 y(p)p 2522 740 42 4 v 73 x Fu(2)o Fr(D)2632 825 y Fq(j)2667 813 y Fr(r)2706 779 y Fq(j)2704 833 y(\024)2747 813 y Fu(\()p Fr(k)2822 825 y Fs(1)2860 813 y Fu(\))p Fr(;)37 b Fu(on)23 b(\()p Fr(H)3164 825 y Fs(0)3220 813 y Fu(+)18 b(1\))3377 779 y Fo(\000)p Fs(1)3466 813 y Fp(D)r Fu(\()p Fr(N)3650 756 y Fi(1)p 3650 765 29 3 v 3650 799 a(2)3693 813 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 152 1117 a(H)228 1083 y Fo(f)p Fs(1)p Fq(;)p Fs(2)p Fo(g)221 1138 y Fq(\024)409 1117 y Fu(=)551 1061 y(1)p 506 1098 132 4 v 506 1174 a(2)p Fr(M)696 1013 y Fq(P)661 1038 y Fl(X)664 1215 y Fq(j)s Fs(=1)795 1117 y Fr(r)834 1083 y Fq(j)832 1138 y(\024)875 1117 y Fu(\()p Fr(k)950 1129 y Fs(1)988 1117 y Fu(\))p Fr(r)1059 1083 y Fq(j)1057 1138 y(\024)1101 1117 y Fu(\()p Fr(k)1176 1129 y Fs(2)1214 1117 y Fu(\))p Fr(;)37 b Fu(on)23 b(\()p Fr(H)1518 1129 y Fs(0)1574 1117 y Fu(+)18 b(1\))1731 1083 y Fo(\000)p Fs(1)1820 1117 y Fp(D)r Fu(\()p Fr(N)9 b Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 152 1344 a(H)228 1309 y Fq(J)221 1364 y(\024)297 1344 y Fu(=)23 b(0)p Fr(;)41 b Fu(for)14 b(all)e Fr(J)22 b Fu(suc)n(h)14 b(that)23 b(#)p Fr(J)31 b Fp(\025)23 b Fu(3)p Fr(:)3790 1038 y @beginspecial @setspecial @endspecial 100 1486 a Fu(Before)h(starting)g (with)i(the)g(pull-through)e(form)n(ula)g(w)n(e)h(will)h(pro)n(v)n(e)d (t)n(w)n(o)h(lemma)i(whic)n(h)f(will)g(b)r(e)h(useful)f(in)h(the)f (sequel.)0 1502 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 143 x Fz(Lemma)30 b(4.4.)40 b Fn(L)l(et)29 b @beginspecial @setspecial @endspecial Fr(r)j Fn(b)l(e)e(an)g(inte)l(ger)g(and)g Fr(z)36 b(=)-52 b Fp(2)24 b Fr(\033)s Fu(\()1646 1624 y(^)1624 1645 y Fr(H)1693 1657 y Fq(\024)1736 1645 y Fu(\))p Fn(,)31 b(then)1135 1800 y @beginspecial @setspecial @endspecial Fu(\()p Fr(H)1236 1812 y Fs(0)1292 1800 y Fu(+)18 b(1\)\()1502 1779 y(^)1481 1800 y Fr(H)1550 1812 y Fq(\024)1611 1800 y Fp(\000)g Fr(z)t Fu(\))1769 1765 y Fo(\000)p Fs(1)1881 1800 y Fu(:)23 b Fp(D)r Fu(\()p Fr(N)2111 1743 y Fj(r)p 2111 1752 30 3 v 2112 1785 a Fi(2)2155 1800 y Fu(\))g Fp(!)g(D)r Fu(\()p Fr(N)2501 1743 y Fj(r)p 2501 1752 V 2502 1785 a Fi(2)2544 1800 y Fu(\))p Fr(;)83 b @beginspecial @setspecial @endspecial 0 1946 a Fn(is)30 b(a)g(bije)l(ctive)i(map.)0 1962 y @beginspecial @setspecial @endspecial 100 2105 a Fz(Pro)s(of.)43 b @beginspecial @setspecial @endspecial 24 w Fu(Since)29 b @beginspecial @setspecial @endspecial(\()p Fr(N)f Fu(+)18 b(1\))928 2075 y Fo(\000)990 2053 y Fj(r)p 990 2062 V 991 2095 a Fi(2)1043 2084 y Fu(^)1033 2105 y Fr(I)1069 2117 y Fq(\024)1112 2105 y Fr(N)1198 2053 y Fj(r)p 1198 2062 V 1199 2095 a Fi(2)1242 2105 y Fu(\()p Fr(H)1343 2117 y Fs(0)1399 2105 y Fu(+)g(1\))1556 2075 y Fo(\000)p Fs(1)1674 2105 y Fu(is)28 b(b)r(ounded)g(for)g @beginspecial @setspecial @endspecial Fr(\024)c(<)f Fp(1)p Fu(,)29 b(w)n(e)e(see)h(that)g(\()3011 2084 y(^)2989 2105 y Fr(H)3058 2117 y Fq(\024)3121 2105 y Fp(\000)18 b Fr(z)t Fu(\)\()p Fr(H)3380 2117 y Fs(0)3436 2105 y Fu(+)g(1\))3593 2075 y Fo(\000)p Fs(1)3710 2105 y Fu(maps)0 2218 y @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)184 2165 y Fj(r)p 184 2174 V 185 2208 a Fi(2)228 2218 y Fu(\))28 b(in)n(to)f Fp(D)r Fu(\()p Fr(N)640 2165 y Fj(r)p 640 2174 V 641 2208 a Fi(2)684 2218 y Fu(\).)37 b(So)27 b(it)h(is)f(enough) g(to)h(sho)n(w)e(that)i @beginspecial @setspecial @endspecial(\()p Fr(H)1931 2230 y Fs(0)1987 2218 y Fu(+)18 b(1\))p Fr(N)2229 2165 y Fj(r)p 2229 2174 V 2230 2208 a Fi(2)2272 2218 y Fu(\()2326 2197 y(^)2304 2218 y Fr(H)2373 2230 y Fq(\024)2435 2218 y Fp(\000)g Fr(z)t Fu(\))2593 2188 y Fo(\000)p Fs(1)2682 2218 y Fu(\(1)g(+)f Fr(N)9 b Fu(\))2974 2162 y Fg(\000)p Fj(r)p 2974 2175 75 3 v 2997 2208 a Fi(2)3090 2218 y Fu(is)28 b(b)r(ounded.)37 b(W)-7 b(e)28 b(de\014ne)0 2326 y @beginspecial @setspecial @endspecial([)p Fr(N)t(;)14 b Fu(i)176 2305 y(^)154 2326 y Fr(H)223 2338 y Fq(\024)266 2326 y Fu(])28 b(as)f(quadratic)f(form)i(on)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)1269 2338 y Fs(0)1307 2326 y Fu(\))461 2553 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial([)p Fr(N)t(;)14 b Fu(i)637 2532 y(^)615 2553 y Fr(H)684 2565 y Fq(\024)727 2553 y Fu(])23 b(=)895 2449 y Fq(P)861 2474 y Fl(X)863 2651 y Fq(j)s Fs(=1)995 2553 y Fr(\036)p Fu(\(i)14 b Fr(v)1156 2519 y Fq(j)1153 2573 y(\024)1192 2581 y Fi(0)1229 2553 y Fu(\))19 b(+)1417 2497 y(1)p 1373 2534 132 4 v 1373 2610 a(2)p Fr(M)1562 2449 y Fq(P)1527 2474 y Fl(X)1530 2651 y Fq(j)s Fs(=1)1661 2553 y Fu(i)p Fr(a)1728 2519 y Fo(\003)1762 2494 y Fi(2)1799 2553 y Fu(\()p Fr(r)1870 2519 y Fq(j)1868 2573 y(\024)1912 2553 y Fu(\))f Fp(\000)g Fu(i)c Fr(a)2126 2519 y Fs(2)2163 2553 y Fu(\()p Fr(r)2234 2519 y Fq(j)2232 2573 y(\024)2276 2553 y Fu(\))19 b(+)f(i)2433 2480 y Fp(p)p 2502 2480 42 4 v 73 x Fu(2)p Fr(D)2613 2565 y Fq(j)2661 2553 y Fr(a)p Fu(\()p Fr(r)2776 2519 y Fq(j)2774 2573 y(\024)2818 2553 y Fu(\))h Fp(\000)f Fu(i)2975 2480 y Fp(p)p 3044 2480 V 73 x Fu(2)o Fr(a)3129 2519 y Fo(\003)3167 2553 y Fu(\()p Fr(r)3238 2519 y Fq(j)3236 2573 y(\024)3280 2553 y Fu(\))p Fr(D)3381 2565 y Fq(j)3416 2553 y Fr(:)p @beginspecial @setspecial @endspecial 0 2804 a Fu(F)-7 b(or)32 b @beginspecial @setspecial @endspecial Fr(\024)e(<)h Fp(1)p Fu(,)j(using)e(Corollary)d(2.4)j(and)g(the)h(fact)f(that)h @beginspecial @setspecial @endspecial Fp(h)p Fr(D)r Fp(i)p Fu(\()p Fr(K)28 b Fu(+)21 b(1\))2293 2774 y Fo(\000)2355 2752 y Fi(1)p 2355 2761 29 3 v 2355 2794 a(2)2429 2804 y Fu(is)33 b(b)r(ounded,)h(w)n(e)e(see)f(that)i @beginspecial @setspecial @endspecial([)p Fr(N)t(;)14 b Fu(i)3512 2783 y(^)3490 2804 y Fr(H)3559 2816 y Fq(\024)3602 2804 y Fu(])33 b(can)f(b)r(e)0 2904 y(de\014ned)c(as)f(a)g(b)r(ounded)h(op)r (erator)e(on)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)1413 2916 y Fs(0)1451 2904 y Fu(\).)0 3012 y(W)-7 b(e)21 b(set)g Fr(ad)346 2982 y Fq(`)346 3035 y(N)409 3012 y Fr(:)j Fu(:=)e([)p Fr(N)t(;)14 b Fu(i)g Fr(ad)821 2976 y Fq(`)p Fo(\000)p Fs(1)821 3036 y Fq(N)938 3012 y Fr(:)p Fu(])22 b(and)e @beginspecial @setspecial @endspecial Fr(ad)1247 2982 y Fs(1)1247 3035 y Fq(N)1310 3012 y Fr(:)k Fu(:=)e([)p Fr(N)t(;)14 b Fu(i)p Fr(:)p Fu(].)35 b(So)21 b Fr(ad)1921 2982 y Fq(`)1921 3035 y(N)2006 2991 y Fu(^)1984 3012 y Fr(H)2053 3024 y Fq(\024)2096 3012 y Fu(,)i(whic)n(h)e(is)f(similar)h(to)g @beginspecial @setspecial @endspecial Fr(ad)2898 2982 y Fs(1)2898 3035 y Fq(N)2982 2991 y Fu(^)2961 3012 y Fr(H)3030 3024 y Fq(\024)3094 3012 y Fu(is)g(de\014ned)g(b)n(y)g (induction)0 3111 y(in)28 b @beginspecial @setspecial @endspecial Fr(`)f Fu(as)g(a)g(b)r(ounded)h(op)r(erator)e(on)h Fp(D)r Fu(\()p Fr(H)1286 3123 y Fs(0)1324 3111 y Fu(\))h(and)g(equal)f (to:)210 3266 y @beginspecial @setspecial @endspecial 166 w Fr(ad)463 3231 y Fq(`)463 3286 y(N)548 3245 y Fu(^)526 3266 y Fr(H)595 3278 y Fq(\024)662 3266 y Fu(=)411 3414 y Fq(P)376 3439 y Fl(X)379 3616 y Fq(j)s Fs(=1)510 3518 y Fr(\036)p Fu(\(i)614 3483 y Fq(`)661 3518 y Fr(v)704 3483 y Fq(j)701 3538 y(\024)740 3546 y Fi(0)776 3518 y Fu(\))19 b(+)965 3461 y(1)p 920 3498 132 4 v 920 3575 a(2)p Fr(M)1109 3414 y Fq(P)1075 3439 y Fl(X)1078 3616 y Fq(j)s Fs(=1)1209 3518 y Fu(i)1232 3483 y Fq(`)1264 3518 y Fu(2)1306 3483 y Fq(`)p Fo(\000)p Fs(1)1422 3518 y Fr(a)1466 3483 y Fo(\003)1500 3458 y Fi(2)1537 3518 y Fu(\()p Fr(r)1608 3483 y Fq(j)1606 3538 y(\024)1649 3518 y Fu(\))g Fp(\000)f Fu(i)1806 3483 y Fq(`)1838 3518 y Fu(\()p Fp(\000)p Fu(2\))2009 3483 y Fq(`)p Fo(\000)p Fs(1)2140 3518 y Fr(a)2184 3483 y Fs(2)2221 3518 y Fu(\()p Fr(r)2292 3483 y Fq(j)2290 3538 y(\024)2334 3518 y Fu(\))g Fp(\000)g Fu(\()p Fp(\000)p Fu(i\))2619 3483 y Fq(`)2651 3445 y Fp(p)p 2721 3445 42 4 v 2721 3518 a Fu(2)o Fr(D)2831 3530 y Fq(j)2880 3518 y Fr(a)p Fu(\()p Fr(r)2995 3483 y Fq(j)2993 3538 y(\024)3036 3518 y Fu(\))h Fp(\000)f Fu(i)3193 3483 y Fq(`)3225 3445 y Fp(p)p 3294 3445 V 73 x Fu(2)p Fr(a)3380 3483 y Fo(\003)3418 3518 y Fu(\()p Fr(r)3489 3483 y Fq(j)3487 3538 y(\024)3531 3518 y Fu(\))p Fr(D)3632 3530 y Fq(j)3667 3518 y Fr(:)0 3769 y Fu(Since)29 b @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)9 b Fu(\))26 b Fp(\033)f(D)r Fu(\()660 3748 y(^)638 3769 y Fr(H)707 3781 y Fq(\024)751 3769 y Fu(\))g(=)g Fp(D)r Fu(\()p Fr(H)1065 3781 y Fs(0)1103 3769 y Fu(\),)30 b(the)f(resolv)n(en)n(t)e @beginspecial @setspecial @endspecial(\()p Fr(z)c Fp(\000)1882 3748 y Fu(^)1860 3769 y Fr(H)1929 3781 y Fq(\024)1972 3769 y Fu(\))2004 3738 y Fo(\000)p Fs(1)2123 3769 y Fu(preserv)n(es)k(the)i(domain)f(of)h Fr(N)9 b Fu(.)41 b(This)29 b(means)f(that)i(the)0 3868 y(follo)n(wing)d(iden)n(tit)n(y)722 4018 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)9 b Fu(\()p Fr(z)21 b Fp(\000)995 3997 y Fu(^)973 4018 y Fr(H)1042 4030 y Fq(\024)1085 4018 y Fu(\))1117 3983 y Fo(\000)p Fs(1)1230 4018 y Fu(=)h(\()p Fr(z)g Fp(\000)1515 3997 y Fu(^)1494 4018 y Fr(H)1563 4030 y Fq(\024)1606 4018 y Fu(\))1638 3983 y Fo(\000)p Fs(1)1727 4018 y Fr(N)27 b Fu(+)18 b(\()p Fr(z)k Fp(\000)2102 3997 y Fu(^)2080 4018 y Fr(H)2149 4030 y Fq(\024)2192 4018 y Fu(\))2224 3983 y Fo(\000)p Fs(1)2314 4018 y Fu([)p Fr(N)t(;)2466 3997 y Fu(^)2445 4018 y Fr(H)2514 4030 y Fq(\024)2557 4018 y Fu(]\()p Fr(z)g Fp(\000)2778 3997 y Fu(^)2756 4018 y Fr(H)2825 4030 y Fq(\024)2868 4018 y Fu(\))2900 3983 y Fo(\000)p Fs(1)2989 4018 y Fr(;)83 b @beginspecial @setspecial @endspecial -3095 w Fu(\(4.3\))0 4172 y(holds)29 b(in)g(the)h(sense)f(of)g(b)r(ounded)h(op)r(erators)d (on)i @beginspecial @setspecial @endspecial Fp(H)q Fu(.)42 b(Using)29 b(rep)r(eatedly)g(\(4.3\))g(w)n(e)f(notice)i(that)f @beginspecial @setspecial @endspecial(\()p Fr(z)23 b Fp(\000)3330 4151 y Fu(^)3308 4172 y Fr(H)3377 4184 y Fq(\024)3420 4172 y Fu(\))3452 4142 y Fo(\000)p Fs(1)3571 4172 y Fu(preserv)n(es)0 4272 y Fp(D)r Fu(\()p Fr(N)174 4241 y Fq(p)213 4272 y Fu(\))28 b(and)f(w)n(e)g(obtain)h(on)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)1104 4241 y Fq(p)1143 4272 y Fu(\))487 4426 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)563 4391 y Fq(p)601 4426 y Fu(\()p Fr(z)22 b Fp(\000)799 4405 y Fu(^)777 4426 y Fr(H)846 4438 y Fq(\024)889 4426 y Fu(\))921 4391 y Fo(\000)p Fs(1)1034 4426 y Fu(=)g Fr(N)1197 4391 y Fq(p)p Fo(\000)p Fs(1)1320 4426 y Fu(\()p Fr(z)g Fp(\000)1518 4405 y Fu(^)1496 4426 y Fr(H)1565 4438 y Fq(\024)1608 4426 y Fu(\))1640 4391 y Fo(\000)p Fs(1)1730 4426 y Fr(N)27 b Fp(\000)18 b Fu(i)p Fr(N)2006 4391 y Fq(p)p Fo(\000)p Fs(1)2129 4426 y Fu(\()p Fr(z)k Fp(\000)2327 4405 y Fu(^)2305 4426 y Fr(H)2374 4438 y Fq(\024)2417 4426 y Fu(\))2449 4391 y Fo(\000)p Fs(1)2539 4426 y Fr(ad)2626 4391 y Fs(1)2626 4446 y Fq(N)2710 4405 y Fu(^)2689 4426 y Fr(H)2758 4438 y Fq(\024)2815 4426 y Fu(\()p Fr(z)g Fp(\000)3012 4405 y Fu(^)2991 4426 y Fr(H)3060 4438 y Fq(\024)3103 4426 y Fu(\))3135 4391 y Fo(\000)p Fs(1)3224 4426 y Fr(:)83 b @beginspecial @setspecial @endspecial -3330 w Fu(\(4.4\))0 4572 y(W)-7 b(e)29 b(mo)n(v)n(e)f(no)n(w)g(all)g(factors)g(of)h @beginspecial @setspecial @endspecial Fr(N)37 b Fu(in)29 b(eac)n(h)f(term)h(to)g(the) g(righ)n(t,)f(w)n(e)h(obtain)f(the)h(follo)n(wing)f(iden)n(tit)n(y)h(b) r(et)n(w)n(een)g(b)r(ounded)0 4671 y(op)r(erators)817 4895 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)893 4861 y Fq(p)945 4895 y Fu(\()p Fr(z)22 b Fp(\000)1142 4874 y Fu(^)1121 4895 y Fr(H)1190 4907 y Fq(\024)1233 4895 y Fu(\))1265 4861 y Fo(\000)p Fs(1)1368 4895 y Fr(N)1444 4861 y Fo(\000)p Fq(p)1557 4895 y Fu(=)h(\()p Fr(z)f Fp(\000)1842 4874 y Fu(^)1821 4895 y Fr(H)1890 4907 y Fq(\024)1933 4895 y Fu(\))1965 4861 y Fo(\000)p Fs(1)2073 4895 y Fu(+)2197 4791 y Fq(k)2156 4816 y Fl(X)2160 4995 y Fq(`)p Fs(=1)2289 4895 y Fu(\()p Fr(z)g Fp(\000)2487 4874 y Fu(^)2465 4895 y Fr(H)2534 4907 y Fq(\024)2577 4895 y Fu(\))2609 4861 y Fo(\000)p Fs(1)2699 4895 y Fr(B)2762 4907 y Fq(`)2794 4895 y Fu(\()p Fr(z)t Fu(\))p Fr(N)2977 4861 y Fo(\000)p Fq(`)3060 4895 y Fr(;)p @beginspecial @setspecial @endspecial 0 5136 a Fu(where)28 b @beginspecial @setspecial @endspecial Fr(B)304 5148 y Fq(`)336 5136 y Fu(\()p Fr(z)t Fu(\))h(is)f(a)g(p)r (olynomial)g(in)43 b Fr(ad)1255 5097 y Fq(j)1255 5161 y(N)1340 5115 y Fu(^)1318 5136 y Fr(H)1387 5148 y Fq(\024)1444 5136 y Fu(\()p Fr(z)22 b Fp(\000)1643 5115 y Fu(^)1621 5136 y Fr(H)1690 5148 y Fq(\024)1733 5136 y Fu(\))1765 5106 y Fo(\000)p Fs(1)1855 5136 y Fr(;)27 b(j)j Fp(\024)24 b Fr(`)p Fu(.)40 b(Using)28 b(Lemma)h @beginspecial @setspecial @endspecial(3.3)e(with)i Fr(s)c Fu(=)f(0,)29 b(w)n(e)f(see)g(that)h @beginspecial @setspecial @endspecial Fr(B)3761 5148 y Fq(`)3793 5136 y Fu(\()p Fr(z)t Fu(\))0 5248 y(is)37 b(b)r(ounded)h(for)f Fr(\024)i(<)g Fp(1)p Fu(.)67 b(Hence)38 b(\()p Fr(z)28 b Fp(\000)1410 5227 y Fu(^)1389 5248 y Fr(H)1458 5260 y Fq(\024)1501 5248 y Fu(\))1533 5218 y Fo(\000)p Fs(1)1622 5248 y Fu(\()p Fr(H)1723 5260 y Fs(0)1786 5248 y Fu(+)c(1\))38 b(is)f(a)g(bijectiv)n(e)h(map)f(from)g @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)3073 5218 y Fq(p)3112 5248 y Fu(\))h(in)n(to)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)3534 5218 y Fq(p)3573 5248 y Fu(\).)66 b(Using)0 5347 y(Hadamard's)26 b(three)i(lines)f(lemma)h([23)o(])g(for)542 5506 y @beginspecial @setspecial @endspecial Fr(f)9 b Fu(\()p Fr(\020)d Fu(\))24 b(:=)f(\(\()p Fr(z)f Fp(\000)1063 5485 y Fu(^)1041 5506 y Fr(H)1110 5518 y Fq(\024)1153 5506 y Fu(\))1185 5472 y Fo(\000)p Fs(1)1274 5506 y Fr(H)1343 5518 y Fs(0)1381 5506 y Fr(N)1457 5472 y Fo(\000)p Fq(\020)1546 5506 y Fr( )s(;)14 b(N)1722 5456 y Fs(\026)1716 5472 y Fq(\020)1754 5506 y Fr(\036)p Fu(\))p Fr(;)38 b Fu(in)23 b Fm(S)g Fu(:=)g Fp(f)p Fr(\020)29 b Fp(2)23 b Fk(C)15 b Fr(;)f(p)29 b Fp(\024)23 b Fu(Im\()p Fr(\020)6 b Fu(\))24 b Fp(\024)e Fr(p)d Fu(+)f(1)p Fp(g)p Fr(;)82 b @beginspecial @setspecial @endspecial eop %%Page: 16 16 16 15 bop 0 226 a Fs(16)1695 b(Z.)29 b(AMMARI)0 489 y Fu(where)e @beginspecial @setspecial @endspecial Fr( )s(;)14 b(\036)24 b Fp(2)f(K)d(\012)e Fu(\000)703 501 y Fs(\014n)781 489 y Fu(\()p Fm(h)p Fu(\).)38 b Fr(f)9 b Fu(\()p Fr(\020)d Fu(\))28 b(is)g(a)f(b)r(ounded)h(con)n(tin)n(uous)f(analytic)f (function)j(on)e @beginspecial @setspecial @endspecial Fm(S)p Fu(,)g(satisfying)1441 639 y @beginspecial @setspecial @endspecial 166 w Fp(j)p Fr(f)9 b Fu(\()p Fr(p)p Fu(\))p Fp(j)23 b(\024)g Fr(c)g Fp(jj)p Fr(\036)p Fp(jj)h(jj)p Fr( )s Fp(jj)p Fr(;)1607 814 y Fp(j)p Fr(f)9 b Fu(\()p Fr(p)18 b Fu(+)g(1\))p Fp(j)23 b(\024)g Fr(c)g Fp(jj)p Fr(\036)p Fp(jj)h(jj)p Fr( )s Fp(jj)p Fr(:)0 965 y Fu(W)-7 b(e)28 b(obtain)1315 1115 y @beginspecial @setspecial @endspecial Fp(j)p Fr(f)9 b Fu(\()p Fr(\020)d Fu(\))p Fp(j)23 b(\024)g Fr(c)g Fp(jj)p Fr(\036)p Fp(jj)h(jj)p Fr( )s Fp(jj)p Fr(;)37 b Fu(for)23 b Fr(\020)29 b Fp(2)23 b Fm(S)p Fr(:)83 b @beginspecial @setspecial @endspecial 0 1273 a Fu(Let)30 b @beginspecial @setspecial @endspecial(2)p Fr(p)c Fp(\024)g Fr(r)k Fp(\024)c Fu(2)p Fr(p)19 b Fu(+)h(2.)43 b(Since)30 b Fp(K)21 b(\012)f Fu(\000)1246 1285 y Fs(\014n)1324 1273 y Fu(\()p Fm(h)p Fu(\))31 b(is)e(a)h(core)e(for)i @beginspecial @setspecial @endspecial Fr(N)2011 1221 y Fj(r)p 2011 1230 30 3 v 2012 1263 a Fi(2)2085 1273 y Fu(then)g(\()p Fr(z)23 b Fp(\000)2477 1252 y Fu(^)2455 1273 y Fr(H)2524 1285 y Fq(\024)2567 1273 y Fu(\))2599 1243 y Fo(\000)p Fs(1)2688 1273 y Fr(H)2757 1285 y Fs(0)2795 1273 y Fu(\(1)c(+)h Fr(N)9 b Fu(\))3081 1243 y Fo(\000)3143 1221 y Fj(r)p 3143 1230 V 3144 1263 a Fi(2)3186 1273 y Fr( )30 b Fp(2)d(D)r Fu(\()p Fr(N)3536 1221 y Fj(r)p 3536 1230 V 3537 1263 a Fi(2)3580 1273 y Fu(\))j(and)f(w)n(e)0 1373 y(ha)n(v)n(e)d(also)793 1523 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)925 1467 y Fj(r)p 925 1476 V 926 1509 a Fi(2)969 1523 y Fu(\()p Fr(z)c Fp(\000)1166 1502 y Fu(^)1145 1523 y Fr(H)1214 1535 y Fq(\024)1257 1523 y Fu(\))1289 1489 y Fo(\000)p Fs(1)1378 1523 y Fr(H)1447 1535 y Fs(0)1484 1523 y Fu(\(1)d(+)f Fr(N)9 b Fu(\))1768 1489 y Fo(\000)1829 1467 y Fj(r)p 1829 1476 V 1830 1509 a Fi(2)1873 1523 y Fr( )s Fp(jj)23 b(\024)g Fr(c)g Fp(jj)p Fr( )s Fp(jj)p Fr(;)37 b( )26 b Fp(2)e(K)19 b(\012)f Fu(\000)2731 1535 y Fs(\014n)2810 1523 y Fu(\()p Fm(h)p Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 1681 a Fu(Then)28 b(w)n(e)f(ha)n(v)n(e)f @beginspecial @setspecial @endspecial(\()p Fr(H)631 1693 y Fs(0)687 1681 y Fu(+)18 b(1\))p Fr(N)930 1629 y Fj(r)p 930 1638 V 931 1671 a Fi(2)973 1681 y Fu(\()1027 1660 y(^)1005 1681 y Fr(H)1074 1693 y Fq(\024)1136 1681 y Fp(\000)g Fr(z)t Fu(\))1294 1651 y Fo(\000)p Fs(1)1383 1681 y Fu(\(1)g(+)g Fr(N)9 b Fu(\))1676 1629 y Fj(r)p 1676 1638 V 1677 1671 a Fi(2)1747 1681 y Fu(b)r(ounded.)38 b @beginspecial @setspecial @endspecial 1716 w Fh(\004)0 1702 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 138 x Fz(Lemma)30 b(4.5.)40 b Fn(Ther)l(e)30 b(exist)g @beginspecial @setspecial @endspecial Fr(\017)22 b(>)h Fu(0)p Fr(;)14 b(b)22 b(<)h Fu(0)29 b Fn(and)h Fr(c)g Fn(indep)l(endent)g(for)h Fr(\024)725 2223 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 766 2067 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))14 b Fp(jj)p Fr(R)959 1996 y Fi(1)p 959 2005 29 3 v 959 2038 a(2)948 2092 y Fq(`)1002 2067 y Fu(\()p Fr(b)p Fu(\))p Fr(H)1178 2032 y Fo(f)p Fs(1)p Fo(g)1171 2087 y Fq(\024)1297 2067 y Fr(R)1371 1996 y Fi(1)p 1370 2005 V 1370 2038 a(2)1360 2092 y Fq(`)p Fs(+1)1476 2067 y Fu(\()p Fr(b)p Fu(\))p Fp(jj)23 b(\024)g Fr(c)p Fu(\()1835 1963 y Fq(P)1801 1988 y Fl(X)1804 2165 y Fq(j)s Fs(=1)1935 2067 y Fp(j)p Fr(v)2001 2032 y Fq(j)1998 2087 y(\024)2037 2095 y Fi(0)2073 2067 y Fu(\()p Fr(k)2148 2079 y Fs(1)2186 2067 y Fu(\))p Fp(j)c Fu(+)f Fp(j)p Fr(r)2405 2032 y Fq(j)2403 2087 y(\024)2446 2067 y Fu(\()p Fr(k)2521 2079 y Fs(1)2559 2067 y Fu(\))p Fr(!)s Fu(\()p Fr(k)2721 2079 y Fs(1)2759 2067 y Fu(\))2791 2032 y Fo(\000)p Fq(\017)2875 2067 y Fp(j)p Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 766 2371 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))14 b Fp(jj)p Fr(R)986 2300 y Fi(1)p 986 2309 V 986 2342 a(2)975 2396 y Fq(`)1028 2371 y Fu(\()p Fr(b)p Fu(\))p Fr(H)1204 2337 y Fo(f)p Fs(1)p Fq(;)p Fs(2)p Fo(g)1197 2392 y Fq(\024)1376 2371 y Fr(R)1450 2300 y Fi(1)p 1450 2309 V 1450 2342 a(2)1439 2396 y Fq(`)p Fs(+1)1555 2371 y Fu(\()p Fr(b)p Fu(\))p Fp(jj)24 b(\024)e Fr(c)p Fu(\()1915 2267 y Fq(P)1880 2292 y Fl(X)1883 2469 y Fq(j)s Fs(=1)2014 2371 y Fp(j)p Fr(r)2076 2337 y Fq(j)2074 2392 y(\024)2118 2371 y Fu(\()p Fr(k)2193 2383 y Fs(1)2230 2371 y Fu(\))p Fr(!)s Fu(\()p Fr(k)2392 2383 y Fs(1)2430 2371 y Fu(\))2462 2337 y Fo(\000)p Fq(\017)2546 2371 y Fr(r)2585 2337 y Fq(j)2583 2392 y(\024)2627 2371 y Fu(\()p Fr(k)2702 2383 y Fs(2)2739 2371 y Fu(\))p Fr(!)s Fu(\()p Fr(k)2901 2383 y Fs(2)2939 2371 y Fu(\))2971 2337 y Fo(\000)p Fq(\017)3055 2371 y Fp(j)p Fu(\))p Fr(:)3175 2223 y @beginspecial @setspecial @endspecial 100 2626 a Fz(Pro)s(of.)56 b @beginspecial @setspecial @endspecial 30 w Fu(Let)32 b Fp(D)j Fu(b)r(e)d(a)g(dense)f(set)h(of)g(analytic)f(v)n(ectors)g(for) g Fr(K)6 b Fu(.)49 b @beginspecial @setspecial @endspecial Fr(H)2314 2595 y Fq(J)2351 2603 y Fj(i)2307 2646 y Fq(\024)2413 2626 y Fu(can)31 b(b)r(e)i(de\014ned)f(on)g Fp(D)23 b(\012)e Fu(\000)3322 2638 y Fs(\014n)3400 2626 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)3575 2595 y Fs(3)3618 2626 y Fu(\)\),)34 b(then)0 2752 y @beginspecial @setspecial @endspecial Fr(H)76 2722 y Fq(J)113 2730 y Fj(i)69 2773 y Fq(\024)143 2752 y Fr(R)217 2681 y Fi(1)p 217 2690 V 217 2724 a(2)206 2777 y Fq(`)p Fs(+1)322 2752 y Fu(\()p Fr(b)p Fu(\))c(is)f(w)n(ell)g (de\014ned)g(on)g @beginspecial @setspecial @endspecial Fp(D)1175 2764 y Fq(\024)1218 2752 y Fu(\()p Fr(b)p Fu(\))d(:=)g(\() 1512 2731 y(^)1490 2752 y Fr(H)1559 2764 y Fq(\024)1622 2752 y Fp(\000)19 b Fr(b)p Fu(\))1784 2700 y Fi(1)p 1784 2709 V 1784 2742 a(2)1826 2752 y Fp(D)j(\012)d Fu(\000)2048 2764 y Fs(\014n)2126 2752 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)2301 2722 y Fs(3)2344 2752 y Fu(\)\).)42 b(F)-7 b(urthermore)29 b(since)g @beginspecial @setspecial @endspecial Fp(D)22 b(\012)d Fu(\000)3384 2764 y Fs(\014n)3462 2752 y Fu(\()p Fr(S)5 b Fu(\()p Fk(R)3636 2722 y Fs(3)3679 2752 y Fu(\)\))30 b(is)f(a)0 2872 y(core)e(for)h @beginspecial @setspecial @endspecial(\()358 2851 y(^)336 2872 y Fr(H)405 2884 y Fq(\024)467 2872 y Fp(\000)18 b Fr(b)p Fu(\),)29 b Fp(D)734 2884 y Fq(\024)777 2872 y Fu(\()p Fr(b)p Fu(\))f(is)g(dense) h(in)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\(\()1467 2851 y(^)1444 2872 y Fr(H)1513 2884 y Fq(\024)1576 2872 y Fp(\000)18 b Fr(b)p Fu(\))1737 2819 y Fi(1)p 1737 2828 V 1737 2862 a(2)1779 2872 y Fu(\),)29 b(whic)n(h)f(is)h(dense)f(in)g Fp(H)q Fu(.)39 b(Hence)28 b(it)h(is)f(enough)f(to)i(sho)n(w)e @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))19 b Fp(\000)j Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))0 2998 y(in)26 b Fp(D)159 3010 y Fq(\024)202 2998 y Fu(\()p Fr(b)p Fu(\).)36 b(Lemma)25 b(3.3)g(and)g(Lemma)g @beginspecial @setspecial @endspecial(2.8)g(giv)n(e)f(the)i(b)r(ounds)g (uniformly)f(in)h Fr(\024)p Fu(.)36 b(Then)25 b @beginspecial @setspecial @endspecial Fr(R)2834 2927 y Fi(1)p 2834 2936 V 2834 2970 a(2)2823 3023 y Fq(`)2876 2998 y Fu(\()p Fr(b)p Fu(\))14 b Fr(H)3066 2968 y Fq(J)3059 3019 y(\024)3126 2998 y Fr(R)3200 2927 y Fi(1)p 3200 2936 V 3200 2970 a(2)3189 3023 y Fq(`)p Fs(+1)3305 2998 y Fu(\()p Fr(b)p Fu(\))26 b(extends)f(from)0 3101 y @beginspecial @setspecial @endspecial Fp(D)64 3113 y Fq(\024)107 3101 y Fu(\()p Fr(b)p Fu(\))j(to)f(a)h(b)r(ounded)g(op)r(erator)d(on)j @beginspecial @setspecial @endspecial Fp(H)q Fu(.)2546 b Fh(\004)100 3201 y Fu(The)27 b(follo)n(wing)g(lemma)g(is)h(the)g Fn(gener)l(alize)l(d)j(pul)t(l)f(thr)l(ough)g(formula)p Fu(.)0 3217 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 142 x Fz(Lemma)g(4.6.)40 b Fn(The)30 b(fol)t(lowing)i(identity)f(holds)g(for)g(al)t(l)f Fr(\036)24 b Fp(2)f(D)r Fu(\()p Fr(N)2153 3307 y Fj(r)p 2153 3316 30 3 v 2154 3349 a Fi(2)2196 3359 y Fu(\))p Fn(:)169 3576 y @beginspecial @setspecial @endspecial 206 3472 a Fq(r)170 3497 y Fl(Y)169 3674 y Fq(i)p Fs(=1)290 3576 y Fr(a)p Fu(\()p Fr(k)409 3588 y Fq(i)437 3576 y Fu(\)\()p Fr(z)f Fp(\000)667 3555 y Fu(^)645 3576 y Fr(H)714 3588 y Fq(\024)757 3576 y Fu(\))789 3542 y Fo(\000)p Fs(1)879 3576 y Fr(\036)83 b Fu(=)g Fr(R)1222 3588 y Fs(1)1259 3576 y Fu(\()p Fr(z)t Fu(\))1417 3472 y Fq(r)1380 3497 y Fl(Y)1380 3674 y Fq(i)p Fs(=1)1501 3576 y Fr(a)p Fu(\()p Fr(k)1620 3588 y Fq(i)1648 3576 y Fu(\))p Fr(\036)19 b Fu(+)1842 3497 y Fl(X)1831 3671 y Fs(part)p Fq(:)1987 3576 y Fr(R)2050 3588 y Fs(1)2087 3576 y Fu(\()p Fr(z)t Fu(\))p Fr(H)2270 3542 y Fq(J)2307 3550 y Fi(1)2263 3596 y Fq(\024)2343 3576 y Fr(R)2406 3588 y Fs(2)2443 3576 y Fu(\()p Fr(z)t Fu(\))p Fr(:::R)2682 3588 y Fq(`)p Fo(\000)p Fs(1)2799 3576 y Fu(\()p Fr(z)t Fu(\))p Fr(H)2982 3542 y Fq(J)3019 3551 y Fj(`)p Fg(\000)p Fi(1)2975 3596 y Fq(\024)3125 3576 y Fr(R)3188 3588 y Fq(`)3220 3576 y Fu(\()p Fr(z)t Fu(\))3358 3497 y Fl(Y)3341 3675 y Fq(j)s Fo(2)p Fq(J)3453 3684 y Fj(`)3495 3576 y Fr(a)p Fu(\()p Fr(k)3614 3588 y Fq(j)3650 3576 y Fu(\))p Fr(\036)0 3720 y @beginspecial @setspecial @endspecial 1159 3866 a Fu(+)1248 3787 y Fl(X)1238 3961 y Fs(part)p Fq(:)1392 3866 y Fr(R)1455 3878 y Fs(1)1493 3866 y Fu(\()p Fr(z)t Fu(\))p Fr(H)1676 3831 y Fq(J)1713 3839 y Fi(1)1669 3886 y Fq(\024)1749 3866 y Fr(R)1812 3878 y Fs(2)1849 3866 y Fu(\()p Fr(z)t Fu(\))p Fr(:::R)2088 3878 y Fq(`)p Fo(\000)p Fs(1)2205 3866 y Fu(\()p Fr(z)t Fu(\))p Fr(H)2388 3831 y Fq(J)2425 3840 y Fj(`)2381 3886 y Fq(\024)2457 3866 y Fr(R)q Fu(\()p Fr(z)t Fu(\))14 b Fr(\036:)0 4102 y Fn(The)31 b(sum)e(in)h(r.h.s.)40 b(is)30 b(taken)g(over)g(al)t(l)h(the)f(p)l(artitions)g(of)h(the)f(set) f @beginspecial @setspecial @endspecial Fp(f)p Fu(1)p Fr(;)14 b(:::;)g(r)r Fp(g)29 b Fn(into)h(or)l(der)l(e)l(d)h(subsets.)0 4123 y @beginspecial @setspecial @endspecial 100 4260 a Fz(Pro)s(of.)88 b @beginspecial @setspecial @endspecial 45 w Fu(W)-7 b(e)41 b(pro)n(v)n(e)e(this)j(lemma)f(b)n(y)f(induction)i (on)e @beginspecial @setspecial @endspecial Fr(r)r Fu(.)78 b(By)41 b(Lemma)f(4.4,)k(w)n(e)c(kno)n(w)g(that)i(there)e(exist)h @beginspecial @setspecial @endspecial Fr( )49 b Fp(2)0 4368 y Fu(\()p Fr(H)101 4380 y Fs(0)157 4368 y Fu(+)18 b(1\))314 4334 y Fo(\000)p Fs(1)403 4368 y Fp(D)r Fu(\()p Fr(N)587 4312 y Fi(1)p 587 4321 29 3 v 587 4354 a(2)630 4368 y Fu(\))28 b(suc)n(h)f(that)h Fr(\036)23 b Fu(=)g(\()p Fr(z)f Fp(\000)1415 4347 y Fu(^)1393 4368 y Fr(H)1462 4380 y Fq(\024)1505 4368 y Fu(\))p Fr( )s Fu(.)37 b(W)-7 b(e)28 b(consider)f @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)2240 4380 y Fs(1)2277 4368 y Fu(\))h(as)f(b)r (ounded)h(op)r(erator)685 4531 y @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)804 4543 y Fs(1)841 4531 y Fu(\))c(:)f(\()p Fr(H)1044 4543 y Fs(0)1100 4531 y Fu(+)18 b(1\))1257 4497 y Fo(\000)p Fs(1)1346 4531 y Fp(D)r Fu(\()p Fr(N)1530 4475 y Fi(1)p 1530 4484 V 1530 4517 a(2)1572 4531 y Fu(\))24 b Fp(!)f Fu(\()p Fr(H)1835 4543 y Fs(0)1891 4531 y Fu(+)18 b Fr(!)s Fu(\()p Fr(k)2104 4543 y Fs(1)2141 4531 y Fu(\))h(+)f(1\))2349 4497 y Fo(\000)p Fs(1)2438 4531 y Fr(L)2495 4497 y Fs(2)2532 4531 y Fu(\()p Fk(R)2618 4497 y Fs(3)2661 4531 y Fr(;)c(dk)2784 4543 y Fs(1)2822 4531 y Fu(\))k Fp(\012)g(H)q Fr(:)83 b @beginspecial @setspecial @endspecial 0 4681 a Fu(Then)28 b(w)n(e)f(can)g(write)h (for)f @beginspecial @setspecial @endspecial Fr(\024)c(<)f Fp(1)971 4832 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)1090 4844 y Fs(1)1128 4832 y Fu(\))p Fr( )k Fu(=)d(\()p Fr(z)f Fp(\000)c Fr(!)s Fu(\()p Fr(k)1634 4844 y Fs(1)1671 4832 y Fu(\))h Fp(\000)1827 4811 y Fu(^)1805 4832 y Fr(H)1874 4844 y Fq(\024)1917 4832 y Fu(\))1949 4797 y Fo(\000)p Fs(1)2038 4832 y Fu(\()p Fr(z)j Fp(\000)c Fr(!)s Fu(\()p Fr(k)2344 4844 y Fs(1)2382 4832 y Fu(\))h Fp(\000)2537 4811 y Fu(^)2516 4832 y Fr(H)2585 4844 y Fq(\024)2628 4832 y Fu(\))p Fr(a)p Fu(\()p Fr(k)2779 4844 y Fs(1)2817 4832 y Fu(\))p Fr( )s(:)p @beginspecial @setspecial @endspecial 0 4998 a Fu(By)27 b(the)h(justi\014cation)f(in)h(the)g(b)r(eginning)f (of)g(this)h(subsection,)f(w)n(e)g(see)g(that)h @beginspecial @setspecial @endspecial Fr(H)2540 4955 y Fo(f)p Fs(1)p Fo(g)2533 5008 y Fq(\024)2645 4998 y Fr( )e Fp(2)d Fr(L)2860 4968 y Fs(2)2897 4998 y Fu(\()p Fk(R)2983 4968 y Fs(3)3027 4998 y Fr(;)14 b(dk)3150 5010 y Fs(1)3187 4998 y Fu(\))k Fp(\012)g(H)q Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(the)0 5098 y(follo)n(wing)h(iden)n(tit)n(y)g(on)h @beginspecial @setspecial @endspecial Fr(L)830 5068 y Fs(2)866 5098 y Fu(\()p Fk(R)953 5068 y Fs(3)996 5098 y Fr(;)14 b(dk)1119 5110 y Fs(1)1156 5098 y Fu(\))19 b Fp(\012)f(H)913 5256 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(z)k Fp(\000)c Fr(!)s Fu(\()p Fr(k)1219 5268 y Fs(1)1257 5256 y Fu(\))h Fp(\000)1412 5235 y Fu(^)1391 5256 y Fr(H)1460 5268 y Fq(\024)1503 5256 y Fu(\))p Fr(a)p Fu(\()p Fr(k)1654 5268 y Fs(1)1692 5256 y Fu(\))p Fr( )26 b Fu(=)d Fr(a)p Fu(\()p Fr(k)2011 5268 y Fs(1)2048 5256 y Fu(\)\()p Fr(z)g Fp(\000)2278 5235 y Fu(^)2257 5256 y Fr(H)2326 5268 y Fq(\024)2369 5256 y Fu(\))p Fr( )e Fu(+)d Fr(H)2635 5222 y Fo(f)p Fs(1)p Fo(g)2628 5277 y Fq(\024)2740 5256 y Fr( )s(:)84 b @beginspecial @setspecial @endspecial -2904 w Fu(\(4.5\))0 5406 y(This)28 b(pro)n(v)n(es)d(the)j(pull)g(through)f(form)n(ula)g (for)g @beginspecial @setspecial @endspecial Fr(r)f Fu(=)c(1.)0 5506 y(The)28 b(form)n(ula)e(\()p @beginspecial @setspecial @endspecial(4.5\))i(can)f(b)r(e)h(generalized)e(b)n(y)h(induction)h(in) g Fn(r)p Fu(.)p eop %%Page: 17 17 17 16 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(17)100 543 y Fu(W)-7 b(e)28 b(claim)f(that)h (for)f @beginspecial @setspecial @endspecial Fr( )f Fp(2)d Fu(\()p Fr(H)1030 555 y Fs(0)1086 543 y Fu(+)18 b(1\))1243 509 y Fo(\000)p Fs(1)1346 543 y Fp(D)r Fu(\()p Fr(N)1531 486 y Fj(r)p 1531 495 30 3 v 1532 529 a Fi(2)1574 543 y Fu(\),)28 b @beginspecial @setspecial @endspecial Fr(I)1693 555 y Fq(r)1753 543 y Fu(:=)23 b Fp(f)p Fu(1)p Fr(:::r)r Fp(g)p Fu(,)k(w)n(e)h(ha)n(v)n(e)e(in)i Fr(L)2616 509 y Fs(2)2653 543 y Fu(\()p Fk(R)2739 509 y Fs(3)p Fq(r)2815 543 y Fr(;)2889 439 y Fq(r)2852 464 y Fl(Y)2852 641 y Fq(i)p Fs(=1)2973 543 y Fr(dk)3059 555 y Fq(i)3087 543 y Fu(\))19 b Fp(\012)f(H)q Fu(:)228 782 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\()p Fr(z)k Fp(\000)409 703 y Fl(X)404 881 y Fq(i)p Fo(2)p Fq(I)501 889 y Fj(r)549 782 y Fr(!)s Fu(\()p Fr(k)679 794 y Fq(i)707 782 y Fu(\))c Fp(\000)862 761 y Fu(^)840 782 y Fr(H)909 794 y Fq(\024)952 782 y Fu(\))1011 703 y Fl(Y)998 881 y Fq(i)p Fo(2)p Fq(I)1095 889 y Fj(r)1143 782 y Fr(a)p Fu(\()p Fr(k)1262 794 y Fq(i)1291 782 y Fu(\))p Fr( )26 b Fu(=)1503 703 y Fl(Y)1491 881 y Fq(i)p Fo(2)p Fq(I)1588 889 y Fj(r)1636 782 y Fr(a)p Fu(\()p Fr(k)1755 794 y Fq(i)1783 782 y Fu(\)\()p Fr(z)c Fp(\000)2013 761 y Fu(^)1991 782 y Fr(H)2060 794 y Fq(\024)2103 782 y Fu(\))p Fr( )g Fu(+)c Fr(H)2370 748 y Fq(I)2399 756 y Fj(r)2363 803 y Fq(\024)2436 782 y Fr( )k Fu(+)2697 703 y Fl(X)2595 885 y Fq(I)2624 893 y Fj(r)2658 885 y Fs(=\()p Fq(J)2772 893 y Fi(1)2804 885 y Fq(;J)2861 893 y Fi(2)2893 885 y Fs(\))2933 782 y Fr(H)3009 748 y Fq(J)3046 756 y Fi(1)3002 803 y Fq(\024)3111 703 y Fl(Y)3095 881 y Fq(i)p Fo(2)p Fq(J)3200 889 y Fi(2)3247 782 y Fr(a)p Fu(\()p Fr(k)3366 794 y Fq(i)3394 782 y Fu(\))p Fr( )s(;)83 b @beginspecial @setspecial @endspecial -3589 w Fu(\(4.6\))0 1016 y(where)36 b(the)i(sum)f(is)g(o)n(v)n(er)e(all)i(partitions)f @beginspecial @setspecial @endspecial(\()p Fr(J)1458 1028 y Fs(1)1496 1016 y Fr(;)14 b(J)1579 1028 y Fs(2)1616 1016 y Fu(\))37 b(of)g Fr(I)1825 1028 y Fq(r)1863 1016 y Fu(.)65 b(Let)37 b(us)g(pro)n(v)n(e)e(\()p @beginspecial @setspecial @endspecial(4.6\),)40 b(b)n(y)c(induction)i(on)e Fr(r)r Fu(.)66 b(W)-7 b(e)38 b(ha)n(v)n(e)e(for)0 1133 y Fr( )26 b Fp(2)e Fu(\()p Fr(H)260 1145 y Fs(0)316 1133 y Fu(+)18 b(1\))473 1098 y Fo(\000)p Fs(1)561 1133 y Fp(D)r Fu(\()p Fr(N)746 1073 y Fj(r)q Fi(+1)p 746 1086 101 3 v 782 1118 a(2)860 1133 y Fu(\):)95 1286 y @beginspecial @setspecial @endspecial(\()p Fr(z)k Fp(\000)312 1207 y Fl(X)271 1385 y Fq(i)p Fo(2)p Fq(I)368 1393 y Fj(r)q Fi(+1)487 1286 y Fr(!)s Fu(\()p Fr(k)617 1298 y Fq(i)645 1286 y Fu(\))c Fp(\000)800 1265 y Fu(^)778 1286 y Fr(H)847 1298 y Fq(\024)890 1286 y Fu(\))984 1207 y Fl(Y)936 1385 y Fq(i)p Fo(2)p Fq(I)1033 1393 y Fj(r)q Fi(+1)1152 1286 y Fr(a)p Fu(\()p Fr(k)1271 1298 y Fq(i)1299 1286 y Fu(\))p Fr( )87 b @beginspecial @setspecial @endspecial Fu(=)82 b Fr(a)p Fu(\()p Fr(k)1738 1298 y Fq(r)r Fs(+1)1860 1286 y Fu(\)\()p Fr(z)22 b Fp(\000)2073 1207 y Fl(X)2068 1385 y Fq(i)p Fo(2)p Fq(I)2165 1393 y Fj(r)2213 1286 y Fr(!)s Fu(\()p Fr(k)2343 1298 y Fq(i)2371 1286 y Fu(\))c Fp(\000)2526 1265 y Fu(^)2504 1286 y Fr(H)2573 1298 y Fq(\024)2616 1286 y Fu(\))2675 1207 y Fl(Y)2662 1385 y Fq(i)p Fo(2)p Fq(I)2759 1393 y Fj(r)2807 1286 y Fr(a)p Fu(\()p Fr(k)2926 1298 y Fq(i)2955 1286 y Fu(\))p Fr( )j Fu(+)d Fr(H)3221 1252 y Fo(f)p Fq(r)r Fs(+1)p Fo(g)3214 1307 y Fq(\024)3436 1207 y Fl(Y)3424 1385 y Fq(i)p Fo(2)p Fq(I)3521 1393 y Fj(r)3569 1286 y Fr(a)p Fu(\()p Fr(k)3688 1298 y Fq(i)3716 1286 y Fu(\))p Fr( )0 1434 y @beginspecial @setspecial @endspecial 1472 1566 a Fu(=)1667 1488 y Fl(Y)1619 1666 y Fq(i)p Fo(2)p Fq(I)1716 1674 y Fj(r)q Fi(+1)1835 1566 y Fr(a)p Fu(\()p Fr(k)1954 1578 y Fq(i)1982 1566 y Fu(\)\()p Fr(z)23 b Fp(\000)2212 1545 y Fu(^)2191 1566 y Fr(H)2260 1578 y Fq(\024)2303 1566 y Fu(\))p Fr( )e Fu(+)d Fr(H)2569 1532 y Fo(f)p Fq(r)r Fs(+1)p Fo(g)2562 1587 y Fq(\024)2784 1488 y Fl(Y)2772 1666 y Fq(i)p Fo(2)p Fq(I)2869 1674 y Fj(r)2917 1566 y Fr(a)p Fu(\()p Fr(k)3036 1578 y Fq(i)3064 1566 y Fu(\))p Fr( )0 1714 y @beginspecial @setspecial @endspecial 1619 1847 a Fu(+)p Fr(a)p Fu(\()p Fr(k)1803 1859 y Fq(r)r Fs(+1)1924 1847 y Fu(\))2072 1768 y Fl(X)1970 1950 y Fq(I)1999 1958 y Fj(r)2033 1950 y Fs(=\()p Fq(J)2147 1958 y Fi(1)2179 1950 y Fq(;J)2236 1958 y Fi(2)2268 1950 y Fs(\))2308 1847 y Fr(H)2384 1813 y Fq(J)2421 1821 y Fi(1)2377 1867 y Fq(\024)2471 1768 y Fl(Y)2489 1946 y Fq(J)2526 1954 y Fi(2)2591 1847 y Fr(a)p Fu(\()p Fr(k)2710 1859 y Fq(i)2738 1847 y Fu(\))p Fr( )j Fu(+)d Fr(a)p Fu(\()p Fr(k)3047 1859 y Fq(r)r Fs(+1)3169 1847 y Fu(\))p Fr(H)3277 1813 y Fq(I)3306 1821 y Fj(r)3270 1867 y Fq(\024)3344 1847 y Fr( )s(:)0 1998 y @beginspecial @setspecial @endspecial 1504 2043 a @beginspecial @setspecial @endspecial 0 2180 a Fu(Mo)n(ving)26 b Fr(a)p Fu(\()p Fr(k)416 2192 y Fq(r)r Fs(+1)538 2180 y Fu(\))i(through)f Fr(H)988 2150 y Fq(J)1025 2158 y Fi(1)981 2201 y Fq(\024)1088 2180 y Fu(and)h @beginspecial @setspecial @endspecial Fr(H)1326 2150 y Fq(I)1355 2158 y Fj(r)1319 2201 y Fq(\024)1420 2180 y Fu(and)g(using)f(the)h(iden)n (tit)n(y)646 2341 y @beginspecial @setspecial @endspecial Fr(a)p Fu(\()p Fr(k)765 2353 y Fq(r)r Fs(+1)886 2341 y Fu(\))p Fr(H)994 2306 y Fq(J)1031 2314 y Fi(1)987 2361 y Fq(\024)1091 2341 y Fu(=)22 b Fr(H)1254 2306 y Fq(J)1291 2314 y Fi(1)1247 2361 y Fq(\024)1327 2341 y Fr(a)p Fu(\()p Fr(k)1446 2353 y Fq(r)r Fs(+1)1567 2341 y Fu(\))d(+)f Fr(H)1777 2306 y Fq(J)1814 2314 y Fi(1)1846 2306 y Fo([f)p Fq(r)r Fs(+1)p Fo(g)1770 2361 y Fq(\024)2080 2341 y Fr(;)37 b Fu(on)22 b(\()p Fr(H)2351 2353 y Fs(0)2407 2341 y Fu(+)c(1\))2564 2306 y Fo(\000)p Fs(1)2653 2341 y Fp(D)r Fu(\()p Fr(N)2838 2275 y Fi(#)p Fj(J)2916 2287 y Fi(1)2948 2275 y(+1)p 2838 2293 181 3 v 2914 2326 a(2)3033 2341 y Fu(\))p Fr(;)83 b @beginspecial @setspecial @endspecial 0 2478 a Fu(w)n(e)27 b(obtain)h(\()p @beginspecial @setspecial @endspecial(4.6\).)0 2577 y(No)n(w)f(assume)g(that)h(the)g(pull)g(through)f(form)n(ula)f (holds)i(for)f @beginspecial @setspecial @endspecial Fr(r)r Fu(,)h(and)g(let)g(us)f(pro)n(v)n(e)f(it)i(for)f Fr(r)21 b Fu(+)d(1.)37 b(W)-7 b(e)28 b(ha)n(v)n(e:)351 2726 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 399 2647 a Fl(Y)351 2825 y Fq(i)p Fo(2)p Fq(I)448 2833 y Fj(r)q Fi(+1)567 2726 y Fr(a)p Fu(\()p Fr(k)686 2738 y Fq(i)714 2726 y Fu(\)\()p Fr(z)22 b Fp(\000)944 2705 y Fu(^)922 2726 y Fr(H)991 2738 y Fq(\024)1034 2726 y Fu(\))1066 2692 y Fo(\000)p Fs(1)1155 2726 y Fr(\036)i Fu(=)f(\()p Fr(z)f Fp(\000)1533 2647 y Fl(X)1492 2825 y Fq(i)p Fo(2)p Fq(I)1589 2833 y Fj(r)q Fi(+1)1707 2726 y Fr(!)s Fu(\()p Fr(k)1837 2738 y Fq(i)1865 2726 y Fu(\))d Fp(\000)2021 2705 y Fu(^)1999 2726 y Fr(H)2068 2738 y Fq(\024)2111 2726 y Fu(\))2143 2692 y Fo(\000)p Fs(1)2232 2726 y Fu(\()p Fr(z)j Fp(\000)2449 2647 y Fl(X)2408 2825 y Fq(i)p Fo(2)p Fq(I)2505 2833 y Fj(r)q Fi(+1)2624 2726 y Fr(!)s Fu(\()p Fr(k)2754 2738 y Fq(i)2782 2726 y Fu(\))d Fp(\000)2938 2705 y Fu(^)2916 2726 y Fr(H)2985 2738 y Fq(\024)3028 2726 y Fu(\))3122 2647 y Fl(Y)3074 2825 y Fq(i)p Fo(2)p Fq(I)3171 2833 y Fj(r)q Fi(+1)3290 2726 y Fr(a)p Fu(\()p Fr(k)3409 2738 y Fq(i)3437 2726 y Fu(\))p Fr( )s(:)p @beginspecial @setspecial @endspecial 0 2957 a Fu(Using)27 b(no)n(w)g(the)h(iterated)g(form)n(ula)e(\()p @beginspecial @setspecial @endspecial(4.6\),)i(w)n(e)f(obtain:)218 3110 y @beginspecial @setspecial @endspecial 266 3031 a Fl(Y)218 3209 y Fq(i)p Fo(2)p Fq(I)315 3217 y Fj(r)q Fi(+1)434 3110 y Fr(a)p Fu(\()p Fr(k)553 3122 y Fq(i)581 3110 y Fu(\)\()p Fr(z)22 b Fp(\000)811 3089 y Fu(^)789 3110 y Fr(H)858 3122 y Fq(\024)901 3110 y Fu(\))933 3076 y Fo(\000)p Fs(1)1023 3110 y Fr(\036)83 b Fu(=)g @beginspecial @setspecial @endspecial(\()p Fr(z)22 b Fp(\000)1520 3031 y Fl(X)1479 3209 y Fq(i)p Fo(2)p Fq(I)1576 3217 y Fj(r)q Fi(+1)1695 3110 y Fr(!)s Fu(\()p Fr(k)1825 3122 y Fq(i)1852 3110 y Fu(\))d Fp(\000)2008 3089 y Fu(^)1986 3110 y Fr(H)2055 3122 y Fq(\024)2098 3110 y Fu(\))2130 3076 y Fo(\000)p Fs(1)2281 3031 y Fl(Y)2234 3209 y Fq(i)p Fo(2)p Fq(I)2331 3217 y Fj(r)q Fi(+1)2449 3110 y Fr(a)p Fu(\()p Fr(k)2568 3122 y Fq(i)2596 3110 y Fu(\))p Fr(\036)0 3258 y @beginspecial @setspecial @endspecial 1303 3391 a Fu(+\()p Fr(z)i Fp(\000)1584 3312 y Fl(X)1543 3490 y Fq(i)p Fo(2)p Fq(I)1640 3498 y Fj(r)q Fi(+1)1759 3391 y Fr(!)s Fu(\()p Fr(k)1889 3403 y Fq(i)1917 3391 y Fu(\))e Fp(\000)2073 3370 y Fu(^)2051 3391 y Fr(H)2120 3403 y Fq(\024)2163 3391 y Fu(\))2195 3356 y Fo(\000)p Fs(1)2284 3391 y Fr(H)2360 3356 y Fq(I)2389 3364 y Fj(r)q Fi(+1)2353 3411 y Fq(\024)2498 3391 y Fu(\()p Fr(z)j Fp(\000)2696 3370 y Fu(^)2674 3391 y Fr(H)2743 3403 y Fq(\024)2786 3391 y Fu(\))2818 3356 y Fo(\000)p Fs(1)2907 3391 y Fr(\036)0 3538 y @beginspecial @setspecial @endspecial 1303 3671 a Fu(+\()p Fr(z)f Fp(\000)1584 3592 y Fl(X)1543 3770 y Fq(i)p Fo(2)p Fq(I)1640 3778 y Fj(r)q Fi(+1)1759 3671 y Fr(!)s Fu(\()p Fr(k)1889 3683 y Fq(i)1917 3671 y Fu(\))e Fp(\000)2073 3650 y Fu(^)2051 3671 y Fr(H)2120 3683 y Fq(\024)2163 3671 y Fu(\))2195 3637 y Fo(\000)p Fs(1)2435 3592 y Fl(X)2298 3774 y Fq(I)2327 3782 y Fj(r)q Fi(+1)2432 3774 y Fs(=\()p Fq(J)2546 3782 y Fi(1)2578 3774 y Fq(;J)2635 3782 y Fi(2)2667 3774 y Fs(\))2707 3671 y Fr(H)2783 3637 y Fq(J)2820 3645 y Fi(1)2776 3692 y Fq(\024)2885 3592 y Fl(Y)2869 3770 y Fq(i)p Fo(2)p Fq(J)2974 3778 y Fi(2)3021 3671 y Fr(a)p Fu(\()p Fr(k)3140 3683 y Fq(i)3168 3671 y Fu(\)\()p Fr(z)j Fp(\000)3398 3650 y Fu(^)3376 3671 y Fr(H)3445 3683 y Fq(\024)3488 3671 y Fu(\))3520 3637 y Fo(\000)p Fs(1)3609 3671 y Fr(\036:)0 3901 y Fu(Using)27 b(the)h(induction)g(h)n(yp)r(othesis)f(w)n(e)h (obtain:)226 4050 y @beginspecial @setspecial @endspecial 364 3971 a Fl(X)226 4153 y Fq(I)255 4161 y Fj(r)q Fi(+1)360 4153 y Fs(=\()p Fq(J)474 4161 y Fi(1)506 4153 y Fq(;J)563 4161 y Fi(2)595 4153 y Fs(\))635 4050 y Fr(H)711 4016 y Fq(J)748 4024 y Fi(1)704 4070 y Fq(\024)813 3971 y Fl(Y)798 4149 y Fq(i)p Fo(2)p Fq(J)903 4157 y Fi(2)949 4050 y Fr(a)p Fu(\()p Fr(k)1068 4062 y Fq(i)1096 4050 y Fu(\)\()p Fr(z)22 b Fp(\000)1326 4029 y Fu(^)1304 4050 y Fr(H)1373 4062 y Fq(\024)1416 4050 y Fu(\))1448 4016 y Fo(\000)p Fs(1)1561 4050 y Fu(=)1659 3971 y Fl(X)1648 4145 y Fs(part)p Fq(:)1804 4050 y Fr(R)1867 4062 y Fs(1)1904 4050 y Fu(\()p Fr(z)t Fu(\))p Fr(H)2087 4016 y Fq(J)2124 4024 y Fi(1)2080 4070 y Fq(\024)2160 4050 y Fr(R)2223 4062 y Fs(2)2260 4050 y Fu(\()p Fr(z)t Fu(\))p Fr(:::R)2499 4062 y Fq(`)p Fo(\000)p Fs(1)2616 4050 y Fu(\()p Fr(z)t Fu(\))p Fr(H)2799 4016 y Fq(J)2836 4025 y Fj(`)p Fg(\000)p Fi(1)2792 4070 y Fq(\024)2942 4050 y Fr(R)3005 4062 y Fq(`)3037 4050 y Fu(\()p Fr(z)t Fu(\))3171 3971 y Fl(Y)3158 4149 y Fq(i)p Fo(2)p Fq(J)3263 4158 y Fj(`)3305 4050 y Fr(a)p Fu(\()p Fr(k)3424 4062 y Fq(i)3452 4050 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 100 4280 a Fu(This)27 b(completes)g(the)h(pro)r(of.)p @beginspecial @setspecial @endspecial 2807 w Fh(\004)0 4296 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 134 x Fz(Theorem)i(4.7.)40 b Fn(L)l(et)30 b Fr(\027)e Fp(\024)23 b Fu(1)p Fr(;)14 b Fu(0)22 b Fp(\024)g Fr(\034)40 b Fn(and)30 b Fr(r)c Fp(2)d Fk(N)t Fn(.)45 b(Then)953 4594 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1085 4523 y Fi(1)p 1085 4532 29 3 v 1085 4566 a(2)1066 4604 y Fq(\027)1127 4594 y Fr(N)1213 4520 y Fj(r)q Fg(\000)p Fi(1)p 1213 4533 103 3 v 1250 4566 a(2)1194 4615 y Fo(\000)p Fq(\034)1330 4594 y Fr( )s Fp(jj)23 b(\024)36 b Fr(c)14 b Fp(jj)p Fu(\()1707 4573 y(^)1685 4594 y Fr(H)1754 4606 y Fq(\024)1816 4594 y Fu(+)k Fr(b)p Fu(\))1977 4538 y Fj(r)p 1977 4547 30 3 v 1978 4580 a Fi(2)2020 4594 y Fr( )s Fp(jj)p Fr(;)24 b @beginspecial @setspecial @endspecial 43 w Fn(for)g Fr( )i Fp(2)e(D)r Fu(\()2618 4573 y(^)2596 4594 y Fr(H)2682 4523 y Fj(r)p 2682 4532 V 2683 4566 a Fi(2)2665 4604 y Fq(\024)2725 4594 y Fu(\))p Fr(;)84 b @beginspecial @setspecial @endspecial -2864 w Fu(\(4.7\))0 4731 y Fn(wher)l(e)30 b(c,b)h(ar)l(e)f(c)l(onstants)f(indep)l(endent)h(of)h @beginspecial @setspecial @endspecial Fr(\024)p Fn(.)0 4747 y @beginspecial @setspecial @endspecial 100 4882 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 23 w Fu(W)-7 b(e)27 b(pro)n(v)n(e)e(the)j(theorem)e(b)n(y)g(induction)h (on)g Fr(r)r Fu(.)37 b(The)27 b(case)f Fr(r)g Fu(=)c(1)27 b(follo)n(ws)e(from)i(Lemma)f @beginspecial @setspecial @endspecial(4.2.)36 b(Assume)27 b(that)0 4990 y(\()p @beginspecial @setspecial @endspecial(4.7\))j(holds)g(for)f(all)h @beginspecial @setspecial @endspecial Fr(j)j Fp(\024)27 b Fr(r)r Fu(.)45 b(Let)31 b Fr(\036)d Fp(2)f Fu(\()p Fr(H)1346 5002 y Fs(0)1404 4990 y Fu(+)20 b(1\))1563 4960 y Fo(\000)1625 4937 y Fi(1)p 1625 4946 29 3 v 1625 4980 a(2)1667 4990 y Fp(D)r Fu(\()p Fr(N)1851 4934 y Fj(r)q Fg(\000)p Fi(1)p 1851 4947 103 3 v 1888 4980 a(2)1968 4990 y Fu(\))28 b Fp(\032)f(D)r Fu(\()p Fr(N)2304 4937 y Fj(r)p 2304 4946 30 3 v 2305 4980 a Fi(2)2347 4990 y Fu(\))k(and)f @beginspecial @setspecial @endspecial Fr( )g Fu(:=)e Fr(R)q Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fu(.)44 b(Since)31 b @beginspecial @setspecial @endspecial Fr(\036)d Fp(2)f(D)r Fu(\()p Fr(N)3683 4937 y Fj(r)p 3683 4946 V 3684 4980 a Fi(2)3726 4990 y Fu(\),)k(b)n(y)0 5122 y(Lemma)c @beginspecial @setspecial @endspecial(4.4,)g(w)n(e)g (see)h(that)f Fr( )g Fp(2)c(D)r Fu(\()p Fr(N)1232 5051 y Fi(1)p 1232 5060 29 3 v 1232 5093 a(2)1213 5132 y Fq(\027)1275 5122 y Fr(N)1360 5051 y Fj(r)p 1360 5060 30 3 v 1361 5093 a Fi(2)1342 5143 y Fo(\000)p Fq(\034)1435 5122 y Fu(\).)37 b(Clearly)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e)f(also)h @beginspecial @setspecial @endspecial Fr(\036)c Fp(2)h(D)r Fu(\()p Fr(N)2647 5051 y Fi(1)p 2647 5060 29 3 v 2647 5093 a(2)2628 5132 y Fq(\027)2690 5122 y Fr(N)2775 5047 y Fj(j)r Fg(\000)p Fi(2)p 2775 5061 101 3 v 2811 5093 a(2)2757 5143 y Fo(\000)p Fq(\034)2889 5122 y Fu(\))p Fr(;)k(j)g Fp(\024)23 b Fr(r)r Fu(.)100 5222 y(W)-7 b(e)28 b(ha)n(v)n(e:)640 5422 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)772 5351 y Fi(1)p 772 5360 29 3 v 772 5393 a(2)753 5431 y Fq(\027)814 5422 y Fr(N)900 5351 y Fj(r)p 900 5360 30 3 v 901 5393 a Fi(2)881 5442 y Fo(\000)p Fq(\034)974 5422 y Fr( )s Fp(jj)1077 5387 y Fs(2)1198 5422 y @beginspecial @setspecial @endspecial Fu(=)1389 5318 y Fq(r)1345 5343 y Fl(X)1348 5520 y Fq(j)s Fs(=1)1479 5309 y Fl(Z)1576 5422 y Fr(P)1641 5386 y Fo(\000)p Fq(\034)1629 5445 y(r)n(;j)1734 5422 y Fp(jj)p Fu(\()p Fr(N)1879 5434 y Fq(\027)1939 5422 y Fu(+)2067 5314 y Fq(j)2022 5343 y Fl(X)2028 5520 y Fq(i)p Fs(=1)2156 5422 y Fr(!)s Fu(\()p Fr(k)2286 5434 y Fq(i)2314 5422 y Fu(\))2346 5387 y Fq(\027)2387 5422 y Fu(\))2429 5365 y Fi(1)p 2430 5374 29 3 v 2430 5407 a(2)2524 5314 y Fq(j)2487 5343 y Fl(Y)2486 5520 y Fq(i)p Fs(=1)2607 5422 y Fr(a)p Fu(\()p Fr(k)2726 5434 y Fq(i)2754 5422 y Fu(\))p Fr(R)q Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)3110 5387 y Fs(2)3148 5422 y Fr(dk)s(:)-3260 b Fu(\(4.8\))p eop %%Page: 18 18 18 17 bop 0 226 a Fs(18)1695 b(Z.)29 b(AMMARI)0 489 y Fu(\()p @beginspecial @setspecial @endspecial(4.8\))j(follo)n(ws)g(b)n (y)g(Corollary)e @beginspecial @setspecial @endspecial(2.2.)51 b(W)-7 b(e)32 b(recall)g(that)h Fr(H)1789 459 y Fq(J)1782 509 y(\024)1867 489 y Fu(is)g(an)f(op)r(erator)f(v)-5 b(alued)32 b(function)h(in)g(v)-5 b(ariables)31 b @beginspecial @setspecial @endspecial Fr(k)3505 501 y Fq(i)3533 489 y Fr(;)14 b(i)31 b Fp(2)g Fr(J)8 b Fu(.)52 b(If)0 616 y(w)n(e)27 b(denote)f(b)n(y)h @beginspecial @setspecial @endspecial Fr(dJ)k Fu(:=)23 b Fr(dk)820 628 y Fq(i)843 636 y Fi(1)880 616 y Fr(:::dk)1035 628 y Fq(i)1058 636 y Fj(p)1126 616 y Fu(where)j @beginspecial @setspecial @endspecial Fr(J)31 b Fu(=)23 b Fp(f)p Fr(i)1601 628 y Fs(1)1637 616 y Fr(;)14 b(:::;)g(i)1809 628 y Fq(p)1847 616 y Fp(g)p Fu(,)27 b(then)g(b)n(y)g(Lemma)f(4.5)g(w)n(e)h(ha)n(v)n(e) f Fp(jj)p Fr(R)3103 545 y Fi(1)p 3102 554 29 3 v 3102 587 a(2)3092 641 y Fq(`)3145 616 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(H)3386 586 y Fq(J)3379 636 y(\024)3432 616 y Fr(R)3506 545 y Fi(1)p 3505 554 V 3505 587 a(2)3495 641 y Fq(`)p Fs(+1)3611 616 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fp(jj)d(2)0 810 y Fr(L)57 780 y Fs(2)94 810 y Fu(\()p Fk(R)180 780 y Fs(3)p Fq(p)257 810 y Fr(;)14 b(dJ)8 b Fu(\).)36 b(Using)21 b(the)g(pull)g(through)g(form)n(ula)f(in) h(the)h(r.h.s.)34 b(of)21 b(\()p @beginspecial @setspecial @endspecial(4.8\))g(and)g(the)g(fact)g(that)h @beginspecial @setspecial @endspecial(\()p Fr(N)3053 822 y Fq(\027)3099 810 y Fu(+)3214 703 y Fq(j)3169 732 y Fl(X)3176 908 y Fq(i)p Fs(=1)3303 810 y Fr(!)s Fu(\()p Fr(k)3433 822 y Fq(i)3461 810 y Fu(\))3493 776 y Fq(\027)3535 810 y Fu(\))3577 754 y Fi(1)p 3577 763 V 3577 796 a(2)3619 810 y Fr(R)3693 739 y Fi(1)p 3693 748 V 3693 782 a(2)3682 833 y Fs(1)3735 810 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))0 983 y(is)27 b(b)r(ounded,)i(w)n(e)e(obtain:)63 1222 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)195 1151 y Fi(1)p 195 1160 V 195 1193 a(2)176 1231 y Fq(\027)237 1222 y Fr(N)323 1151 y Fj(r)p 323 1160 30 3 v 324 1193 a Fi(2)304 1242 y Fo(\000)p Fq(\034)397 1222 y Fr( )s Fp(jj)500 1187 y Fs(2)632 1222 y @beginspecial @setspecial @endspecial Fp(\024)94 b Fr(c)898 1118 y Fq(r)855 1143 y Fl(X)857 1320 y Fq(j)s Fs(=1)988 1109 y Fl(Z)1085 1222 y Fr(P)1150 1186 y Fo(\000)p Fq(\034)1138 1245 y(r)n(;j)1244 1222 y Fp(jj)p Fr(R)1364 1151 y Fi(1)p 1363 1160 29 3 v 1363 1193 a(2)1353 1244 y Fs(1)1406 1222 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))1623 1114 y Fq(j)1585 1143 y Fl(Y)1585 1320 y Fq(i)p Fs(=1)1706 1222 y Fr(a)p Fu(\()p Fr(k)1825 1234 y Fq(i)1853 1222 y Fu(\))p Fr(\036)p Fp(jj)1980 1187 y Fs(2)2018 1222 y Fr(dk)0 1364 y @beginspecial @setspecial @endspecial 791 1585 a Fu(+)p Fr(c)949 1481 y Fq(r)905 1506 y Fl(X)908 1683 y Fq(j)s Fs(=1)1050 1506 y Fl(X)1039 1680 y Fs(part)p Fq(:)1194 1472 y Fl(Z)1291 1585 y Fr(P)1356 1549 y Fo(\000)p Fq(\034)1344 1608 y(r)n(;j)1502 1481 y(d)1466 1506 y Fl(Y)1463 1685 y Fq(`)p Fs(=1)1589 1585 y Fp(jj)p Fr(R)1709 1514 y Fi(1)p 1709 1523 V 1709 1556 a(2)1698 1610 y Fq(`)1751 1585 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(H)1992 1536 y Fq(J)2034 1508 y Fj(j)2029 1556 y(`)1985 1595 y Fq(\024)2083 1585 y Fr(R)2157 1514 y Fi(1)p 2157 1523 V 2157 1556 a(2)2146 1610 y Fq(`)p Fs(+1)2262 1585 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fp(jj)2473 1551 y Fs(2)2528 1585 y Fp(\002)19 b(jj)p Fr(R)2732 1514 y Fi(1)p 2731 1523 V 2731 1556 a(2)2721 1610 y Fq(d)p Fs(+1)2843 1585 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))3074 1506 y Fl(Y)3022 1700 y Fq(i)p Fo(2)p Fq(J)3132 1672 y Fj(j)3127 1720 y(d)p Fi(+1)3247 1585 y Fr(a)p Fu(\()p Fr(k)3366 1597 y Fq(i)3394 1585 y Fu(\))p Fr(\036)p Fp(jj)3521 1551 y Fs(2)3573 1481 y Fq(d)p Fs(+1)3579 1506 y Fl(Y)3576 1685 y Fq(`)p Fs(=1)3705 1585 y Fr(dJ)3802 1545 y Fq(j)3794 1610 y(`)0 1760 y @beginspecial @setspecial @endspecial 791 1989 a Fu(+)p Fr(c)963 1885 y Fq(r)919 1910 y Fl(X)922 2087 y Fq(j)s Fs(=1)1064 1910 y Fl(X)1053 2084 y Fs(part)p Fq(:)1208 1876 y Fl(Z)1305 1989 y Fr(P)1370 1953 y Fo(\000)p Fq(\034)1358 2012 y(r)n(;j)1505 1885 y(d)1540 1860 y Fg(0)1480 1910 y Fl(Y)1477 2089 y Fq(`)p Fs(=1)1603 1989 y Fp(jj)p Fr(R)1723 1918 y Fi(1)p 1723 1927 V 1723 1960 a(2)1712 2014 y Fq(`)1765 1989 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(H)2006 1940 y Fq(J)2048 1912 y Fj(j)2043 1960 y(`)1999 1999 y Fq(\024)2097 1989 y Fr(R)2171 1918 y Fi(1)p 2170 1927 V 2170 1960 a(2)2160 2014 y Fq(`)p Fs(+1)2276 1989 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fp(jj)2487 1955 y Fs(2)2542 1989 y Fp(\002)f(jj)p Fr(R)2745 1932 y Fi(1)p 2745 1941 V 2745 1975 a(2)2787 1989 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)3047 1955 y Fs(2)3126 1885 y Fq(d)3161 1860 y Fg(0)3102 1910 y Fl(Y)3099 2089 y Fq(`)p Fs(=1)3225 1989 y Fr(dJ)3322 1949 y Fq(j)3314 2014 y(`)0 2131 y @beginspecial @setspecial @endspecial 620 2256 a Fu(=:)83 b Fr(I)26 b Fu(+)18 b Fr(I)7 b(I)25 b Fu(+)18 b Fr(I)7 b(I)g(I)g(:)0 2486 y Fu(W)-7 b(e)28 b(recall)f(that)g @beginspecial @setspecial @endspecial Fr(P)609 2451 y Fo(\000)p Fq(\034)597 2509 y(j;j)726 2486 y Fu(=)852 2379 y Fq(j)814 2407 y Fl(Y)813 2584 y Fq(i)p Fs(=1)935 2486 y Fr(!)s Fu(\()p Fr(k)1065 2498 y Fq(i)1093 2486 y Fu(\))1125 2452 y Fo(\000)p Fq(\034)1246 2486 y Fu(and)g(b)n(y)h(\()p @beginspecial @setspecial @endspecial(2.7\))f(w)n(e)g(notice)h(that)1617 2732 y Fr(P)1682 2697 y Fo(\000)p Fq(\034)1670 2755 y(r)n(;j)1798 2732 y Fp(\024)23 b Fr(c)14 b(P)2001 2697 y Fo(\000)p Fq(\034)1989 2755 y(j;j)2094 2732 y Fr(:)83 b @beginspecial @setspecial @endspecial -2200 w Fu(\(4.9\))0 2940 y(In)28 b @beginspecial @setspecial @endspecial(\()p Fr(I)7 b(I)g Fu(\))28 b(using)f(\(4.9\))h(w)n(e)f(can)g(separate)f(the)i(in)n (tegral)f(in)g(v)-5 b(ariables)27 b @beginspecial @setspecial @endspecial Fr(k)2230 2952 y Fq(i)2290 2940 y Fr(=)-51 b Fp(2)23 b Fr(J)2413 2900 y Fq(j)2405 2965 y(d)p Fs(+1)2556 2940 y Fu(and)k Fr(k)2760 2952 y Fq(i)2811 2940 y Fp(2)c Fr(J)2943 2900 y Fq(j)2935 2965 y(d)p Fs(+1)3058 2940 y Fu(.)37 b(Since)28 b @beginspecial @setspecial @endspecial Fp(jj)p Fr(R)3455 2869 y Fi(1)p 3455 2878 V 3455 2911 a(2)3444 2965 y Fq(`)3497 2940 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(H)3738 2891 y Fq(J)3780 2863 y Fj(j)3775 2911 y(`)3731 2949 y Fq(\024)0 3078 y Fr(R)74 3007 y Fi(1)p 74 3016 V 74 3049 a(2)63 3103 y Fq(`)p Fs(+1)179 3078 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fp(jj)23 b(2)g Fr(L)548 3047 y Fs(2)585 3078 y Fu(\()p Fk(R)671 3047 y Fs(3)p Fq(p)749 3078 y Fr(;)14 b(dJ)8 b Fu(\),)28 b(then)g(w)n(e)f(obtain:)680 3315 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(I)7 b(I)30 b Fp(\024)23 b Fr(c)984 3211 y Fq(r)941 3236 y Fl(X)943 3413 y Fq(j)s Fs(=1)1085 3236 y Fl(X)1074 3410 y Fs(part)p Fq(:)1243 3202 y Fl(Z)1393 3236 y(Y)1340 3430 y Fq(i)p Fo(2)p Fq(J)1450 3402 y Fj(j)1445 3450 y(d)p Fi(+1)1565 3315 y Fr(!)s Fu(\()p Fr(k)1695 3327 y Fq(i)1723 3315 y Fu(\))1755 3281 y Fo(\000)p Fq(\034)1871 3315 y Fp(jj)p Fr(R)1991 3244 y Fi(1)p 1991 3253 V 1991 3286 a(2)1980 3340 y Fq(d)p Fs(+1)2103 3315 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))2334 3236 y Fl(Y)2282 3430 y Fq(i)p Fo(2)p Fq(J)2392 3402 y Fj(j)2387 3450 y(d)p Fi(+1)2506 3315 y Fr(a)p Fu(\()p Fr(k)2625 3327 y Fq(i)2653 3315 y Fu(\))p Fr(\036)p Fp(jj)2780 3281 y Fs(2)2819 3315 y Fr(dJ)2916 3275 y Fq(j)2908 3340 y(d)p Fs(+1)3031 3315 y Fr(:)83 b @beginspecial @setspecial @endspecial -3137 w Fu(\(4.10\))0 3612 y(No)n(w)27 b(reordering)e(the)j (terms)g(in)g(the)g(r.h.s.)36 b(of)28 b(\()p @beginspecial @setspecial @endspecial(4.10\))e(and)i(taking)f(in)n(to)g(accoun)n(t)g (the)h(fact)g(that)g @beginspecial @setspecial @endspecial Fr(J)3186 3572 y Fq(j)3178 3637 y(d)p Fs(+1)3323 3612 y Fp(6)p Fu(=)23 b Fp(;)p Fu(,)k(w)n(e)h(ha)n(v)n(e:)797 3867 y Fr(I)7 b(I)30 b Fp(\024)23 b Fr(c)1059 3763 y Fq(r)r Fo(\000)p Fs(1)1058 3788 y Fl(X)1060 3965 y Fq(j)s Fs(=1)1191 3754 y Fl(Z)1288 3867 y Fr(P)1353 3832 y Fo(\000)p Fq(\034)1341 3890 y(j;j)1446 3867 y Fp(jj)p Fu(\()1547 3846 y(^)1524 3867 y Fr(H)1593 3879 y Fq(\024)1655 3867 y Fu(+)1783 3759 y Fq(j)1738 3788 y Fl(X)1745 3965 y Fq(i)p Fs(=1)1872 3867 y Fr(!)s Fu(\()p Fr(k)2002 3879 y Fq(i)2030 3867 y Fu(\))2062 3833 y Fq(\027)2122 3867 y Fu(+)18 b Fr(b)p Fu(\))2273 3833 y Fo(\000)2335 3810 y Fi(1)p 2335 3819 V 2335 3853 a(2)2429 3759 y Fq(j)2392 3788 y Fl(Y)2391 3965 y Fq(i)p Fs(=1)2512 3867 y Fr(a)p Fu(\()p Fr(k)2631 3879 y Fq(i)2659 3867 y Fu(\))p Fr(\036)p Fp(jj)2786 3833 y Fs(2)2825 3867 y Fr(dk)s(:)83 b @beginspecial @setspecial @endspecial -3020 w Fu(\(4.11\))0 4115 y(Doing)27 b(the)h(same)f(thing)h(for)f @beginspecial @setspecial @endspecial(\()p Fr(I)7 b(I)g(I)g Fu(\),)28 b(w)n(e)g(obtain:)1481 4288 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(I)7 b(I)g(I)30 b Fp(\024)23 b Fr(c)14 b Fp(jj)p Fr(R)1891 4231 y Fi(1)p 1890 4240 V 1890 4273 a(2)1932 4288 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)2192 4253 y Fs(2)2230 4288 y Fr(:)83 b @beginspecial @setspecial @endspecial -2336 w Fu(\(4.12\))0 4447 y(Collecting)27 b(\()p @beginspecial @setspecial @endspecial(4.11\)-\(4.12\))f(w)n(e)h(obtain)307 4691 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)439 4620 y Fi(1)p 439 4629 V 439 4662 a(2)420 4701 y Fq(\027)482 4691 y Fr(N)567 4620 y Fj(r)p 567 4629 30 3 v 568 4662 a Fi(2)549 4712 y Fo(\000)p Fq(\034)642 4691 y Fr( )s Fp(jj)745 4657 y Fs(2)865 4691 y @beginspecial @setspecial @endspecial Fp(\024)83 b Fr(c)1063 4624 y Fl(\000)1158 4587 y Fq(r)1114 4612 y Fl(X)1117 4789 y Fq(j)s Fs(=1)1248 4578 y Fl(Z)1345 4691 y Fr(P)1410 4656 y Fo(\000)p Fq(\034)1398 4714 y(r)n(;j)1503 4691 y Fp(jj)p Fu(\()1603 4670 y(^)1581 4691 y Fr(H)1650 4703 y Fq(\024)1712 4691 y Fu(+)1840 4583 y Fq(j)1795 4612 y Fl(X)1801 4789 y Fq(i)p Fs(=1)1929 4691 y Fr(!)s Fu(\()p Fr(k)2059 4703 y Fq(i)2087 4691 y Fu(\))2119 4657 y Fq(\027)2179 4691 y Fu(+)18 b Fr(b)p Fu(\))2330 4657 y Fo(\000)2392 4634 y Fi(1)p 2392 4643 29 3 v 2392 4677 a(2)2486 4583 y Fq(j)2448 4612 y Fl(Y)2448 4789 y Fq(i)p Fs(=1)2569 4691 y Fr(a)p Fu(\()p Fr(k)2688 4703 y Fq(i)2716 4691 y Fu(\))p Fr(\036)p Fp(jj)2843 4657 y Fs(2)2881 4691 y Fr(dk)0 4833 y @beginspecial @setspecial @endspecial 1013 5056 a Fu(+)1092 4953 y Fq(r)r Fo(\000)p Fs(1)1091 4977 y Fl(X)1094 5154 y Fq(j)s Fs(=1)1225 4943 y Fl(Z)1322 5056 y Fr(P)1387 5021 y Fo(\000)p Fq(\034)1375 5079 y(j;j)1480 5056 y Fp(jj)p Fu(\()1580 5035 y(^)1558 5056 y Fr(H)1627 5068 y Fq(\024)1689 5056 y Fu(+)1816 4949 y Fq(j)1772 4977 y Fl(X)1778 5154 y Fq(i)p Fs(=1)1906 5056 y Fr(!)s Fu(\()p Fr(k)2036 5068 y Fq(i)2063 5056 y Fu(\))2095 5022 y Fq(\027)2155 5056 y Fu(+)g Fr(b)p Fu(\))2306 5022 y Fo(\000)2368 5000 y Fi(1)p 2368 5009 V 2368 5042 a(2)2463 4949 y Fq(j)2425 4977 y Fl(Y)2424 5154 y Fq(i)p Fs(=1)2546 5056 y Fr(a)p Fu(\()p Fr(k)2665 5068 y Fq(i)2693 5056 y Fu(\))p Fr(\036)p Fp(jj)2820 5022 y Fs(2)2858 5056 y Fr(dk)j Fu(+)e Fp(jj)p Fr(R)3169 5000 y Fi(1)p 3168 5009 V 3168 5042 a(2)3211 5056 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)3471 5022 y Fs(2)3508 4989 y Fl(\001)0 5199 y @beginspecial @setspecial @endspecial 865 5422 a Fp(\024)83 b Fr(c)1063 5354 y Fl(\000)1158 5318 y Fq(r)1114 5343 y Fl(X)1117 5520 y Fq(j)s Fs(=1)1248 5309 y Fl(Z)1345 5422 y Fr(P)1410 5386 y Fo(\000)p Fq(\034)1398 5445 y(j;j)1503 5422 y Fp(jj)p Fu(\()1603 5401 y(^)1581 5422 y Fr(H)1650 5434 y Fq(\024)1712 5422 y Fu(+)1840 5314 y Fq(j)1795 5343 y Fl(X)1801 5520 y Fq(i)p Fs(=1)1929 5422 y Fr(!)s Fu(\()p Fr(k)2059 5434 y Fq(i)2087 5422 y Fu(\))2119 5387 y Fq(\027)2179 5422 y Fu(+)18 b Fr(b)p Fu(\))2330 5387 y Fo(\000)2392 5365 y Fi(1)p 2392 5374 V 2392 5407 a(2)2486 5314 y Fq(j)2448 5343 y Fl(Y)2448 5520 y Fq(i)p Fs(=1)2569 5422 y Fr(a)p Fu(\()p Fr(k)2688 5434 y Fq(i)2716 5422 y Fu(\))p Fr(\036)p Fp(jj)2843 5387 y Fs(2)2881 5422 y Fr(dk)k Fu(+)c Fp(jj)p Fr(R)3192 5365 y Fi(1)p 3192 5374 V 3192 5407 a(2)3234 5422 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)3494 5387 y Fs(2)3532 5354 y Fl(\001)3570 5422 y Fr(:)-3593 b Fu(\(4.13\))p eop %%Page: 19 19 19 18 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(19)0 560 y Fu(\()p @beginspecial @setspecial @endspecial(4.13\))28 b(follo)n(ws)f(b)n(y)h(\(4.9\).)39 b(No)n(w)28 b(using)g(the)h(fact)f(that)h @beginspecial @setspecial @endspecial(\()1814 539 y(^)1792 560 y Fr(H)1861 572 y Fq(\024)1923 560 y Fu(+)2052 453 y Fq(j)2007 482 y Fl(X)2013 658 y Fq(i)p Fs(=1)2141 560 y Fr(!)s Fu(\()p Fr(k)2271 572 y Fq(i)2299 560 y Fu(\))2331 526 y Fq(\027)2391 560 y Fu(+)19 b Fr(b)p Fu(\))2543 526 y Fo(\000)2604 504 y Fi(1)p 2604 513 29 3 v 2604 546 a(2)2647 560 y Fu(\()p Fr(H)2748 572 y Fs(0)2804 560 y Fu(+)2932 453 y Fq(j)2888 482 y Fl(X)2894 658 y Fq(i)p Fs(=1)3022 560 y Fr(!)s Fu(\()p Fr(k)3152 572 y Fq(i)3180 560 y Fu(\))3212 526 y Fq(\027)3272 560 y Fu(+)g Fr(b)p Fu(\))3434 504 y Fi(1)p 3434 513 V 3434 546 a(2)3504 560 y Fu(is)29 b(b)r(ounded)0 733 y(uniformly)e(in)h @beginspecial @setspecial @endspecial Fr(k)518 745 y Fq(i)574 733 y Fu(and)f(Corollary)e(2.2,)i (w)n(e)g(obtain)734 941 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)866 870 y Fi(1)p 866 879 V 866 913 a(2)847 951 y Fq(\027)909 941 y Fr(N)994 870 y Fj(r)p 994 879 30 3 v 995 913 a Fi(2)976 962 y Fo(\000)p Fq(\034)1069 941 y Fr(R)q Fu(\()p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)1328 907 y Fs(2)1448 941 y @beginspecial @setspecial @endspecial Fp(\024)83 b Fr(c)23 b Fu(\()1731 838 y Fq(r)1687 862 y Fl(X)1690 1039 y Fq(j)s Fs(=1)1821 941 y Fp(jj)p Fu(\()p Fr(H)1968 953 y Fs(0)2024 941 y Fu(+)18 b Fr(b)p Fu(\))2175 907 y Fo(\000)2237 885 y Fi(1)p 2237 894 29 3 v 2237 927 a(2)2279 941 y Fr(N)2366 866 y Fj(j)p 2365 880 V 2365 913 a Fi(2)2346 962 y Fo(\000)p Fq(\034)2439 941 y Fr(\036)p Fp(jj)2534 907 y Fs(2)2591 941 y Fu(+)g Fp(jj)p Fr(R)2794 885 y Fi(1)p 2793 894 V 2793 927 a(2)2836 941 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)3096 907 y Fs(2)3133 941 y Fu(\))-3165 b(\(4.14\))0 1084 y @beginspecial @setspecial @endspecial 1448 1276 a Fp(\024)83 b Fr(c)23 b Fu(\()1731 1173 y Fq(r)1687 1198 y Fl(X)1690 1374 y Fq(j)s Fs(=1)1821 1276 y Fp(jj)p Fr(N)1953 1206 y Fi(1)p 1953 1215 V 1953 1248 a(2)1934 1286 y Fq(\027)1995 1276 y Fr(N)2081 1201 y Fj(j)r Fg(\000)p Fi(2)p 2081 1215 101 3 v 2117 1248 a(2)2062 1297 y Fo(\000)p Fq(\034)2195 1276 y Fr(\036)p Fp(jj)2290 1242 y Fs(2)2346 1276 y Fu(+)18 b Fp(jj)p Fr(R)2549 1220 y Fi(1)p 2549 1229 29 3 v 2549 1262 a(2)2592 1276 y Fu(\()p Fp(\000)p Fr(b)p Fu(\))p Fr(\036)p Fp(jj)2852 1242 y Fs(2)2889 1276 y Fu(\))p Fr(:)-2944 b Fu(\(4.15\))100 1529 y(In)26 b(fact,)h(using)f(the)h(fact) g(that)f @beginspecial @setspecial @endspecial Fr(N)1154 1541 y Fo(\000)p Fq(\034)1247 1529 y Fu(\()p Fr(H)1348 1541 y Fs(0)1402 1529 y Fu(+)16 b Fr(b)p Fu(\))1551 1499 y Fo(\000)1612 1476 y Fi(1)p 1612 1485 V 1612 1519 a(2)1655 1529 y Fu(\(1)g(+)f Fr(N)1892 1541 y Fq(\027)1933 1529 y Fu(\))1965 1499 y Fo(\000)2027 1476 y Fi(1)p 2027 1485 V 2027 1519 a(2)2096 1529 y Fu(is)27 b(b)r(ounded)f(,)h(w)n(e)f(see)g (that)h(the)g(r.h.s.)36 b(in)27 b(\()p @beginspecial @setspecial @endspecial(4.14\))e(is)i(less)0 1641 y(than)j(\()p @beginspecial @setspecial @endspecial(4.15\),)f(whic)n(h)h(holds)g(for) f(all)g Fr(\036)e Fp(2)g Fu(\()p Fr(H)1426 1653 y Fs(0)1484 1641 y Fu(+)19 b(1\))1642 1611 y Fo(\000)1704 1589 y Fi(1)p 1704 1598 V 1704 1631 a(2)1746 1641 y Fp(D)r Fu(\()p Fr(N)1931 1585 y Fj(r)q Fg(\000)p Fi(1)p 1931 1598 103 3 v 1968 1631 a(2)2047 1641 y Fu(\).)44 b(Since)30 b @beginspecial @setspecial @endspecial Fp(D)22 b(\012)e Fu(\000)2588 1653 y Fs(\014n)2666 1641 y Fu(\()p Fr(C)2763 1611 y Fo(1)2757 1662 y Fs(0)2834 1641 y Fu(\()p Fk(R)2920 1611 y Fs(3)2963 1641 y Fu(\)\))28 b Fp(\032)e Fu(\()p Fr(H)3247 1653 y Fs(0)3304 1641 y Fu(+)20 b(1\))3463 1611 y Fo(\000)3524 1589 y Fi(1)p 3524 1598 29 3 v 3524 1631 a(2)3567 1641 y Fp(D)r Fu(\()p Fr(N)3751 1585 y Fj(r)q Fg(\000)p Fi(1)p 3751 1598 103 3 v 3788 1631 a(2)3868 1641 y Fu(\))0 1772 y(is)28 b(a)f(core)g(for)g @beginspecial @setspecial @endspecial Fr(N)542 1701 y Fi(1)p 542 1710 29 3 v 542 1744 a(2)523 1782 y Fq(\027)584 1772 y Fr(N)670 1698 y Fj(r)q Fg(\000)p Fi(2)p 670 1711 103 3 v 707 1744 a(2)651 1793 y Fo(\000)p Fq(\034)787 1772 y Fu(,)h(w)n(e)f(see)g(that)h @beginspecial @setspecial @endspecial(\()p Fr(H)1375 1784 y Fs(0)1432 1772 y Fu(+)18 b(1\))1589 1742 y Fo(\000)1650 1720 y Fi(1)p 1650 1729 29 3 v 1650 1762 a(2)1693 1772 y Fp(D)r Fu(\()p Fr(N)1877 1716 y Fj(r)q Fg(\000)p Fi(1)p 1877 1729 103 3 v 1914 1762 a(2)1994 1772 y Fu(\))28 b(is)g(dense)f(in)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(N)2646 1701 y Fi(1)p 2646 1710 29 3 v 2646 1744 a(2)2626 1782 y Fq(\027)2688 1772 y Fr(N)2774 1698 y Fj(r)q Fg(\000)p Fi(2)p 2774 1711 103 3 v 2811 1744 a(2)2755 1793 y Fo(\000)p Fq(\034)2890 1772 y Fu(\))h(and)e(hence) h(\()p @beginspecial @setspecial @endspecial(4.15\))f(holds)h(for)0 1910 y(all)g @beginspecial @setspecial @endspecial Fr(\036)d Fp(2)f(D)r Fu(\()p Fr(N)453 1839 y Fi(1)p 453 1848 29 3 v 453 1881 a(2)434 1920 y Fq(\027)496 1910 y Fr(N)581 1836 y Fj(r)q Fg(\000)p Fi(2)p 581 1849 103 3 v 619 1881 a(2)563 1931 y Fo(\000)p Fq(\034)698 1910 y Fu(\).)39 b(No)n(w)28 b(let)g @beginspecial @setspecial @endspecial Fr(\036)d Fu(:=)f(\()1342 1889 y(^)1320 1910 y Fr(H)1389 1922 y Fq(\024)1451 1910 y Fu(+)18 b Fr(b)p Fu(\))p Fr( )s(;)c( )27 b Fp(2)e(D)r Fu(\(\()2010 1889 y(^)1987 1910 y Fr(H)2056 1922 y Fq(\024)2119 1910 y Fu(+)18 b Fr(b)p Fu(\))2280 1854 y Fj(r)q Fi(+1)p 2280 1867 101 3 v 2316 1900 a(2)2394 1910 y Fu(\))29 b(and)f @beginspecial @setspecial @endspecial Fr(b)c(>)f Fu(0.)39 b(Then)28 b Fr(\036)d Fp(2)f(D)r Fu(\(\()3392 1889 y(^)3369 1910 y Fr(H)3438 1922 y Fq(\024)3501 1910 y Fu(+)19 b Fr(b)p Fu(\))3663 1854 y Fj(r)q Fg(\000)p Fi(1)p 3662 1867 103 3 v 3700 1900 a(2)3779 1910 y Fu(\))24 b Fp(\032)0 2044 y(D)r Fu(\()p Fr(N)184 1973 y Fi(1)p 184 1982 29 3 v 184 2015 a(2)165 2053 y Fq(\027)227 2044 y Fr(N)312 1988 y Fj(r)q Fg(\000)p Fi(2)p 312 2001 103 3 v 350 2033 a(2)429 2044 y Fu(\).)37 b(The)28 b(induction)g(h)n(yp)r (othesis)f(and)h(\()p @beginspecial @setspecial @endspecial(4.15\))e (giv)n(e)865 2218 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)996 2147 y Fi(1)p 996 2156 29 3 v 996 2189 a(2)978 2227 y Fq(\027)1039 2218 y Fr(N)1124 2147 y Fj(r)p 1124 2156 30 3 v 1125 2189 a Fi(2)1106 2238 y Fo(\000)p Fq(\034)1199 2218 y Fr( )s Fp(jj)d(\024)g Fr(c)g Fp(jj)p Fu(\()1572 2197 y(^)1550 2218 y Fr(H)1619 2230 y Fq(\024)1681 2218 y Fu(+)18 b Fr(b)p Fu(\))1842 2158 y Fj(r)q Fi(+1)p 1841 2171 101 3 v 1877 2203 a(2)1956 2218 y Fr( )s Fp(jj)p Fr(;)37 b( )26 b Fp(2)d(D)r Fu(\(\()2430 2197 y(^)2407 2218 y Fr(H)2476 2230 y Fq(\024)2539 2218 y Fu(+)18 b Fr(b)p Fu(\))2700 2158 y Fj(r)q Fi(+1)p 2700 2171 V 2736 2203 a(2)2814 2218 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 2364 a @beginspecial @setspecial @endspecial 3835 w Fh(\004)-3900 b @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 157 x Fz(Corollary)33 b(4.8.)40 b Fn(L)l(et)29 b Fr(\015)f Fp(\025)23 b Fu(0)29 b Fn(and)h Fr(\017)23 b(<)1305 2488 y Fs(1)p 1305 2502 34 4 v 1305 2549 a Fq(r)1348 2521 y Fn(,)30 b(wher)l(e)g @beginspecial @setspecial @endspecial Fr(r)c Fp(2)d Fk(N)t Fn(.)45 b(Then)1707 2684 y Fr(N)1783 2649 y Fq(r)1774 2704 y(\017)1857 2684 y Fp(\024)36 b Fr(c)23 b Fu(\()2071 2663 y(^)2049 2684 y Fr(H)2118 2696 y Fq(\024)2180 2684 y Fu(+)18 b Fr(b)p Fu(\))2331 2649 y Fq(r)2368 2684 y Fr(;)96 b @beginspecial @setspecial @endspecial -2487 w Fu(\(4.16\))1329 2863 y @beginspecial @setspecial @endspecial Fr(H)1405 2824 y Fs(1)p Fo(\000)p Fq(\015)1398 2886 y Fs(0)1547 2863 y Fr(N)1623 2829 y Fq(r)r Fo(\000)p Fs(1+)p Fq(\015)1870 2863 y Fp(\024)37 b Fr(c)23 b Fu(\()2085 2842 y(^)2063 2863 y Fr(H)2132 2875 y Fq(\024)2194 2863 y Fu(+)18 b Fr(b)p Fu(\))2345 2829 y Fq(r)2381 2863 y Fr(;)83 b @beginspecial @setspecial @endspecial -2487 w Fu(\(4.17\))0 3010 y Fn(wher)l(e)30 b(c)g(and)h(b)e(ar)l(e)i(c)l(onstants)d(indep)l(endent)j(of)f @beginspecial @setspecial @endspecial Fr(\024)p Fn(.)0 3026 y @beginspecial @setspecial @endspecial 100 3167 a Fz(Pro)s(of.)52 b @beginspecial @setspecial @endspecial 28 w Fu(The)30 b(inequalities)h(follo)n(w)e(from)i(Lemma)f(2.5)g(and)g (Thm.)46 b @beginspecial @setspecial @endspecial(4.7.)f(\(4.16\))30 b(follo)n(ws)f(with)i Fr(\027)j Fu(=)27 b(1)p Fr(;)14 b(\034)37 b Fu(=)3645 3134 y Fs(1)p Fo(\000)p Fq(\017r)p 3645 3148 146 4 v 3659 3195 a Fs(1)p Fo(\000)p Fq(r)3831 3167 y Fu(in)0 3281 y(Thm.)50 b @beginspecial @setspecial @endspecial(4.7)31 b(and)h Fr(b)d Fu(=)h Fr(!)s(;)14 b(\034)840 3293 y Fs(1)907 3281 y Fu(=)30 b Fr(\017;)14 b(\034)1109 3293 y Fs(2)1176 3281 y Fu(=)30 b(1)p Fr(;)14 b(\034)1386 3293 y Fs(3)1453 3281 y Fu(=)30 b Fp(\000)1623 3248 y Fs(1)p Fo(\000)p Fq(\017r)p 1622 3262 V 1636 3309 a Fs(1)p Fo(\000)p Fq(r)1777 3281 y Fr(;)14 b(\013)1867 3293 y Fs(1)1935 3281 y Fu(=)29 b Fr(r)n(;)14 b(\013)2154 3293 y Fs(2)2222 3281 y Fu(=)30 b(1)p Fr(;)14 b(\013)2449 3293 y Fs(3)2516 3281 y Fu(=)29 b Fr(r)24 b Fp(\000)d Fu(1)32 b(in)g(Lemma)f @beginspecial @setspecial @endspecial(2.5.)49 b(\()p @beginspecial @setspecial @endspecial(4.17\))31 b(follo)n(ws)0 3387 y(with)d Fr(\027)h Fu(=)23 b(1)p Fr(;)14 b(\034)32 b Fu(=)631 3350 y Fq(\015)p 592 3368 118 4 v 592 3415 a(r)r Fo(\000)p Fs(1)747 3387 y Fu(in)c(Thm.)37 b @beginspecial @setspecial @endspecial(4.7)27 b(and)h Fr(b)23 b Fu(=)f Fr(!)s(;)14 b(\034)1649 3399 y Fs(1)1710 3387 y Fu(=)23 b(0)p Fr(;)14 b(\034)1913 3399 y Fs(2)1973 3387 y Fu(=)23 b(1)p Fr(;)14 b(\034)2176 3399 y Fs(3)2236 3387 y Fu(=)23 b Fp(\000)2438 3350 y Fq(\015)p 2398 3368 V 2398 3415 a(r)r Fo(\000)p Fs(1)2526 3387 y Fr(;)14 b(\013)2616 3399 y Fs(1)2676 3387 y Fu(=)23 b Fr(r)e Fp(\000)d Fu(1)g(+)g Fr(\015)5 b(;)14 b(\013)3186 3399 y Fs(2)3246 3387 y Fu(=)23 b Fr(\015)5 b(;)14 b(\013)3472 3399 y Fs(3)3532 3387 y Fu(=)23 b Fr(r)e Fp(\000)e Fu(1)27 b(in)0 3508 y(Lemma)g @beginspecial @setspecial @endspecial(2.5)g(and)h (the)g(fact)f(that)h(d\000\()p Fr(!)s Fu(\))1295 3478 y Fq(\015)1361 3508 y Fp(\024)23 b Fr(H)1525 3468 y Fq(\015)t Fo(\000)p Fs(1)1518 3530 y(0)1666 3508 y Fu(d\000\()p Fr(!)s Fu(\).)p @beginspecial @setspecial @endspecial 1929 w Fh(\004)100 3607 y Fu(Using)i(the)h(principle)f(of)g(the)h (cuto\013)g(indep)r(endence,)h(form)n(ulated)d(in)i(Lemma)f @beginspecial @setspecial @endspecial(4.1,)g(w)n(e)g(deduce)h(similar)e (estimates)i(for)22 3690 y(^)0 3711 y Fr(H)69 3723 y Fo(1)139 3711 y Fu(.)0 3723 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 144 x Fz(Theorem)k(4.9.)40 b Fn(L)l(et)30 b Fr(\015)d Fp(\025)c Fu(0)29 b Fn(and)i Fr(\017)22 b(<)1284 3834 y Fs(1)p 1284 3848 34 4 v 1284 3896 a Fq(r)1327 3867 y Fn(,)30 b(wher)l(e)h @beginspecial @setspecial @endspecial Fr(r)25 b Fp(2)f Fk(N)t Fn(.)44 b Fr(c;)14 b(b)30 b Fn(ar)l(e)g(p)l(ositive)h (c)l(onstants.)38 b(Then)1357 4098 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1399 4030 a @beginspecial @setspecial @endspecial Fr(N)1475 3995 y Fq(r)1466 4050 y(\017)1548 4030 y Fp(\024)f Fr(c)23 b Fu(\()1763 4009 y(^)1741 4030 y Fr(H)1810 4042 y Fo(1)1899 4030 y Fu(+)18 b Fr(b)p Fu(\))2050 3995 y Fq(r)2087 4030 y Fr(;)p @beginspecial @setspecial @endspecial 1399 4173 a(H)1475 4133 y Fs(1)p Fo(\000)p Fq(\015)1468 4195 y Fs(0)1616 4173 y Fr(N)1692 4139 y Fq(r)r Fo(\000)p Fs(1+)p Fq(\015)1940 4173 y Fp(\024)36 b Fr(c)23 b Fu(\()2154 4152 y(^)2132 4173 y Fr(H)2201 4185 y Fo(1)2290 4173 y Fu(+)18 b Fr(b)p Fu(\))2441 4139 y Fq(r)2478 4173 y Fr(:)2543 4098 y @beginspecial @setspecial @endspecial 100 4330 a Fu(Thm.)37 b @beginspecial @setspecial @endspecial(4.9)27 b(for)g Fr(\017)22 b Fu(=)h(0)k(and)h(Hadamard's)e(three)h(lines)h (lemma)f(in)h @beginspecial @setspecial @endspecial([22)o(])g(giv)n(e)f (the)h(follo)n(wing)e(corollary)-7 b(.)0 4351 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 136 x Fz(Corollary)33 b(4.10.)40 b Fn(F)-6 b(or)30 b Fr(r)c Fp(2)d Fk(R)1014 4457 y Fs(+)1105 4487 y Fn(ther)l(e)30 b(ar)l(e)g(c,b)g(p)l(ositive)h(c)l(onstants)e(such)h(that)1408 4637 y @beginspecial @setspecial @endspecial Fu(\()p Fr(N)d Fu(+)18 b(1\))1691 4603 y Fq(r)1765 4637 y Fp(\024)46 b Fr(c)14 b Fu(\()1979 4616 y(^)1958 4637 y Fr(H)2027 4649 y Fo(1)2115 4637 y Fu(+)k Fr(b)p Fu(\))2266 4603 y Fq(r)2303 4637 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4794 a @beginspecial @setspecial @endspecial Fz(Corollary)33 b(4.11.)40 b Fn(F)-6 b(or)30 b Fr(r)c Fp(2)d Fk(R)1014 4764 y Fs(+)1105 4794 y Fn(ther)l(e)30 b(ar)l(e)g(c,b)g(p)l(ositive)h(c)l(onstants)e(such)h(that)1440 4941 y @beginspecial @setspecial @endspecial Fu(\()p Fr(N)d Fu(+)18 b(1\))1723 4906 y Fq(r)1797 4941 y Fp(\024)45 b Fr(c)14 b Fu(\()p Fr(H)26 b Fu(+)18 b Fr(b)p Fu(\))2235 4906 y Fq(r)2271 4941 y Fr(:)83 b @beginspecial @setspecial @endspecial 100 5097 a Fz(Pro)s(of.)54 b @beginspecial @setspecial @endspecial 30 w Fu(If)32 b(w)n(e)f(pro)n(v)n(e)e(that)j Fr(N)1143 5067 y Fq(r)1179 5097 y Fr(U)1236 5109 y Fo(1)1307 5097 y Fu(\()p Fr(N)e Fu(+)20 b(1\))1595 5067 y Fo(\000)p Fq(r)1715 5097 y Fu(is)31 b(b)r(ounded)h(for)f @beginspecial @setspecial @endspecial Fr(r)j Fu(p)r(ositiv)n(e)d(in)n(teger,)g(the)h (corollary)d(follo)n(ws)h(from)0 5197 y(Corollary)25 b @beginspecial @setspecial @endspecial(4.10.)0 5296 y(W)-7 b(e)28 b(ha)n(v)n(e)e(on)i Fp(D)r Fu(\()p Fr(N)9 b Fu(\))28 b(the)g(iden)n(tit)n(y)1185 5482 y @beginspecial @setspecial @endspecial Fr(U)1251 5448 y Fo(\003)1242 5503 y(1)1312 5482 y Fr(N)9 b(U)1445 5494 y Fo(1)1538 5482 y Fu(=)22 b Fr(N)27 b Fp(\000)18 b Fu(i)p Fr(\036)p Fu(\(i)p Fr(G)1994 5494 y Fo(1)2066 5482 y Fu(\))h Fp(\000)2210 5426 y Fu(1)p 2210 5463 42 4 v 2210 5539 a(2)2261 5482 y Fp(jj)p Fr(G)2372 5494 y Fo(1)2443 5482 y Fp(jj)2489 5448 y Fs(2)2526 5482 y Fr(:)83 b @beginspecial @setspecial @endspecial eop %%Page: 20 20 20 19 bop 0 226 a Fs(20)1695 b(Z.)29 b(AMMARI)0 489 y Fu(So)e @beginspecial @setspecial @endspecial Fr(U)172 501 y Fo(1)270 489 y Fu(preserv)n(es)f Fp(D)r Fu(\()p Fr(N)9 b Fu(\).)37 b(By)27 b(iteration)g(w)n(e)h(ha)n(v)n(e)e(on)h Fp(D)r Fu(\()p Fr(N)1963 459 y Fq(r)2001 489 y Fu(\))1116 664 y @beginspecial @setspecial @endspecial Fr(U)1182 630 y Fo(\003)1173 685 y(1)1243 664 y Fr(N)1319 630 y Fq(r)1355 664 y Fr(U)1412 676 y Fo(1)1505 664 y Fu(=)c(\()p Fr(N)k Fp(\000)18 b Fu(i)p Fr(\036)p Fu(\(i)p Fr(G)1994 676 y Fo(1)2066 664 y Fu(\))h Fp(\000)2210 608 y Fu(1)p 2210 645 42 4 v 2210 721 a(2)2261 664 y Fp(jj)p Fr(G)2372 676 y Fo(1)2443 664 y Fp(jj)2489 630 y Fs(2)2526 664 y Fu(\))2558 630 y Fq(r)2595 664 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 832 a Fu(Then)28 b(w)n(e)f(obtain)g(the)h(b)r (oundness)g(of)f(the)h(op)r(erator)e @beginspecial @setspecial @endspecial Fr(N)1793 802 y Fq(r)1830 832 y Fr(U)1887 844 y Fo(1)1957 832 y Fu(\()p Fr(N)h Fu(+)18 b(1\))2240 802 y Fo(\000)p Fq(r)2329 832 y Fu(,)27 b(since)1251 1008 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\()p Fr(N)h Fp(\000)18 b Fu(i)p Fr(\036)p Fu(\(i)p Fr(G)1653 1020 y Fo(1)1724 1008 y Fu(\))h Fp(\000)1868 952 y Fu(1)p 1868 989 V 1868 1065 a(2)1920 1008 y Fp(jj)p Fr(G)2031 1020 y Fo(1)2101 1008 y Fp(jj)2147 974 y Fs(2)2185 1008 y Fu(\))2217 974 y Fq(r)2254 1008 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))2537 974 y Fo(\000)p Fq(r)2626 1008 y Fr(;)p @beginspecial @setspecial @endspecial 0 1167 a Fu(is)27 b(b)r(ounded.)p @beginspecial @setspecial @endspecial 3418 w Fh(\004)0 1221 y @beginspecial @setspecial @endspecial 100 x Fu(4.2.)47 b Fz(Num)m(b)s(er-energy)30 b(estimates.)0 1471 y(Lemma)g(4.12.)39 b Fn(We)30 b(have)h(uniformly)g(in)f @beginspecial @setspecial @endspecial Fr(\024)f Fn(for)i Fr(z)i Fn(in)d(a)g(c)l(omp)l(act)g(set)g (of)g Fk(C)51 b Fn(and)30 b @beginspecial @setspecial @endspecial Fr(m)23 b Fp(2)g Fk(N)t Fr(;)14 b(\015)34 b Fp(\025)23 b Fu(0)p Fn(:)594 1756 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 636 1621 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)o Fu(\)\()p Fr(H)795 1633 y Fs(0)852 1621 y Fu(+)18 b(1\))1019 1560 y Fi(1)p Fg(\000)p Fj(\015)p 1018 1574 108 3 v 1058 1607 a Fi(2)1140 1621 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))1433 1560 y Fj(\015)p 1433 1574 34 3 v 1436 1607 a Fi(2)1477 1587 y Fs(+)p Fq(m)1605 1621 y Fu(\()p Fr(z)k Fp(\000)1802 1600 y Fu(^)1781 1621 y Fr(H)1850 1633 y Fq(\024)1893 1621 y Fu(\))1925 1587 y Fo(\000)p Fq(k)2032 1621 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))2315 1587 y Fo(\000)p Fq(m)p Fs(+)p Fq(k)q Fo(\000)p Fs(1)2626 1621 y Fp(2)23 b Fr(O)r Fu(\()p Fp(j)p Fu(Im)q(\()p Fr(z)t Fu(\))p Fp(j)3054 1587 y Fq(c)3084 1596 y Fj(m;k)3194 1621 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 640 1775 a Fu(_)-27 b Fr({)t Fu(_)f Fr({)p Fu(\)\()p Fr(N)28 b Fu(+)18 b(1\))1005 1740 y Fq(m)1082 1775 y Fu(\()p Fr(z)k Fp(\000)1279 1754 y Fu(^)1258 1775 y Fr(H)1327 1787 y Fq(\024)1370 1775 y Fu(\))1402 1740 y Fo(\000)p Fq(k)1509 1775 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))1792 1740 y Fo(\000)p Fq(m)p Fs(+)p Fq(k)q Fs(+)2051 1713 y Fj(\015)s Fg(\000)p Fi(2)p 2051 1727 108 3 v 2091 1760 a(2)2172 1775 y Fu(\()p Fr(H)2273 1787 y Fs(0)2329 1775 y Fu(+)g(1\))2496 1713 y Fi(1)p Fg(\000)p Fj(\015)p 2496 1727 V 2536 1760 a Fi(2)2641 1775 y Fp(2)23 b Fr(O)r Fu(\()p Fp(j)p Fu(Im)q(\()p Fr(z)t Fu(\))p Fp(j)3069 1740 y Fq(c)3099 1749 y Fj(m;k)3209 1775 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 640 1909 a Fu(_)-27 b Fr({)t Fu(_)f Fr({)t Fu(_)h Fr({)p Fu(\))14 b Fr(Let)27 b(\037)c Fp(2)g Fr(C)1133 1875 y Fo(1)1127 1930 y Fs(0)1204 1909 y Fu(\()p Fk(R)p Fu(\))14 b Fr(and)29 b(n;)14 b(q)26 b Fp(2)d Fk(N)33 b Fu(:)3306 1756 y @beginspecial @setspecial @endspecial 1565 2099 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1687 2065 y Fq(n)1732 2099 y Fr(\037)p Fu(\()1838 2078 y(^)1816 2099 y Fr(H)1885 2111 y Fq(\024)1928 2099 y Fu(\))p Fr(N)2036 2065 y Fq(q)2073 2099 y Fp(jj)23 b Fr(<)f Fp(1)p Fr(:)p @beginspecial @setspecial @endspecial 100 2250 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(recall)e(the)i(iden)n(tit)n(y)g(\(4.4\))f(on)g Fp(D)r Fu(\()p Fr(N)1725 2220 y Fq(k)1767 2250 y Fu(\),)h(whic)n(h)f (had)h(b)r(een)g(pro)n(v)n(ed)e(in)i(the)g(pro)r(of)f(of)g(Lemma)h @beginspecial @setspecial @endspecial(4.4)566 2394 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)642 2360 y Fq(k)683 2394 y Fu(\()p Fr(z)22 b Fp(\000)881 2373 y Fu(^)859 2394 y Fr(H)928 2406 y Fq(\024)971 2394 y Fu(\))1003 2360 y Fo(\000)p Fs(1)1115 2394 y Fu(=)h Fr(N)1279 2360 y Fq(k)q Fo(\000)p Fs(1)1405 2394 y Fu(\()p Fr(z)f Fp(\000)1602 2373 y Fu(^)1581 2394 y Fr(H)1650 2406 y Fq(\024)1693 2394 y Fu(\))1725 2360 y Fo(\000)p Fs(1)1814 2394 y Fr(N)27 b Fp(\000)18 b Fu(i)p Fr(N)2090 2360 y Fq(k)q Fo(\000)p Fs(1)2216 2394 y Fu(\()p Fr(z)k Fp(\000)2414 2373 y Fu(^)2392 2394 y Fr(H)2461 2406 y Fq(\024)2504 2394 y Fu(\))2536 2360 y Fo(\000)p Fs(1)2625 2394 y Fr(ad)2712 2360 y Fs(1)2712 2415 y Fq(N)2797 2373 y Fu(^)2775 2394 y Fr(H)2844 2406 y Fq(\024)2901 2394 y Fu(\()p Fr(z)g Fp(\000)3099 2373 y Fu(^)3077 2394 y Fr(H)3146 2406 y Fq(\024)3189 2394 y Fu(\))3221 2360 y Fo(\000)p Fs(1)3311 2394 y Fr(:)p @beginspecial @setspecial @endspecial 0 2531 a Fu(W)-7 b(e)29 b(mo)n(v)n(e)f(no)n(w)g(all)g (factors)g(of)h @beginspecial @setspecial @endspecial Fr(N)37 b Fu(in)29 b(eac)n(h)f(term)h(to)g(the)g(righ)n(t,)f(w)n(e)h (obtain)f(the)h(follo)n(wing)f(iden)n(tit)n(y)h(b)r(et)n(w)n(een)g(b)r (ounded)0 2630 y(op)r(erators)594 2844 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)670 2809 y Fq(k)725 2844 y Fu(\()p Fr(z)22 b Fp(\000)923 2823 y Fu(^)901 2844 y Fr(H)970 2856 y Fq(\024)1013 2844 y Fu(\))1045 2809 y Fo(\000)p Fs(1)1148 2844 y Fr(N)1224 2809 y Fo(\000)p Fq(k)1340 2844 y Fu(=)g(\()p Fr(z)g Fp(\000)1625 2823 y Fu(^)1603 2844 y Fr(H)1672 2856 y Fq(\024)1715 2844 y Fu(\))1747 2809 y Fo(\000)p Fs(1)1855 2844 y Fu(+)1980 2740 y Fq(k)1938 2765 y Fl(X)1942 2944 y Fq(`)p Fs(=1)2072 2844 y Fu(\()p Fr(z)g Fp(\000)2270 2823 y Fu(^)2248 2844 y Fr(H)2317 2856 y Fq(\024)2360 2844 y Fu(\))2392 2809 y Fo(\000)2454 2787 y Fi(1)p 2454 2796 29 3 v 2454 2829 a(2)2497 2844 y Fr(B)2560 2856 y Fq(`)2592 2844 y Fu(\()p Fr(z)t Fu(\)\()p Fr(z)f Fp(\000)2896 2823 y Fu(^)2874 2844 y Fr(H)2943 2856 y Fq(\024)2986 2844 y Fu(\))3018 2809 y Fo(\000)3081 2787 y Fi(1)p 3081 2796 V 3081 2829 a(2)3123 2844 y Fr(N)3199 2809 y Fo(\000)p Fq(`)3283 2844 y Fr(;)p @beginspecial @setspecial @endspecial 0 3076 a Fu(where)d @beginspecial @setspecial @endspecial Fr(B)294 3088 y Fq(`)326 3076 y Fu(\()p Fr(z)t Fu(\))h(is)f(a)g(p)r (olynomial)g(in)h(\()p Fr(z)t Fp(\000)1256 3055 y Fu(^)1235 3076 y Fr(H)1304 3088 y Fq(\024)1347 3076 y Fu(\))1379 3046 y Fo(\000)1441 3024 y Fi(1)p 1441 3033 V 1441 3066 a(2)1497 3076 y Fr(ad)1584 3036 y Fq(j)1584 3100 y(N)1669 3055 y Fu(^)1647 3076 y Fr(H)1716 3088 y Fq(\024)1773 3076 y Fu(\()p Fr(z)t Fp(\000)1935 3055 y Fu(^)1914 3076 y Fr(H)1983 3088 y Fq(\024)2025 3076 y Fu(\))2057 3046 y Fo(\000)2120 3024 y Fi(1)p 2120 3033 V 2120 3066 a(2)2162 3076 y Fr(;)28 b(j)g Fp(\024)22 b Fr(`)p Fu(.)34 b(Using)18 b(Lemma)h @beginspecial @setspecial @endspecial(3.3)f(and)g(Lemma)g (4.2,)i(w)n(e)e(see)0 3212 y(that)i @beginspecial @setspecial @endspecial Fp(jj)p Fr(B)281 3224 y Fq(`)313 3212 y Fu(\()p Fr(z)t Fu(\))p Fp(jj)j(\024)f Fr(c)p Fp(j)p Fu(Im\()p Fr(z)t Fu(\))p Fp(j)864 3182 y Fo(\000)p Fq(c)946 3191 y Fj(`)979 3212 y Fu(,)f(uniformly)e(in)h @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)34 b(Using)19 b(Corollary)e(4.8,)j(\()p @beginspecial @setspecial @endspecial (4.17\),)g(w)n(e)f(see)g(that)g Fp(jj)p Fr(H)3120 3136 y Fi(1)p Fg(\000)p Fj(\015)p 3120 3150 108 3 v 3160 3183 a Fi(2)3103 3234 y Fs(0)3241 3212 y Fr(N)3327 3155 y Fj(\015)p 3327 3169 34 3 v 3330 3202 a Fi(2)3375 3212 y Fu(\()p Fr(z)5 b Fp(\000)3539 3191 y Fu(^)3518 3212 y Fr(H)3587 3224 y Fq(\024)3630 3212 y Fu(\))3662 3182 y Fo(\000)3724 3159 y Fi(1)p 3724 3168 29 3 v 3724 3202 a(2)3766 3212 y Fp(jj)23 b(\024)0 3322 y(j)p Fu(Im\()p Fr(z)t Fu(\))p Fp(j)252 3291 y Fo(\000)314 3269 y Fi(1)p 314 3278 V 314 3311 a(2)356 3322 y Fu(,)28 b(whic)n(h)g(pro)n(v)n(es)d @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))28 b(for)f Fr(k)f Fu(=)d(1.)36 b(F)-7 b(or)27 b Fr(k)f Fp(6)p Fu(=)d(1,)k(w)n(e)g(write)176 3540 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(H)262 3464 y Fi(1)p Fg(\000)p Fj(\015)p 262 3478 108 3 v 302 3511 a Fi(2)245 3562 y Fs(0)383 3540 y Fr(N)459 3505 y Fq(m)p Fs(+)579 3479 y Fj(\015)p 579 3493 34 3 v 582 3525 a Fi(2)627 3540 y Fu(\()p Fr(z)22 b Fp(\000)825 3519 y Fu(^)803 3540 y Fr(H)872 3552 y Fq(\024)915 3540 y Fu(\))947 3505 y Fo(\000)p Fq(k)1040 3540 y Fr(N)1116 3505 y Fo(\000)p Fq(m)p Fs(+)p Fq(k)q Fo(\000)p Fs(1)1426 3540 y Fu(=)h Fr(H)1600 3464 y Fi(1)p Fg(\000)p Fj(\015)p 1600 3478 108 3 v 1639 3511 a Fi(2)1583 3562 y Fs(0)1721 3540 y Fr(N)1797 3505 y Fq(m)p Fs(+)1917 3479 y Fj(\015)p 1917 3493 34 3 v 1920 3525 a Fi(2)1965 3540 y Fu(\()p Fr(z)f Fp(\000)2162 3519 y Fu(^)2141 3540 y Fr(H)2210 3552 y Fq(\024)2253 3540 y Fu(\))2285 3505 y Fo(\000)p Fs(1)2374 3540 y Fr(N)2450 3505 y Fo(\000)p Fq(m)2616 3436 y(k)2581 3461 y Fl(Y)2579 3640 y Fq(`)p Fs(=0)2718 3540 y Fr(N)2794 3505 y Fq(m)p Fo(\000)p Fq(`)2937 3540 y Fu(\()p Fr(z)g Fp(\000)3134 3519 y Fu(^)3113 3540 y Fr(H)3182 3552 y Fq(\024)3225 3540 y Fu(\))3257 3505 y Fo(\000)p Fs(1)3346 3540 y Fr(N)3422 3505 y Fo(\000)p Fq(m)p Fs(+)p Fq(`)p Fo(\000)p Fs(1)3701 3540 y Fr(:)p @beginspecial @setspecial @endspecial 0 3755 a Fu(This)33 b(pro)n(v)n(es)e @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))34 b(for)e Fr(k)j Fp(6)p Fu(=)d(1.)53 b(The)34 b(pro)r(of)e(of)37 b(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))33 b(is)g(similar)g(to)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\).)58 b(_)-28 b Fr({)t Fu(_)h Fr({)t Fu(_)g Fr({)p Fu(\))33 b(follo)n(ws)f(from)h(the)h (higher)e(order)g(estimates)h(in)0 3855 y(Thm.)k @beginspecial @setspecial @endspecial(4.7)27 b(with)h @beginspecial @setspecial @endspecial Fr(\034)33 b Fu(=)22 b Fr(\027)29 b Fu(=)22 b(0.)2899 b Fh(\004)0 3954 y Fu(W)-7 b(e)28 b(set:)1008 4091 y @beginspecial @setspecial @endspecial Fp(H)1079 4056 y Fs(ext)1256 4091 y Fu(:=)83 b Fp(H)19 b(\012)f Fu(\000\()p Fm(h)p Fu(\))p Fr(;)1864 4070 y Fu(^)1842 4091 y Fr(H)1918 4056 y Fs(ext)1911 4111 y Fq(\024)2035 4091 y Fu(=)2145 4070 y(^)2123 4091 y Fr(H)2192 4103 y Fq(\024)2253 4091 y Fp(\012)g Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\))p Fr(;)1069 4215 y @beginspecial @setspecial @endspecial(N)1136 4227 y Fs(0)1256 4215 y Fu(:=)83 b Fr(N)27 b Fp(\012)18 b Fu(1)-15 b(l)o Fr(;)60 b(N)1803 4227 y Fo(1)1897 4215 y Fu(:=)22 b(1)-15 b(l)18 b Fp(\012)g Fr(N)t(:)0 4366 y @beginspecial @setspecial @endspecial Fz(Lemma)30 b(4.13.)39 b Fn(We)30 b(have)h(uniformly)g(in)f @beginspecial @setspecial @endspecial Fr(\024)f Fn(and)h(z)g(in)g(a)g(c)l(omp)l(act)g(set)g(of)g Fk(C)51 b Fn(and)30 b @beginspecial @setspecial @endspecial Fr(m)23 b Fp(2)h Fk(N)567 4576 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 608 4506 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))23 b(\()p Fr(N)789 4518 y Fs(0)845 4506 y Fu(+)18 b Fr(N)995 4518 y Fo(1)1065 4506 y Fu(\))1097 4471 y Fq(m)1174 4506 y Fu(\()p Fr(z)k Fp(\000)1372 4485 y Fu(^)1350 4506 y Fr(H)1426 4471 y Fs(ext)1419 4526 y Fq(\024)1521 4506 y Fu(\))1553 4471 y Fo(\000)p Fs(1)1656 4506 y Fu(\()p Fr(N)1755 4518 y Fs(0)1811 4506 y Fu(+)c Fr(N)1961 4518 y Fo(1)2031 4506 y Fu(\))2063 4471 y Fo(\000)p Fq(m)p Fs(+1)2285 4506 y Fp(2)23 b Fr(O)r Fu(\()p Fp(j)p Fu(Im)r(\()p Fr(z)t Fu(\))p Fp(j)2714 4471 y Fo(\000)p Fq(c)2796 4479 y Fj(m)2854 4506 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 608 4654 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))19 b(\()p Fr(H)821 4619 y Fs(ext)814 4674 y(0)934 4654 y Fu(+)f(1\))1101 4597 y Fi(1)p 1100 4606 29 3 v 1100 4639 a(2)1157 4654 y Fu(\()p Fr(N)1256 4666 y Fs(0)1311 4654 y Fu(+)g Fr(N)1461 4666 y Fo(1)1531 4654 y Fu(\))1563 4619 y Fq(m)1640 4654 y Fu(\()p Fr(z)k Fp(\000)1838 4633 y Fu(^)1817 4654 y Fr(H)1893 4619 y Fs(ext)1886 4674 y Fq(\024)1987 4654 y Fu(\))2019 4619 y Fo(\000)p Fs(1)2122 4654 y Fu(\()p Fr(N)2221 4666 y Fs(0)2277 4654 y Fu(+)c Fr(N)2427 4666 y Fo(1)2497 4654 y Fu(\))2529 4619 y Fo(\000)p Fq(m)2667 4654 y Fp(2)23 b Fr(O)r Fu(\()p Fp(j)p Fu(Im)r(\()p Fr(z)t Fu(\))p Fp(j)3096 4619 y Fo(\000)p Fq(c)3178 4627 y Fj(m)3236 4654 y Fu(\))p Fr(:)3333 4576 y @beginspecial @setspecial @endspecial 100 4800 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(The)27 b(pro)r(of)g(is)h(analogous)d(to)i(the)h(pro)r(of)f(of)h(Lemma)f(4.12.) 36 b(In)28 b(fact)f(w)n(e)h(ha)n(v)n(e)700 5149 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 742 4940 a @beginspecial @setspecial @endspecial Fr(ad)829 4910 y Fq(`)829 4963 y(N)882 4971 y Fi(0)914 4963 y Fs(+)p Fq(N)1018 4971 y Fg(1)1104 4919 y Fu(^)1082 4940 y Fr(H)1158 4910 y Fs(ext)1151 4961 y Fq(\024)1276 4940 y Fu(=)22 b Fr(ad)1450 4910 y Fq(`)1450 4963 y(N)1535 4919 y Fu(^)1513 4940 y Fr(H)1582 4952 y Fq(\024)1644 4940 y Fp(\012)c Fu(1)-15 b(l)o Fr(;)p @beginspecial @setspecial @endspecial 742 5085 a Fu(\()p Fr(N)841 5097 y Fs(0)896 5085 y Fu(+)18 b Fr(N)1046 5097 y Fo(1)1117 5085 y Fu(\))1149 5051 y Fq(m)1226 5085 y Fu(\()p Fr(z)k Fp(\000)1423 5064 y Fu(^)1402 5085 y Fr(H)1478 5051 y Fs(ext)1471 5105 y Fq(\024)1572 5085 y Fu(\))1604 5051 y Fo(\000)p Fs(1)1707 5085 y Fu(\()p Fr(N)1806 5097 y Fs(0)1862 5085 y Fu(+)c Fr(N)2012 5097 y Fo(1)2082 5085 y Fu(\))2114 5051 y Fo(\000)p Fq(m)2252 5085 y Fu(=)p @beginspecial @setspecial @endspecial 760 5287 a(\()p Fr(z)k Fp(\000)958 5266 y Fu(^)936 5287 y Fr(H)1012 5252 y Fs(ext)1005 5307 y Fq(\024)1107 5287 y Fu(\))1139 5252 y Fo(\000)p Fs(1)1246 5287 y Fu(+)1360 5183 y Fq(m)1329 5208 y Fl(X)1333 5387 y Fq(`)p Fs(=1)1449 5287 y Fu(\()p Fr(z)g Fp(\000)1647 5266 y Fu(^)1625 5287 y Fr(H)1701 5252 y Fs(ext)1694 5307 y Fq(\024)1796 5287 y Fu(\))1828 5252 y Fo(\000)1890 5230 y Fi(1)p 1890 5239 V 1890 5272 a(2)1932 5287 y Fr(B)1995 5299 y Fq(`)2027 5287 y Fu(\()p Fr(z)t Fu(\))14 b(\()p Fr(z)22 b Fp(\000)2345 5266 y Fu(^)2324 5287 y Fr(H)2400 5252 y Fs(ext)2393 5307 y Fq(\024)2494 5287 y Fu(\))2526 5252 y Fo(\000)2588 5230 y Fi(1)p 2588 5239 V 2588 5272 a(2)2644 5287 y Fu(\()p Fr(N)2743 5299 y Fs(0)2799 5287 y Fu(+)c Fr(N)2949 5299 y Fo(1)3019 5287 y Fu(\))3051 5252 y Fo(\000)p Fq(`)3135 5287 y Fr(;)3200 5149 y @beginspecial @setspecial @endspecial 0 5506 a Fu(where)27 b @beginspecial @setspecial @endspecial Fr(B)303 5518 y Fq(`)335 5506 y Fu(\()p Fr(z)t Fu(\))c Fp(2)g Fr(O)r Fu(\()p Fp(j)p Fu(Im)q Fr(z)t Fp(j)829 5476 y Fo(\000)p Fq(c)911 5485 y Fj(`)943 5506 y Fu(\).)p @beginspecial @setspecial @endspecial 2837 w Fh(\004)p eop %%Page: 21 21 21 20 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(21)0 489 y @beginspecial @setspecial @endspecial Fu(4.3.)47 b Fz(Comm)m(utator)34 b(estimates.)39 b Fu(Let)31 b Fr(q)i Fp(2)d Fr(C)1577 459 y Fo(1)1571 510 y Fs(0)1647 489 y Fu(\()p Fk(R)1734 459 y Fs(3)1777 489 y Fu(\))p Fr(;)14 b Fu(0)29 b Fp(\024)g Fr(q)k Fp(\024)c Fu(1)p Fr(;)14 b(q)32 b Fu(=)d(1)i(near)f(0.)48 b(W)-7 b(e)32 b(set)g @beginspecial @setspecial @endspecial Fr(q)3112 459 y Fq(R)3196 489 y Fu(:=)d Fr(q)s Fu(\()3401 456 y Fq(x)p 3395 470 51 4 v 3395 518 a(R)3455 489 y Fu(\).)49 b(W)-7 b(e)32 b(recall)0 593 y(that)c(w)n(e)f(consider)g @beginspecial @setspecial @endspecial Fm(h)h Fu(in)f(its)h(momen)n(tum) g(represen)n(tation)e @beginspecial @setspecial @endspecial Fr(L)1945 563 y Fs(2)1982 593 y Fu(\()p Fk(R)2068 563 y Fs(3)2111 593 y Fr(;)14 b(dk)s Fu(\).)0 693 y(W)-7 b(e)28 b(use)f(the)h(follo)n(wing)f(functional)h(calculus)f(form)n (ula,)f(see)i @beginspecial @setspecial @endspecial([10)o(],)g(for)f Fr(\037)c Fp(2)g Fr(C)2471 663 y Fo(1)2465 714 y Fs(0)2542 693 y Fu(\()p Fk(R)p Fu(\))34 b(and)27 b Fr(A)h Fu(a)f(selfadjoin)n(t)h (op)r(erator:)1140 876 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()p Fr(A)p Fu(\))c(=)1474 820 y(i)p 1439 857 92 4 v 1439 933 a(2)p Fr(\031)1555 763 y Fl(Z)1601 952 y Fe(C)1661 876 y Fr(@)1709 888 y Fs(\026)-37 b Fq(z)1753 876 y Fu(\025)-52 b Fr(\037)p Fu(\()p Fr(z)t Fu(\))23 b(\()p Fr(z)f Fp(\000)c Fr(A)p Fu(\))2195 842 y Fo(\000)p Fs(1)2308 876 y Fr(dz)k Fp(^)d Fr(d)5 b Fu(\026)-47 b Fr(z)s(;)83 b @beginspecial @setspecial @endspecial -2677 w Fu(\(4.18\))0 1059 y(where)27 b @beginspecial @setspecial @endspecial 10 w(\025)-52 b Fr(\037)28 b Fu(is)f(an)g(almost)g(analytic)g(extension)h(of)f @beginspecial @setspecial @endspecial Fr(\037)p Fu(,)h(suc)n(h)f(that) 1265 1200 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 10 w(\025)-52 b Fr(\037)1317 1215 y Fo(j)p Fe(R)1406 1200 y Fu(=)23 b Fr(\037;)37 b Fp(j)p Fr(@)1677 1212 y Fs(\026)-37 b Fq(z)1721 1200 y Fu(\025)-52 b Fr(\037)p Fu(\()p Fr(z)t Fu(\))p Fp(j)23 b(\024)g Fr(c)2040 1212 y Fq(n)2108 1200 y Fp(j)p Fu(Im)p Fr(z)t Fp(j)2296 1166 y Fq(n)2340 1200 y Fr(;)37 b(n)23 b Fp(2)h Fk(N)t Fr(:)p @beginspecial @setspecial @endspecial 0 1341 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 153 x Fz(Lemma)30 b(4.14.)39 b Fn(L)l(et)30 b Fr(\037)23 b Fp(2)g Fr(C)938 1464 y Fo(1)932 1515 y Fs(0)1009 1494 y Fu(\()p Fk(R)p Fu(\))p Fn(,)36 b(then)30 b(one)g(has)g(uniformly)h(for)g @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(\024)f(1)1348 1643 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(N)1424 1609 y Fq(n)1469 1643 y Fu([)p Fr(\037)p Fu(\()1598 1622 y(^)1576 1643 y Fr(H)1645 1655 y Fq(\024)1688 1643 y Fu(\))p Fr(;)14 b Fu(\000\()p Fr(q)1881 1609 y Fq(R)1936 1643 y Fu(\)])p Fr(N)2067 1609 y Fq(m)2167 1643 y Fp(2)24 b Fr(O)r Fu(\()p Fr(R)2407 1609 y Fo(\000)p Fs(1)2497 1643 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 1805 a Fz(Pro)s(of.)66 b @beginspecial @setspecial @endspecial 34 w Fu(Comm)n(utation)35 b(relations)e(allo)n(w)g(to)i(compute)g @beginspecial @setspecial @endspecial([)2057 1784 y(^)2036 1805 y Fr(H)2105 1817 y Fq(\024)2148 1805 y Fr(;)14 b Fu(\000\()p Fr(q)2309 1775 y Fq(R)2363 1805 y Fu(\)])35 b(as)f(quadratic)g(form)g(on)g Fr(D)r Fu(\()p Fr(H)3438 1817 y Fs(0)3476 1805 y Fu(\),)j(whic)n(h)d(b) n(y)0 1904 y @beginspecial @setspecial @endspecial Fr(N)67 1916 y Fq(\034)108 1904 y Fp(\000)p Fu(estimates)27 b(is)h(a)f(b)r (ounded)h(op)r(erator)e(on)h Fr(D)r Fu(\()p Fr(H)1652 1916 y Fs(0)1689 1904 y Fu(\),)i(when)e Fr(\024)c(<)g Fp(1)p Fu(.)0 2004 y(W)-7 b(e)28 b(ha)n(v)n(e)752 2145 y @beginspecial @setspecial @endspecial([)p Fr(H)844 2157 y Fs(0)881 2145 y Fr(;)14 b Fu(\000\()p Fr(q)1042 2111 y Fq(R)1097 2145 y Fu(\)])83 b(=)g(d\000\()p Fr(q)1553 2111 y Fq(R)1608 2145 y Fr(;)14 b Fu([)p Fr(!)s(;)g(q)1800 2111 y Fq(R)1854 2145 y Fu(]\))p Fr(;)0 2191 y @beginspecial @setspecial @endspecial 629 2314 a Fu([)p Fr(\036)p Fu(\()p Fr(v)773 2326 y Fq(\024)812 2334 y Fi(0)849 2314 y Fu(\))p Fr(;)g Fu(\000\()p Fr(q)1042 2280 y Fq(R)1097 2314 y Fu(\)])83 b @beginspecial @setspecial @endspecial(=)g Fr(a)1427 2280 y Fo(\003)1465 2314 y Fu(\(\(1)18 b Fp(\000)g Fr(q)1712 2280 y Fq(R)1767 2314 y Fu(\))p Fr(v)1839 2326 y Fq(\024)1878 2334 y Fi(0)1915 2314 y Fu(\)\000\()p Fr(q)2071 2280 y Fq(R)2126 2314 y Fu(\))h Fp(\000)f Fu(\000\()p Fr(q)2384 2280 y Fq(R)2438 2314 y Fu(\))p Fr(a)p Fu(\(\(1)h Fp(\000)f Fr(q)2762 2280 y Fq(R)2817 2314 y Fu(\))p Fr(v)2889 2326 y Fq(\024)2928 2334 y Fi(0)2965 2314 y Fu(\))p Fr(;)0 2360 y @beginspecial @setspecial @endspecial 632 2483 a Fu([)p Fr(a)699 2449 y Fs(2)736 2483 y Fu(\()p Fr(r)807 2449 y Fq(j)805 2504 y(\024)849 2483 y Fu(\))p Fr(;)c Fu(\000\()p Fr(q)1042 2449 y Fq(R)1097 2483 y Fu(\)])83 b(=)g Fp(\000)p Fu(\000\()p Fr(q)1572 2449 y Fq(R)1626 2483 y Fu(\))p Fr(a)p Fu(\(\(1)19 b Fp(\000)f Fr(q)1950 2449 y Fq(R)2004 2483 y Fu(\))p Fr(r)2075 2449 y Fq(j)2073 2504 y(\024)2117 2483 y Fu(\))p Fr(a)p Fu(\(\(1)h(+)f Fr(q)2441 2449 y Fq(R)2495 2483 y Fu(\))p Fr(r)2566 2449 y Fq(j)2564 2504 y(\024)2608 2483 y Fu(\))p Fr(;)0 2529 y @beginspecial @setspecial @endspecial 599 2665 a Fu([)p Fr(a)666 2631 y Fo(\003)700 2605 y Fi(2)736 2665 y Fu(\()p Fr(r)807 2631 y Fq(j)805 2685 y(\024)849 2665 y Fu(\))p Fr(;)c Fu(\000\()p Fr(q)1042 2631 y Fq(R)1097 2665 y Fu(\)])83 b @beginspecial @setspecial @endspecial(=)g Fr(a)1427 2631 y Fo(\003)1465 2665 y Fu(\(\(1)18 b Fp(\000)g Fr(q)1712 2631 y Fq(R)1767 2665 y Fu(\))p Fr(r)1838 2631 y Fq(j)1836 2685 y(\024)1880 2665 y Fu(\))p Fr(a)1956 2631 y Fo(\003)1994 2665 y Fu(\(\(1)h(+)f Fr(q)2242 2631 y Fq(R)2296 2665 y Fu(\))p Fr(r)2367 2631 y Fq(j)2365 2685 y(\024)2409 2665 y Fu(\)\000\()p Fr(q)2565 2631 y Fq(R)2620 2665 y Fu(\))p Fr(;)-2675 b Fu(\(4.19\))0 2710 y @beginspecial @setspecial @endspecial 442 2834 a @beginspecial @setspecial @endspecial([)p Fr(a)509 2800 y Fo(\003)547 2834 y Fu(\()p Fr(r)618 2800 y Fq(j)616 2855 y(\024)660 2834 y Fu(\))p Fr(a)p Fu(\()p Fr(r)807 2800 y Fq(j)805 2855 y(\024)849 2834 y Fu(\))p Fr(;)14 b Fu(\000\()p Fr(q)1042 2800 y Fq(R)1097 2834 y Fu(\)])83 b(=)g @beginspecial @setspecial @endspecial Fp(\000)p Fr(a)1492 2800 y Fo(\003)1529 2834 y Fu(\()p Fr(r)1600 2800 y Fq(j)1598 2855 y(\024)1642 2834 y Fu(\)\000\()p Fr(q)1798 2800 y Fq(R)1853 2834 y Fu(\))p Fr(a)p Fu(\(\(1)19 b Fp(\000)f Fr(q)2177 2800 y Fq(R)2231 2834 y Fu(\))p Fr(r)2302 2800 y Fq(j)2300 2855 y(\024)2344 2834 y Fu(\))h(+)32 b Fr(a)2536 2800 y Fo(\003)2574 2834 y Fu(\(\(1)18 b Fp(\000)g Fr(q)2821 2800 y Fq(R)2876 2834 y Fu(\))p Fr(r)2947 2800 y Fq(j)2945 2855 y(\024)2989 2834 y Fu(\)\000\()p Fr(q)3145 2800 y Fq(R)3200 2834 y Fu(\))p Fr(a)p Fu(\()p Fr(r)3347 2800 y Fq(j)3345 2855 y(\024)3389 2834 y Fu(\))p Fr(;)0 2880 y @beginspecial @setspecial @endspecial 527 3003 a Fu([)p Fr(a)594 2969 y Fo(\003)633 3003 y Fu(\()p Fr(r)704 2969 y Fq(j)702 3024 y(\024)745 3003 y Fu(\))p Fr(D)846 3015 y Fq(j)881 3003 y Fr(;)c Fu(\000\()p Fr(q)1042 2969 y Fq(R)1097 3003 y Fu(\)])83 b @beginspecial @setspecial @endspecial(=)g Fr(a)1427 2969 y Fo(\003)1465 3003 y Fu(\(\(1)18 b Fp(\000)g Fr(q)1712 2969 y Fq(R)1767 3003 y Fu(\))p Fr(r)1838 2969 y Fq(j)1836 3024 y(\024)1880 3003 y Fu(\)\000\()p Fr(q)2036 2969 y Fq(R)2091 3003 y Fu(\))p Fr(D)2192 3015 y Fq(j)2227 3003 y Fr(;)0 3052 y @beginspecial @setspecial @endspecial 566 3176 a Fu([)p Fr(D)658 3188 y Fq(j)692 3176 y Fr(a)p Fu(\()p Fr(r)807 3141 y Fq(j)805 3196 y(\024)849 3176 y Fu(\))p Fr(;)c Fu(\000\()p Fr(q)1042 3141 y Fq(R)1097 3176 y Fu(\)])83 b @beginspecial @setspecial @endspecial(=)g Fp(\000)p Fr(D)1517 3188 y Fq(j)1551 3176 y Fu(\000\()p Fr(q)1675 3141 y Fq(R)1730 3176 y Fu(\))p Fr(a)p Fu(\(\(1)19 b Fp(\000)f Fr(q)2054 3141 y Fq(R)2108 3176 y Fu(\))p Fr(r)2179 3141 y Fq(j)2177 3196 y(\024)2221 3176 y Fu(\))p Fr(:)0 3317 y Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(\037)201 3329 y Fs(1)261 3317 y Fp(2)23 b Fr(C)404 3286 y Fo(1)398 3337 y Fs(0)475 3317 y Fu(\()p Fk(R)p Fu(\))34 b(suc)n(h)27 b(that)h Fr(\037)1046 3329 y Fs(1)1084 3317 y Fr(\037)23 b Fu(=)f Fr(\037)p Fu(.)37 b(Using)27 b(\()p @beginspecial @setspecial @endspecial(4.18\),)g(w)n(e)h(ha)n(v)n (e:)644 3466 y @beginspecial @setspecial @endspecial 84 w Fr(N)804 3431 y Fq(n)848 3466 y Fu([)p Fr(\037)p Fu(\()977 3445 y(^)955 3466 y Fr(H)1024 3478 y Fq(\024)1068 3466 y Fu(\))p Fr(;)14 b Fu(\000\()p Fr(q)1261 3431 y Fq(R)1316 3466 y Fu(\)])p Fr(N)1447 3431 y Fq(m)644 3640 y @beginspecial @setspecial @endspecial 84 w Fu(=)22 b Fr(N)891 3605 y Fq(n)936 3640 y Fr(\037)988 3652 y Fs(1)1025 3640 y Fu(\()1079 3619 y(^)1057 3640 y Fr(H)1126 3652 y Fq(\024)1170 3640 y Fu(\)[)p Fr(\037)p Fu(\()1331 3619 y(^)1309 3640 y Fr(H)1378 3652 y Fq(\024)1421 3640 y Fu(\))p Fr(;)14 b Fu(\000\()p Fr(q)1614 3605 y Fq(R)1669 3640 y Fu(\)])p Fr(N)1800 3605 y Fq(m)1882 3640 y Fu(+)k Fr(N)2041 3605 y Fq(n)2086 3640 y Fu([)p Fr(\037)2161 3652 y Fs(1)2198 3640 y Fu(\()2252 3619 y(^)2230 3640 y Fr(H)2299 3652 y Fq(\024)2342 3640 y Fu(\))p Fr(;)c Fu(\000\()p Fr(q)2535 3605 y Fq(R)2590 3640 y Fu(\)])p Fr(\037)p Fu(\()2751 3619 y(^)2729 3640 y Fr(H)2798 3652 y Fq(\024)2842 3640 y Fu(\))p Fr(N)2950 3605 y Fq(m)0 3685 y @beginspecial @setspecial @endspecial 644 3848 a @beginspecial @setspecial @endspecial 84 w Fu(=)859 3792 y(i)p 825 3829 V 825 3905 a(2)p Fr(\031)941 3735 y Fl(Z)987 3923 y Fe(C)1047 3848 y Fr(@)1095 3860 y Fs(\026)-37 b Fq(z)1139 3848 y Fu(\025)-52 b Fr(\037)p Fu(\()p Fr(z)t Fu(\))p Fr(N)1364 3814 y Fq(n)1409 3848 y Fr(\037)1461 3860 y Fs(1)1498 3848 y Fu(\()1552 3827 y(^)1530 3848 y Fr(H)1599 3860 y Fq(\024)1642 3848 y Fu(\)\()p Fr(z)23 b Fp(\000)1872 3827 y Fu(^)1851 3848 y Fr(H)1920 3860 y Fq(\024)1963 3848 y Fu(\))1995 3814 y Fo(\000)p Fs(1)2098 3848 y Fu([)2143 3827 y(^)2121 3848 y Fr(H)2190 3860 y Fq(\024)2233 3848 y Fr(;)14 b Fu(\000\()p Fr(q)2394 3814 y Fq(R)2449 3848 y Fu(\)]\()p Fr(z)22 b Fp(\000)2702 3827 y Fu(^)2680 3848 y Fr(H)2749 3860 y Fq(\024)2792 3848 y Fu(\))2824 3814 y Fo(\000)p Fs(1)2913 3848 y Fr(N)2989 3814 y Fq(m)3052 3848 y Fr(dz)g Fp(^)d Fr(d)5 b Fu(\026)-47 b Fr(z)-3312 b Fu(\(4.20\))0 3948 y @beginspecial @setspecial @endspecial 728 4111 a(+)836 4055 y(i)p 803 4092 V 803 4168 a(2)p Fr(\031)918 3998 y Fl(Z)964 4187 y Fe(C)1024 4111 y Fr(@)1072 4123 y Fs(\026)-37 b Fq(z)1116 4111 y Fu(\025)-52 b Fr(\037)1158 4123 y Fs(1)1195 4111 y Fu(\()p Fr(z)t Fu(\))p Fr(N)1378 4077 y Fq(n)1446 4111 y Fu(\()p Fr(z)22 b Fp(\000)1644 4090 y Fu(^)1622 4111 y Fr(H)1691 4123 y Fq(\024)1734 4111 y Fu(\))1766 4077 y Fo(\000)p Fs(1)1869 4111 y Fu([)1914 4090 y(^)1892 4111 y Fr(H)1961 4123 y Fq(\024)2005 4111 y Fr(;)14 b Fu(\000\()p Fr(q)2166 4077 y Fq(R)2220 4111 y Fu(\)]\()p Fr(z)22 b Fp(\000)2473 4090 y Fu(^)2451 4111 y Fr(H)2520 4123 y Fq(\024)2563 4111 y Fu(\))2595 4077 y Fo(\000)p Fs(1)2685 4111 y Fr(\037)p Fu(\()2791 4090 y(^)2769 4111 y Fr(H)2838 4123 y Fq(\024)2881 4111 y Fu(\))p Fr(N)2989 4077 y Fq(m)3052 4111 y Fr(dz)g Fp(^)d Fr(d)5 b Fu(\026)-47 b Fr(z)s(:)0 4315 y Fu(Mo)n(ving)25 b(the)h(p)r(o)n(w)n(er)e(of)i @beginspecial @setspecial @endspecial Fr(N)35 b Fu(to)n(w)n(ard)24 b Fr(\037)p Fu(\()1253 4294 y(^)1231 4315 y Fr(H)1300 4327 y Fq(\024)1343 4315 y Fu(\))p Fr(;)14 b(\037)1464 4327 y Fs(1)1501 4315 y Fu(\()1555 4294 y(^)1533 4315 y Fr(H)1602 4327 y Fq(\024)1646 4315 y Fu(\))26 b(and)f(then)h(using)g (Lemma)f @beginspecial @setspecial @endspecial(4.12)g(and)g(Corollary)f @beginspecial @setspecial @endspecial(4.8,)h(w)n(e)g(see)h(that)g(it)0 4423 y(is)g(enough)g(to)g(sho)n(w)g(that)h(for)f Fr(b)c(>)h Fu(0)p Fr(;)36 b Fu(\()p Fr(N)26 b Fu(+)15 b(1\))1503 4393 y Fo(\000)p Fq(n)1600 4423 y Fu(\()p Fr(H)1701 4435 y Fs(0)1755 4423 y Fu(+)h Fr(b)p Fu(\))1904 4393 y Fo(\000)1965 4371 y Fi(1)p 1965 4380 29 3 v 1965 4413 a(2)2008 4423 y Fu([)2053 4402 y(^)2031 4423 y Fr(H)2100 4435 y Fq(\024)2143 4423 y Fr(;)e Fu(\000\()p Fr(q)2304 4393 y Fq(R)2358 4423 y Fu(\)]\()p Fr(H)2514 4435 y Fs(0)2568 4423 y Fu(+)i Fr(b)p Fu(\))2717 4393 y Fo(\000)2779 4371 y Fi(1)p 2779 4380 V 2779 4413 a(2)2844 4423 y Fp(2)24 b Fr(O)r Fu(\()p Fr(R)3084 4393 y Fo(\000)p Fs(1)3173 4423 y Fu(\),)k(uniformly)e(in)h @beginspecial @setspecial @endspecial Fr(\024)p Fu(,)f(to)0 4523 y(ha)n(v)n(e)g(the)i(lemma.)37 b(Using)28 b(no)n(w)f(Lemma)g @beginspecial @setspecial @endspecial(3.4,)g(w)n(e)g(obtain)462 4672 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 166 w Fp(jj)p Fu(\()p Fr(N)g Fu(+)18 b(1\))957 4638 y Fo(\000)p Fq(n)1054 4672 y Fu(\()p Fr(H)1155 4684 y Fs(0)1211 4672 y Fu(+)g Fr(b)p Fu(\))1362 4638 y Fo(\000)1424 4615 y Fi(1)p 1424 4624 V 1424 4658 a(2)1466 4672 y Fu([)1511 4651 y(^)1489 4672 y Fr(H)1558 4684 y Fq(\024)1601 4672 y Fr(;)c Fu(\000\()p Fr(q)1762 4638 y Fq(R)1817 4672 y Fu(\)]\()p Fr(H)1973 4684 y Fs(0)2029 4672 y Fu(+)k Fr(b)p Fu(\))2180 4638 y Fo(\000)2242 4615 y Fi(1)p 2242 4624 V 2242 4658 a(2)2285 4672 y Fp(jj)23 b(\024)0 4718 y @beginspecial @setspecial @endspecial 462 4855 a @beginspecial @setspecial @endspecial 166 w Fr(c)p Fu(\()p Fp(jj)p Fu(\(1)18 b Fp(\000)g Fr(q)957 4821 y Fq(R)1012 4855 y Fu(\))c(\()p Fr(V)38 b Fu(+)18 b(1\))1333 4821 y Fo(\000)1394 4798 y Fi(1)p 1394 4807 V 1394 4841 a(2)1437 4855 y Fr(v)1477 4867 y Fq(\024)1516 4875 y Fi(0)1552 4855 y Fp(jj)h Fu(+)f Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2021 4821 y Fo(\000)2082 4798 y Fj(s)p 2082 4807 V 2082 4841 a Fi(2)2125 4855 y Fr(!)2190 4795 y Fj(s)p Fg(\000)p Fi(1)p 2189 4808 102 3 v 2226 4841 a(4)2318 4855 y Fu(\(1)g Fp(\000)g Fr(q)2533 4821 y Fq(R)2588 4855 y Fu(\))p Fr(r)2657 4867 y Fq(\024)2701 4855 y Fp(jj)23 b(jj)p Fr(!)2881 4795 y Fj(s)p Fg(\000)p Fi(1)p 2881 4808 V 2917 4841 a(4)3010 4855 y Fu(\(1)18 b(+)g Fr(q)3225 4821 y Fq(R)3279 4855 y Fu(\))p Fr(r)3348 4867 y Fq(\024)3392 4855 y Fp(jj)0 4900 y @beginspecial @setspecial @endspecial 628 5038 a Fu(+)p Fp(jj)p Fr(N)815 5003 y Fo(\000)p Fs(1)903 5038 y Fu(d\000\()p Fr(q)1073 5003 y Fq(R)1128 5038 y Fr(;)c Fu([)p Fr(!)s(;)g(q)1320 5003 y Fq(R)1374 5038 y Fu(]\))p Fp(jj)19 b Fu(+)f Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))1898 5003 y Fo(\000)1960 4981 y Fj(s)p 1959 4990 29 3 v 1959 5023 a Fi(2)2002 5038 y Fr(!)2067 4978 y Fj(s)p Fg(\000)p Fi(1)p 2066 4991 102 3 v 2103 5023 a(2)2195 5038 y Fu(\(1)h Fp(\000)f Fr(q)2411 5003 y Fq(R)2465 5038 y Fu(\))p Fr(r)2534 5050 y Fq(\024)2578 5038 y Fp(jj)23 b(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2968 5003 y Fo(\000)3030 4981 y Fj(s)p 3030 4990 29 3 v 3030 5023 a Fi(2)3072 5038 y Fr(!)3137 4978 y Fj(s)p Fg(\000)p Fi(1)p 3137 4991 102 3 v 3173 5023 a(2)3252 5038 y Fr(r)3289 5050 y Fq(\024)3332 5038 y Fp(jj)-3378 b Fu(\(4.21\))0 5083 y @beginspecial @setspecial @endspecial 545 5221 a @beginspecial @setspecial @endspecial 83 w(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1013 5186 y Fo(\000)1075 5164 y Fj(s)p 1075 5173 29 3 v 1075 5206 a Fi(2)1117 5221 y Fr(!)1182 5161 y Fj(s)p Fg(\000)p Fi(1)p 1182 5174 102 3 v 1218 5206 a(2)1297 5221 y Fu(\(1)g Fp(\000)g Fr(q)1512 5186 y Fq(R)1567 5221 y Fu(\))p Fr(r)1636 5233 y Fq(\024)1679 5221 y Fp(jj)24 b(jj)p Fu(\()p Fr(K)g Fu(+)18 b(i\))2060 5186 y Fo(\000)2122 5164 y Fi(1)p 2122 5173 29 3 v 2122 5206 a(2)2164 5221 y Fr(D)r Fp(jj)p Fu(\))p Fr(:)0 5362 y Fu(Using)27 b(the)h(inequalit)n(y)f(\()p @beginspecial @setspecial @endspecial(2.3\))h(recalled)e(in)i(Subsect.) 37 b @beginspecial @setspecial @endspecial(2.1,)1322 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1444 5472 y Fo(\000)p Fs(1)1547 5506 y Fu(d\000\()p Fr(q)1717 5472 y Fq(R)1772 5506 y Fr(;)14 b Fu([)p Fr(!)s(;)g(q)1964 5472 y Fq(R)2018 5506 y Fu(]\))p Fp(jj)24 b(\024)e(jj)p Fu([)p Fr(!)s(;)14 b(q)2431 5472 y Fq(R)2485 5506 y Fu(])p Fp(jj)p Fr(;)p @beginspecial @setspecial @endspecial eop %%Page: 22 22 22 21 bop 0 226 a Fs(22)1695 b(Z.)29 b(AMMARI)0 489 y Fu(and)h(the)g(fact)h(that)f @beginspecial @setspecial @endspecial([)p Fr(!)s(;)14 b(q)813 459 y Fq(R)867 489 y Fu(])27 b Fp(2)h Fr(O)r Fu(\()p Fr(R)1161 459 y Fo(\000)p Fs(1)1251 489 y Fu(\))i(,)h(w)n(e)f(see)g(that)g @beginspecial @setspecial @endspecial Fp(jj)p Fr(N)1933 459 y Fo(\000)p Fs(1)2022 489 y Fu(d\000\()p Fr(q)2192 459 y Fq(R)2247 489 y Fr(;)14 b Fu([)p Fr(!)s(;)g(q)2439 459 y Fq(R)2493 489 y Fu(]\))p Fp(jj)28 b(2)f Fr(O)r Fu(\()p Fr(R)2865 459 y Fo(\000)p Fs(1)2955 489 y Fu(\).)45 b(No)n(w)29 b(for)h(the)g(other)g(kind)0 589 y(of)37 b(terms)g(w)n(e)g(will)g(pro)r (ceed)f(as)h(follo)n(ws.)64 b(Since)37 b @beginspecial @setspecial @endspecial Fr(V)58 b Fp(\025)38 b(h)p Fu(x)p Fp(i)1947 558 y Fq(\013)1995 589 y Fr(;)28 b(\013)39 b(>)g Fu(2,)g(w)n(e)d(can)h(pic)n(k)g @beginspecial @setspecial @endspecial Fr(\026)i(>)f Fu(0)f(and)g Fr(s)i(<)f Fu(1,)h(suc)n(h)e (that)0 688 y(\()p Fr(V)h Fu(+)18 b(1\))275 658 y Fo(\000)337 636 y Fj(s)p 336 645 29 3 v 336 678 a Fi(2)379 688 y Fp(h)p Fu(x)p Fp(i)487 658 y Fs(1+)p Fq(\026)643 688 y Fu(is)28 b(b)r(ounded.)37 b(Then)28 b(using)f(Lemma)g @beginspecial @setspecial @endspecial(A.2,)h(w)n(e)f(obtain)1164 843 y @beginspecial @setspecial @endspecial 166 w Fp(jjh)p Fu(x)p Fp(i)1484 808 y Fs(1+)p Fq(\026)1614 843 y Fr(!)1679 783 y Fj(s)p Fg(\000)p Fi(1)p 1678 796 102 3 v 1715 828 a(2)1793 843 y Fu(\(1)19 b Fp(\000)f Fr(q)2009 808 y Fq(R)2063 843 y Fu(\))p Fr(r)2132 855 y Fq(\024)2176 843 y Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(R)2484 808 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2666 843 y Fu(\))p Fr(;)0 888 y @beginspecial @setspecial @endspecial 1247 1022 a @beginspecial @setspecial @endspecial 83 w Fp(jjh)p Fu(x)p Fp(i)1484 987 y Fs(1+)p Fq(\026)1614 1022 y Fr(!)1679 962 y Fj(s)p Fg(\000)p Fi(1)p 1678 975 V 1715 1007 a(4)1793 1022 y Fu(\(1)c Fp(\000)f Fr(q)2009 987 y Fq(R)2063 1022 y Fu(\))p Fr(r)2132 1034 y Fq(\024)2176 1022 y Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(R)2484 987 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2666 1022 y Fu(\))p Fr(:)0 1159 y Fu(Hence)28 b(w)n(e)f(ha)n(v)n(e:)865 1296 y Fp(jj)p Fu(\()p Fr(N)g Fu(+)19 b(1\))1195 1262 y Fo(\000)p Fq(n)1291 1296 y Fu(\()p Fr(H)1392 1308 y Fs(0)1448 1296 y Fu(+)f Fr(b)p Fu(\))1599 1262 y Fo(\000)1661 1240 y Fi(1)p 1661 1249 29 3 v 1661 1282 a(2)1704 1296 y Fu([)1737 1275 y(^)1727 1296 y Fr(I)1763 1308 y Fq(\024)1806 1296 y Fr(;)c Fu(\000\()p Fr(q)1967 1262 y Fq(R)2022 1296 y Fu(\)]\()p Fr(H)2178 1308 y Fs(0)2234 1296 y Fu(+)k Fr(b)p Fu(\))2385 1262 y Fo(\000)2447 1240 y Fi(1)p 2447 1249 V 2447 1282 a(2)2489 1296 y Fp(jj)23 b(2)h Fr(O)r Fu(\()p Fr(R)2798 1262 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2980 1296 y Fu(\))p Fr(:)-3035 b Fu(\(4.22\))0 1444 y(Then)28 b(the)g(in)n(tegrand)e(in)i(\()p @beginspecial @setspecial @endspecial(4.20\))f(is)g Fp(j)p Fu(Im)q(\()p Fr(z)t Fu(\))p Fp(j)1404 1414 y Fo(\000)p Fs(2)1493 1444 y Fr(O)r Fu(\()p Fr(R)1654 1414 y Fo(\000)p Fs(1)1743 1444 y Fu(\).)38 b(This)27 b(ends)h(the)g(pro)r(of.)p @beginspecial @setspecial @endspecial 1264 w Fh(\004)100 1543 y @beginspecial @setspecial @endspecial 100 x Fu(Let)i @beginspecial @setspecial @endspecial Fr(j)285 1655 y Fs(0)349 1643 y Fp(2)e Fr(C)497 1613 y Fo(1)491 1664 y Fs(0)568 1643 y Fu(\()p Fk(R)654 1613 y Fs(3)697 1643 y Fu(\))p Fr(;)g(j)814 1655 y Fo(1)912 1643 y Fp(2)f Fr(C)1059 1613 y Fo(1)1130 1643 y Fu(\()p Fk(R)1216 1613 y Fs(3)1260 1643 y Fu(\))p Fr(;)h Fu(0)e Fp(\024)h Fr(j)1537 1655 y Fs(0)1574 1643 y Fr(;)h Fu(0)f Fp(\024)g Fr(j)1820 1655 y Fo(1)1904 1643 y Fr(;)14 b(j)1980 1613 y Fs(2)1975 1664 y(0)2037 1643 y Fu(+)20 b Fr(j)2161 1613 y Fs(2)2156 1663 y Fo(1)2254 1643 y Fp(\024)26 b Fu(1)p Fr(;)i(j)2472 1655 y Fs(0)2536 1643 y Fu(=)f(1)44 b(near)29 b @beginspecial @setspecial @endspecial(0.)44 b(W)-7 b(e)30 b(set)g(for)g @beginspecial @setspecial @endspecial Fr(R)e Fp(\025)f Fu(1)p Fr(;)g(j)3731 1613 y Fq(R)3812 1643 y Fu(:=)0 1743 y(\()p Fr(j)71 1712 y Fq(R)66 1763 y Fs(0)126 1743 y Fr(;)14 b(j)202 1712 y Fq(R)197 1763 y Fo(1)267 1743 y Fu(\),)28 b(where)f @beginspecial @setspecial @endspecial Fr(j)629 1712 y Fq(R)624 1763 y Fs(0)707 1743 y Fu(:=)22 b Fr(j)851 1755 y Fs(0)889 1743 y Fu(\()937 1710 y Fq(x)p 931 1724 51 4 v 931 1771 a(R)991 1743 y Fu(\))p Fr(;)14 b(j)1099 1712 y Fq(R)1094 1763 y Fo(1)1188 1743 y Fu(:=)23 b Fr(j)1333 1755 y Fo(1)1403 1743 y Fu(\()1452 1710 y Fq(x)p 1446 1724 V 1446 1771 a(R)1506 1743 y Fu(\).)37 b(W)-7 b(e)28 b(set)g @beginspecial @setspecial @endspecial Fr(j)g Fu(:=)22 b Fr(j)2082 1712 y Fs(1)0 1771 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 123 x Fz(Lemma)30 b(4.15.)39 b Fn(One)30 b(has)g(uniformly)h(for)f @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(\024)g(1)795 2152 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 836 2034 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))19 b Fr(\037)p Fu(\()1020 2013 y(^)998 2034 y Fr(H)1074 2000 y Fs(ext)1067 2055 y Fq(\024)1168 2034 y Fu(\))14 b Fr(I)1257 2000 y Fo(\003)1296 2034 y Fu(\()p Fr(j)1367 2000 y Fq(R)1421 2034 y Fu(\))19 b Fp(\000)f Fr(I)1598 2000 y Fo(\003)1636 2034 y Fu(\()p Fr(j)1707 2000 y Fq(R)1762 2034 y Fu(\))c Fr(\037)p Fu(\()1914 2013 y(^)1892 2034 y Fr(H)1961 2046 y Fq(\024)2004 2034 y Fu(\))37 b Fp(2)24 b Fr(O)r Fu(\()p Fr(R)2313 2000 y Fo(\000)p Fs(1)2403 2034 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 840 2169 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))32 b Fr(Let)k(\037)23 b Fp(2)h Fr(C)1335 2134 y Fo(1)1329 2189 y Fs(0)1405 2169 y Fu(\()p Fk(R)q Fu(\))43 b Fr(then)13 b @beginspecial @setspecial @endspecial 836 2277 a Fu(\()p Fr(N)935 2289 y Fs(0)991 2277 y Fu(+)18 b Fr(N)1141 2289 y Fo(1)1211 2277 y Fu(\))1243 2247 y Fq(n)1302 2277 y Fu(\()p Fr(\037)p Fu(\()1441 2256 y(^)1418 2277 y Fr(H)1494 2247 y Fs(ext)1487 2298 y Fq(\024)1589 2277 y Fu(\))c Fr(I)1678 2247 y Fo(\003)1716 2277 y Fu(\()p Fr(j)1787 2247 y Fq(R)1842 2277 y Fu(\))19 b Fp(\000)f Fr(I)2019 2247 y Fo(\003)2057 2277 y Fu(\()p Fr(j)2128 2247 y Fq(R)2183 2277 y Fu(\))c Fr(\037)p Fu(\()2335 2256 y(^)2313 2277 y Fr(H)2382 2289 y Fq(\024)2425 2277 y Fu(\)\))g Fr(N)2579 2247 y Fq(m)2679 2277 y Fp(2)24 b Fr(O)r Fu(\()p Fr(R)2919 2247 y Fo(\000)p Fs(1)3008 2277 y Fu(\))p Fr(:)3105 2152 y @beginspecial @setspecial @endspecial 100 2429 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(The)27 b(pro)r(of)g(is)h(similar)e(to)i (the)g(previous)e(one.)37 b(Instead)27 b(of)h(\()p @beginspecial @setspecial @endspecial(4.19\),)f(w)n(e)g(use)g(the)h(iden)n(tities:) 672 2570 y @beginspecial @setspecial @endspecial Fr(H)748 2535 y Fs(ext)741 2590 y(0)843 2570 y Fr(I)886 2535 y Fo(\003)924 2570 y Fu(\()p Fr(j)995 2535 y Fq(R)1049 2570 y Fu(\))19 b Fp(\000)f Fr(I)1226 2535 y Fo(\003)1264 2570 y Fu(\()p Fr(j)1335 2535 y Fq(R)1390 2570 y Fu(\))p Fr(H)1491 2582 y Fs(0)1612 2570 y @beginspecial @setspecial @endspecial Fu(=)82 b @beginspecial @setspecial @endspecial(d)p Fr(I)1848 2535 y Fo(\003)1887 2570 y Fu(\()p Fr(j)1958 2535 y Fq(R)2012 2570 y Fr(;)14 b Fu([)p Fr(!)s(;)g(j)2203 2535 y Fq(R)2257 2570 y Fu(]\))p Fr(;)37 b Fu(where)23 b([)p Fr(!)s(;)14 b(j)2762 2535 y Fq(R)2816 2570 y Fu(])23 b(=)g(\([)p Fr(!)s(;)14 b(j)3136 2535 y Fq(R)3131 2590 y Fs(0)3190 2570 y Fu(])p Fr(;)g Fu([)p Fr(!)s(;)g(j)3404 2535 y Fq(R)3399 2590 y Fo(1)3469 2570 y Fu(]\))p Fr(;)0 2615 y @beginspecial @setspecial @endspecial 339 2735 a(\036)p Fu(\()p Fr(v)460 2747 y Fq(\024)499 2755 y Fi(0)536 2735 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)o Fr(I)762 2701 y Fo(\003)800 2735 y Fu(\()p Fr(j)871 2701 y Fq(R)926 2735 y Fu(\))19 b Fp(\000)f Fr(I)1103 2701 y Fo(\003)1141 2735 y Fu(\()p Fr(j)1212 2701 y Fq(R)1267 2735 y Fu(\))p Fr(\036)p Fu(\()p Fr(v)1420 2747 y Fq(\024)1459 2755 y Fi(0)1496 2735 y Fu(\))84 b @beginspecial @setspecial @endspecial(=)e @beginspecial @setspecial @endspecial Fr(\036)p Fu(\(\(1)19 b Fp(\000)f Fr(j)2055 2701 y Fq(R)2050 2756 y Fs(0)2110 2735 y Fu(\))p Fr(v)2182 2747 y Fq(\024)2221 2755 y Fi(0)2258 2735 y Fu(\))g Fp(\012)g Fu(1)-15 b(l)p Fr(I)2484 2701 y Fo(\003)2522 2735 y Fu(\()p Fr(j)2593 2701 y Fq(R)2648 2735 y Fu(\))18 b Fp(\000)g Fu(1)-15 b(l)2842 2722 y(~)2831 2735 y Fp(\012)o Fr(\036)p Fu(\()p Fr(j)3015 2701 y Fq(R)3010 2756 y Fo(1)3082 2735 y Fr(v)3122 2747 y Fq(\024)3161 2755 y Fi(0)3197 2735 y Fu(\))p Fr(I)3272 2701 y Fo(\003)3311 2735 y Fu(\()p Fr(j)3382 2701 y Fq(R)3436 2735 y Fu(\))p Fr(;)171 3010 y @beginspecial @setspecial @endspecial(a)215 2975 y Fo(\003)253 3010 y Fu(\()p Fr(r)324 2975 y Fq(j)322 3030 y(\024)366 3010 y Fu(\))p Fr(D)467 3022 y Fq(j)520 3010 y Fp(\012)19 b Fu(1)-15 b(l)o Fr(I)696 2975 y Fo(\003)734 3010 y Fu(\()p Fr(j)805 2975 y Fq(R)860 3010 y Fu(\))18 b Fp(\000)g Fr(I)1036 2975 y Fo(\003)1075 3010 y Fu(\()p Fr(j)1146 2975 y Fq(R)1200 3010 y Fu(\))p Fr(a)1276 2975 y Fo(\003)1315 3010 y Fu(\()p Fr(r)1386 2975 y Fq(j)1384 3030 y(\024)1427 3010 y Fu(\))p Fr(D)1528 3022 y Fq(j)1646 3010 y Fu(=)83 b @beginspecial @setspecial @endspecial Fr(a)1838 2975 y Fo(\003)1876 3010 y Fu(\(\(1)19 b Fp(\000)f Fr(j)2123 2975 y Fq(R)2118 3030 y Fs(0)2177 3010 y Fu(\))p Fr(r)2248 2975 y Fq(j)2246 3030 y(\024)2290 3010 y Fu(\))p Fr(D)2391 3022 y Fq(j)2444 3010 y Fp(\012)g Fu(1)-15 b(l)p Fr(I)2620 2975 y Fo(\003)2658 3010 y Fu(\()p Fr(j)2729 2975 y Fq(R)2784 3010 y Fu(\))18 b Fp(\000)g Fu(1)-15 b(l)2978 2997 y(~)2967 3010 y Fp(\012)o Fr(a)3075 2975 y Fo(\003)3113 3010 y Fu(\()p Fr(j)3184 2975 y Fq(R)3179 3030 y Fo(1)3250 3010 y Fr(r)3289 2975 y Fq(j)3287 3030 y(\024)3331 3010 y Fu(\))p Fr(D)3432 3022 y Fq(j)3467 3010 y Fr(I)3510 2975 y Fo(\003)3548 3010 y Fu(\()p Fr(j)3619 2975 y Fq(R)3673 3010 y Fu(\))p @beginspecial @setspecial @endspecial(,)220 3178 y Fr(D)289 3190 y Fq(j)337 3178 y Fr(a)p Fu(\()p Fr(r)452 3144 y Fq(j)450 3199 y(\024)494 3178 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)o Fr(I)720 3144 y Fo(\003)758 3178 y Fu(\()p Fr(j)829 3144 y Fq(R)884 3178 y Fu(\))19 b Fp(\000)f Fr(I)1061 3144 y Fo(\003)1099 3178 y Fu(\()p Fr(j)1170 3144 y Fq(R)1225 3178 y Fu(\))p Fr(D)1326 3190 y Fq(j)1375 3178 y Fr(a)p Fu(\()p Fr(r)1490 3144 y Fq(j)1488 3199 y(\024)1531 3178 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)g Fr(D)1863 3190 y Fq(j)1912 3178 y Fr(a)p Fu(\(\(1)18 b Fp(\000)g Fr(j)2202 3144 y Fq(R)2197 3199 y Fs(0)2256 3178 y Fu(\))p Fr(r)2327 3144 y Fq(j)2325 3199 y(\024)2369 3178 y Fu(\))h Fp(\012)f Fu(1)-15 b(l)o Fr(I)2595 3144 y Fo(\003)2634 3178 y Fu(\()p Fr(j)2705 3144 y Fq(R)2759 3178 y Fu(\))19 b Fp(\000)f Fu(1)-15 b(l)2954 3165 y(~)2942 3178 y Fp(\012)p Fr(D)3076 3190 y Fq(j)3111 3178 y Fr(a)p Fu(\()p Fr(j)3226 3144 y Fq(R)3221 3199 y Fo(1)3291 3178 y Fr(r)3330 3144 y Fq(j)3328 3199 y(\024)3372 3178 y Fu(\))p Fr(I)3447 3144 y Fo(\003)3485 3178 y Fu(\()p Fr(j)3556 3144 y Fq(R)3611 3178 y Fu(\))p Fr(;)343 3453 y @beginspecial @setspecial @endspecial(a)387 3418 y Fs(2)424 3453 y Fu(\()p Fr(r)495 3418 y Fq(j)493 3473 y(\024)537 3453 y Fu(\))18 b Fp(\012)g Fu(1)-15 b(l)p Fr(I)763 3418 y Fo(\003)801 3453 y Fu(\()p Fr(j)872 3418 y Fq(R)927 3453 y Fu(\))18 b Fp(\000)g Fr(I)1103 3418 y Fo(\003)1142 3453 y Fu(\()p Fr(j)1213 3418 y Fq(R)1267 3453 y Fu(\))p Fr(a)1343 3418 y Fs(2)1381 3453 y Fu(\()p Fr(r)1452 3418 y Fq(j)1450 3473 y(\024)1494 3453 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)f Fp(\000)p Fr(I)1864 3418 y Fo(\003)1902 3453 y Fu(\()p Fr(j)1973 3418 y Fq(R)2028 3453 y Fu(\))p Fr(a)p Fu(\(\(1)19 b Fp(\000)f Fr(j)2351 3418 y Fq(R)2346 3473 y Fs(0)2405 3453 y Fu(\))p Fr(r)2476 3418 y Fq(j)2474 3473 y(\024)2518 3453 y Fu(\))p Fr(a)p Fu(\(\(1)g(+)g Fr(j)2840 3418 y Fq(R)2835 3473 y Fs(0)2895 3453 y Fu(\))p Fr(r)2966 3418 y Fq(j)2964 3473 y(\024)3008 3453 y Fu(\))p Fr(;)0 3498 y @beginspecial @setspecial @endspecial 276 3630 a(a)320 3596 y Fo(\003)354 3571 y Fi(2)391 3630 y Fu(\()p Fr(r)462 3596 y Fq(j)460 3651 y(\024)503 3630 y Fu(\))h Fp(\012)f Fu(1)-15 b(l)p Fr(I)730 3596 y Fo(\003)768 3630 y Fu(\()p Fr(j)839 3596 y Fq(R)893 3630 y Fu(\))19 b Fp(\000)f Fr(I)1070 3596 y Fo(\003)1108 3630 y Fu(\()p Fr(j)1179 3596 y Fq(R)1234 3630 y Fu(\))p Fr(a)1310 3596 y Fo(\003)1344 3571 y Fi(2)1381 3630 y Fu(\()p Fr(r)1452 3596 y Fq(j)1450 3651 y(\024)1494 3630 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)f(\()p Fp(\000)p Fu(2)p Fr(a)1939 3596 y Fo(\003)1977 3630 y Fu(\()p Fr(j)2048 3596 y Fq(R)2043 3651 y Fs(0)2103 3630 y Fr(r)2142 3596 y Fq(j)2140 3651 y(\024)2183 3630 y Fu(\))19 b Fp(\012)f Fr(a)2361 3596 y Fo(\003)2399 3630 y Fu(\()p Fr(j)2470 3596 y Fq(R)2465 3651 y Fo(1)2536 3630 y Fr(r)2575 3596 y Fq(j)2573 3651 y(\024)2616 3630 y Fu(\))h Fp(\000)f Fu(1)-15 b(l)2811 3617 y(~)2799 3630 y Fp(\012)p Fr(a)2908 3596 y Fo(\003)2942 3571 y Fi(2)2978 3630 y Fu(\()p Fr(j)3049 3596 y Fq(R)3044 3651 y Fo(1)3115 3630 y Fr(r)3154 3596 y Fq(j)3152 3651 y(\024)3196 3630 y Fu(\)\))19 b Fp(\002)f Fr(I)3405 3596 y Fo(\003)3443 3630 y Fu(\()p Fr(j)3514 3596 y Fq(R)3568 3630 y Fu(\))p Fr(;)408 3905 y @beginspecial @setspecial @endspecial(a)452 3870 y Fo(\003)490 3905 y Fu(\()p Fr(r)561 3870 y Fq(j)559 3925 y(\024)603 3905 y Fu(\))p Fr(a)p Fu(\()p Fr(r)750 3870 y Fq(j)748 3925 y(\024)792 3905 y Fu(\))h Fp(\012)f Fu(1)-15 b(l)o Fr(I)1018 3870 y Fo(\003)1056 3905 y Fu(\()p Fr(j)1127 3870 y Fq(R)1182 3905 y Fu(\))19 b Fp(\000)f Fr(I)1359 3870 y Fo(\003)1397 3905 y Fu(\()p Fr(j)1468 3870 y Fq(R)1522 3905 y Fu(\))p Fr(a)1598 3870 y Fo(\003)1637 3905 y Fu(\()p Fr(r)1708 3870 y Fq(j)1706 3925 y(\024)1750 3905 y Fu(\))p Fr(a)p Fu(\()p Fr(r)1897 3870 y Fq(j)1895 3925 y(\024)1939 3905 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)f Fr(a)2245 3870 y Fo(\003)2283 3905 y Fu(\()p Fr(r)2354 3870 y Fq(j)2352 3925 y(\024)2396 3905 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)o Fr(I)2622 3870 y Fo(\003)2661 3905 y Fu(\()p Fr(j)2732 3870 y Fq(R)2786 3905 y Fu(\))p Fr(a)p Fu(\(\()p Fp(\000)p Fu(1)18 b(+)g Fr(j)3173 3870 y Fq(R)3168 3925 y Fs(0)3228 3905 y Fu(\))p Fr(r)3299 3870 y Fq(j)3297 3925 y(\024)3341 3905 y Fu(\))g(+)0 3950 y @beginspecial @setspecial @endspecial 2201 4070 a(\()p Fr(a)2277 4036 y Fo(\003)2316 4070 y Fu(\(\(1)g Fp(\000)g Fr(j)2562 4036 y Fq(R)2557 4091 y Fs(0)2617 4070 y Fu(\))p Fr(r)2688 4036 y Fq(j)2686 4091 y(\024)2729 4070 y Fu(\))h Fp(\012)f Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)3075 4057 y(~)3063 4070 y Fp(\012)p Fr(a)3172 4036 y Fo(\003)3210 4070 y Fu(\()p Fr(j)3281 4036 y Fq(R)3276 4091 y Fo(1)3347 4070 y Fr(r)3386 4036 y Fq(j)3384 4091 y(\024)3427 4070 y Fu(\)\))0 4116 y @beginspecial @setspecial @endspecial 2201 4236 a Fp(\002)p Fr(I)2309 4202 y Fo(\003)2347 4236 y Fu(\()p Fr(j)2418 4202 y Fq(R)2473 4236 y Fu(\))p Fr(a)p Fu(\()p Fr(r)2620 4202 y Fq(j)2618 4256 y(\024)2662 4236 y Fu(\))p Fr(:)0 4373 y Fu(W)-7 b(e)28 b(notice)f(that)h(for)f @beginspecial @setspecial @endspecial Fr(v)g Fp(2)c(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)20 b(\012)e Fm(h)p Fu(\))p Fr(;)c Fm(h)1351 4385 y Fs(1)1412 4373 y Fu(=)22 b Fm(h)1542 4385 y Fs(2)1603 4373 y Fu(=)h Fm(h)p Fu(,)28 b(w)n(e)f(de\014ne)1041 4514 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-15 b(l)1102 4501 y(~)1091 4514 y Fp(\012)o Fr(a)1199 4480 y Fq(])1230 4514 y Fu(\()p Fr(v)s Fu(\))24 b(:)f Fp(K)d(\012)e Fu(\000\()p Fm(h)1700 4526 y Fs(1)1738 4514 y Fu(\))g Fp(\012)g Fu(\000\()p Fm(h)1998 4526 y Fs(2)2036 4514 y Fu(\))23 b Fp(!)g(K)d(\012)e Fu(\000\()p Fm(h)2490 4526 y Fs(1)2528 4514 y Fu(\))h Fp(\012)f Fu(\000\()p Fm(h)2789 4526 y Fs(2)2827 4514 y Fu(\))p @beginspecial @setspecial @endspecial 1352 4693 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-15 b(l)1413 4680 y(~)1401 4693 y Fp(\012)p Fr(a)1510 4659 y Fq(])1541 4693 y Fu(\()p Fr(v)s Fu(\))23 b(:=)g Fr(T)i Fu(1)-15 b(l)1905 4722 y Fs(\000\()p Fm(h)2015 4730 y Fi(1)2048 4722 y Fs(\))2097 4693 y Fp(\012)18 b Fr(a)2224 4659 y Fq(])2254 4693 y Fu(\()p Fr(v)s Fu(\))c Fr(T)2436 4659 y Fo(\000)p Fs(1)2525 4693 y Fr(;)p @beginspecial @setspecial @endspecial 0 4841 a Fu(where)27 b @beginspecial @setspecial @endspecial Fr(T)39 b Fu(is)27 b(the)h(natural)f(iden)n(ti\014cation:)1148 4979 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(T)34 b Fu(:)23 b Fp(K)d(\012)e Fu(\000\()p Fm(h)1570 4991 y Fs(1)1608 4979 y Fu(\))h Fp(\012)f Fu(\000\()p Fm(h)1869 4991 y Fs(2)1906 4979 y Fu(\))24 b Fp(!)f Fu(\000\()p Fm(h)2195 4991 y Fs(1)2232 4979 y Fu(\))c Fp(\012)f(K)i(\012)e Fu(\000\()p Fm(h)2659 4991 y Fs(2)2697 4979 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 5215 a Fu(Using)26 b(\()p @beginspecial @setspecial @endspecial(2.4\))g(with)g Fr(j)655 5185 y Fs(2)650 5236 y(0)708 5215 y Fu(+)15 b Fr(j)827 5185 y Fs(2)822 5236 y Fo(1)915 5215 y Fp(\024)23 b Fu(1,)j(w)n(e)f(see)h (that)g @beginspecial @setspecial @endspecial(\()p Fr(H)1633 5185 y Fs(ext)1626 5236 y(0)1728 5215 y Fr(I)1771 5185 y Fo(\003)1809 5215 y Fu(\()p Fr(j)1880 5185 y Fq(R)1935 5215 y Fu(\))15 b Fp(\000)g Fr(I)2105 5185 y Fo(\003)2143 5215 y Fu(\()p Fr(j)2214 5185 y Fq(R)2269 5215 y Fu(\))p Fr(H)2370 5227 y Fs(0)2408 5215 y Fu(\)\()p Fr(N)24 b Fu(+)15 b(1\))2717 5185 y Fo(\000)p Fs(1)2832 5215 y Fu(is)26 b(b)r(ounded,)h(whic)n(h)f(sho)n(ws)f(that)0 5315 y @beginspecial @setspecial @endspecial Fr(I)43 5285 y Fo(\003)81 5315 y Fu(\()p Fr(j)152 5285 y Fq(R)207 5315 y Fu(\))e(:)g Fp(D)r Fu(\()p Fr(H)475 5327 y Fs(0)513 5315 y Fu(\))h Fp(!)f(D)r Fu(\()p Fr(H)849 5285 y Fs(ext)842 5336 y(0)944 5315 y Fu(\).)37 b(W)-7 b(e)28 b(obtain)f(the)h(follo)n (wing)f(iden)n(tit)n(y)g(on)h @beginspecial @setspecial @endspecial Fp(H)824 5460 y @beginspecial @setspecial @endspecial Fr(C)6 b Fu(\()p Fr(z)t Fu(\))83 b(:=)f(\()p Fr(z)22 b Fp(\000)1447 5439 y Fu(^)1425 5460 y Fr(H)1501 5426 y Fs(ext)1494 5481 y Fq(\024)1596 5460 y Fu(\))1628 5426 y Fo(\000)p Fs(1)1717 5460 y Fr(I)1760 5426 y Fo(\003)1798 5460 y Fu(\()p Fr(j)1869 5426 y Fq(R)1924 5460 y Fu(\))d Fp(\000)f Fr(I)2101 5426 y Fo(\003)2139 5460 y Fu(\()p Fr(j)2210 5426 y Fq(R)2265 5460 y Fu(\)\()p Fr(z)k Fp(\000)2495 5439 y Fu(^)2473 5460 y Fr(H)2542 5472 y Fq(\024)2585 5460 y Fu(\))2617 5426 y Fo(\000)p Fs(1)0 5506 y @beginspecial @setspecial @endspecial eop %%Page: 23 23 23 22 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(23)1090 489 y Fu(=)94 b(\()p Fr(z)22 b Fp(\000)1447 468 y Fu(^)1425 489 y Fr(H)1501 455 y Fs(ext)1494 509 y Fq(\024)1596 489 y Fu(\))1628 455 y Fo(\000)p Fs(1)1717 489 y Fu(\()1771 468 y(^)1749 489 y Fr(H)1825 455 y Fs(ext)1818 509 y Fq(\024)1920 489 y Fr(I)1963 455 y Fo(\003)2001 489 y Fu(\()p Fr(j)2072 455 y Fq(R)2127 489 y Fu(\))c Fp(\000)g Fr(I)2303 455 y Fo(\003)2341 489 y Fu(\()p Fr(j)2412 455 y Fq(R)2467 489 y Fu(\))2521 468 y(^)2499 489 y Fr(H)2568 501 y Fq(\024)2612 489 y Fu(\)\()p Fr(z)k Fp(\000)2842 468 y Fu(^)2820 489 y Fr(H)2889 501 y Fq(\024)2932 489 y Fu(\))2964 455 y Fo(\000)p Fs(1)3053 489 y Fr(:)0 630 y Fu(Using)27 b(\()p @beginspecial @setspecial @endspecial(4.18\),)g(w)n(e)h(obtain:)766 813 y @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()872 792 y(^)850 813 y Fr(H)926 779 y Fs(ext)919 834 y Fq(\024)1021 813 y Fu(\))14 b Fr(I)1110 779 y Fo(\003)1148 813 y Fu(\()p Fr(j)1219 779 y Fq(R)1274 813 y Fu(\))k Fp(\000)g Fr(I)1450 779 y Fo(\003)1489 813 y Fu(\()p Fr(j)1560 779 y Fq(R)1614 813 y Fu(\))c Fr(\037)p Fu(\()1766 792 y(^)1744 813 y Fr(H)1813 825 y Fq(\024)1857 813 y Fu(\))23 b(=)2044 757 y(i)p 2010 794 92 4 v 2010 870 a(2)p Fr(\031)2125 700 y Fl(Z)2171 889 y Fe(C)2231 813 y Fr(@)2279 825 y Fs(\026)-37 b Fq(z)2324 813 y Fu(\025)-52 b Fr(\037)p Fu(\()p Fr(z)t Fu(\))23 b Fr(C)6 b Fu(\()p Fr(z)t Fu(\))14 b Fr(dz)21 b Fp(^)e Fr(d)5 b Fu(\026)-47 b Fr(z)t(:)83 b @beginspecial @setspecial @endspecial 0 996 a Fu(Using)27 b(Corollary)f @beginspecial @setspecial @endspecial(4.8)g(and)i(Lemma)f(3.4)g(w)n(e)g(obtain)g(for)g @beginspecial @setspecial @endspecial Fr(b)c(<)g Fu(0)338 1149 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(C)6 b Fu(\()p Fr(b)p Fu(\))p Fp(jj)83 b(\024)g Fr(c)p Fu(\()p Fp(jj)p Fu(\()p Fr(N)1039 1161 y Fo(1)1128 1149 y Fu(+)18 b Fr(N)1278 1161 y Fs(0)1315 1149 y Fu(\))1347 1115 y Fo(\000)p Fs(1)1437 1149 y Fu(d)p Fr(I)1526 1115 y Fo(\003)1564 1149 y Fu(\()p Fr(j)1635 1115 y Fq(R)1690 1149 y Fr(;)c Fu([)p Fr(j)1789 1115 y Fq(R)1843 1149 y Fr(;)g(!)s Fu(]\))p Fp(jj)k Fu(+)g Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2458 1115 y Fo(\000)2520 1093 y Fj(s)p 2520 1102 29 3 v 2520 1135 a Fi(2)2562 1149 y Fr(!)2627 1089 y Fj(s)p Fg(\000)p Fi(1)p 2627 1102 102 3 v 2663 1135 a(2)2742 1149 y Fu(\(1)g Fp(\000)g Fr(j)2956 1115 y Fq(R)2951 1170 y Fs(0)3011 1149 y Fu(\))p Fr(v)3083 1161 y Fq(\024)3122 1169 y Fi(0)3159 1149 y Fp(jj)0 1195 y @beginspecial @setspecial @endspecial 711 1332 a @beginspecial @setspecial @endspecial 115 w Fu(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1211 1298 y Fo(\000)1273 1275 y Fj(s)p 1273 1284 29 3 v 1273 1318 a Fi(2)1315 1332 y Fr(!)1380 1272 y Fj(s)p Fg(\000)p Fi(1)p 1380 1285 102 3 v 1416 1318 a(2)1495 1332 y Fr(j)1534 1298 y Fq(R)1529 1353 y Fo(1)1600 1332 y Fr(v)1640 1344 y Fq(\024)1679 1352 y Fi(0)1715 1332 y Fp(jj)h Fu(+)f Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2184 1298 y Fo(\000)2246 1275 y Fj(s)p 2245 1284 29 3 v 2245 1318 a Fi(2)2288 1332 y Fr(!)2353 1272 y Fj(s)p Fg(\000)p Fi(1)p 2352 1285 102 3 v 2389 1318 a(4)2467 1332 y Fu(\(1)h Fp(\000)f Fr(j)2682 1298 y Fq(R)2677 1353 y Fs(0)2736 1332 y Fu(\))p Fr(r)2805 1344 y Fq(\024)2849 1332 y Fp(jj)c(jj)p Fr(!)3020 1272 y Fj(s)p Fg(\000)p Fi(1)p 3020 1285 V 3056 1318 a(4)3135 1332 y Fu(\(1)k(+)g Fr(j)3349 1298 y Fq(R)3344 1353 y Fs(0)3403 1332 y Fu(\))p Fr(r)3472 1344 y Fq(\024)3516 1332 y Fp(jj)0 1378 y @beginspecial @setspecial @endspecial 711 1515 a @beginspecial @setspecial @endspecial 115 w Fu(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1211 1481 y Fo(\000)1273 1458 y Fj(s)p 1273 1467 29 3 v 1273 1501 a Fi(2)1315 1515 y Fr(!)1380 1455 y Fj(s)p Fg(\000)p Fi(1)p 1380 1468 102 3 v 1416 1501 a(4)1495 1515 y Fr(j)1534 1481 y Fq(R)1529 1535 y Fo(1)1600 1515 y Fr(r)1637 1527 y Fq(\024)1680 1515 y Fp(jj)c(jj)p Fr(!)1851 1455 y Fj(s)p Fg(\000)p Fi(1)p 1851 1468 V 1887 1501 a(4)1966 1515 y Fr(j)2005 1481 y Fq(R)2000 1535 y Fs(0)2059 1515 y Fr(r)2096 1527 y Fq(\024)2140 1515 y Fp(jj)k Fu(+)g Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2608 1481 y Fo(\000)2670 1458 y Fj(s)p 2670 1467 29 3 v 2670 1501 a Fi(4)2712 1515 y Fr(!)2777 1455 y Fj(s)p Fg(\000)p Fi(1)p 2777 1468 102 3 v 2813 1501 a(4)2892 1515 y Fr(j)2931 1481 y Fq(R)2926 1535 y Fo(1)2997 1515 y Fr(r)3034 1527 y Fq(\024)3077 1515 y Fp(jj)3123 1481 y Fs(2)0 1515 y Fu(\(4.23\))0 1561 y @beginspecial @setspecial @endspecial 595 1698 a @beginspecial @setspecial @endspecial 231 w(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1211 1663 y Fo(\000)1273 1641 y Fj(s)p 1273 1650 29 3 v 1273 1683 a Fi(2)1315 1698 y Fr(!)1380 1638 y Fj(s)p Fg(\000)p Fi(1)p 1380 1651 102 3 v 1416 1683 a(2)1495 1698 y Fr(r)1532 1710 y Fq(\024)1576 1698 y Fp(jj)g(\002)g(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2044 1663 y Fo(\000)2106 1641 y Fj(s)p 2106 1650 29 3 v 2106 1683 a Fi(2)2148 1698 y Fr(!)2213 1638 y Fj(s)p Fg(\000)p Fi(1)p 2213 1651 102 3 v 2249 1683 a(2)2328 1698 y Fu(\(1)g Fp(\000)g Fr(j)2542 1663 y Fq(R)2537 1718 y Fs(0)2596 1698 y Fu(\))p Fr(r)2665 1710 y Fq(\024)2709 1698 y Fp(jj)0 1743 y @beginspecial @setspecial @endspecial 826 1880 a @beginspecial @setspecial @endspecial Fu(+\()p Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))1244 1846 y Fo(\000)1306 1824 y Fj(s)p 1305 1833 29 3 v 1305 1866 a Fi(2)1348 1880 y Fr(!)1413 1821 y Fj(s)p Fg(\000)p Fi(1)p 1412 1834 102 3 v 1449 1866 a(2)1527 1880 y Fu(\(1)h Fp(\000)f Fr(j)1742 1846 y Fq(R)1737 1901 y Fs(0)1796 1880 y Fu(\))p Fr(r)1865 1892 y Fq(\024)1909 1880 y Fp(jj)41 b Fu(+)19 b Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))2400 1846 y Fo(\000)2462 1824 y Fj(s)p 2462 1833 29 3 v 2462 1866 a Fi(2)2504 1880 y Fr(!)2569 1821 y Fj(s)p Fg(\000)p Fi(1)p 2569 1834 102 3 v 2605 1866 a(2)2684 1880 y Fr(j)2723 1846 y Fq(R)2718 1901 y Fo(1)2789 1880 y Fr(r)2826 1892 y Fq(\024)2869 1880 y Fp(jj)p Fu(\))24 b Fp(jj)p Fu(\()p Fr(K)g Fu(+)18 b(i\))3282 1846 y Fo(\000)3344 1824 y Fi(1)p 3344 1833 29 3 v 3344 1866 a(2)3386 1880 y Fr(D)r Fp(jj)0 1926 y @beginspecial @setspecial @endspecial 826 2063 a Fu(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1211 2029 y Fo(\000)1273 2007 y Fj(s)p 1273 2016 V 1273 2049 a Fi(2)1315 2063 y Fr(!)1380 2003 y Fj(s)p Fg(\000)p Fi(1)p 1380 2016 102 3 v 1416 2049 a(2)1495 2063 y Fr(r)1532 2075 y Fq(\024)1576 2063 y Fp(jj)g(\002)g(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2044 2029 y Fo(\000)2106 2007 y Fj(s)p 2106 2016 29 3 v 2106 2049 a Fi(2)2148 2063 y Fr(!)2213 2003 y Fj(s)p Fg(\000)p Fi(1)p 2213 2016 102 3 v 2249 2049 a(2)2328 2063 y Fr(j)2367 2029 y Fq(R)2362 2084 y Fo(1)2432 2063 y Fr(r)2469 2075 y Fq(\024)2513 2063 y Fp(jj)p Fu(\))p Fr(:)0 2204 y Fu(Applying)28 b(\()p @beginspecial @setspecial @endspecial(2.4\))f(with)h([)p Fr(j)809 2174 y Fq(R)804 2225 y Fs(0)864 2204 y Fr(;)14 b(!)s Fu(])979 2174 y Fs(2)1034 2204 y Fu(+)k([)p Fr(j)1179 2174 y Fq(R)1174 2225 y Fo(1)1245 2204 y Fr(;)c(!)s Fu(])1360 2174 y Fs(2)1420 2204 y Fp(2)23 b Fr(O)r Fu(\()p Fr(R)1659 2174 y Fo(\000)p Fs(2)1749 2204 y Fu(\),)28 b(w)n(e)f(obtain)808 2353 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(b)18 b Fp(\000)1045 2332 y Fu(^)1023 2353 y Fr(H)1099 2319 y Fs(ext)1092 2374 y Fq(\024)1193 2353 y Fu(\))1225 2319 y Fo(\000)p Fs(1)1315 2353 y Fu(\()p Fr(H)1423 2319 y Fs(ext)1416 2374 y(0)1517 2353 y Fr(I)1560 2319 y Fo(\003)1599 2353 y Fu(\()p Fr(j)1670 2319 y Fq(R)1724 2353 y Fu(\))h Fp(\000)f Fr(I)1901 2319 y Fo(\003)1939 2353 y Fu(\()p Fr(j)2010 2319 y Fq(R)2065 2353 y Fu(\))p Fr(H)2166 2365 y Fs(0)2203 2353 y Fu(\)\()p Fr(b)h Fp(\000)2427 2332 y Fu(^)2405 2353 y Fr(H)2474 2365 y Fq(\024)2517 2353 y Fu(\))2549 2319 y Fo(\000)p Fs(1)2639 2353 y Fp(jj)k(2)g Fr(O)r Fu(\()p Fr(R)2947 2319 y Fo(\000)p Fs(1)3037 2353 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 2494 a Fu(F)-7 b(or)18 b(the)h(other)f(terms)h(of)g @beginspecial @setspecial @endspecial Fr(C)6 b Fu(\()p Fr(b)p Fu(\),)21 b(w)n(e)d(use)h(\(4.23\),)h(the)f(fact)g(that)g(w)n(e)f(can)g(pic)n(k)h @beginspecial @setspecial @endspecial Fr(\026)k(>)f Fu(0)d(and)f Fr(s)23 b(<)g Fu(1)18 b(suc)n(h)h(that)g @beginspecial @setspecial @endspecial(\()p Fr(V)h Fu(+)q(1\))3560 2464 y Fo(\000)3621 2441 y Fj(s)p 3621 2450 29 3 v 3621 2484 a Fi(2)3663 2494 y Fp(h)p Fu(x)p Fp(i)3771 2464 y Fs(1+)p Fq(\026)0 2594 y Fu(is)27 b(b)r(ounded)h(and)g(Lemma)f @beginspecial @setspecial @endspecial(A.2.)37 b(W)-7 b(e)28 b(obtain:)1347 2734 y Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1667 2700 y Fo(\000)1729 2678 y Fj(s)p 1729 2687 V 1729 2720 a Fi(2)1771 2734 y Fr(!)1836 2674 y Fj(s)p Fg(\000)p Fi(1)p 1836 2687 102 3 v 1872 2720 a(2)1951 2734 y Fr(f)2001 2700 y Fq(R)1992 2755 y(\017)2055 2734 y Fr(r)2092 2746 y Fq(\024)2136 2734 y Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(R)2444 2700 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2626 2734 y Fu(\))p Fr(;)0 2780 y @beginspecial @setspecial @endspecial 1347 2917 a Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1667 2883 y Fo(\000)1729 2861 y Fj(s)p 1729 2870 29 3 v 1729 2903 a Fi(2)1771 2917 y Fr(!)1836 2857 y Fj(s)p Fg(\000)p Fi(1)p 1836 2870 102 3 v 1872 2903 a(4)1951 2917 y Fr(f)2001 2883 y Fq(R)1992 2938 y(\017)2055 2917 y Fr(r)2092 2929 y Fq(\024)2136 2917 y Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(R)2444 2883 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2626 2917 y Fu(\))p Fr(;)-2681 b Fu(\(4.24\))0 2963 y @beginspecial @setspecial @endspecial 1347 3100 a Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1667 3066 y Fo(\000)1729 3043 y Fj(s)p 1729 3052 29 3 v 1729 3086 a Fi(4)1771 3100 y Fr(!)1836 3040 y Fj(s)p Fg(\000)p Fi(1)p 1836 3053 102 3 v 1872 3086 a(4)1951 3100 y Fr(j)1990 3066 y Fq(R)1985 3121 y Fo(1)2056 3100 y Fr(r)2093 3112 y Fq(\024)2136 3100 y Fp(jj)2182 3066 y Fs(2)2243 3100 y Fp(2)23 b Fr(O)r Fu(\()p Fr(R)2482 3066 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2664 3100 y Fu(\))p Fr(;)0 3241 y Fu(Where)k @beginspecial @setspecial @endspecial Fr(f)315 3211 y Fq(R)306 3261 y(\017)397 3241 y Fu(denotes)g @beginspecial @setspecial @endspecial Fr(j)736 3211 y Fq(R)731 3261 y Fo(1)830 3241 y Fu(or)f(1)18 b Fp(\000)g Fr(j)1113 3211 y Fq(R)1108 3261 y Fs(0)1168 3241 y Fu(.)37 b(Then)27 b(w)n(e)h(ha)n(v)n(e)655 3390 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(b)18 b Fp(\000)892 3369 y Fu(^)870 3390 y Fr(H)946 3355 y Fs(ext)939 3410 y Fq(\024)1041 3390 y Fu(\))1073 3355 y Fo(\000)p Fs(1)1162 3390 y Fu(\()1204 3369 y(^)1194 3390 y Fr(I)1230 3402 y Fq(\024)1292 3390 y Fp(\012)g Fu(1)-15 b(l)14 b Fr(I)1482 3355 y Fo(\003)1520 3390 y Fu(\()p Fr(j)1591 3355 y Fq(R)1645 3390 y Fu(\))19 b Fp(\000)f Fr(I)1822 3355 y Fo(\003)1860 3390 y Fu(\()p Fr(j)1931 3355 y Fq(R)1986 3390 y Fu(\))2028 3369 y(^)2018 3390 y Fr(I)2054 3402 y Fq(\024)2098 3390 y Fu(\)\()p Fr(b)h Fp(\000)2321 3369 y Fu(^)2300 3390 y Fr(H)2369 3402 y Fq(\024)2412 3390 y Fu(\))2444 3355 y Fo(\000)p Fs(1)2533 3390 y Fp(jj)k(2)h Fr(O)r Fu(\()p Fr(R)2842 3355 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)3024 3390 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial -3162 w Fu(\(4.25\))0 3530 y(Hence)28 b(w)n(e)f(ha)n(v)n(e)f @beginspecial @setspecial @endspecial Fp(jj)p Fr(C)6 b Fu(\()p Fr(z)t Fu(\))p Fp(jj)24 b(2)f(j)p Fu(Im\()p Fr(z)t Fu(\))p Fp(j)1178 3500 y Fo(\000)p Fs(2)1267 3530 y Fr(O)r Fu(\()p Fr(R)1428 3500 y Fo(\000)p Fs(1)1518 3530 y Fu(\).)37 b(This)28 b(pro)n(v)n(es)e @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)o Fu(\).)0 3630 y(Let)28 b Fr(\037)201 3642 y Fs(1)261 3630 y Fp(2)23 b Fr(C)404 3600 y Fo(1)398 3651 y Fs(0)475 3630 y Fu(\()p Fk(R)p Fu(\))34 b(suc)n(h)27 b(that)h Fr(\037)1046 3642 y Fs(1)1084 3630 y Fr(\037)23 b Fu(=)f Fr(\037)p Fu(.)37 b(As)28 b(in)g(the)g(previous)e(lemma,)i(w)n(e)f(ha)n(v)n(e)f(using)i (\()p @beginspecial @setspecial @endspecial(4.18\):)827 4142 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 869 3779 a @beginspecial @setspecial @endspecial(\()p Fr(N)968 3791 y Fs(0)1023 3779 y Fu(+)19 b Fr(N)1174 3791 y Fo(1)1244 3779 y Fu(\))1276 3745 y Fq(n)1321 3779 y Fr(\037)p Fu(\()1427 3758 y(^)1405 3779 y Fr(H)1481 3745 y Fs(ext)1474 3800 y Fq(\024)1576 3779 y Fu(\))p Fr(I)1651 3745 y Fo(\003)1689 3779 y Fu(\()p Fr(j)1760 3745 y Fq(R)1815 3779 y Fu(\))f Fp(\000)g Fr(I)1991 3745 y Fo(\003)2030 3779 y Fu(\()p Fr(j)2101 3745 y Fq(R)2155 3779 y Fu(\))p Fr(\037)p Fu(\()2294 3758 y(^)2271 3779 y Fr(H)2340 3791 y Fq(\024)2384 3779 y Fu(\))p Fr(N)2492 3745 y Fq(m)2555 3779 y @beginspecial @setspecial @endspecial 869 3922 a Fu(=)k(\()p Fr(N)1055 3934 y Fs(0)1111 3922 y Fu(+)c Fr(N)1261 3934 y Fo(1)1331 3922 y Fu(\))1363 3888 y Fq(n)1409 3922 y Fr(\037)1461 3934 y Fs(1)1498 3922 y Fu(\()1552 3901 y(^)1530 3922 y Fr(H)1606 3888 y Fs(ext)1599 3943 y Fq(\024)1700 3922 y Fu(\)\()p Fr(\037)p Fu(\()1871 3901 y(^)1848 3922 y Fr(H)1924 3888 y Fs(ext)1917 3943 y Fq(\024)2019 3922 y Fu(\))p Fr(I)2094 3888 y Fo(\003)2133 3922 y Fu(\()p Fr(j)2204 3888 y Fq(R)2259 3922 y Fu(\))g Fp(\000)g Fr(I)2435 3888 y Fo(\003)2474 3922 y Fu(\()p Fr(j)2545 3888 y Fq(R)2599 3922 y Fu(\))p Fr(\037)p Fu(\()2737 3901 y(^)2715 3922 y Fr(H)2784 3934 y Fq(\024)2828 3922 y Fu(\)\))p Fr(N)2968 3888 y Fq(m)3031 3922 y @beginspecial @setspecial @endspecial 869 4066 a Fu(+\()p Fr(N)1033 4078 y Fs(0)1088 4066 y Fu(+)g Fr(N)1238 4078 y Fo(1)1308 4066 y Fu(\))1340 4032 y Fq(n)1386 4066 y Fu(\()p Fr(\037)1470 4078 y Fs(1)1507 4066 y Fu(\()1561 4045 y(^)1539 4066 y Fr(H)1615 4032 y Fs(ext)1608 4086 y Fq(\024)1710 4066 y Fu(\))p Fr(I)1785 4032 y Fo(\003)1823 4066 y Fu(\()p Fr(j)1894 4032 y Fq(R)1949 4066 y Fu(\))h Fp(\000)f Fr(I)2126 4032 y Fo(\003)2164 4066 y Fu(\()p Fr(j)2235 4032 y Fq(R)2289 4066 y Fu(\))p Fr(\037)2373 4078 y Fs(1)2411 4066 y Fu(\()2465 4045 y(^)2443 4066 y Fr(H)2512 4078 y Fq(\024)2555 4066 y Fu(\)\))p Fr(\037)p Fu(\()2726 4045 y(^)2703 4066 y Fr(H)2772 4078 y Fq(\024)2816 4066 y Fu(\))p Fr(N)2924 4032 y Fq(m)2987 4066 y @beginspecial @setspecial @endspecial 869 4244 a Fu(=)976 4187 y(i)p 966 4224 42 4 v 966 4301 a(2)1032 4131 y Fl(Z)1078 4319 y Fe(C)1138 4244 y Fr(@)1186 4256 y Fs(\026)-37 b Fq(z)1230 4244 y Fu(\025)-52 b Fr(\037)p Fu(\()p Fr(z)t Fu(\)\()p Fr(N)1478 4256 y Fs(0)1534 4244 y Fu(+)18 b Fr(N)1684 4256 y Fo(1)1754 4244 y Fu(\))1786 4209 y Fq(n)1831 4244 y Fr(\037)1883 4256 y Fs(1)1920 4244 y Fu(\()1974 4223 y(^)1952 4244 y Fr(H)2028 4209 y Fs(ext)2021 4264 y Fq(\024)2123 4244 y Fu(\))p Fr(C)6 b Fu(\()p Fr(z)t Fu(\))p Fr(N)2403 4209 y Fq(m)2466 4244 y Fr(dz)22 b Fp(^)d Fr(d)5 b Fu(\026)-47 b Fr(z)s @beginspecial @setspecial @endspecial 869 4467 a Fu(+)953 4411 y(i)p 944 4448 V 944 4524 a(2)1009 4354 y Fl(Z)1055 4543 y Fe(C)1115 4467 y Fr(@)1163 4479 y Fs(\026)-37 b Fq(z)1207 4467 y Fu(\025)-52 b Fr(\037)1249 4479 y Fs(1)1286 4467 y Fu(\()p Fr(z)t Fu(\)\()p Fr(N)1492 4479 y Fs(0)1548 4467 y Fu(+)18 b Fr(N)1698 4479 y Fo(1)1768 4467 y Fu(\))1800 4433 y Fq(n)1845 4467 y Fr(C)6 b Fu(\()p Fr(z)t Fu(\))p Fr(\037)p Fu(\()2123 4446 y(^)2101 4467 y Fr(H)2170 4479 y Fq(\024)2214 4467 y Fu(\))p Fr(N)2322 4433 y Fq(m)2385 4467 y Fr(dz)22 b Fp(^)d Fr(d)5 b Fu(\026)-47 b Fr(z)s(:)3073 4142 y @beginspecial @setspecial @endspecial 0 4671 a Fu(Mo)n(ving)24 b @beginspecial @setspecial @endspecial(\()p Fr(N)394 4683 y Fs(0)446 4671 y Fu(+)13 b Fr(N)591 4683 y Fo(1)661 4671 y Fu(\))693 4641 y Fq(n)764 4671 y Fu(\(resp)p Fr(:)h(N)1058 4641 y Fq(m)1121 4671 y Fu(\))25 b(to)n(w)n(ard)f @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1559 4650 y(^)1537 4671 y Fr(H)1606 4683 y Fq(\024)1650 4671 y Fu(\))p Fr(;)k Fu(\(resp)p Fr(:)13 b(\037)2002 4683 y Fs(1)2039 4671 y Fu(\()2093 4650 y(^)2071 4671 y Fr(H)2147 4641 y Fs(ext)2140 4691 y Fq(\024)2242 4671 y Fu(\)\))26 b(in)g(the)f(last)g(expression)f(and)i(then)g(using)f @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)o Fu(\))26 b(and)0 4770 y(Lemma)h(4.13)g(w)n(e)g(pro)n(v)n(e)f @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\).)2909 b Fh(\004)921 4958 y @beginspecial @setspecial @endspecial Fu(5.)45 b Fv(Spectral)31 b(theor)-6 b(y)32 b(f)n(or)g(the)g(Nelson)e(Hamil)-6 b(tonian)100 5107 y Fu(W)f(e)30 b(study)g(in)g(this)g(section)g(the)g(sp)r(ectral)f (prop)r(erties)g(of)h(b)r(oth)h(Nelson)e(and)h(mo)r(di\014ed)g (Hamiltonians.)44 b(In)30 b(Subsect.)0 5207 y @beginspecial @setspecial @endspecial(5.1)35 b(w)n(e)g(pro)n(v)n(e)f(the)i(existence) f(of)h(ground)e(state)h(for)g(the)h(Nelson)g(Hamiltonian.)60 b(W)-7 b(e)36 b(use)g(essen)n(tially)e(the)i(fact)g(that)0 5307 y @beginspecial @setspecial @endspecial Fr(H)69 5319 y Fq(\024)131 5307 y Fp(\000)18 b Fr(E)275 5319 y Fq(\024)346 5307 y Fu(are)26 b(P)n(auli-Fierz)g(Hamiltonian)i(whic)n (h)f(con)n(v)n(erge)f(in)i(the)g(norm)f(resolv)n(en)n(t)f(sense)h(to)h @beginspecial @setspecial @endspecial Fr(H)7 b Fu(.)37 b(This)27 b(is)h(the)g(sub)5 b(ject)0 5406 y(of)31 b(Thm.)49 b @beginspecial @setspecial @endspecial(5.1.)e(In)31 b(Subsect.)48 b(5.2)31 b(w)n(e)g(pro)n(v)n(e)e(a)i(Mourre)f(estimate)h (for)g(the)h(mo)r(di\014ed)f(Hamiltonian,)h(whic)n(h)g(giv)n(es)e(that) 0 5506 y(pure)d(p)r(oin)n(t)h(sp)r(ectrum)g(is)f(lo)r(cally)g(\014nite) h(outside)g(its)g(thresholds.)p eop %%Page: 24 24 24 23 bop 0 226 a Fs(24)1695 b(Z.)29 b(AMMARI)0 406 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 83 x Fu(5.1.)47 b Fz(HVZ)32 b(theorem.)-787 b @beginspecial @setspecial @endspecial 153 x(Theorem)30 b(5.1.)40 b Fn(One)30 b(has)1391 782 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\033)1438 794 y Fs(ess)1525 782 y Fu(\()p Fr(H)7 b Fu(\))23 b(=)g([inf)d Fr(\033)s Fu(\()p Fr(H)7 b Fu(\))20 b(+)e Fr(m;)c Fu(+)p Fp(1)p Fu([)p Fr(;)p @beginspecial @setspecial @endspecial 0 922 a Fn(and)30 b @beginspecial @setspecial @endspecial Fu(inf)21 b Fr(\033)s Fu(\()p Fr(H)7 b Fu(\))31 b Fn(is)f(a)g(discr)l(ete)g(eigenvalue)h(of)g Fr(H)7 b Fn(.)0 943 y @beginspecial @setspecial @endspecial 100 1075 a Fz(Pro)s(of.)49 b @beginspecial @setspecial @endspecial 27 w Fr(H)504 1087 y Fq(\024)577 1075 y Fu(is)30 b(an)f(example)h(of)g(a)f(P)n(auli-Fierz)f(Hamiltonian,)j(see)e @beginspecial @setspecial @endspecial([11)o(,)i(Sect.)44 b(3].)f(The)30 b(HVZ)g(theorem)f(pro)n(v)n(ed)g(in)0 1174 y([11)o(])f(for)f(the)h(P)n(auli-Fierz)e(Hamiltonians)h(giv)n(es)f (for)h @beginspecial @setspecial @endspecial Fr(\024)c(<)g Fp(1)1041 1322 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\033)1088 1334 y Fs(ess)1175 1322 y Fu(\()1229 1301 y(^)1207 1322 y Fr(H)1276 1334 y Fq(\024)1319 1322 y Fu(\))h(=)e Fr(\033)1509 1334 y Fs(ess)1596 1322 y Fu(\()p Fr(H)1697 1334 y Fq(\024)1759 1322 y Fp(\000)c Fr(E)1903 1334 y Fq(\024)1946 1322 y Fu(\))23 b(=)g([inf)e Fr(\033)s Fu(\()2331 1301 y(^)2309 1322 y Fr(H)2378 1334 y Fq(\024)2422 1322 y Fu(\))d(+)g Fr(m;)c Fu(+)p Fp(1)p Fu([)p Fr(:)p @beginspecial @setspecial @endspecial 0 1471 a Fu(Using)27 b(the)h(fact)g(that)g @beginspecial @setspecial @endspecial(\()p Fr(z)22 b Fp(\000)918 1450 y Fu(^)896 1471 y Fr(H)965 1483 y Fq(\024)1008 1471 y Fu(\))1040 1440 y Fo(\000)p Fs(1)1153 1471 y Fp(!)h Fu(\()p Fr(z)f Fp(\000)1456 1450 y Fu(^)1435 1471 y Fr(H)1504 1483 y Fo(1)1574 1471 y Fu(\))1606 1440 y Fo(\000)p Fs(1)1723 1471 y Fu(and)27 b([23,)55 b(Thm.)37 b(VI)r(I)r(I.23],)28 b(w)n(e)f(see)g(that)1315 1619 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 28 w(lim)1315 1669 y Fq(\024)p Fo(!1)1509 1619 y Fu(\(inf)22 b Fr(\033)s Fu(\()1761 1598 y(^)1739 1619 y Fr(H)1808 1631 y Fq(\024)1851 1619 y Fu(\)\))i(=)f(inf)d Fr(\033)s Fu(\()2246 1598 y(^)2223 1619 y Fr(H)2292 1631 y Fo(1)2363 1619 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial -2502 w Fu(\(5.1\))0 1804 y(Let)30 b @beginspecial @setspecial @endspecial Fr(\037)e Fp(2)g Fr(C)379 1774 y Fo(1)373 1825 y Fs(0)449 1804 y Fu(\(])21 b Fp(\000)f(1)p Fr(;)14 b Fu(inf)20 b Fr(\033)s Fu(\()949 1783 y(^)926 1804 y Fr(H)995 1816 y Fo(1)1067 1804 y Fu(\))g(+)g Fr(m)p Fu([\),)31 b(then)g @beginspecial @setspecial @endspecial Fr(\037)c Fp(2)h Fr(C)1805 1774 y Fo(1)1799 1825 y Fs(0)1876 1804 y Fu(\(])20 b Fp(\000)g(1)p Fr(;)14 b Fu(inf)21 b Fr(\033)s Fu(\()2375 1783 y(^)2353 1804 y Fr(H)2422 1816 y Fq(\024)2461 1824 y Fj(n)2506 1804 y Fu(\))g(+)f Fr(m)p Fu([\))30 b(for)g(a)g(sequence)g @beginspecial @setspecial @endspecial Fr(\024)3399 1816 y Fq(n)3471 1804 y Fp(!)e Fu(+)p Fp(1)p Fu(.)44 b(By)0 1912 y(the)28 b(HVZ)g(theorem)f(for)g @beginspecial @setspecial @endspecial 818 1891 a(^)796 1912 y Fr(H)865 1924 y Fq(\024)904 1932 y Fj(n)949 1912 y Fu(,)h Fr(\037)p Fu(\()1106 1891 y(^)1084 1912 y Fr(H)1153 1924 y Fq(\024)1192 1932 y Fj(n)1237 1912 y Fu(\))g(is)f(compact)g(and) h(b)n(y)f(Thm.)37 b @beginspecial @setspecial @endspecial(3.6)27 b Fr(\037)p Fu(\()2464 1891 y(^)2442 1912 y Fr(H)2511 1924 y Fo(1)2581 1912 y Fu(\))h(is)g(compact.)36 b(So)28 b(w)n(e)f(obtain)1327 2060 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\033)1374 2072 y Fs(ess)1461 2060 y Fu(\()1515 2039 y(^)1493 2060 y Fr(H)1562 2072 y Fo(1)1633 2060 y Fu(\))c Fp(\032)g Fu([inf)d Fr(\033)s Fu(\()2018 2039 y(^)1995 2060 y Fr(H)2064 2072 y Fo(1)2136 2060 y Fu(\))e(+)g Fr(m;)c Fu(+)p Fp(1)p Fu([)p Fr(:)p @beginspecial @setspecial @endspecial 0 2208 a Fu(Let)32 b(us)g(no)n(w)g(sho)n(w)f(that)h @beginspecial @setspecial @endspecial([inf)21 b Fr(\033)s Fu(\()1079 2187 y(^)1056 2208 y Fr(H)1125 2220 y Fo(1)1196 2208 y Fu(\))h(+)f Fr(m;)14 b Fu(+)p Fp(1)p Fu([)p Fp(\032)30 b Fr(\033)1759 2220 y Fs(ess)1846 2208 y Fu(\()1900 2187 y(^)1878 2208 y Fr(H)1947 2220 y Fo(1)2017 2208 y Fu(\).)51 b(Let)32 b @beginspecial @setspecial @endspecial Fr(\025)f(>)f Fu(inf)21 b Fr(\033)s Fu(\()2670 2187 y(^)2647 2208 y Fr(H)2716 2220 y Fo(1)2787 2208 y Fu(\))h(+)f Fr(m)p Fu(.)51 b(By)31 b(\()p @beginspecial @setspecial @endspecial(5.1\))h (and)g(the)h(HVZ)0 2316 y(theorem)24 b(for)466 2295 y(^)444 2316 y Fr(H)513 2328 y Fq(\024)556 2316 y Fu(,)i(there)e(exists)g(a)h (sequence)f @beginspecial @setspecial @endspecial Fr(\024)1496 2328 y Fq(n)1564 2316 y Fp(!)f Fu(+)p Fp(1)h Fu(suc)n(h)h(that)g Fr(\025)e Fp(2)h Fr(\033)2401 2328 y Fs(ess)2487 2316 y Fu(\()2541 2295 y(^)2519 2316 y Fr(H)2588 2328 y Fq(\024)2627 2336 y Fj(n)2672 2316 y Fu(\),)i(or)e(equiv)-5 b(alen)n(tly)24 b @beginspecial @setspecial @endspecial Fr(\026)f Fu(:=)g(\()p Fr(\025)13 b Fu(+)g Fr(c)p Fu(\))3733 2286 y Fo(\000)p Fs(1)3845 2316 y Fp(2)0 2424 y Fr(\033)47 2436 y Fs(ess)134 2356 y Fl(\000)172 2424 y Fu(\()226 2403 y(^)204 2424 y Fr(H)273 2436 y Fq(\024)312 2444 y Fj(n)375 2424 y Fu(+)18 b Fr(c)p Fu(\))526 2394 y Fo(\000)p Fs(1)616 2356 y Fl(\001)654 2424 y Fr(;)27 b(c)c(>>)g Fu(1.)0 2540 y(Let)28 b(us)f(sho)n(w)g(that)h @beginspecial @setspecial @endspecial Fr(\026)23 b Fp(2)g Fr(\033)839 2552 y Fs(ess)926 2472 y Fl(\000)964 2540 y Fu(\()1018 2519 y(^)996 2540 y Fr(H)1065 2552 y Fo(1)1154 2540 y Fu(+)18 b Fr(c)p Fu(\))1305 2510 y Fo(\000)p Fs(1)1395 2472 y Fl(\001)1433 2540 y Fu(,)27 b(i.e.:)37 b Fr(\025)24 b Fp(2)f Fr(\033)1846 2552 y Fs(ess)1933 2540 y Fu(\()1987 2519 y(^)1965 2540 y Fr(H)2034 2552 y Fo(1)2105 2540 y Fu(\):)0 2656 y(Assume)30 b(the)g(con)n(trary)e(and)h(let)h @beginspecial @setspecial @endspecial Fr(\037)d Fp(2)g Fr(C)1301 2621 y Fo(1)1295 2676 y Fs(0)1371 2656 y Fu(\()p Fk(R)p Fu(\))q(,)36 b(suc)n(h)29 b(that)h @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()p Fr(\026)p Fu(\))e(=)e(1)p Fr(;)14 b(\037)p Fu(\(\()2421 2635 y(^)2400 2656 y Fr(H)2469 2668 y Fo(1)2559 2656 y Fu(+)19 b Fr(c)p Fu(\))2711 2626 y Fo(\000)p Fs(1)2801 2656 y Fu(\))30 b(compact.)42 b(Let)30 b @beginspecial @setspecial @endspecial Fr(')3438 2668 y Fq(\024)3477 2676 y Fj(n)3518 2668 y Fq(;j)3603 2656 y Fu(b)r(e)g(W)-7 b(eyl)0 2766 y(sequences)27 b(for)g @beginspecial @setspecial @endspecial(\()558 2745 y(^)536 2766 y Fr(H)605 2778 y Fq(\024)644 2786 y Fj(n)707 2766 y Fu(+)18 b Fr(c)p Fu(\))858 2736 y Fo(\000)p Fs(1)975 2766 y Fu(at)28 b Fr(\026)g Fu(suc)n(h)f(that)682 2915 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(')782 2927 y Fq(\024)821 2935 y Fj(n)862 2927 y Fq(;j)917 2915 y Fp(jj)c Fu(=)f(1)p Fr(;)38 b Fu(lim)1152 2967 y Fq(j)s Fo(!1)1315 2915 y Fu(\(\()1401 2894 y(^)1379 2915 y Fr(H)1448 2927 y Fq(\024)1487 2935 y Fj(n)1551 2915 y Fu(+)18 b Fr(c)p Fu(\))1702 2880 y Fo(\000)p Fs(1)1810 2915 y Fp(\000)g Fr(\026)p Fu(\))p Fr(')2029 2927 y Fq(\024)2068 2935 y Fj(n)2109 2927 y Fq(;j)2187 2915 y Fu(=)k(0)p Fr(;)27 b Fu(and)14 b Fr(w)21 b Fp(\000)42 b Fu(lim)2677 2967 y Fq(j)s Fo(!1)2854 2915 y Fr(')2908 2927 y Fq(\024)2947 2935 y Fj(n)2988 2927 y Fq(;j)3066 2915 y Fu(=)23 b(0)p Fr(:)p @beginspecial @setspecial @endspecial 0 3093 a Fu(One)k(has)584 3288 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 625 3220 a @beginspecial @setspecial @endspecial Fp(jj)p Fr(\037)p Fu(\(\()810 3199 y(^)787 3220 y Fr(H)856 3232 y Fo(1)946 3220 y Fu(+)18 b Fr(c)p Fu(\))1097 3186 y Fo(\000)p Fs(1)1186 3220 y Fu(\))p Fr(')1272 3232 y Fq(\024)1311 3240 y Fj(n)1352 3232 y Fq(;j)1425 3220 y Fp(\000)g Fr(')1562 3232 y Fq(\024)1601 3240 y Fj(n)1642 3232 y Fq(;j)1697 3220 y Fp(jj)83 b @beginspecial @setspecial @endspecial(\024)g(jj)p Fr(\037)p Fu(\(\()2158 3199 y(^)2136 3220 y Fr(H)2205 3232 y Fq(\024)2244 3240 y Fj(n)2308 3220 y Fu(+)18 b Fr(c)p Fu(\))2459 3190 y Fo(\000)p Fs(1)2548 3220 y Fu(\))h Fp(\000)f Fr(\037)p Fu(\(\()2820 3199 y(^)2798 3220 y Fr(H)2867 3232 y Fo(1)2956 3220 y Fu(+)g Fr(c)p Fu(\))3107 3190 y Fo(\000)p Fs(1)3196 3220 y Fu(\))p Fp(jj)p @beginspecial @setspecial @endspecial 1974 3364 a Fu(+)p Fp(jj)p Fr(\037)p Fu(\(\()2223 3343 y(^)2201 3364 y Fr(H)2270 3376 y Fq(\024)2309 3384 y Fj(n)2372 3364 y Fu(+)g Fr(c)p Fu(\))2523 3330 y Fo(\000)p Fs(1)2612 3364 y Fu(\))p Fr(')2698 3376 y Fq(\024)2737 3384 y Fj(n)2779 3376 y Fq(;j)2852 3364 y Fp(\000)g Fr(')2989 3376 y Fq(\024)3028 3384 y Fj(n)3069 3376 y Fq(;j)3123 3364 y Fp(jj)p Fr(:)3316 3288 y @beginspecial @setspecial @endspecial 0 3521 a Fu(Since)28 b @beginspecial @setspecial @endspecial Fr(\037)p Fu(\(\()355 3500 y(^)333 3521 y Fr(H)402 3533 y Fo(1)491 3521 y Fu(+)18 b Fr(c)p Fu(\))642 3491 y Fo(\000)p Fs(1)731 3521 y Fu(\))28 b(compact,)g(there)f(exists)g (for)g Fr(\017)c(>)g Fu(0,)k(a)g @beginspecial @setspecial @endspecial Fr(\024)2069 3533 y Fs(1)2134 3521 y Fu(and)g Fr(j)2329 3533 y Fs(1)2394 3521 y Fu(suc)n(h)h(that)1176 3669 y @beginspecial @setspecial @endspecial 166 w Fp(jj)p Fr(\037)p Fu(\(\()1526 3648 y(^)1504 3669 y Fr(H)1573 3681 y Fo(1)1662 3669 y Fu(+)18 b Fr(c)p Fu(\))1813 3635 y Fo(\000)p Fs(1)1903 3669 y Fu(\))p Fr(')1989 3681 y Fq(\024)2028 3689 y Fi(1)2061 3681 y Fq(;j)2108 3689 y Fi(1)2163 3669 y Fp(\000)g Fr(')2300 3681 y Fq(\024)2339 3689 y Fi(1)2371 3681 y Fq(;j)2418 3689 y Fi(1)2455 3669 y Fp(jj)23 b(\024)g Fu(2)p Fr(\017;)0 3718 y @beginspecial @setspecial @endspecial 1342 3845 a Fp(jj)p Fr(\037)p Fu(\(\()1526 3824 y(^)1504 3845 y Fr(H)1573 3857 y Fo(1)1662 3845 y Fu(+)18 b Fr(c)p Fu(\))1813 3811 y Fo(\000)p Fs(1)1903 3845 y Fu(\))p Fr(')1989 3857 y Fq(\024)2028 3865 y Fi(1)2061 3857 y Fq(;j)2108 3865 y Fi(1)2144 3845 y Fp(jj)24 b(\024)36 b Fr(\017:)0 3988 y Fu(W)-7 b(e)28 b(obtain)f @beginspecial @setspecial @endspecial Fp(jj)p Fr(')501 4000 y Fq(\024)540 4008 y Fi(1)573 4000 y Fq(;j)620 4008 y Fi(1)657 3988 y Fp(jj)c(\024)g Fu(3)p Fr(\017)p Fu(,)j(this)i(giv)n(es)f(a)g(con)n (tradiction,)f(if)i(w)n(e)g(c)n(ho)r(ose)e @beginspecial @setspecial @endspecial Fr(\017)d(<)2518 3955 y Fs(1)p 2518 3969 34 4 v 2518 4017 a(3)2561 3988 y Fu(.)p @beginspecial @setspecial @endspecial 1251 w Fh(\004)0 4078 y @beginspecial @setspecial @endspecial 71 x Fu(5.2.)47 b Fz(Mourre)32 b(estimate.)38 b Fu(W)-7 b(e)28 b(denote)g(b)n(y)f Fr(b)h Fu(the)g(op)r(erator)d(acting)i(on)h @beginspecial @setspecial @endspecial Fm(h)g Fu(de\014ned)g(b)n(y)1139 4323 y Fr(b)23 b Fu(:=)1318 4267 y(1)p 1318 4304 42 4 v 1318 4380 a(2)1370 4323 y(\()p Fp(r)p Fr(!)17 b(:)d(D)1646 4335 y Fq(k)1705 4323 y Fu(+)k Fr(D)1857 4335 y Fq(k)1911 4323 y Fr(:)c Fp(r)p Fr(!)s Fu(\))p Fr(;)37 b Fu(on)23 b Fr(C)2340 4289 y Fo(1)2334 4344 y Fs(0)2410 4323 y Fu(\()p Fk(R)2497 4289 y Fs(3)2540 4323 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 4491 a Fu([1,)30 b(Prop.)c(4.2.3])j(yields)g (that)i(the)f(closure)f(of)h Fr(b)f Fu(is)h(the)h(in\014nitesimal)f (generator)e(of)i(the)g(strongly)f(con)n(tin)n(uous)g(unitary)0 4591 y(group)d @beginspecial @setspecial @endspecial Fr(U)292 4603 y Fq(t)349 4591 y Fu(asso)r(ciated)g(to)i(the)g(v)n (ector)e(\014eld)i Fp(r)p Fr(!)i Fu(in)e(the)g(follo)n(wing)f(sense) 1031 4739 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(U)1088 4751 y Fq(t)1117 4739 y Fr(F)35 b Fu(:=)22 b([det)q Fp(r)p Fr(\036)1572 4751 y Fo(\000)p Fq(t)1654 4739 y Fu(\()p Fr(x)p Fu(\)])1798 4683 y Fi(1)p 1799 4692 29 3 v 1799 4725 a(2)1864 4739 y Fr(F)12 b Fu(\()p Fr(\036)2010 4751 y Fo(\000)p Fq(t)2092 4739 y Fu(\()p Fr(x)p Fu(\)\))p Fr(;)38 b(F)d Fp(2)23 b Fr(S)2518 4705 y Fo(0)2541 4739 y Fu(\()p Fk(R)2628 4705 y Fs(3)2671 4739 y Fu(\))83 b @beginspecial @setspecial @endspecial -2786 w(\(5.2\))0 4879 y(where)28 b @beginspecial @setspecial @endspecial Fr(\036)290 4891 y Fq(t)348 4879 y Fu(is)g(the)g(\015o)n(w)g(of)g(the)g(v)n(ector)f(\014eld)h Fp(r)p Fr(!)s Fu(.)39 b(Moreo)n(v)n(er)25 b @beginspecial @setspecial @endspecial Fr(C)2033 4849 y Fo(1)2027 4900 y Fs(0)2104 4879 y Fu(\()p Fk(R)2190 4849 y Fs(3)2233 4879 y Fu(\))k(is)f(a)f(core)h(for)f Fr(b)p Fu(.)38 b(W)-7 b(e)29 b(denote)f(in)g(the)h(sequel)f(b)n(y)f @beginspecial @setspecial @endspecial Fr(b)0 4979 y Fu(its)h(closure.)35 b(W)-7 b(e)28 b(set)g Fr(B)f Fu(:=)c(d\000\()p Fr(b)p Fu(\).)37 b(Clearly)27 b @beginspecial @setspecial @endspecial Fr(B)k Fu(is)d(essen)n(tially)f(selfadjoin)n(t)g(on)g @beginspecial @setspecial @endspecial(\000)2588 4991 y Fs(\014n)2667 4979 y Fu(\()p Fr(C)2764 4949 y Fo(1)2758 5000 y Fs(0)2834 4979 y Fu(\()p Fk(R)2921 4949 y Fs(3)2964 4979 y Fu(\)\).)0 5079 y(W)-7 b(e)28 b(denote)f(b)n(y)h @beginspecial @setspecial @endspecial Fr(\034)37 b Fu(the)28 b(set)g(of)f Fr(thr)r(eshol)r(ds)p Fu(,)1548 5222 y Fr(\034)33 b Fu(:=)23 b Fr(\033)1775 5234 y Fs(pp)1852 5222 y Fu(\()1906 5201 y(^)1884 5222 y Fr(H)1953 5234 y Fo(1)2024 5222 y Fu(\))c(+)f Fr(m)p Fk(N)2285 5188 y Fo(\003)2329 5222 y Fr(:)p @beginspecial @setspecial @endspecial 0 5362 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(S)k Fu(b)r(e)c(a)f(selfadjoin)n(t)h(op)r(erator)e(on)h Fp(H)q Fu(,)h(w)n(e)f(sa)n(y)f(that)i Fr(S)k Fu(is)c(of)g(class)e @beginspecial @setspecial @endspecial Fr(C)2352 5332 y Fs(1)2390 5362 y Fu(\()p Fr(B)t Fu(\),)i(see)f([1],)h(if)g(the)g(map) 1512 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(t)23 b Fp(7!)g Fu(e)1708 5472 y Fs(i)p Fq(tB)1809 5506 y Fu(\()p Fr(S)g Fp(\000)18 b Fr(z)t Fu(\))2073 5472 y Fo(\000)p Fs(1)2176 5506 y Fu(e)2213 5472 y Fo(\000)p Fs(i)p Fq(tB)2365 5506 y Fr(;)p @beginspecial @setspecial @endspecial eop %%Page: 25 25 25 24 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(25)0 489 y Fu(is)30 b(strongly)g @beginspecial @setspecial @endspecial Fr(C)476 459 y Fs(1)544 489 y Fu(for)g(some)g Fr(z)h Fp(2)e Fk(C)41 b Fp(n)20 b Fr(\033)s Fu(\()p Fr(S)5 b Fu(\).)47 b(By)30 b @beginspecial @setspecial @endspecial([1)o(,)i(Lemma)e(6.2.9])g Fr(S)i Fp(2)d Fr(C)2429 459 y Fs(1)2466 489 y Fu(\()p Fr(B)t Fu(\))i(if)h(and)e(only)g(if)h(the)g(sesquilinear)e(form)0 589 y @beginspecial @setspecial @endspecial([)p Fr(B)t(;)14 b Fu(\()p Fr(z)22 b Fp(\000)c Fr(S)5 b Fu(\))391 558 y Fo(\000)p Fs(1)480 589 y Fu(])28 b(on)f Fp(D)r Fu(\()p Fr(B)t Fu(\))i(is)e(con)n(tin)n(uous)g(for)g(the)h(top)r(ology)e(of)i @beginspecial @setspecial @endspecial Fp(H)h Fu(i.e:)716 727 y Fp(j)p Fu(\(\()p Fr(S)23 b Fp(\000)18 b Fr(z)t Fu(\))1035 693 y Fo(\000)p Fs(1)1124 727 y Fr(';)c(B)t(')p Fu(\))20 b Fp(\000)e Fu(\()p Fr(B)t(';)c Fu(\()p Fr(S)24 b Fp(\000)18 b Fr(z)t Fu(\))1926 693 y Fo(\000)p Fs(1)2014 727 y Fr(')p Fu(\))p Fp(j)24 b(\024)f Fr(c)14 b Fp(jj)p Fr(')p Fp(jj)2431 693 y Fs(2)2528 727 y Fu(for)f Fr(')24 b Fp(2)f(D)r Fu(\()p Fr(B)t Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial -3101 w Fu(\(5.3\))100 865 y(W)-7 b(e)28 b(recall)e(here)h(a)h(w)n(ell)f(kno)n(wn)g(theorem,)g(see)g @beginspecial @setspecial @endspecial([1,)g(Thm.)38 b(6.2.10].)0 886 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 130 x Fz(Theorem)30 b(5.2.)40 b Fn(L)l(et)30 b Fr(S;)14 b(B)34 b Fn(two)29 b(selfadjoint)j(op)l(er)l (ators)f(acting)f(on)g(Hilb)l(ert)g(sp)l(ac)l(e.)39 b(If)31 b @beginspecial @setspecial @endspecial Fr(S)c Fp(2)d Fr(C)3040 986 y Fs(1)3077 1016 y Fu(\()p Fr(B)t Fu(\))31 b Fn(then)1082 1290 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1124 1153 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))14 b Fp(D)r Fu(\()p Fr(S)5 b Fu(\))19 b Fp(\\)f(D)r Fu(\()p Fr(B)t Fu(\))c Fr(is)g(dense)g(in)22 b Fp(D)r Fu(\()p Fr(S)5 b Fu(\))p @beginspecial @setspecial @endspecial 1128 1289 a(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))14 b(\([)p Fr(B)t(;)g(S)5 b Fu(])p Fr(u;)14 b(u)p Fu(\))23 b Fp(\024)f Fr(c)h Fp(jj)p Fr(S)5 b(u)p Fp(jj)1991 1255 y Fs(2)2028 1289 y Fr(;)14 b(u)23 b Fp(2)g(D)r Fu(\()p Fr(S)5 b Fu(\))19 b Fp(\\)g(D)r Fu(\()p Fr(B)t Fu(\))p @beginspecial @setspecial @endspecial 1128 1426 a(_)-27 b Fr({)t Fu(_)g Fr({)s Fu(_)g Fr({)p Fu(\))14 b([)p Fr(B)t(;)g Fu(\()p Fr(z)22 b Fp(\000)c Fr(S)5 b Fu(\))1641 1391 y Fo(\000)p Fs(1)1730 1426 y Fu(])24 b(=)e(\()p Fr(z)g Fp(\000)c Fr(S)5 b Fu(\))2128 1391 y Fo(\000)p Fs(1)2217 1426 y Fu([)p Fr(B)t(;)14 b(S)5 b Fu(]\()p Fr(z)22 b Fp(\000)c Fr(S)5 b Fu(\))2687 1391 y Fo(\000)p Fs(1)2818 1290 y @beginspecial @setspecial @endspecial 0 1564 a Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(is)g(understo)l(o)l(d)g(as)g(identity)g(b)l(etwe)l (en)g(b)l(ounde)l(d)g(op)l(er)l(ators)g(in)g(the)g(fol)t(lowing)i (sense:)1256 1746 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(H)1350 1692 y Fs(\()p Fq(z)r Fo(\000)p Fq(S)s Fs(\))1532 1667 y Fg(\000)p Fi(1)1413 1746 y @beginspecial @setspecial @endspecial Fp(\000)-14 b(!)86 b(D)r Fu(\()p Fr(S)5 b Fu(\))1842 1692 y Fs([)p Fq(B)s(;S)s Fs(])1853 1746 y @beginspecial @setspecial @endspecial Fp(\000)-15 b(!)34 b(D)r Fu(\()p Fr(S)5 b Fu(\))2206 1712 y Fo(\003)2267 1692 y Fs(\()p Fq(z)r Fo(\000)p Fq(S)s Fs(\))2449 1667 y Fg(\000)p Fi(1)2330 1746 y @beginspecial @setspecial @endspecial Fp(\000)-14 b(!)86 b(H)q Fr(:)p @beginspecial @setspecial @endspecial 0 1906 a @beginspecial @setspecial @endspecial Fz(Lemma)30 b(5.3.)550 1885 y Fu(^)529 1906 y Fr(H)598 1918 y Fq(\024)670 1906 y Fn(is)g(of)h(class)f Fr(C)1121 1876 y Fs(1)1159 1906 y Fu(\()p Fr(B)t Fu(\))g Fn(if)h Fr(\024)23 b(<)g Fp(1)p Fn(.)0 1926 y @beginspecial @setspecial @endspecial 100 2057 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(w)n(e)27 b(will)h(pro)n(v)n(e)664 2262 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 705 2189 a @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))23 b(e)824 2159 y Fs(i)p Fq(tB)948 2189 y Fu(preserv)n(es)e Fp(D)r Fu(\()p Fr(H)1467 2201 y Fs(0)1505 2189 y Fu(\))p Fr(:)709 2332 y Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))23 b Fp(j)p Fu(\()891 2311 y(^)869 2332 y Fr(H)938 2344 y Fq(\024)981 2332 y Fr(u;)14 b(B)t(u)p Fu(\))k Fp(\000)g Fu(\()p Fr(B)t(u;)1520 2311 y Fu(^)1498 2332 y Fr(H)1567 2344 y Fq(\024)1610 2332 y Fr(u)p Fu(\))p Fp(j)23 b(\024)g Fr(c)14 b Fp(jj)p Fr(H)1989 2344 y Fs(0)2026 2332 y Fr(u)p Fp(jj)2120 2298 y Fs(2)2175 2332 y Fu(+)k Fp(jj)p Fr(u)p Fp(jj)2398 2298 y Fs(2)2435 2332 y Fr(;)37 b(u)23 b Fp(2)g(D)r Fu(\()p Fr(H)2811 2344 y Fs(0)2849 2332 y Fu(\))c Fp(\\)g(D)r Fu(\()p Fr(B)t Fu(\))p Fr(:)3236 2262 y @beginspecial @setspecial @endspecial 0 2487 a @beginspecial @setspecial @endspecial Fu([1,)27 b(Thm.)38 b(6.3.4])26 b(and)h([1,)h(Prop.)35 b(6.3.5])27 b(giv)n(e)f(that)i @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))p Fr(;)18 b Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(implies)2170 2466 y(^)2148 2487 y Fr(H)2217 2499 y Fq(\024)2284 2487 y Fp(2)23 b Fr(C)2427 2457 y Fs(1)2465 2487 y Fu(\()p Fr(B)t Fu(\).)0 2587 y(Let)i(us)f(pro)n(v)n(e)j(_)-27 b Fr({)p Fu(\).)36 b(It)25 b(is)f(enough)g(to)h(sho)n(w)e(that)i @beginspecial @setspecial @endspecial(e)1553 2557 y Fs(i)p Fq(tB)1654 2587 y Fr(H)1723 2599 y Fs(0)1760 2587 y Fu(e)1797 2557 y Fo(\000)p Fs(i)p Fq(tB)1950 2587 y Fu(\()p Fr(H)2051 2599 y Fs(0)2101 2587 y Fu(+)12 b(1\))2252 2557 y Fo(\000)p Fs(1)2366 2587 y Fu(is)24 b(b)r(ounded)h(to)f(ha)n(v)n(e)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\).)36 b(W)-7 b(e)25 b(ha)n(v)n(e)e(using)h(\()p @beginspecial @setspecial @endspecial(5.2\))1278 2728 y(e)1315 2694 y Fs(i)p Fq(tB)1416 2728 y Fu(d\000\()p Fr(!)s Fu(\)e)1670 2694 y Fo(\000)p Fs(i)p Fq(tB)1906 2728 y @beginspecial @setspecial @endspecial Fu(=)82 b(d\000\(e)2220 2694 y Fs(i)p Fq(tb)2298 2728 y Fr(!)s Fu(e)2390 2694 y Fo(\000)p Fs(i)p Fq(tb)2518 2728 y Fu(\))0 2774 y @beginspecial @setspecial @endspecial 1906 2891 a(=)g(d\000\()p Fr(!)s Fu(\()p Fr(\036)2319 2910 y Fo(\000)2384 2888 y Fj(t)p 2382 2897 29 3 v 2382 2930 a Fi(2)2424 2891 y Fu(\()p Fr(k)s Fu(\)\)\))p Fr(:)0 3039 y Fu(Since)29 b @beginspecial @setspecial @endspecial Fp(r)p Fr(!)i Fu(is)e(a)f(b)r(ounded)h (complete)g @beginspecial @setspecial @endspecial Fr(C)1282 3009 y Fo(1)1381 3039 y Fu(v)n(ector)e(\014eld)i(w)n(e)f(ha)n(v)n(e)g @beginspecial @setspecial @endspecial Fp(j)p Fr(\036)2200 3051 y Fq(t)2230 3039 y Fu(\()p Fr(k)s Fu(\))19 b Fp(\000)g Fr(k)s Fp(j)25 b(\024)f Fr(c)14 b Fp(j)p Fr(t)p Fp(j)28 b Fu(uniformly)h(in)g Fr(k)s Fu(.)40 b(So)28 b(this)h(implies)0 3220 y @beginspecial @setspecial @endspecial Fp(j)67 3116 y Fq(N)37 3141 y Fl(X)80 3317 y Fs(1)171 3220 y Fr(!)s Fu(\()p Fr(\036)307 3232 y Fq(t)336 3220 y Fu(\()p Fr(k)411 3232 y Fq(i)439 3220 y Fu(\)\))19 b Fp(\000)f Fr(!)s Fu(\()p Fr(k)735 3232 y Fq(i)763 3220 y Fu(\))p Fp(j)24 b(\024)e Fr(c)14 b Fp(j)p Fr(t)p Fp(j)g Fr(N)9 b Fu(,)28 b(and)f(hence)h @beginspecial @setspecial @endspecial(e)1625 3189 y Fs(i)p Fq(tB)1726 3220 y Fr(H)1795 3232 y Fs(0)1832 3220 y Fu(e)1869 3189 y Fo(\000)p Fs(i)p Fq(tB)2022 3220 y Fu(\()p Fr(H)2123 3232 y Fs(0)2179 3220 y Fu(+)18 b(1\))2336 3189 y Fo(\000)p Fs(1)2452 3220 y Fu(is)28 b(b)r(ounded.)37 b(This)28 b(pro)n(v)n(e)d @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\).)0 3412 y(Let)28 b(us)f(pro)n(v)n(e)f @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\).)37 b(W)-7 b(e)28 b(compute)g([)1149 3391 y(^)1128 3412 y Fr(H)1197 3424 y Fq(\024)1240 3412 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])23 b(:)763 3847 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 804 3557 a @beginspecial @setspecial @endspecial(\(i)p Fr(B)t(u;)1033 3536 y Fu(^)1011 3557 y Fr(H)1080 3569 y Fq(\024)1123 3557 y Fr(u)p Fu(\))18 b Fp(\000)g Fu(\(i)1381 3536 y(^)1359 3557 y Fr(H)1428 3569 y Fq(\024)1472 3557 y Fr(u;)c(B)t(u)p Fu(\))22 b(=)804 3774 y(\(d\000\()p Fp(jr)p Fr(!)s Fp(j)1136 3740 y Fs(2)1174 3774 y Fu(\))p Fr(u;)14 b(u)p Fu(\))k Fp(\000)1506 3670 y Fq(P)1472 3695 y Fl(X)1474 3872 y Fq(j)s Fs(=1)1592 3774 y Fu(\()p Fr(\036)p Fu(\(i)p Fr(bv)1807 3740 y Fq(j)1804 3794 y(\024)1843 3802 y Fi(0)1880 3774 y Fu(\))p Fr(u;)c(u)p Fu(\))k(+)2233 3718 y(1)p 2188 3755 132 4 v 2188 3831 a(2)p Fr(M)2377 3670 y Fq(P)2343 3695 y Fl(X)2345 3872 y Fq(j)s Fs(=1)2463 3774 y Fu(\(i)p Fr(a)p Fu(\()p Fr(r)2633 3740 y Fq(j)2631 3794 y(\024)2675 3774 y Fu(\))p Fr(a)p Fu(\()p Fr(br)2858 3740 y Fq(j)2856 3794 y(\024)2899 3774 y Fu(\))p Fr(u;)c(u)p Fu(\))p @beginspecial @setspecial @endspecial 804 3999 a Fp(\000)p Fu(\(i)p Fr(a)968 3965 y Fo(\003)1006 3999 y Fu(\()p Fr(r)1077 3965 y Fq(j)1075 4020 y(\024)1119 3999 y Fu(\))p Fr(a)1195 3965 y Fo(\003)1233 3999 y Fu(\()p Fr(br)1340 3965 y Fq(j)1338 4020 y(\024)1381 3999 y Fu(\))p Fr(u;)g(u)p Fu(\))k(+)g(\(i)p Fr(a)1778 3965 y Fo(\003)1817 3999 y Fu(\()p Fr(r)1888 3965 y Fq(j)1886 4020 y(\024)1930 3999 y Fu(\))p Fr(a)p Fu(\()p Fr(br)2113 3965 y Fq(j)2111 4020 y(\024)2154 3999 y Fu(\))h Fp(\000)f Fu(i)p Fr(a)2355 3965 y Fo(\003)2393 3999 y Fu(\()p Fr(br)2500 3965 y Fq(j)2498 4020 y(\024)2541 3999 y Fu(\))p Fr(a)p Fu(\()p Fr(r)2688 3965 y Fq(j)2686 4020 y(\024)2730 3999 y Fu(\))p Fr(u;)c(u)p Fu(\))p @beginspecial @setspecial @endspecial 804 4143 a Fp(\000)869 4071 y(p)p 938 4071 42 4 v 72 x Fu(2)o(\(i)p Fr(D)1103 4155 y Fq(j)1138 4143 y Fr(a)p Fu(\()p Fr(br)1289 4109 y Fq(j)1287 4164 y(\024)1331 4143 y Fu(\))p Fr(u;)g(u)p Fu(\))k(+)1629 4071 y Fp(p)p 1698 4071 V 72 x Fu(2)o(\(i)p Fr(a)1838 4109 y Fo(\003)1877 4143 y Fu(\()p Fr(br)1984 4109 y Fq(j)1982 4164 y(\024)2025 4143 y Fu(\))p Fr(D)2126 4155 y Fq(j)2161 4143 y Fr(u;)c(u)p Fu(\))p Fr(:)3137 3847 y @beginspecial @setspecial @endspecial 0 4277 a Fu(A)28 b(simple)g(computation)f(yields)834 4415 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 171 w(_)-28 b Fr({)p Fu(\))p Fp(h)p Fu(x)p Fp(i)1167 4380 y Fo(\000)p Fs(1)1257 4415 y Fr(bv)1333 4427 y Fq(\024)1372 4435 y Fi(0)1431 4415 y Fp(2)24 b(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)19 b(\012)f Fm(h)p Fu(\))p Fr(;)0 4460 y @beginspecial @setspecial @endspecial 917 4578 a @beginspecial @setspecial @endspecial 88 w Fu(_)-28 b Fr({)t Fu(_)h Fr({)p Fu(\))p Fp(h)p Fu(x)p Fp(i)1194 4544 y Fo(\000)p Fs(1)1284 4578 y Fr(br)1357 4590 y Fq(\024)1423 4578 y Fp(2)23 b(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)20 b(\012)e Fm(h)p Fu(\))24 b(for)e Fr(\024)h(<)g Fp(1)p Fr(;)-2344 b Fu(\(5.4\))0 4624 y @beginspecial @setspecial @endspecial 917 4742 a @beginspecial @setspecial @endspecial 88 w(_)-28 b Fr({)t Fu(_)h Fr({)t Fu(_)g Fr({)p Fu(\))p Fp(h)p Fr(k)s Fp(i)1223 4708 y Fo(\000)p Fq(\017)1307 4742 y Fp(h)p Fu(x)p Fp(i)1415 4708 y Fo(\000)p Fs(1)1505 4742 y Fr(br)1578 4754 y Fq(\024)1644 4742 y Fp(2)23 b(B)s Fu(\()p Fp(K)q Fr(;)14 b Fp(K)20 b(\012)e Fm(h)p Fu(\))24 b(for)e Fr(\017)h(>)f Fu(0)p Fr(;)28 b Fu(uniformly)14 b(in)24 b Fr(\024:)0 4879 y Fu(No)n(w)j(using)g (Lemma)h @beginspecial @setspecial @endspecial(3.4)e(with)j Fr(\014)e Fu(=)c(1)k(and)g(\()p @beginspecial @setspecial @endspecial(5.4\))32 b(_)-27 b Fr({)o Fu(\))19 b Fp(\000)j Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(w)n(e)f(obtain)g(for)g Fr(u)c Fp(2)g(D)r Fu(\()p Fr(H)2714 4891 y Fs(0)2752 4879 y Fu(\))c Fp(\\)g(D)r Fu(\()p Fr(B)t Fu(\))378 5030 y @beginspecial @setspecial @endspecial Fp(j)p Fu(\(i)p Fr(B)t(u;)630 5009 y Fu(^)608 5030 y Fr(H)677 5042 y Fq(\024)720 5030 y Fr(u)p Fu(\))g Fp(\000)f Fu(\(i)979 5009 y(^)957 5030 y Fr(H)1026 5042 y Fq(\024)1069 5030 y Fr(u;)c(B)t(u)p Fu(\))p Fp(j)82 b @beginspecial @setspecial @endspecial(\024)h Fr(c)p Fu(\()p Fp(jj)p Fr(r)1705 5042 y Fq(\024)1749 5030 y Fp(jj)23 b(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2139 4995 y Fo(\000)2200 4973 y Fi(1)p 2200 4982 29 3 v 2200 5015 a(2)2243 5030 y Fr(br)2316 5042 y Fq(\024)2359 5030 y Fp(jj)g Fu(+)g Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2827 4995 y Fo(\000)2889 4973 y Fi(1)p 2889 4982 V 2889 5015 a(2)2931 5030 y Fr(bv)3007 5042 y Fq(\024)3046 5050 y Fi(0)3083 5030 y Fp(jj)0 5075 y @beginspecial @setspecial @endspecial 1554 5205 a Fu(+)p Fp(jj)p Fu(\()p Fr(K)24 b Fu(+)18 b(i\))1930 5171 y Fo(\000)1992 5148 y Fi(1)p 1992 5157 V 1992 5191 a(2)2034 5205 y Fr(D)r Fp(jj)24 b(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))2495 5171 y Fo(\000)2557 5148 y Fi(1)p 2557 5157 V 2557 5191 a(2)2599 5205 y Fr(br)2672 5217 y Fq(\024)2716 5205 y Fp(jj)p Fu(\))g Fp(\002)g Fu(\()p Fp(jj)p Fr(H)3042 5217 y Fs(0)3080 5205 y Fr(u)p Fp(jj)3174 5171 y Fs(2)3229 5205 y Fu(+)g Fp(jj)p Fr(u)p Fp(jj)3452 5171 y Fs(2)3489 5205 y Fu(\))0 5343 y(Since)28 b @beginspecial @setspecial @endspecial Fr(V)42 b Fp(\025)394 5280 y Fl(P)482 5367 y Fq(i)510 5343 y Fp(h)p Fu(x)586 5355 y Fq(i)613 5343 y Fp(i)645 5312 y Fq(\013)693 5343 y Fr(;)28 b(\013)23 b(>)g Fu(2,)k(w)n(e)g(pro)n(v)n(e)j(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\).)37 b(This)28 b(completes)f(the)h(pro)r(of.)p @beginspecial @setspecial @endspecial 1414 w Fh(\004)0 5367 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 139 x Fz(Lemma)i(5.4.)40 b @beginspecial @setspecial @endspecial 550 5485 a Fu(^)529 5506 y Fr(H)598 5518 y Fo(1)698 5506 y Fn(is)30 b(of)g(class)h Fr(C)1149 5476 y Fs(1)1186 5506 y Fu(\()p Fr(B)t Fu(\))p Fn(.)0 5527 y @beginspecial @setspecial @endspecial eop %%Page: 26 26 26 25 bop 0 226 a Fs(26)1695 b(Z.)29 b(AMMARI)100 489 y Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Since)662 468 y(^)640 489 y Fr(H)709 501 y Fq(\024)780 489 y Fu(is)27 b(of)h(class)f Fr(C)1218 459 y Fs(1)1255 489 y Fu(\()p Fr(B)t Fu(\))i(w)n(e)e(kno)n(w)g(b)n(y)g([1)o(,)h(Thm.)37 b(6.2.10])26 b(that)i @beginspecial @setspecial @endspecial(\()p Fr(z)22 b Fp(\000)2860 468 y Fu(^)2838 489 y Fr(H)2907 501 y Fq(\024)2950 489 y Fu(\))2982 459 y Fo(\000)p Fs(1)3095 489 y Fu(:)h Fp(D)r Fu(\()p Fr(B)t Fu(\))h Fp(!)f(D)r Fu(\()p Fr(B)t Fu(\))29 b(and)836 635 y @beginspecial @setspecial @endspecial(\()p Fr(z)22 b Fp(\000)1034 614 y Fu(^)1012 635 y Fr(H)1081 647 y Fq(\024)1124 635 y Fu(\))1156 600 y Fo(\000)p Fs(1)1245 635 y Fu([)1290 614 y(^)1268 635 y Fr(H)1337 647 y Fq(\024)1380 635 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]\()p Fr(z)23 b Fp(\000)1728 614 y Fu(^)1707 635 y Fr(H)1776 647 y Fq(\024)1819 635 y Fu(\))1851 600 y Fo(\000)p Fs(1)1963 635 y Fu(=)g([\()p Fr(z)f Fp(\000)2272 614 y Fu(^)2250 635 y Fr(H)2319 647 y Fq(\024)2362 635 y Fu(\))2394 600 y Fo(\000)p Fs(1)2483 635 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(;)37 b Fu(on)23 b Fp(H)q Fr(:)83 b @beginspecial @setspecial @endspecial 0 772 a Fu(F)-7 b(or)27 b @beginspecial @setspecial @endspecial Fr(\036)d Fp(2)f Fr(D)r Fu(\()p Fr(B)t Fu(\),)28 b(one)g(has)518 918 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\(i\()628 897 y(^)605 918 y Fr(H)674 930 y Fq(\024)737 918 y Fu(+)18 b(i\))875 883 y Fo(\000)p Fs(1)964 918 y Fr(\036;)c(B)t(\036)p Fu(\))20 b Fp(\000)e Fu(\(i)p Fr(B)t(\036;)c Fu(\()1564 897 y(^)1541 918 y Fr(H)1610 930 y Fq(\024)1672 918 y Fp(\000)k Fu(i\))1810 883 y Fo(\000)p Fs(1)1900 918 y Fr(\036)p Fu(\))24 b(=)e(\(\()2179 897 y(^)2156 918 y Fr(H)2225 930 y Fq(\024)2287 918 y Fu(+)c(i\))2425 883 y Fo(\000)p Fs(1)2515 918 y Fr(\036;)c Fu([)2646 897 y(^)2624 918 y Fr(H)2693 930 y Fq(\024)2736 918 y Fr(;)g Fu(i)p Fr(B)t Fu(]\()2941 897 y(^)2918 918 y Fr(H)2987 930 y Fq(\024)3049 918 y Fp(\000)k Fu(i\))3187 883 y Fo(\000)p Fs(1)3277 918 y Fr(\036)p Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 1055 a Fu(Using)27 b(Lemma)h @beginspecial @setspecial @endspecial(3.3)e(and)i(\(4.17\))f (w)n(e)g(obtain)787 1340 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 828 1201 a @beginspecial @setspecial @endspecial Fp(j)p Fu(\(\()938 1180 y(^)915 1201 y Fr(H)984 1213 y Fq(\024)1046 1201 y Fu(+)18 b(i\))1184 1171 y Fo(\000)p Fs(1)1274 1201 y Fr(\036;)c(B)t(\036)p Fu(\))20 b Fp(\000)e Fu(\()p Fr(B)t(\036;)c Fu(\()1850 1180 y(^)1828 1201 y Fr(H)1897 1213 y Fq(\024)1959 1201 y Fp(\000)k Fu(i\))2097 1171 y Fo(\000)p Fs(1)2187 1201 y Fr(\036)p Fu(\))p Fp(j)24 b(\024)p @beginspecial @setspecial @endspecial 842 1337 a Fr(c)14 b Fu(\()p Fp(jj)p Fu(d\000\()p Fp(jr)p Fr(!)s Fp(j)1270 1303 y Fs(2)1308 1337 y Fu(\)\()p Fr(N)28 b Fu(+)18 b(1\))1624 1303 y Fo(\000)p Fs(1)1713 1337 y Fp(jj)g Fu(+)g Fp(jjh)p Fr(k)s Fp(i)2016 1303 y Fo(\000)p Fq(\017)2100 1337 y Fr(r)2137 1349 y Fq(\024)2181 1337 y Fp(jj)23 b(jjh)p Fr(k)s Fp(i)2406 1303 y Fo(\000)p Fq(\017)2490 1337 y Fp(h)p Fu(x)p Fp(i)2598 1303 y Fo(\000)p Fs(1)2688 1337 y Fr(br)2761 1349 y Fq(\024)2804 1337 y Fp(jj)p @beginspecial @setspecial @endspecial 828 1485 a Fu(+)p Fp(jjh)p Fu(x)p Fp(i)1047 1451 y Fo(\000)p Fs(1)1137 1485 y Fr(bv)1213 1497 y Fq(\024)1252 1505 y Fi(0)1288 1485 y Fp(jj)18 b Fu(+)g Fp(jj)p Fu(\()p Fr(K)25 b Fu(+)18 b(i\))1747 1451 y Fo(\000)1809 1429 y Fi(1)p 1809 1438 29 3 v 1809 1471 a(2)1851 1485 y Fr(D)r Fp(jj)24 b(jjh)p Fr(k)s Fp(i)2148 1451 y Fo(\000)p Fq(\017)2232 1485 y Fp(h)p Fu(x)p Fp(i)2340 1451 y Fo(\000)p Fs(1)2429 1485 y Fr(br)2502 1497 y Fq(\024)2545 1485 y Fp(jj)p Fu(\))p Fp(jj)p Fr(\036)p Fp(jj)2764 1451 y Fs(2)2803 1485 y Fr(;)37 b(\017)22 b(>)h Fu(0)p Fr(;)3113 1340 y @beginspecial @setspecial @endspecial 0 1619 a Fu(where)k(c)h(is)f(indep)r(enden)n(t) i(from)e @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)37 b(Then)27 b(letting)h Fr(\024)23 b Fp(!)g(1)28 b Fu(and)f(using)h(Thm.)37 b @beginspecial @setspecial @endspecial(3.6)27 b(w)n(e)g(obtain)730 1760 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(j)p Fu(\(\()840 1739 y(^)817 1760 y Fr(H)886 1772 y Fo(1)976 1760 y Fu(+)18 b(i\))1114 1726 y Fo(\000)p Fs(1)1203 1760 y Fr(\036;)c Fu(i)p Fr(B)t(\036)p Fu(\))20 b Fp(\000)e Fu(\(i)p Fr(B)t(\036;)c Fu(\()1826 1739 y(^)1803 1760 y Fr(H)1872 1772 y Fo(1)1962 1760 y Fp(\000)k Fu(i\))2100 1726 y Fo(\000)p Fs(1)2189 1760 y Fr(\036)p Fu(\))p Fp(j)24 b(\024)37 b Fr(c)14 b Fp(jj)p Fr(\036)p Fp(jj)2610 1726 y Fs(2)2671 1760 y Fu(for)22 b Fr(\036)i Fp(2)f Fr(D)r Fu(\()p Fr(B)t Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 1906 a Fu(This)28 b(implies)f @beginspecial @setspecial @endspecial 493 1885 a(^)471 1906 y Fr(H)540 1918 y Fo(1)634 1906 y Fp(2)c Fr(C)777 1876 y Fs(1)815 1906 y Fu(\()p Fr(B)t Fu(\).)2866 b Fh(\004)0 1927 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 130 x Fz(Lemma)30 b(5.5.)40 b Fn(We)29 b(have)i(for)g Fr(\037)23 b Fp(2)g Fr(C)1221 2027 y Fo(1)1215 2078 y Fs(0)1292 2057 y Fu(\()p Fk(R)p Fu(\))867 2203 y @beginspecial @setspecial @endspecial 171 w(_)-27 b Fr({)o Fu(\))14 b Fr(w)22 b Fp(\000)46 b Fu(lim)1270 2252 y Fq(\024)p Fo(!1)1441 2203 y Fu([\()1518 2182 y(^)1496 2203 y Fr(H)1565 2215 y Fq(\024)1627 2203 y Fp(\000)18 b Fu(i\))1765 2168 y Fo(\000)p Fs(1)1854 2203 y Fr(;)c Fu(i)p Fr(B)t Fu(])23 b(=)g([\()2192 2182 y(^)2170 2203 y Fr(H)2239 2215 y Fo(1)2328 2203 y Fp(\000)18 b Fu(i\))2466 2168 y Fo(\000)p Fs(1)2556 2203 y Fr(;)c Fu(i)p Fr(B)t Fu(])p Fr(;)0 2286 y @beginspecial @setspecial @endspecial 1038 2411 a Fu(_)-27 b Fr({)s Fu(_)g Fr({)p Fu(\))56 b(lim)1147 2460 y Fq(\024)p Fo(!1)1332 2411 y Fr(\037)p Fu(\()1438 2390 y(^)1416 2411 y Fr(H)1485 2423 y Fq(\024)1528 2411 y Fu(\)[)1605 2390 y(^)1583 2411 y Fr(H)1652 2423 y Fq(\024)1696 2411 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1952 2390 y(^)1930 2411 y Fr(H)1999 2423 y Fq(\024)2042 2411 y Fu(\))24 b(=)e Fr(\037)p Fu(\()2291 2390 y(^)2269 2411 y Fr(H)2338 2423 y Fo(1)2409 2411 y Fu(\)[)2486 2390 y(^)2464 2411 y Fr(H)2533 2423 y Fo(1)2603 2411 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2860 2390 y(^)2837 2411 y Fr(H)2906 2423 y Fo(1)2977 2411 y Fu(\))p Fr(:)100 2591 y Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 28 w Fu(_)-27 b Fr({)o Fu(\))28 b(follo)n(ws)f(from)g(the)h(pro)r(of)f (of)h(Lemma)f(5.4.)36 b(Let)28 b(us)f(pro)n(v)n(e)f @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\):)1151 2736 y Fr(\037)p Fu(\()1257 2715 y(^)1235 2736 y Fr(H)1304 2748 y Fq(\024)1347 2736 y Fu(\)[)1424 2715 y(^)1402 2736 y Fr(H)1471 2748 y Fq(\024)1515 2736 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1771 2715 y(^)1749 2736 y Fr(H)1818 2748 y Fq(\024)1861 2736 y Fu(\))19 b Fp(\000)f Fr(\037)p Fu(\()2101 2715 y(^)2079 2736 y Fr(H)2148 2748 y Fq(\024)2187 2732 y Fg(0)2213 2736 y Fu(\)[)2291 2715 y(^)2268 2736 y Fr(H)2337 2748 y Fq(\024)2376 2732 y Fg(0)2403 2736 y Fr(;)c Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2659 2715 y(^)2637 2736 y Fr(H)2706 2748 y Fq(\024)2745 2732 y Fg(0)2772 2736 y Fu(\))24 b(=)0 2782 y @beginspecial @setspecial @endspecial 1151 2907 a(\()p Fr(\037)p Fu(\()1289 2886 y(^)1267 2907 y Fr(H)1336 2919 y Fq(\024)1379 2907 y Fu(\))19 b Fp(\000)f Fr(\037)p Fu(\()1619 2886 y(^)1597 2907 y Fr(H)1666 2919 y Fq(\024)1705 2903 y Fg(0)1732 2907 y Fu(\)\)[)1841 2886 y(^)1819 2907 y Fr(H)1888 2919 y Fq(\024)1932 2907 y Fr(;)c Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2188 2886 y(^)2166 2907 y Fr(H)2235 2919 y Fq(\024)2278 2907 y Fu(\))0 2953 y @beginspecial @setspecial @endspecial 1151 3077 a(+)p Fr(\037)p Fu(\()1321 3056 y(^)1300 3077 y Fr(H)1369 3089 y Fq(\024)1408 3073 y Fg(0)1434 3077 y Fu(\)\([)1544 3056 y(^)1521 3077 y Fr(H)1590 3089 y Fq(\024)1634 3077 y Fr(;)g Fu(i)p Fr(B)t Fu(])k Fp(\000)g Fu([)1930 3056 y(^)1908 3077 y Fr(H)1977 3089 y Fq(\024)2016 3073 y Fg(0)2043 3077 y Fr(;)c Fu(i)p Fr(B)t Fu(]\))p Fr(\037)p Fu(\()2331 3056 y(^)2309 3077 y Fr(H)2378 3089 y Fq(\024)2422 3077 y Fu(\))-2454 b(\(5.5\))0 3123 y @beginspecial @setspecial @endspecial 1151 3248 a(+)p Fr(\037)p Fu(\()1321 3227 y(^)1300 3248 y Fr(H)1369 3260 y Fq(\024)1412 3248 y Fu(\)[)1489 3227 y(^)1467 3248 y Fr(H)1536 3260 y Fq(\024)1575 3244 y Fg(0)1602 3248 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]\()p Fr(\037)p Fu(\()1890 3227 y(^)1868 3248 y Fr(H)1937 3260 y Fq(\024)1980 3248 y Fu(\))19 b Fp(\000)f Fr(\037)p Fu(\()2220 3227 y(^)2198 3248 y Fr(H)2267 3260 y Fq(\024)2306 3244 y Fg(0)2333 3248 y Fu(\)\))p Fr(:)0 3385 y Fu(W)-7 b(e)28 b(\014rst)f(claim)h(that) 1337 3529 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 28 w(lim)1337 3579 y Fq(\024)p Fo(!1)1522 3529 y Fr(\037)p Fu(\()1628 3508 y(^)1606 3529 y Fr(H)1675 3541 y Fq(\024)1719 3529 y Fu(\))p Fr(H)1837 3458 y Fi(1)p 1837 3467 V 1837 3501 a(2)1820 3551 y Fs(0)1902 3529 y Fu(=)23 b Fr(\037)p Fu(\()2096 3508 y(^)2074 3529 y Fr(H)2143 3541 y Fo(1)2213 3529 y Fu(\))p Fr(H)2331 3458 y Fi(1)p 2331 3467 V 2331 3501 a(2)2314 3551 y Fs(0)2374 3529 y Fr(:)83 b @beginspecial @setspecial @endspecial -2480 w Fu(\(5.6\))0 3696 y(T)-7 b(o)27 b(ha)n(v)n(e)g(\()p @beginspecial @setspecial @endspecial(5.6\),)g(w)n(e)g(see)g(using)h (the)g(functional)f(calculus)h(form)n(ula)e(\()p @beginspecial @setspecial @endspecial(4.18\),)h(that)h(it)g(is)f(enough)g(to)h(sho)n (w)f(that)1191 3861 y(lim)1163 3910 y Fq(\024)p Fo(!1)1335 3861 y Fu(\()1389 3840 y(^)1367 3861 y Fr(H)1436 3873 y Fq(\024)1497 3861 y Fp(\000)18 b Fr(z)t Fu(\))1655 3826 y Fo(\000)p Fs(1)1744 3861 y Fr(H)1830 3790 y Fi(1)p 1830 3799 V 1830 3832 a(2)1813 3883 y Fs(0)1895 3861 y Fu(=)23 b(\()2037 3840 y(^)2015 3861 y Fr(H)2084 3873 y Fo(1)2173 3861 y Fp(\000)18 b Fr(z)t Fu(\))2331 3826 y Fo(\000)p Fs(1)2420 3861 y Fr(H)2505 3790 y Fi(1)p 2505 3799 V 2505 3832 a(2)2489 3883 y Fs(0)2548 3861 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4027 a Fu(F)-7 b(or)27 b Fr(\024;)14 b(\024)282 3997 y Fo(0)328 4027 y Fr(<)23 b Fp(1)p Fu(,)k(w)n(e)h(ha)n(v)n(e)e(the)i(follo)n(wing) e(op)r(erator)g(iden)n(tit)n(y)i(on)f @beginspecial @setspecial @endspecial Fp(H)702 4168 y Fu(\()756 4147 y(^)734 4168 y Fr(H)803 4180 y Fq(\024)864 4168 y Fp(\000)18 b Fr(z)t Fu(\))1022 4134 y Fo(\000)p Fs(1)1130 4168 y Fp(\000)g Fu(\()1267 4147 y(^)1245 4168 y Fr(H)1314 4180 y Fq(\024)1353 4164 y Fg(0)1398 4168 y Fp(\000)g Fr(z)t Fu(\))1556 4134 y Fo(\000)p Fs(1)1668 4168 y Fu(=)k(\()1809 4147 y(^)1787 4168 y Fr(H)1856 4180 y Fq(\024)1918 4168 y Fp(\000)c Fr(z)t Fu(\))2076 4134 y Fo(\000)p Fs(1)2165 4168 y Fu(\()2219 4147 y(^)2197 4168 y Fr(H)2266 4180 y Fq(\024)2305 4164 y Fg(0)2350 4168 y Fp(\000)2455 4147 y Fu(^)2433 4168 y Fr(H)2502 4180 y Fq(\024)2545 4168 y Fu(\)\()2631 4147 y(^)2609 4168 y Fr(H)2678 4180 y Fq(\024)2717 4164 y Fg(0)2763 4168 y Fp(\000)g Fr(z)t Fu(\))2921 4134 y Fo(\000)p Fs(1)3009 4168 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4306 a Fu(It)28 b(follo)n(ws)f(from)g(the)h(pro)r(of)f (of)g(Thm.)38 b @beginspecial @setspecial @endspecial(3.6)26 b(that:)1090 4452 y(lim)1022 4505 y Fq(\024;\024)1120 4488 y Fg(0)1142 4505 y Fo(!1)1288 4452 y Fu(\()p Fr(H)1389 4464 y Fs(0)1445 4452 y Fu(+)18 b(1\))1602 4417 y Fo(\000)1664 4395 y Fi(1)p 1664 4404 V 1664 4437 a(2)1706 4452 y Fu(\()1760 4431 y(^)1738 4452 y Fr(H)1807 4464 y Fq(\024)1846 4447 y Fg(0)1891 4452 y Fp(\000)1996 4431 y Fu(^)1974 4452 y Fr(H)2043 4464 y Fq(\024)2086 4452 y Fu(\)\()p Fr(H)2219 4464 y Fs(0)2276 4452 y Fu(+)g(1\))2433 4417 y Fo(\000)2494 4395 y Fi(1)p 2494 4404 V 2494 4437 a(2)2560 4452 y Fu(=)k(0)p Fr(:)83 b @beginspecial @setspecial @endspecial 0 4633 a Fu(Then)28 b(w)n(e)f(obtain)g(\()p @beginspecial @setspecial @endspecial(5.6\).)37 b(W)-7 b(e)28 b(also)e(claim)i(that:)228 4783 y(lim)200 4833 y Fq(\024)p Fo(!1)385 4783 y Fu(\()p Fr(H)486 4795 y Fs(0)542 4783 y Fu(+)18 b(1\))699 4749 y Fo(\000)761 4727 y Fi(1)p 761 4736 V 761 4769 a(2)803 4783 y Fu([)848 4762 y(^)826 4783 y Fr(H)895 4795 y Fq(\024)938 4783 y Fr(;)c Fu(i)p Fr(B)t Fu(]\()p Fr(H)1189 4795 y Fs(0)1245 4783 y Fu(+)19 b(1\))1403 4749 y Fo(\000)1464 4727 y Fi(1)p 1464 4736 V 1464 4769 a(2)1507 4783 y Fu(\()p Fr(N)27 b Fu(+)18 b(1\))1790 4749 y Fo(\000)p Fs(1)1902 4783 y Fu(=)23 b(\()p Fr(H)2091 4795 y Fs(0)2147 4783 y Fu(+)18 b(1\))2304 4749 y Fo(\000)2365 4727 y Fi(1)p 2365 4736 V 2365 4769 a(2)2408 4783 y Fu([)2453 4762 y(^)2431 4783 y Fr(H)2500 4795 y Fo(1)2570 4783 y Fr(;)c Fu(i)p Fr(B)t Fu(]\()p Fr(H)2821 4795 y Fs(0)2877 4783 y Fu(+)k(1\))3034 4749 y Fo(\000)3096 4727 y Fi(1)p 3096 4736 V 3096 4769 a(2)3138 4783 y Fu(\()p Fr(N)28 b Fu(+)18 b(1\))3422 4749 y Fo(\000)p Fs(1)3511 4783 y Fr(:)83 b @beginspecial @setspecial @endspecial -3617 w Fu(\(5.7\))0 4955 y(In)28 b(fact,)g(using)f(Lemma)g @beginspecial @setspecial @endspecial(3.4)g(and)g(the)h(fact)g(that)g @beginspecial @setspecial @endspecial Fr(V)42 b Fp(\025)23 b Fr(c)1827 4893 y Fl(P)1915 4980 y Fq(i)1942 4955 y Fp(h)p Fr(x)2021 4967 y Fq(i)2050 4955 y Fp(i)2082 4925 y Fs(2)2119 4955 y Fu(,)28 b(w)n(e)f(get:)298 5110 y @beginspecial @setspecial @endspecial 83 w Fp(jj)p Fu(\()p Fr(H)528 5122 y Fs(0)584 5110 y Fu(+)18 b(i\))722 5076 y Fo(\000)785 5053 y Fi(1)p 785 5062 V 785 5096 a(2)827 5110 y Fr(a)871 5076 y Fo(\003)909 5110 y Fu(\()p Fr(br)1014 5122 y Fq(\024)1053 5106 y Fg(0)1098 5110 y Fp(\000)g Fr(br)1254 5122 y Fq(\024)1298 5110 y Fu(\))p Fr(a)1374 5076 y Fo(\003)1412 5110 y Fu(\()p Fr(r)1481 5122 y Fq(\024)1520 5106 y Fg(0)1547 5110 y Fu(\)\()p Fr(H)1680 5122 y Fs(0)1736 5110 y Fu(+)g(i\))1874 5076 y Fo(\000)1937 5053 y Fi(1)p 1937 5062 V 1937 5096 a(2)1993 5110 y Fr(N)2069 5076 y Fo(\000)p Fs(1)2158 5110 y Fp(jj)23 b(\024)f(jjh)p Fr(k)s Fp(i)2470 5076 y Fo(\000)p Fq(\017)2555 5110 y Fp(h)p Fu(x)p Fp(i)2663 5076 y Fo(\000)p Fs(1)2752 5110 y Fu(\()p Fr(br)2857 5122 y Fq(\024)2896 5106 y Fg(0)2942 5110 y Fp(\000)c Fr(br)3098 5122 y Fq(\024)3141 5110 y Fu(\))p Fp(jj)23 b(jjh)p Fr(k)s Fp(i)3398 5076 y Fo(\000)p Fq(\017)3482 5110 y Fr(r)3519 5122 y Fq(\024)3558 5106 y Fg(0)3585 5110 y Fp(jj)p Fr(;)0 5156 y @beginspecial @setspecial @endspecial 381 5285 a Fp(jj)p Fu(\()p Fr(H)528 5297 y Fs(0)584 5285 y Fu(+)18 b(i\))722 5251 y Fo(\000)785 5228 y Fi(1)p 785 5237 V 785 5271 a(2)827 5285 y Fr(a)p Fu(\()p Fr(r)940 5297 y Fq(\024)979 5281 y Fg(0)1025 5285 y Fp(\000)g Fr(r)1145 5297 y Fq(\024)1188 5285 y Fu(\))p Fr(a)p Fu(\()p Fr(br)1369 5297 y Fq(\024)1408 5281 y Fg(0)1435 5285 y Fu(\)\()p Fr(H)1568 5297 y Fs(0)1624 5285 y Fu(+)g(i\))1762 5251 y Fo(\000)1825 5228 y Fi(1)p 1825 5237 V 1825 5271 a(2)1881 5285 y Fr(N)1957 5251 y Fo(\000)p Fs(1)2046 5285 y Fp(jj)23 b(\024)g(jjh)p Fr(k)s Fp(i)2359 5251 y Fo(\000)p Fq(\017)2443 5285 y Fu(\()p Fr(r)2512 5297 y Fq(\024)2551 5281 y Fg(0)2596 5285 y Fp(\000)18 b Fr(r)2716 5297 y Fq(\024)2760 5285 y Fu(\))p Fp(jj)23 b(jjh)p Fr(k)s Fp(i)3017 5251 y Fo(\000)p Fq(\017)3102 5285 y Fp(h)p Fu(x)p Fp(i)3210 5251 y Fo(\000)p Fs(1)3299 5285 y Fr(br)3372 5297 y Fq(\024)3411 5281 y Fg(0)3438 5285 y Fp(jj)p Fr(;)0 5331 y @beginspecial @setspecial @endspecial 381 5460 a Fp(jj)p Fu(\()p Fr(H)528 5472 y Fs(0)584 5460 y Fu(+)18 b(i\))722 5426 y Fo(\000)785 5404 y Fi(1)p 785 5413 V 785 5446 a(2)827 5460 y Fr(a)871 5426 y Fo(\003)909 5460 y Fu(\()p Fr(br)1014 5472 y Fq(\024)1053 5456 y Fg(0)1098 5460 y Fp(\000)g Fr(br)1254 5472 y Fq(\024)1298 5460 y Fu(\))p Fr(a)p Fu(\()p Fr(r)1443 5472 y Fq(\024)1482 5456 y Fg(0)1509 5460 y Fu(\)\()p Fr(H)1642 5472 y Fs(0)1698 5460 y Fu(+)g(i\))1836 5426 y Fo(\000)1898 5404 y Fi(1)p 1898 5413 V 1898 5446 a(2)1955 5460 y Fr(N)2031 5426 y Fo(\000)p Fs(1)2119 5460 y Fp(jj)24 b(\024)e(jjh)p Fr(k)s Fp(i)2432 5426 y Fo(\000)p Fq(\017)2517 5460 y Fp(h)p Fu(x)p Fp(i)2625 5426 y Fo(\000)p Fs(1)2714 5460 y Fu(\()p Fr(br)2819 5472 y Fq(\024)2858 5456 y Fg(0)2903 5460 y Fp(\000)c Fr(br)3059 5472 y Fq(\024)3103 5460 y Fu(\))p Fp(jj)23 b(jjh)p Fr(k)s Fp(i)3360 5426 y Fo(\000)p Fq(\017)3444 5460 y Fr(r)3481 5472 y Fq(\024)3520 5456 y Fg(0)3547 5460 y Fp(jj)p Fr(;)0 5506 y @beginspecial @setspecial @endspecial eop %%Page: 27 27 27 26 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(27)381 489 y Fp(jj)p Fu(\()p Fr(H)528 501 y Fs(0)584 489 y Fu(+)18 b(i\))722 455 y Fo(\000)785 433 y Fi(1)p 785 442 29 3 v 785 475 a(2)827 489 y Fr(a)871 455 y Fo(\003)909 489 y Fu(\()p Fr(br)1014 501 y Fq(\024)1053 485 y Fg(0)1098 489 y Fp(\000)g Fr(br)1254 501 y Fq(\024)1298 489 y Fu(\))p Fr(D)1399 501 y Fq(j)1434 489 y Fu(\()p Fr(H)1535 501 y Fs(0)1591 489 y Fu(+)g(i\))1729 455 y Fo(\000)1791 433 y Fi(1)p 1791 442 V 1791 475 a(2)1847 489 y Fr(N)1923 455 y Fo(\000)p Fs(1)2012 489 y Fp(jj)23 b(\024)g(jjh)p Fr(k)s Fp(i)2325 455 y Fo(\000)p Fq(\017)2409 489 y Fp(h)p Fu(x)p Fp(i)2517 455 y Fo(\000)p Fs(1)2606 489 y Fu(\()p Fr(br)2711 501 y Fq(\024)2750 485 y Fg(0)2796 489 y Fp(\000)18 b Fr(br)2952 501 y Fq(\024)2995 489 y Fu(\))p Fp(jj)23 b(jj)p Fu(\()p Fr(K)i Fu(+)18 b(1\))3427 455 y Fo(\000)3488 433 y Fi(1)p 3488 442 V 3488 475 a(2)3531 489 y Fr(D)r Fp(jj)p Fr(;)0 538 y @beginspecial @setspecial @endspecial 381 689 a Fp(jj)p Fu(\()p Fr(H)528 701 y Fs(0)584 689 y Fu(+)g(i\))722 655 y Fo(\000)785 633 y Fi(1)p 785 642 V 785 675 a(2)827 689 y Fr(D)896 701 y Fq(j)931 689 y Fr(a)p Fu(\()p Fr(br)1080 701 y Fq(\024)1119 685 y Fg(0)1164 689 y Fp(\000)g Fr(br)1320 701 y Fq(\024)1363 689 y Fu(\)\()p Fr(H)1496 701 y Fs(0)1552 689 y Fu(+)g(i\))1690 655 y Fo(\000)1753 633 y Fi(1)p 1753 642 V 1753 675 a(2)1809 689 y Fr(N)1885 655 y Fo(\000)p Fs(1)1974 689 y Fp(jj)23 b(\024)f(jjh)p Fr(k)s Fp(i)2286 655 y Fo(\000)p Fq(\017)2371 689 y Fp(h)p Fu(x)p Fp(i)2479 655 y Fo(\000)p Fs(1)2568 689 y Fu(\()p Fr(br)2673 701 y Fq(\024)2712 685 y Fg(0)2758 689 y Fp(\000)c Fr(br)2914 701 y Fq(\024)2957 689 y Fu(\))p Fp(jj)23 b(jj)p Fu(\()p Fr(K)h Fu(+)18 b(1\))3388 655 y Fo(\000)3450 633 y Fi(1)p 3450 642 V 3450 675 a(2)3492 689 y Fr(D)r Fp(jj)p Fr(:)0 849 y Fu(Using)27 b(these)h(estimates)f (and)h(\()p @beginspecial @setspecial @endspecial(5.4\),)f(w)n(e)g (obtain)h(\(5.7\).)36 b(No)n(w)27 b(using)h(\()p @beginspecial @setspecial @endspecial(5.5\),\(5.6\))f(and)g(\(5.7\))g(w)n(e)h(obtain) f @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\).)383 b Fh(\004)100 948 y @beginspecial @setspecial @endspecial 100 x Fu(W)-7 b(e)28 b(ha)n(v)n(e)e(to)h(pro)n (v)n(e)f(a)i(lo)r(calization)e(estimate)i(for)f @beginspecial @setspecial @endspecial([)1774 1027 y(^)1752 1048 y Fr(H)1821 1060 y Fq(\024)1864 1048 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])28 b(similar)e(to)i(the)g(one)f(in)h(the)g(Lemma)f(4.15.)0 1069 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 143 x Fz(Lemma)j(5.6.)40 b Fn(We)29 b(have)i(uniformly)g(in)f @beginspecial @setspecial @endspecial Fr(\024)p Fn(:)796 1371 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 838 1375 a @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()944 1354 y(^)922 1375 y Fr(H)998 1345 y Fs(ext)991 1395 y Fq(\024)1092 1375 y Fu(\))1124 1308 y Fl(\000)1162 1375 y Fu([)1207 1354 y(^)1185 1375 y Fr(H)1261 1345 y Fs(ext)1254 1395 y Fq(\024)1356 1375 y Fr(;)14 b Fu(i)p Fr(B)1483 1345 y Fs(ext)1577 1375 y Fu(])p Fr(I)1643 1345 y Fo(\003)1682 1375 y Fu(\()p Fr(j)1753 1345 y Fq(R)1807 1375 y Fu(\))19 b Fp(\000)f Fr(I)1984 1345 y Fo(\003)2022 1375 y Fu(\()p Fr(j)2093 1345 y Fq(R)2148 1375 y Fu(\)[)2225 1354 y(^)2203 1375 y Fr(H)2272 1387 y Fq(\024)2315 1375 y Fr(;)c Fu(i)p Fr(B)t Fu(])2465 1308 y Fl(\001)2504 1375 y Fr(\037)p Fu(\()2609 1354 y(^)2588 1375 y Fr(H)2657 1387 y Fq(\024)2700 1375 y Fu(\))23 b Fp(2)h Fr(o)p Fu(\()p Fr(R)2970 1345 y Fs(0)3007 1375 y Fu(\))p Fr(:)3104 1371 y @beginspecial @setspecial @endspecial 100 1535 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(set)580 1694 y Fr(C)6 b Fu(\()p Fr(z)t Fu(\))23 b(:=)g(\()p Fr(z)f Fp(\000)1084 1673 y Fu(^)1062 1694 y Fr(H)1138 1660 y Fs(ext)1131 1715 y Fq(\024)1232 1694 y Fu(\))1264 1660 y Fo(\000)p Fs(1)1354 1694 y Fu(\([)1431 1673 y(^)1409 1694 y Fr(H)1485 1660 y Fs(ext)1478 1715 y Fq(\024)1579 1694 y Fr(;)14 b Fu(i)p Fr(B)1706 1660 y Fs(ext)1801 1694 y Fu(])p Fr(I)1867 1660 y Fo(\003)1905 1694 y Fu(\()p Fr(j)1976 1660 y Fq(R)2031 1694 y Fu(\))19 b Fp(\000)f Fr(I)2208 1660 y Fo(\003)2246 1694 y Fu(\()p Fr(j)2317 1660 y Fq(R)2371 1694 y Fu(\)[)2449 1673 y(^)2426 1694 y Fr(H)2495 1706 y Fq(\024)2539 1694 y Fr(;)c Fu(i)p Fr(B)t Fu(]\)\()p Fr(z)22 b Fp(\000)2919 1673 y Fu(^)2897 1694 y Fr(H)2966 1706 y Fq(\024)3009 1694 y Fu(\))3041 1660 y Fo(\000)p Fs(1)3131 1694 y Fr(;)83 b @beginspecial @setspecial @endspecial 0 1854 a Fu(where)27 b Fr(B)307 1824 y Fs(ext)425 1854 y Fu(:=)22 b Fr(B)h Fp(\012)18 b Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fr(B)t Fu(.)37 b(A)28 b(simple)g(computation)f(giv)n(es)691 2013 y @beginspecial @setspecial @endspecial([)p Fr(H)790 1979 y Fs(ext)783 2034 y(0)885 2013 y Fr(;)14 b Fu(i)p Fr(B)1012 1979 y Fs(ext)1106 2013 y Fu(])p Fr(I)1172 1979 y Fo(\003)1210 2013 y Fu(\()p Fr(j)1281 1979 y Fq(R)1336 2013 y Fu(\))19 b Fp(\000)f Fr(I)1513 1979 y Fo(\003)1551 2013 y Fu(\()p Fr(j)1622 1979 y Fq(R)1677 2013 y Fu(\)[)p Fr(H)1801 2025 y Fs(0)1838 2013 y Fr(;)c Fu(i)p Fr(B)t Fu(])83 b(=)g @beginspecial @setspecial @endspecial(d)p Fr(I)2308 1979 y Fo(\003)2346 2013 y Fu(\()p Fr(j)2417 1979 y Fq(R)2472 2013 y Fr(;)14 b Fu([)p Fp(jr)p Fr(!)s Fp(j)2702 1979 y Fs(2)2739 2013 y Fr(;)g(j)2815 1979 y Fq(R)2870 2013 y Fu(]\))p Fr(;)431 2201 y(a)p Fu(\()p Fr(r)546 2167 y Fq(j)544 2221 y(\024)588 2201 y Fu(\))p Fr(a)p Fu(\()p Fr(br)771 2167 y Fq(j)769 2221 y(\024)812 2201 y Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)o Fr(I)1038 2167 y Fo(\003)1076 2201 y Fu(\()p Fr(j)1147 2167 y Fq(R)1202 2201 y Fu(\))19 b Fp(\000)f Fr(I)1379 2167 y Fo(\003)1417 2201 y Fu(\()p Fr(j)1488 2167 y Fq(R)1543 2201 y Fu(\))p Fr(a)p Fu(\()p Fr(r)1690 2167 y Fq(j)1688 2221 y(\024)1732 2201 y Fu(\))p Fr(a)p Fu(\()p Fr(br)1915 2167 y Fq(j)1913 2221 y(\024)1956 2201 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)g @beginspecial @setspecial @endspecial Fr(I)2262 2167 y Fo(\003)2300 2201 y Fu(\()p Fr(j)2371 2167 y Fq(R)2426 2201 y Fu(\)\()p Fr(a)p Fu(\(\()p Fr(j)2637 2167 y Fq(R)2632 2221 y Fs(0)2711 2201 y Fp(\000)18 b Fu(1\))p Fr(r)2907 2167 y Fq(j)2905 2221 y(\024)2948 2201 y Fu(\))p Fr(a)p Fu(\()p Fr(j)3095 2167 y Fq(R)3090 2221 y Fs(0)3150 2201 y Fr(br)3225 2167 y Fq(j)3223 2221 y(\024)3266 2201 y Fu(\))2071 2387 y(+)83 b Fr(a)p Fu(\()p Fr(r)2334 2353 y Fq(j)2332 2408 y(\024)2376 2387 y Fu(\))p Fr(a)p Fu(\(\()p Fr(j)2550 2399 y Fs(0)2606 2387 y Fp(\000)18 b Fu(1\))p Fr(br)2838 2353 y Fq(j)2836 2408 y(\024)2879 2387 y Fu(\)\))p Fr(;)0 2433 y @beginspecial @setspecial @endspecial 278 2575 a(a)322 2541 y Fo(\003)360 2575 y Fu(\()p Fr(r)431 2541 y Fq(j)429 2596 y(\024)473 2575 y Fu(\))p Fr(a)549 2541 y Fo(\003)587 2575 y Fu(\()p Fr(br)694 2541 y Fq(j)692 2596 y(\024)736 2575 y Fu(\))g Fp(\012)g Fu(1)-15 b(l)p Fr(I)962 2541 y Fo(\003)1000 2575 y Fu(\()p Fr(j)1071 2541 y Fq(R)1126 2575 y Fu(\))18 b Fp(\000)g Fr(I)1302 2541 y Fo(\003)1341 2575 y Fu(\()p Fr(j)1412 2541 y Fq(R)1466 2575 y Fu(\))p Fr(a)1542 2541 y Fo(\003)1581 2575 y Fu(\()p Fr(r)1652 2541 y Fq(j)1650 2596 y(\024)1693 2575 y Fu(\))p Fr(a)1769 2541 y Fo(\003)1808 2575 y Fu(\()p Fr(br)1915 2541 y Fq(j)1913 2596 y(\024)1956 2575 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)g Fp(\000)p Fu(\()p Fr(a)2360 2541 y Fo(\003)2398 2575 y Fu(\()p Fr(j)2469 2541 y Fq(R)2464 2596 y Fs(0)2524 2575 y Fr(r)2563 2541 y Fq(j)2561 2596 y(\024)2604 2575 y Fu(\))19 b Fp(\012)f Fr(a)2782 2541 y Fo(\003)2820 2575 y Fu(\()p Fr(j)2891 2541 y Fq(R)2886 2596 y Fo(1)2957 2575 y Fr(br)3032 2541 y Fq(j)3030 2596 y(\024)3073 2575 y Fu(\))h(+)f Fr(a)3251 2541 y Fo(\003)3289 2575 y Fu(\()p Fr(j)3360 2541 y Fq(R)3355 2596 y Fs(0)3414 2575 y Fr(br)3489 2541 y Fq(j)3487 2596 y(\024)3530 2575 y Fu(\))0 2621 y @beginspecial @setspecial @endspecial 2219 2763 a Fp(\012)p Fr(a)2328 2729 y Fo(\003)2366 2763 y Fu(\()p Fr(j)2437 2729 y Fq(R)2432 2784 y Fo(1)2503 2763 y Fr(r)2542 2729 y Fq(j)2540 2784 y(\024)2583 2763 y Fu(\))h(+)f(1)-15 b(l)18 b Fp(\012)g Fr(a)2912 2729 y Fo(\003)2950 2763 y Fu(\()p Fr(j)3021 2729 y Fq(R)3016 2784 y Fo(1)3086 2763 y Fr(r)3125 2729 y Fq(j)3123 2784 y(\024)3167 2763 y Fu(\))p Fr(a)3243 2729 y Fo(\003)3281 2763 y Fu(\()p Fr(j)3347 2775 y Fo(1)3418 2763 y Fr(br)3493 2729 y Fq(j)3491 2784 y(\024)3534 2763 y Fu(\)\))p Fr(:)0 2922 y Fu(W)-7 b(e)31 b(ha)n(v)n(e)f(also)f(similar)h(iden)n(tities)h(for)g @beginspecial @setspecial @endspecial Fr(a)1322 2892 y Fo(\003)1360 2922 y Fr(a;)14 b(aD)1554 2934 y Fq(j)1588 2922 y Fr(;)g(D)1694 2934 y Fq(j)1729 2922 y Fr(a)1773 2892 y Fo(\003)1811 2922 y Fu(,)31 b(replacing)f Fr(r)2263 2892 y Fq(j)2261 2943 y(\024)2335 2922 y Fu(b)n(y)h @beginspecial @setspecial @endspecial Fr(br)2529 2892 y Fq(j)2527 2943 y(\024)2570 2922 y Fu(,)g(in)g(the)h(pro)r(of)e(of)g(Lemma)h(4.15.)45 b(As)31 b(in)0 3022 y(the)d(pro)r(of)f(of)h(Lemma)f @beginspecial @setspecial @endspecial(4.15)f(w)n(e)h(use)h(Corollary)d(4.8)i(and)g (Lemma)h @beginspecial @setspecial @endspecial(3.3.)0 3122 y(W)-7 b(e)28 b(ha)n(v)n(e)e(for)h Fr(b)c(<)g Fu(0:)1120 3281 y Fp(jj)p Fr(C)6 b Fu(\()p Fr(b)p Fu(\))p Fp(jj)0 3327 y @beginspecial @setspecial @endspecial 1120 3469 a(\024)22 b Fr(c)p Fu(\()p Fp(jj)p Fu(\()p Fr(N)1420 3481 y Fs(0)1476 3469 y Fu(+)c Fr(N)1626 3481 y Fo(1)1696 3469 y Fu(\))1728 3435 y Fo(\000)p Fs(1)1818 3469 y Fu(d)p Fr(I)1907 3435 y Fo(\003)1945 3469 y Fu(\()p Fr(j)2016 3435 y Fq(R)2071 3469 y Fr(;)c Fu([)p Fp(jr)p Fr(!)s Fp(j)2301 3435 y Fs(2)2338 3469 y Fr(;)g(j)2414 3435 y Fq(R)2468 3469 y Fu(]\))p Fp(jj)0 3515 y @beginspecial @setspecial @endspecial 1120 3657 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 3622 y Fo(\000)p Fq(\017)1424 3657 y Fp(h)p Fu(x)p Fp(i)1532 3622 y Fo(\000)p Fs(1)1622 3657 y Fu(\(1)k Fp(\000)g Fr(j)1836 3622 y Fq(R)1831 3677 y Fs(0)1891 3657 y Fu(\))p Fr(r)1960 3669 y Fq(\024)2003 3657 y Fp(jj)24 b(jjh)p Fr(k)s Fp(i)2229 3622 y Fo(\000)p Fq(\017)2313 3657 y Fp(h)p Fu(x)p Fp(i)2421 3622 y Fo(\000)p Fs(1)2510 3657 y Fr(j)2549 3622 y Fq(R)2544 3677 y Fs(0)2604 3657 y Fr(br)2677 3669 y Fq(\024)2720 3657 y Fp(jj)0 3702 y @beginspecial @setspecial @endspecial 1120 3844 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 3810 y Fo(\000)p Fq(\017)1424 3844 y Fp(h)p Fu(x)p Fp(i)1532 3810 y Fo(\000)p Fs(1)1622 3844 y Fu(\(1)18 b Fp(\000)g Fr(j)1836 3810 y Fq(R)1831 3865 y Fs(0)1891 3844 y Fu(\))p Fr(br)1996 3856 y Fq(\024)2039 3844 y Fp(jj)23 b(jjh)p Fr(k)s Fp(i)2264 3810 y Fo(\000)p Fq(\017)2348 3844 y Fp(h)p Fu(x)p Fp(i)2456 3810 y Fo(\000)p Fs(1)2546 3844 y Fr(j)2585 3810 y Fq(R)2580 3865 y Fs(0)2639 3844 y Fr(r)2676 3856 y Fq(\024)2720 3844 y Fp(jj)0 3890 y @beginspecial @setspecial @endspecial 1120 4032 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 3998 y Fo(\000)p Fq(\017)1424 4032 y Fp(h)p Fu(x)p Fp(i)1532 3998 y Fo(\000)p Fs(1)1622 4032 y Fr(j)1661 3998 y Fq(R)1656 4053 y Fs(0)1715 4032 y Fr(r)1752 4044 y Fq(\024)1796 4032 y Fp(jj)g(jjh)p Fr(k)s Fp(i)2021 3998 y Fo(\000)p Fq(\017)2105 4032 y Fp(h)p Fu(x)p Fp(i)2213 3998 y Fo(\000)p Fs(1)2303 4032 y Fr(j)2342 3998 y Fq(R)2337 4053 y Fo(1)2407 4032 y Fr(br)2480 4044 y Fq(\024)2523 4032 y Fp(jj)0 4078 y @beginspecial @setspecial @endspecial 1120 4220 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 4186 y Fo(\000)p Fq(\017)1424 4220 y Fp(h)p Fu(x)p Fp(i)1532 4186 y Fo(\000)p Fs(1)1622 4220 y Fr(j)1661 4186 y Fq(R)1656 4240 y Fs(0)1715 4220 y Fr(br)1788 4232 y Fq(\024)1831 4220 y Fp(jj)h(jjh)p Fr(k)s Fp(i)2057 4186 y Fo(\000)p Fq(\017)2141 4220 y Fp(h)p Fu(x)p Fp(i)2249 4186 y Fo(\000)p Fs(1)2338 4220 y Fr(j)2377 4186 y Fq(R)2372 4240 y Fo(1)2443 4220 y Fr(r)2480 4232 y Fq(\024)2523 4220 y Fp(jj)0 4266 y @beginspecial @setspecial @endspecial 1120 4408 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 4373 y Fo(\000)p Fq(\017)1424 4408 y Fp(h)p Fu(x)p Fp(i)1532 4373 y Fo(\000)p Fs(1)1622 4408 y Fr(j)1661 4373 y Fq(R)1656 4428 y Fo(1)1726 4408 y Fr(br)1799 4420 y Fq(\024)1843 4408 y Fp(jj)f(jjh)p Fr(k)s Fp(i)2068 4373 y Fo(\000)p Fq(\017)2152 4408 y Fp(h)p Fu(x)p Fp(i)2260 4373 y Fo(\000)p Fs(1)2350 4408 y Fr(j)2389 4373 y Fq(R)2384 4428 y Fo(1)2454 4408 y Fr(r)2491 4420 y Fq(\024)2535 4408 y Fp(jj)0 4453 y @beginspecial @setspecial @endspecial 1120 4595 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 4561 y Fo(\000)p Fq(\017)1424 4595 y Fp(h)p Fu(x)p Fp(i)1532 4561 y Fo(\000)p Fs(1)1622 4595 y Fr(br)1695 4607 y Fq(\024)1738 4595 y Fp(jj)g(jjh)p Fr(k)s Fp(i)1963 4561 y Fo(\000)p Fq(\017)2047 4595 y Fp(h)p Fu(x)p Fp(i)2155 4561 y Fo(\000)p Fs(1)2245 4595 y Fu(\(1)18 b Fp(\000)g Fr(j)2459 4561 y Fq(R)2454 4616 y Fs(0)2514 4595 y Fu(\))p Fr(r)2583 4607 y Fq(\024)2626 4595 y Fp(jj)0 4641 y @beginspecial @setspecial @endspecial 1120 4783 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 4749 y Fo(\000)p Fq(\017)1424 4783 y Fp(h)p Fu(x)p Fp(i)1532 4749 y Fo(\000)p Fs(1)1622 4783 y Fr(r)1659 4795 y Fq(\024)1702 4783 y Fp(jj)24 b(jjh)p Fr(k)s Fp(i)1928 4749 y Fo(\000)p Fq(\017)2012 4783 y Fp(h)p Fu(x)p Fp(i)2120 4749 y Fo(\000)p Fs(1)2209 4783 y Fu(\(1)19 b Fp(\000)f Fr(j)2424 4749 y Fq(R)2419 4804 y Fs(0)2478 4783 y Fu(\))p Fr(br)2583 4795 y Fq(\024)2626 4783 y Fp(jj)0 4829 y @beginspecial @setspecial @endspecial 1120 4971 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 4937 y Fo(\000)p Fq(\017)1424 4971 y Fp(h)p Fu(x)p Fp(i)1532 4937 y Fo(\000)p Fs(1)1622 4971 y Fr(r)1659 4983 y Fq(\024)1702 4971 y Fp(jj)24 b(jjh)p Fr(k)s Fp(i)1928 4937 y Fo(\000)p Fq(\017)2012 4971 y Fp(h)p Fu(x)p Fp(i)2120 4937 y Fo(\000)p Fs(1)2209 4971 y Fr(j)2248 4937 y Fq(R)2243 4992 y Fo(1)2314 4971 y Fr(br)2387 4983 y Fq(\024)2430 4971 y Fp(jj)0 5017 y @beginspecial @setspecial @endspecial 1120 5159 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 5124 y Fo(\000)p Fq(\017)1424 5159 y Fp(h)p Fu(x)p Fp(i)1532 5124 y Fo(\000)p Fs(1)1622 5159 y Fr(br)1695 5171 y Fq(\024)1738 5159 y Fp(jj)f(jjh)p Fr(k)s Fp(i)1963 5124 y Fo(\000)p Fq(\017)2047 5159 y Fp(h)p Fu(x)p Fp(i)2155 5124 y Fo(\000)p Fs(1)2245 5159 y Fr(j)2284 5124 y Fq(R)2279 5179 y Fo(1)2350 5159 y Fr(r)2387 5171 y Fq(\024)2430 5159 y Fp(jj)0 5204 y @beginspecial @setspecial @endspecial 1120 5347 a Fu(+)p Fp(jjh)p Fr(k)s Fp(i)1341 5312 y Fo(\000)p Fq(\017)1424 5347 y Fp(h)p Fu(x)p Fp(i)1532 5312 y Fo(\000)p Fs(1)1622 5347 y Fu(\(1)18 b Fp(\000)g Fr(j)1836 5312 y Fq(R)1831 5367 y Fs(0)1891 5347 y Fu(\))p Fr(br)1996 5359 y Fq(\024)2039 5347 y Fp(jj)h Fu(+)f Fp(jjh)p Fr(k)s Fp(i)2343 5312 y Fo(\000)p Fq(\017)2427 5347 y Fp(h)p Fu(x)p Fp(i)2535 5312 y Fo(\000)p Fs(1)2624 5347 y Fr(j)2663 5312 y Fq(R)2658 5367 y Fo(1)2729 5347 y Fr(br)2802 5359 y Fq(\024)2845 5347 y Fp(jj)p Fu(\))p Fr(:)100 5506 y Fu(Using)27 b(Lemma)g @beginspecial @setspecial @endspecial(A.2,)h(Lemma)f(3.3)g(and)g(5.4)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\))27 b(w)n(e)h(obtain)f Fr(C)6 b Fu(\()p Fr(z)t Fu(\))23 b Fp(2)h(j)p Fu(Im\()p Fr(z)t Fu(\))p Fp(j)2579 5476 y Fo(\000)p Fs(2)2668 5506 y Fr(o)p Fu(\()p Fr(R)2804 5476 y Fs(0)2841 5506 y Fu(\))k(uniformly)g (in)g @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)387 b Fh(\004)0 5527 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial eop %%Page: 28 28 28 27 bop 0 226 a Fs(28)1695 b(Z.)29 b(AMMARI)0 489 y Fz(Theorem)h(5.7.)40 b Fn(The)31 b(fol)t(lowing)h(thr)l(e)l(e)e (assertions)g(hold:)4 589 y Fu(_)-27 b Fr({)p Fu(\))30 b Fn(L)l(et)f @beginspecial @setspecial @endspecial Fr(\025)24 b Fp(2)f Fk(R)h Fp(n)18 b Fr(\034)5 b(:)30 b Fn(Then)h(ther)l(e)e (exist)h Fr(\017)23 b(>)f Fu(0)p Fr(;)14 b(C)1513 601 y Fs(0)1573 589 y Fr(>)23 b Fu(0)29 b Fn(and)h(c)l(omp)l(act)h(op)l(er) l(ator)f @beginspecial @setspecial @endspecial Fr(K)2611 601 y Fs(0)2678 589 y Fn(such)g(that)725 733 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(1)-13 b(l)776 748 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])1075 733 y Fu(\()1129 712 y(^)1107 733 y Fr(H)1176 745 y Fo(1)1246 733 y Fu(\)[)1323 712 y(^)1301 733 y Fr(H)1370 745 y Fo(1)1441 733 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-13 b(l)1643 748 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])1941 733 y Fu(\()1995 712 y(^)1973 733 y Fr(H)2042 745 y Fo(1)2113 733 y Fu(\))23 b Fp(\025)46 b Fr(C)2338 745 y Fs(0)2389 733 y Fu(1)-13 b(l)2440 748 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])2739 733 y Fu(\()2793 712 y(^)2771 733 y Fr(H)2840 745 y Fo(1)2911 733 y Fu(\))18 b(+)g Fr(K)3115 745 y Fs(0)3152 733 y Fr(:)p @beginspecial @setspecial @endspecial 0 869 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b(al)t(l)g Fu([)p Fr(\025)460 881 y Fs(1)498 869 y Fr(;)14 b(\025)583 881 y Fs(2)621 869 y Fu(])29 b Fn(such)h(that)g @beginspecial @setspecial @endspecial Fu([)p Fr(\025)1103 881 y Fs(1)1141 869 y Fr(;)14 b(\025)1226 881 y Fs(2)1263 869 y Fu(])19 b Fp(\\)g Fr(\034)32 b Fu(=)23 b Fp(;)p Fn(,)30 b(one)g(has)1539 1013 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(dim)p Fu(1)-13 b(l)1735 973 y Fs(pp)1735 1041 y([)p Fq(\025)1793 1049 y Fi(1)1826 1041 y Fq(;\025)1885 1049 y Fi(2)1917 1041 y Fs(])1940 1013 y Fu(\()1994 992 y(^)1972 1013 y Fr(H)2041 1025 y Fo(1)2112 1013 y Fu(\))23 b Fr(<)g Fp(1)p Fr(:)p @beginspecial @setspecial @endspecial 0 1179 a Fn(Conse)l(quently)30 b @beginspecial @setspecial @endspecial Fr(\033)550 1191 y Fs(pp)628 1179 y Fu(\()682 1158 y(^)660 1179 y Fr(H)729 1191 y Fo(1)799 1179 y Fu(\))g Fn(c)l(an)g(ac)l(cumulate)g(only)g(at)g Fr(\034)9 b Fn(,)31 b(which)g(is)f(a)g(close)l(d)h(c)l(ountable)f(set.) 0 1289 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(L)l(et)g Fr(\025)23 b Fp(2)h Fk(R)g Fp(n)18 b Fu(\()p Fr(\034)28 b Fp([)19 b Fr(\033)791 1301 y Fs(pp)869 1289 y Fu(\()923 1268 y(^)901 1289 y Fr(H)970 1301 y Fo(1)1041 1289 y Fu(\)\))p Fn(.)39 b(Then)30 b(ther)l(e)g(exists)f @beginspecial @setspecial @endspecial Fr(\017)23 b(>)g Fu(0)p Fr(;)14 b(C)2102 1301 y Fs(0)2162 1289 y Fr(>)23 b Fu(0)29 b Fn(such)h(that)830 1436 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(1)-13 b(l)881 1451 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])1179 1436 y Fu(\()1233 1415 y(^)1211 1436 y Fr(H)1280 1448 y Fo(1)1351 1436 y Fu(\)[)1428 1415 y(^)1406 1436 y Fr(H)1475 1448 y Fo(1)1546 1436 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-13 b(l)1747 1451 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])2046 1436 y Fu(\()2100 1415 y(^)2078 1436 y Fr(H)2147 1448 y Fo(1)2217 1436 y Fu(\))24 b Fp(\025)45 b Fr(C)2442 1448 y Fs(0)2494 1436 y Fu(1)-13 b(l)2545 1451 y Fs([)p Fq(\025)p Fo(\000)p Fq(\017;\025)p Fs(+)p Fq(\017)p Fs(])2843 1436 y Fu(\()2898 1415 y(^)2875 1436 y Fr(H)2944 1448 y Fo(1)3015 1436 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 1587 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(set)615 1768 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(d)p Fu(\()p Fr(\025)p Fu(\))d(:=)d(inf)7 b Fp(f)1087 1665 y Fq(n)1048 1689 y Fl(X)1054 1866 y Fq(i)p Fs(=1)1196 1768 y Fp(jr)p Fr(!)s Fu(\()p Fr(k)1418 1780 y Fq(i)1446 1768 y Fu(\))p Fp(j)1501 1734 y Fs(2)1561 1768 y Fp(j)46 b Fr(\034)29 b Fu(+)1817 1665 y Fq(n)1778 1689 y Fl(X)1784 1866 y Fq(i)p Fs(=1)1925 1768 y Fr(!)s Fu(\()p Fr(k)2055 1780 y Fq(i)2083 1768 y Fu(\))23 b(=)g Fr(\025;)37 b(n)23 b Fu(=)g(1)p Fr(;)14 b Fu(2)p Fr(:::;)36 b(\034)d Fp(2)23 b Fr(\033)2938 1780 y Fs(pp)3016 1768 y Fu(\()3070 1747 y(^)3048 1768 y Fr(H)3117 1780 y Fo(1)3188 1768 y Fu(\))p Fp(g)p Fr(;)p @beginspecial @setspecial @endspecial 597 2092 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 612 2071 a Fu(~)597 2092 y Fr(d)p Fu(\()p Fr(\025)p Fu(\))h(:=)f(inf)7 b Fp(f)1069 1989 y Fq(n)1030 2014 y Fl(X)1036 2190 y Fq(i)p Fs(=1)1177 2092 y Fp(jr)p Fr(!)s Fu(\()p Fr(k)1399 2104 y Fq(i)1427 2092 y Fu(\))p Fp(j)1482 2058 y Fs(2)1543 2092 y Fp(j)46 b Fr(\034)28 b Fu(+)1799 1989 y Fq(n)1759 2014 y Fl(X)1765 2190 y Fq(i)p Fs(=1)1907 2092 y Fr(!)s Fu(\()p Fr(k)2037 2104 y Fq(i)2065 2092 y Fu(\))23 b(=)g Fr(\025;)37 b(n)23 b Fu(=)f(0)p Fr(;)14 b Fu(1)p Fr(;)g(:::;)36 b(\034)d Fp(2)24 b Fr(\033)2957 2104 y Fs(pp)3034 2092 y Fu(\()3089 2071 y(^)3066 2092 y Fr(H)3135 2104 y Fo(1)3206 2092 y Fu(\))p Fp(g)p Fr(:)p @beginspecial @setspecial @endspecial 1442 2344 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\001)1511 2304 y Fq(\026)1511 2369 y(\025)1579 2344 y Fu(:=)f([)p Fr(\025)c Fp(\000)f Fr(\026;)c(\025)k Fu(+)g Fr(\026)p Fu(])p Fr(;)37 b(\026)24 b(>)e Fu(0)p Fr(;)p @beginspecial @setspecial @endspecial 1599 2516 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(d)1642 2482 y Fq(\026)1687 2516 y Fu(\()p Fr(\025)p Fu(\))i(:=)60 b(inf)1934 2576 y Fq(\027)t Fo(2)p Fs(\001)2071 2548 y Fj(\026)2071 2596 y(\025)2124 2516 y Fr(d)p Fu(\()p Fr(\027)5 b Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1599 2763 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1614 2741 a Fu(~)1599 2763 y Fr(d)1642 2729 y Fq(\026)1687 2763 y Fu(\()p Fr(\025)p Fu(\))24 b(:=)60 b(inf)1934 2823 y Fq(\027)t Fo(2)p Fs(\001)2071 2795 y Fj(\026)2071 2843 y(\025)2138 2741 y Fu(~)2124 2763 y Fr(d)p Fu(\()p Fr(\027)5 b Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1638 3010 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(E)1699 3022 y Fs(0)1759 3010 y Fu(:=)23 b(inf)e Fr(\033)s Fu(\()2089 2989 y(^)2067 3010 y Fr(H)2136 3022 y Fo(1)2207 3010 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3146 a Fu(W)-7 b(e)31 b(will)f(follo)n(w)f(the)i(logic)e(of)i(the)f(pro)r(of)g(of)g (Mourre)f(estimate)h(in)h(the)f(case)g(of)g(a)f(P)n(auli-Fierz)g (Hamiltonian)h @beginspecial @setspecial @endspecial([11)o(].)45 b(Let)0 3246 y(us)28 b(recall)e(the)i(statemen)n(ts)f(that)h(w)n(e)g (will)f(pro)n(v)n(e)f(b)n(y)i(induction)g(in)g @beginspecial @setspecial @endspecial Fr(n)p Fu(:)0 3346 y Fr(H)69 3358 y Fs(1)106 3346 y Fu(\()p Fr(n)p Fu(\):)38 b(Let)27 b @beginspecial @setspecial @endspecial Fr(\017)c(>)g Fu(0)k(and)g @beginspecial @setspecial @endspecial Fr(\025)d Fp(2)f Fu([)p Fr(E)1038 3358 y Fs(0)1076 3346 y Fr(;)14 b(E)1174 3358 y Fs(0)1230 3346 y Fu(+)k Fr(nm)p Fu([.)37 b(There)27 b(exists)g(a)g(compact)g(op)r(erator)f @beginspecial @setspecial @endspecial Fr(K)2795 3358 y Fs(0)2832 3346 y Fu(,)h(an)h(in)n(terv)-5 b(al)27 b(\001)c Fp(3)g Fr(\025)29 b Fu(suc)n(h)e(that)976 3490 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-15 b(l)1026 3502 y Fs(\001)1084 3490 y Fu(\()1138 3469 y(^)1116 3490 y Fr(H)1185 3502 y Fo(1)1256 3490 y Fu(\)[)1333 3469 y(^)1311 3490 y Fr(H)1380 3502 y Fo(1)1451 3490 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-15 b(l)1650 3502 y Fs(\001)1709 3490 y Fu(\()1763 3469 y(^)1741 3490 y Fr(H)1810 3502 y Fo(1)1881 3490 y Fu(\))23 b Fp(\025)f Fu(\()p Fr(d)p Fu(\()p Fr(\025)p Fu(\))e Fp(\000)e Fr(\017)p Fu(\)1)-15 b(l)2429 3502 y Fs(\001)2487 3490 y Fu(\()2542 3469 y(^)2519 3490 y Fr(H)2588 3502 y Fo(1)2659 3490 y Fu(\))19 b(+)f Fr(K)2864 3502 y Fs(0)2901 3490 y Fr(:)p @beginspecial @setspecial @endspecial 0 3625 a @beginspecial @setspecial @endspecial(H)69 3637 y Fs(2)106 3625 y Fu(\()p Fr(n)p Fu(\):)38 b(Let)27 b Fr(\017)c(>)g Fu(0)k(and)g Fr(\025)d Fp(2)f Fu([)p Fr(E)1038 3637 y Fs(0)1076 3625 y Fr(;)14 b(E)1174 3637 y Fs(0)1230 3625 y Fu(+)k Fr(nm)p Fu([.)37 b(There)27 b(exists)g(an)g(in)n(terv)-5 b(al)27 b @beginspecial @setspecial @endspecial(\001)c Fp(3)h Fr(\025)k Fu(suc)n(h)f(that)1081 3769 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-15 b(l)1130 3781 y Fs(\001)1189 3769 y Fu(\()1243 3748 y(^)1221 3769 y Fr(H)1290 3781 y Fo(1)1361 3769 y Fu(\)[)1438 3748 y(^)1416 3769 y Fr(H)1485 3781 y Fo(1)1555 3769 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-15 b(l)1755 3781 y Fs(\001)1813 3769 y Fu(\()1868 3748 y(^)1845 3769 y Fr(H)1914 3781 y Fo(1)1985 3769 y Fu(\))23 b Fp(\025)g Fu(\()2175 3747 y(~)2160 3769 y Fr(d)q Fu(\()p Fr(\025)p Fu(\))c Fp(\000)f Fr(\017)p Fu(\)1)-15 b(l)2533 3781 y Fs(\001)2592 3769 y Fu(\()2646 3748 y(^)2624 3769 y Fr(H)2693 3781 y Fo(1)2764 3769 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3905 a @beginspecial @setspecial @endspecial(H)69 3917 y Fs(3)106 3905 y Fu(\()p Fr(n)p Fu(\):)38 b(Let)27 b Fr(\026)c(>)g Fu(0)p Fr(;)14 b(\017)703 3917 y Fs(0)763 3905 y Fr(>)22 b Fu(0.)37 b(There)27 b(exists)g Fr(\016)f(>)d Fu(0)k(suc)n(h)g(that)h(for)f(all)g @beginspecial @setspecial @endspecial Fr(\025)d Fp(2)f Fu([)p Fr(E)2484 3917 y Fs(0)2522 3905 y Fr(;)14 b(E)2620 3917 y Fs(0)2676 3905 y Fu(+)k Fr(nm)g Fp(\000)g Fr(\017)3017 3917 y Fs(0)3054 3905 y Fu(],)28 b(one)f(has)1000 4049 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-15 b(l)1049 4069 y Fs(\001)1104 4049 y Fj(\016)1104 4089 y(\025)1147 4049 y Fu(\()1201 4028 y(^)1179 4049 y Fr(H)1248 4061 y Fo(1)1319 4049 y Fu(\)[)1396 4028 y(^)1374 4049 y Fr(H)1443 4061 y Fo(1)1514 4049 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-15 b(l)1713 4069 y Fs(\001)1768 4049 y Fj(\016)1768 4089 y(\025)1811 4049 y Fu(\()1865 4028 y(^)1843 4049 y Fr(H)1912 4061 y Fo(1)1982 4049 y Fu(\))24 b Fp(\025)e Fu(\()2172 4027 y(~)2157 4049 y Fr(d)2200 4015 y Fq(\026)2245 4049 y Fu(\()p Fr(\025)p Fu(\))e Fp(\000)e Fr(\017)p Fu(\)1)-15 b(l)2575 4069 y Fs(\001)2630 4049 y Fj(\016)2630 4089 y(\025)2673 4049 y Fu(\()2727 4028 y(^)2705 4049 y Fr(H)2774 4061 y Fo(1)2845 4049 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4196 a @beginspecial @setspecial @endspecial(S)51 4208 y Fs(1)88 4196 y Fu(\()p Fr(n)p Fu(\):)38 b Fr(\034)f Fu(is)27 b(closed)g(coun)n(table)g(set)h(in)g([)p Fr(E)1351 4208 y Fs(0)1388 4196 y Fr(;)14 b(E)1486 4208 y Fs(0)1542 4196 y Fu(+)k Fr(nm)p Fu(].)0 4296 y @beginspecial @setspecial @endspecial Fr(S)51 4308 y Fs(2)88 4296 y Fu(\()p Fr(n)p Fu(\):)38 b(F)-7 b(or)26 b(all)i Fr(\025)575 4308 y Fs(1)636 4296 y Fp(\024)22 b Fr(\025)771 4308 y Fs(2)832 4296 y Fp(\024)h Fr(E)981 4308 y Fs(0)1037 4296 y Fu(+)18 b Fr(nm)27 b Fu(with)h @beginspecial @setspecial @endspecial([)p Fr(\025)1530 4308 y Fs(1)1568 4296 y Fr(;)14 b(\025)1653 4308 y Fs(2)1690 4296 y Fu(])19 b Fp(\\)g Fr(\034)32 b Fu(=)23 b Fp(;)p Fu(,)k(one)h(has)1540 4440 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(dim)p Fu(1)-15 b(l)1734 4400 y Fs(pp)1734 4468 y([)p Fq(\025)1792 4476 y Fi(1)1825 4468 y Fq(;\025)1884 4476 y Fi(2)1916 4468 y Fs(])1939 4440 y Fu(\()1993 4419 y(^)1971 4440 y Fr(H)2040 4452 y Fo(1)2111 4440 y Fu(\))23 b Fr(<)g Fp(1)p Fr(:)p @beginspecial @setspecial @endspecial 0 4585 a Fu(The)28 b(sk)n(etc)n(h)e(of)i(the)g(pro)r(of)f(is)h(giv)n(en)e (b)n(y)1421 4985 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1462 4715 a @beginspecial @setspecial @endspecial Fr(S)1513 4727 y Fs(2)1550 4715 y Fu(\()p Fr(n)19 b Fp(\000)f Fu(1\))23 b Fp(\))g Fr(S)1988 4727 y Fs(1)2025 4715 y Fu(\()p Fr(n)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1462 4850 a(S)1513 4862 y Fs(1)1550 4850 y Fu(\()p Fr(n)p Fu(\))p Fr(;)14 b(H)1770 4862 y Fs(3)1808 4850 y Fu(\()p Fr(n)k Fp(\000)g Fu(1\))23 b Fp(\))g Fr(H)2263 4862 y Fs(1)2301 4850 y Fu(\()p Fr(n)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1462 4984 a(H)1531 4996 y Fs(1)1568 4984 y Fu(\()p Fr(n)p Fu(\))h Fp(\))f Fr(H)1881 4996 y Fs(2)1918 4984 y Fu(\()p Fr(n)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1462 5119 a @beginspecial @setspecial @endspecial(H)1531 5131 y Fs(2)1568 5119 y Fu(\()p Fr(n)p Fu(\))h Fp(\))f Fr(H)1881 5131 y Fs(3)1918 5119 y Fu(\()p Fr(n)p Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1462 5253 a @beginspecial @setspecial @endspecial(H)1531 5265 y Fs(1)1568 5253 y Fu(\()p Fr(n)p Fu(\))h Fp(\))f Fr(S)1863 5265 y Fs(2)1900 5253 y Fu(\()p Fr(n)p Fu(\))p Fr(:)2479 4985 y @beginspecial @setspecial @endspecial 0 5406 a @beginspecial @setspecial @endspecial(H)7 b Fu(\(1\))26 b(and)f Fr(S)5 b Fu(\(1\))26 b(are)f(immediate)h(b)r(ecause)f(the)i(sp) r(ectrum)e(of)2018 5385 y(^)1996 5406 y Fr(H)2065 5418 y Fo(1)2161 5406 y Fu(is)h(discrete)f(in)i @beginspecial @setspecial @endspecial([)p Fr(E)2726 5418 y Fs(0)2763 5406 y Fr(;)14 b(E)2861 5418 y Fs(0)2913 5406 y Fu(+)h Fr(m)p Fu([.)36 b Fr(S)3199 5418 y Fs(2)3236 5406 y Fu(\()p Fr(n)15 b Fp(\000)g Fu(1\))23 b Fp(\))g Fr(S)3667 5418 y Fs(1)3704 5406 y Fu(\()p Fr(n)p Fu(\))j(is)0 5506 y(ob)n(vious.)33 b @beginspecial @setspecial @endspecial Fr(H)399 5518 y Fs(1)437 5506 y Fu(\()p Fr(n)p Fu(\))23 b Fp(\))g Fr(H)749 5518 y Fs(2)786 5506 y Fu(\()p Fr(n)p Fu(\))p Fr(;)14 b(H)1006 5518 y Fs(2)1044 5506 y Fu(\()p Fr(n)p Fu(\))23 b Fp(\))g Fr(H)1356 5518 y Fs(3)1394 5506 y Fu(\()p Fr(n)p Fu(\))e(follo)n(w)f(using)g(argumen)n(ts)g(in)h @beginspecial @setspecial @endspecial([9)o(],[20)o(].)35 b(The)21 b(implication)g @beginspecial @setspecial @endspecial Fr(H)3417 5518 y Fs(1)3454 5506 y Fu(\()p Fr(n)p Fu(\))i Fp(\))g Fr(S)3748 5518 y Fs(2)3786 5506 y Fu(\()p Fr(n)p Fu(\))p eop %%Page: 29 29 29 28 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(29)0 489 y Fu(is)33 b(based)f(in)i(the)f(Virial) f(theorem)h(whic)n(h)g(holds)g(since)f @beginspecial @setspecial @endspecial 1839 468 a(^)1817 489 y Fr(H)1886 501 y Fo(1)1989 489 y Fp(2)g Fr(C)2141 459 y Fs(1)2179 489 y Fu(\()p Fr(B)t Fu(\),)j(see)d([1,)i(Prop.)52 b(7.2.10].)g(So)32 b(w)n(e)h(ha)n(v)n(e)f(only)g(to)0 589 y(pro)n(v)n(e)26 b(the)i(implication)f @beginspecial @setspecial @endspecial(\()p Fr(S)883 601 y Fs(1)921 589 y Fu(\()p Fr(n)p Fu(\))p Fr(;)14 b(H)1141 601 y Fs(3)1178 589 y Fu(\()p Fr(n)19 b Fp(\000)f Fu(1\))23 b Fp(\))g Fr(H)1634 601 y Fs(1)1671 589 y Fu(\()p Fr(n)p Fu(\)\).)0 688 y(Let)28 b @beginspecial @setspecial @endspecial Fr(\037)23 b Fp(2)g Fr(C)367 658 y Fo(1)361 709 y Fs(0)438 688 y Fu(\()p Fk(R)p Fu(\))617 858 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()723 837 y(^)701 858 y Fr(H)770 870 y Fq(\024)813 858 y Fu(\))84 b(=)e Fr(I)7 b Fu(\()p Fr(j)1190 824 y Fq(R)1245 858 y Fu(\)1)-15 b(l)1327 873 y Fo(f)p Fs(0)p Fo(g)1432 858 y Fu(\()p Fr(N)1531 870 y Fo(1)1601 858 y Fu(\))p Fr(I)1676 824 y Fo(\003)1715 858 y Fu(\()p Fr(j)1786 824 y Fq(R)1840 858 y Fu(\))p Fr(\037)p Fu(\()1979 837 y(^)1956 858 y Fr(H)2025 870 y Fq(\024)2069 858 y Fu(\))19 b(+)f Fr(I)7 b Fu(\()p Fr(j)2317 824 y Fq(R)2371 858 y Fu(\)1)-15 b(l)2453 873 y Fs([1)p Fq(;)p Fo(1)p Fs([)2614 858 y Fu(\()p Fr(N)2713 870 y Fo(1)2783 858 y Fu(\))p Fr(I)2858 824 y Fo(\003)2896 858 y Fu(\()p Fr(j)2967 824 y Fq(R)3022 858 y Fu(\))p Fr(\037)p Fu(\()3160 837 y(^)3138 858 y Fr(H)3207 870 y Fq(\024)3251 858 y Fu(\))-3283 b(\(5.8\))0 912 y @beginspecial @setspecial @endspecial 846 1061 a @beginspecial @setspecial @endspecial 83 w(=)82 b(\000\()p Fr(q)1200 1027 y Fq(R)1255 1061 y Fu(\))p Fr(\037)p Fu(\()1393 1040 y(^)1371 1061 y Fr(H)1440 1073 y Fq(\024)1483 1061 y Fu(\))19 b(+)f Fr(I)7 b Fu(\()p Fr(j)1731 1027 y Fq(R)1786 1061 y Fu(\)1)-15 b(l)1868 1076 y Fs([1)p Fq(;)p Fo(1)p Fs([)2028 1061 y Fu(\()p Fr(N)2127 1073 y Fo(1)2198 1061 y Fu(\))p Fr(\037)p Fu(\()2336 1040 y(^)2314 1061 y Fr(H)2390 1027 y Fs(ext)2383 1082 y Fq(\024)2484 1061 y Fu(\))p Fr(I)2559 1027 y Fo(\003)2598 1061 y Fu(\()p Fr(j)2669 1027 y Fq(R)2724 1061 y Fu(\))18 b(+)g Fr(o)p Fu(\()p Fr(R)2993 1027 y Fs(0)3031 1061 y Fu(\))p Fr(:)-3086 b Fu(\(5.9\))0 1231 y(\()p @beginspecial @setspecial @endspecial(5.8\))27 b(follo)n(ws)g(from)g(the)h(fact)g(that)g Fr(I)7 b Fu(\()p Fr(j)1268 1201 y Fq(R)1322 1231 y Fu(\))p Fr(I)1397 1201 y Fo(\003)1436 1231 y Fu(\()p Fr(j)1507 1201 y Fq(R)1562 1231 y Fu(\))23 b(=)g(1)-15 b(l)27 b(and)g @beginspecial @setspecial @endspecial Fr(I)7 b Fu(\()p Fr(j)2057 1201 y Fq(R)2112 1231 y Fu(\))28 b(is)f(b)r(ounded.)38 b(\(5.9\))27 b(follo)n(ws)f(from)i(the)g(fact)f(that:)701 1396 y @beginspecial @setspecial @endspecial(\000\()p Fr(q)825 1361 y Fq(R)880 1396 y Fu(\))c(=)g Fr(I)7 b Fu(\()p Fr(j)1137 1361 y Fq(R)1191 1396 y Fu(\)1)-15 b(l)1273 1411 y Fo(f)p Fs(0)p Fo(g)1378 1396 y Fu(\()p Fr(N)1477 1408 y Fo(1)1548 1396 y Fu(\))p Fr(I)1623 1361 y Fo(\003)1661 1396 y Fu(\()p Fr(j)1732 1361 y Fq(R)1787 1396 y Fu(\))p Fr(;)83 b(q)1965 1361 y Fq(R)2043 1396 y Fu(=)22 b(\()p Fr(j)2201 1361 y Fq(R)2256 1396 y Fu(\))2288 1361 y Fs(2)2326 1396 y Fu(;)83 b Fr(I)7 b Fu(\()p Fr(j)2546 1361 y Fq(R)2600 1396 y Fu(\))24 b(=)e(\000\()p Fr(j)2866 1361 y Fq(R)2921 1396 y Fu(\))p Fr(U;)83 b @beginspecial @setspecial @endspecial 0 1575 a Fu(and)25 b(Lemma)g(4.15.)35 b(W)-7 b(e)26 b(notice)f(that)g(the)h(term)f @beginspecial @setspecial @endspecial(\000\()p Fr(q)1681 1544 y Fq(R)1736 1575 y Fu(\))p Fr(\037)p Fu(\()1874 1554 y(^)1852 1575 y Fr(H)1921 1587 y Fq(\024)1964 1575 y Fu(\))h(is)f(compact)g(since)g @beginspecial @setspecial @endspecial(\000\()p Fr(q)2758 1544 y Fq(R)2813 1575 y Fu(\)\()p Fr(H)2946 1587 y Fs(0)2997 1575 y Fu(+)14 b(1\))3150 1544 y Fo(\000)3211 1522 y Fi(1)p 3211 1531 29 3 v 3211 1564 a(2)3279 1575 y Fu(is)25 b(compact)g(whic)n(h)0 1674 y(is)i(pro)n(v)n(ed)g(in)g @beginspecial @setspecial @endspecial([11)o(,)h(Lemma)23 b(4)p Fr(:)p Fu(2].)0 1774 y(Let)31 b Fr(\025)d Fp(2)g Fu([)p Fr(E)395 1786 y Fs(0)433 1774 y Fr(;)14 b(E)531 1786 y Fs(0)589 1774 y Fu(+)20 b Fr(nm)p Fu([.)45 b(Since)31 b @beginspecial @setspecial @endspecial Fr(S)1159 1786 y Fs(2)1196 1774 y Fu(\()p Fr(n)21 b Fp(\000)f Fu(1\))27 b Fp(\))h Fr(S)1647 1786 y Fs(1)1684 1774 y Fu(\()p Fr(n)p Fu(\),)k(the)f(set)g Fr(\034)40 b Fu(is)30 b(closed)g(in)h @beginspecial @setspecial @endspecial([)p Fr(E)2726 1786 y Fs(0)2764 1774 y Fr(;)14 b(E)2862 1786 y Fs(0)2919 1774 y Fu(+)20 b Fr(nm)p Fu(],)31 b(whic)n(h)g(giv)n(es)e Fr(d)p Fu(\()p Fr(\025)p Fu(\))g(=)0 1873 y(sup)125 1894 y Fq(\026>)p Fs(0)268 1873 y Fr(d)311 1843 y Fq(\026)356 1873 y Fu(\()p Fr(\025)p Fu(\).)38 b(So)27 b(w)n(e)h(can)f(c)n(ho)r (ose)f @beginspecial @setspecial @endspecial Fr(\026)i Fu(suc)n(h)f(that)h @beginspecial @setspecial @endspecial Fr(d)1670 1843 y Fq(\026)1715 1873 y Fu(\()p Fr(\025)p Fu(\))c Fp(\025)e Fr(d)p Fu(\()p Fr(\025)p Fu(\))e Fp(\000)2209 1841 y Fq(\017)p 2206 1855 34 4 v 2206 1902 a Fs(3)2249 1873 y Fu(.)37 b Fr(H)2378 1885 y Fs(3)2415 1873 y Fu(\()p Fr(n)19 b Fp(\000)f Fu(1\))27 b(giv)n(es)470 2067 y @beginspecial @setspecial @endspecial(1)-15 b(l)520 2087 y Fs(\001)575 2067 y Fj(\016)575 2108 y(\025)610 2120 y Fi(1)650 2067 y Fu(\()704 2046 y(^)682 2067 y Fr(H)751 2079 y Fo(1)822 2067 y Fu(\)[)899 2046 y(^)877 2067 y Fr(H)946 2079 y Fo(1)1016 2067 y Fr(;)14 b Fu(i)p Fr(B)t Fu(]1)-15 b(l)1216 2087 y Fs(\001)1271 2067 y Fj(\016)1271 2108 y(\025)1306 2120 y Fi(1)1346 2067 y Fu(\()1400 2046 y(^)1378 2067 y Fr(H)1447 2079 y Fo(1)1517 2067 y Fu(\))24 b Fp(\025)e Fu(\()1707 2045 y(~)1692 2067 y Fr(d)1735 2033 y Fq(\026)1780 2067 y Fu(\()p Fr(\025)1860 2079 y Fs(1)1898 2067 y Fu(\))d Fp(\000)2046 2011 y Fr(\017)p 2042 2048 42 4 v 2042 2124 a Fu(3)2094 2067 y(\)1)-15 b(l)2175 2087 y Fs(\001)2230 2067 y Fj(\016)2230 2108 y(\025)2273 2067 y Fu(\()2327 2046 y(^)2305 2067 y Fr(H)2374 2079 y Fo(1)2445 2067 y Fu(\))p Fr(;)14 b(\025)2562 2079 y Fs(1)2623 2067 y Fr(<)22 b(E)2771 2079 y Fs(0)2827 2067 y Fu(+)c(\()p Fr(n)h Fp(\000)f Fu(1\))p Fr(m:)83 b @beginspecial @setspecial @endspecial 0 2257 a Fu(Replacing)27 b Fr(\025)432 2269 y Fs(1)497 2257 y Fu(with)h Fr(\025)19 b Fp(\000)f Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\),)29 b(w)n(e)e(obtain)348 2712 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 390 2426 a @beginspecial @setspecial @endspecial(1)-15 b(l)439 2446 y Fs(\001)494 2426 y Fj(\016)494 2467 y(\025)537 2426 y Fu(\()591 2405 y(^)569 2426 y Fr(H)638 2438 y Fo(1)727 2426 y Fu(+)18 b(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\)\([)1398 2405 y(^)1375 2426 y Fr(H)1444 2438 y Fo(1)1516 2426 y Fr(;)c Fu(i)p Fr(B)t Fu(])k(+)g(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\)\)1)-15 b(l)2217 2446 y Fs(\001)2272 2426 y Fj(\016)2272 2467 y(\025)2315 2426 y Fu(\()2369 2405 y(^)2347 2426 y Fr(H)2416 2438 y Fo(1)2505 2426 y Fu(+)18 b(1)-15 b(l)17 b Fp(\012)h Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\)1)-15 b(l)3148 2441 y Fs([1)p Fq(;)p Fo(1)p Fs([)3309 2426 y Fu(\()p Fr(N)3408 2438 y Fo(1)3478 2426 y Fu(\))p @beginspecial @setspecial @endspecial 390 2593 a Fp(\025)22 b Fu(1)-15 b(l)527 2614 y Fs(\001)582 2594 y Fj(\016)582 2634 y(\025)625 2593 y Fu(\()679 2572 y(^)657 2593 y Fr(H)726 2605 y Fo(1)815 2593 y Fu(+)18 b(1)-15 b(l)17 b Fp(\012)i Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\)\()1455 2571 y(~)1440 2593 y Fr(d)1483 2559 y Fq(\026)1528 2593 y Fu(\()p Fr(\025)h Fp(\000)e Fu(1)-15 b(l)17 b Fp(\012)h Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\))i(+)e(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fp(jr)p Fr(!)s Fp(j)2774 2559 y Fs(2)2811 2593 y Fu(\))h Fp(\000)2959 2537 y Fr(\017)p 2955 2574 V 2955 2650 a Fu(3)3006 2593 y(\)1)-15 b(l)3088 2608 y Fs([1)p Fq(;)p Fo(1)p Fs([)3249 2593 y Fu(\()p Fr(N)3348 2605 y Fo(1)3418 2593 y Fu(\))p @beginspecial @setspecial @endspecial 390 2777 a Fp(\025)22 b Fu(\()p Fr(d)552 2743 y Fq(\026)598 2777 y Fu(\()p Fr(\025)p Fu(\))d Fp(\000)826 2721 y Fr(\017)p 822 2758 V 822 2834 a Fu(3)873 2777 y(\)1)-15 b(l)955 2797 y Fs(\001)1010 2777 y Fj(\016)1010 2817 y(\025)1053 2777 y Fu(\()1107 2756 y(^)1085 2777 y Fr(H)1154 2789 y Fo(1)1243 2777 y Fu(+)18 b(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\)\()1891 2756 y(^)1868 2777 y Fr(H)1937 2789 y Fo(1)2027 2777 y Fu(+)g(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\)\)1)-15 b(l)2559 2792 y Fs([1)p Fq(;)p Fo(1)p Fs([)2720 2777 y Fu(\()p Fr(N)2819 2789 y Fo(1)2890 2777 y Fu(\))p @beginspecial @setspecial @endspecial 390 2979 a Fp(\025)22 b Fu(\()p Fr(d)552 2944 y Fq(\026)598 2979 y Fu(\()p Fr(\025)p Fu(\))d Fp(\000)822 2922 y Fu(2)p Fr(\017)p 822 2959 76 4 v 839 3036 a Fu(3)907 2979 y(\)1)-15 b(l)989 2999 y Fs(\001)1044 2979 y Fj(\016)1044 3019 y(\025)1086 2979 y Fu(\()1141 2958 y(^)1118 2979 y Fr(H)1187 2991 y Fo(1)1277 2979 y Fu(+)18 b(1)-15 b(l)17 b Fp(\012)h Fu(d\000\()p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\)\()1924 2958 y(^)1901 2979 y Fr(H)1970 2991 y Fo(1)2060 2979 y Fu(+)g(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\)\)1)-15 b(l)2593 2994 y Fs([1)p Fq(;)p Fo(1)p Fs([)2754 2979 y Fu(\()p Fr(N)2853 2991 y Fo(1)2923 2979 y Fu(\))3552 2712 y @beginspecial @setspecial @endspecial 100 3168 a(Let)27 b @beginspecial @setspecial @endspecial Fr(\037)c Fp(2)h Fr(C)467 3138 y Fo(1)461 3189 y Fs(0)537 3168 y Fu(\()p Fk(R)q Fu(\))p Fr(;)14 b(\037)745 3180 y Fs(1)811 3168 y Fp(2)23 b Fr(C)954 3138 y Fo(1)948 3189 y Fs(0)1025 3168 y Fu(\()p Fk(R)p Fu(\))34 b(suc)n(h)28 b(that)f @beginspecial @setspecial @endspecial Fr(\037)1596 3180 y Fs(1)1634 3168 y Fr(\037)c Fu(=)f Fr(\037)p Fu(.)37 b(One)27 b(has)h(uniformly)f(in)h Fr(\024)p Fu(:)677 3338 y @beginspecial @setspecial @endspecial 166 w Fr(\037)p Fu(\()949 3317 y(^)927 3338 y Fr(H)996 3350 y Fq(\024)1040 3338 y Fu(\)[)1117 3317 y(^)1095 3338 y Fr(H)1164 3350 y Fq(\024)1207 3338 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1463 3317 y(^)1441 3338 y Fr(H)1510 3350 y Fq(\024)1554 3338 y Fu(\))843 3532 y(=)23 b(\000\()p Fr(q)1055 3498 y Fq(R)1110 3532 y Fu(\))p Fr(\037)p Fu(\()1248 3511 y(^)1226 3532 y Fr(H)1295 3544 y Fq(\024)1338 3532 y Fu(\)[)1415 3511 y(^)1393 3532 y Fr(H)1462 3544 y Fq(\024)1506 3532 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1762 3511 y(^)1740 3532 y Fr(H)1809 3544 y Fq(\024)1852 3532 y Fu(\))0 3578 y @beginspecial @setspecial @endspecial 843 3727 a(+)p Fr(I)951 3692 y Fo(\003)989 3727 y Fu(\()p Fr(j)1060 3692 y Fq(R)1115 3727 y Fu(\)1)-15 b(l)1197 3742 y Fs([1)p Fq(;)p Fs(+)p Fo(1)p Fs([)1408 3727 y Fu(\()p Fr(N)1507 3739 y Fo(1)1578 3727 y Fu(\))p Fr(\037)p Fu(\()1716 3706 y(^)1694 3727 y Fr(H)1770 3692 y Fs(ext)1763 3747 y Fq(\024)1864 3727 y Fu(\))p Fr(I)1939 3692 y Fo(\003)1978 3727 y Fu(\()p Fr(j)2049 3692 y Fq(R)2103 3727 y Fu(\))p Fr(\037)2187 3739 y Fs(1)2225 3727 y Fu(\()2279 3706 y(^)2257 3727 y Fr(H)2326 3739 y Fq(\024)2369 3727 y Fu(\)[)2446 3706 y(^)2424 3727 y Fr(H)2493 3739 y Fq(\024)2537 3727 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2793 3706 y(^)2771 3727 y Fr(H)2840 3739 y Fq(\024)2883 3727 y Fu(\))19 b(+)f Fr(o)p Fu(\()p Fr(R)3153 3692 y Fs(0)3190 3727 y Fu(\))-3222 b(\(5.10\))0 3781 y @beginspecial @setspecial @endspecial 843 3930 a(=)23 b(\000\()p Fr(q)1055 3896 y Fq(R)1110 3930 y Fu(\))p Fr(\037)p Fu(\()1248 3909 y(^)1226 3930 y Fr(H)1295 3942 y Fq(\024)1338 3930 y Fu(\)[)1415 3909 y(^)1393 3930 y Fr(H)1462 3942 y Fq(\024)1506 3930 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1762 3909 y(^)1740 3930 y Fr(H)1809 3942 y Fq(\024)1852 3930 y Fu(\))0 3976 y @beginspecial @setspecial @endspecial 843 4124 a(+)p Fr(I)951 4090 y Fo(\003)989 4124 y Fu(\()p Fr(j)1060 4090 y Fq(R)1115 4124 y Fu(\)1)-15 b(l)1197 4139 y Fs([1)p Fq(;)p Fs(+)p Fo(1)p Fs([)1408 4124 y Fu(\()p Fr(N)1507 4136 y Fo(1)1578 4124 y Fu(\))p Fr(\037)p Fu(\()1716 4103 y(^)1694 4124 y Fr(H)1770 4090 y Fs(ext)1763 4145 y Fq(\024)1864 4124 y Fu(\))p Fr(I)1939 4090 y Fo(\003)1978 4124 y Fu(\()p Fr(j)2049 4090 y Fq(R)2103 4124 y Fu(\)[)2181 4103 y(^)2158 4124 y Fr(H)2227 4136 y Fq(\024)2271 4124 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2527 4103 y(^)2505 4124 y Fr(H)2574 4136 y Fq(\024)2617 4124 y Fu(\))19 b(+)f Fr(o)p Fu(\()p Fr(R)2887 4090 y Fs(0)2925 4124 y Fu(\))-2957 b(\(5.11\))0 4179 y @beginspecial @setspecial @endspecial 843 4328 a(=)23 b(\000\()p Fr(q)1055 4293 y Fq(R)1110 4328 y Fu(\))p Fr(\037)p Fu(\()1248 4307 y(^)1226 4328 y Fr(H)1295 4340 y Fq(\024)1338 4328 y Fu(\)[)1415 4307 y(^)1393 4328 y Fr(H)1462 4340 y Fq(\024)1506 4328 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1762 4307 y(^)1740 4328 y Fr(H)1809 4340 y Fq(\024)1852 4328 y Fu(\))0 4373 y @beginspecial @setspecial @endspecial 843 4522 a(+)p Fr(I)951 4488 y Fo(\003)989 4522 y Fu(\()p Fr(j)1060 4488 y Fq(R)1115 4522 y Fu(\)1)-15 b(l)1197 4537 y Fs([1)p Fq(;)p Fs(+)p Fo(1)p Fs([)1408 4522 y Fu(\()p Fr(N)1507 4534 y Fo(1)1578 4522 y Fu(\))p Fr(\037)p Fu(\()1716 4501 y(^)1694 4522 y Fr(H)1770 4488 y Fs(ext)1763 4543 y Fq(\024)1864 4522 y Fu(\)[)1941 4501 y(^)1919 4522 y Fr(H)1995 4488 y Fs(ext)1988 4543 y Fq(\024)2090 4522 y Fr(;)14 b Fu(i)p Fr(B)2217 4488 y Fs(ext)2312 4522 y Fu(])p Fr(\037)p Fu(\()2441 4501 y(^)2419 4522 y Fr(H)2488 4534 y Fq(\024)2531 4522 y Fu(\))p Fr(I)2606 4488 y Fo(\003)2644 4522 y Fu(\()p Fr(j)2715 4488 y Fq(R)2770 4522 y Fu(\))19 b(+)f Fr(o)p Fu(\()p Fr(R)3040 4488 y Fs(0)3077 4522 y Fu(\))p Fr(:)-3132 b Fu(\(5.12\))100 4684 y(\()p @beginspecial @setspecial @endspecial(5.10\))26 b(follo)n(ws)h(b)n(y)g(\(5.9\).)37 b(Lemma)27 b @beginspecial @setspecial @endspecial(4.15)k(_)-27 b Fr({)o Fu(\))28 b(giv)n(es)f(\(5.11\))g(and)g(\(5.12\))g(follo)n(ws)f(b)n(y)i(Lemma)f @beginspecial @setspecial @endspecial(5.6.)0 4784 y(Letting)h Fr(\024)23 b Fp(!)g(1)k Fu(in)h(the)g(expression)e(\()p @beginspecial @setspecial @endspecial(5.12\))31 b(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\),)28 b(using)f(Lemma)g @beginspecial @setspecial @endspecial(5.5,)g(w)n(e)g(obtain:)178 4950 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 220 4954 a @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()326 4933 y(^)304 4954 y Fr(H)373 4966 y Fo(1)443 4954 y Fu(\)[)520 4933 y(^)498 4954 y Fr(H)567 4966 y Fo(1)638 4954 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()894 4933 y(^)872 4954 y Fr(H)941 4966 y Fo(1)1012 4954 y Fu(\))23 b(=)g Fr(K)1226 4966 y Fs(1)1262 4954 y Fu(\()p Fr(R)q Fu(\))c(+)f Fr(I)1535 4919 y Fo(\003)1573 4954 y Fu(\()p Fr(j)1644 4919 y Fq(R)1699 4954 y Fu(\)1)-15 b(l)1781 4969 y Fs([1)p Fq(;)p Fs(+)p Fo(1)p Fs([)1992 4954 y Fu(\()p Fr(N)2091 4966 y Fo(1)2162 4954 y Fu(\))p Fr(\037)p Fu(\()2300 4933 y(^)2278 4954 y Fr(H)2354 4919 y Fs(ext)2347 4974 y Fo(1)2448 4954 y Fu(\)[)2526 4933 y(^)2503 4954 y Fr(H)2579 4919 y Fs(ext)2572 4974 y Fo(1)2674 4954 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()2930 4933 y(^)2908 4954 y Fr(H)2984 4919 y Fs(ext)2977 4974 y Fo(1)3079 4954 y Fu(\))p Fr(I)3154 4919 y Fo(\003)3192 4954 y Fu(\()p Fr(j)3263 4919 y Fq(R)3318 4954 y Fu(\))19 b(+)f Fr(o)p Fu(\()p Fr(R)3588 4919 y Fs(0)3625 4954 y Fu(\))p Fr(;)3722 4950 y @beginspecial @setspecial @endspecial 0 5116 a Fu(where)27 b @beginspecial @setspecial @endspecial Fr(K)311 5128 y Fs(1)348 5116 y Fu(\()p Fr(R)q Fu(\))h(is)f(a)g(compact)h(op)r(erator.)35 b(This)27 b(giv)n(es)g(for)g Fr(\037)g Fu(suc)n(h)h(that)f @beginspecial @setspecial @endspecial(supp)q Fr(\037)c Fp(\032)f Fu([)p Fr(\025)d Fp(\000)f Fr(\016)o(;)c(\025)k Fu(+)g Fr(\016)s Fu([)793 5317 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()899 5296 y(^)877 5317 y Fr(H)946 5329 y Fo(1)1017 5317 y Fu(\)[)1094 5296 y(^)1072 5317 y Fr(H)1141 5329 y Fo(1)1212 5317 y Fr(;)c Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1468 5296 y(^)1446 5317 y Fr(H)1515 5329 y Fo(1)1585 5317 y Fu(\))24 b Fp(\025)e Fu(\()p Fr(d)p Fu(\()p Fr(\025)p Fu(\))e Fp(\000)2028 5260 y Fu(2)p Fr(\017)p 2028 5297 V 2045 5373 a Fu(3)2113 5317 y(\))p Fr(\037)2197 5282 y Fs(2)2235 5317 y Fu(\()2289 5296 y(^)2267 5317 y Fr(H)2336 5329 y Fo(1)2407 5317 y Fu(\))e(+)g Fr(K)2611 5329 y Fs(1)2648 5317 y Fu(\()p Fr(R)q Fu(\))h(+)f Fr(o)p Fu(\()p Fr(R)3014 5282 y Fs(0)3051 5317 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 5506 a Fu(Cho)r(osing)27 b @beginspecial @setspecial @endspecial Fr(R)h Fu(large)e(enough,)h(w)n (e)g(obtain)h Fr(H)1416 5518 y Fs(1)1453 5506 y Fu(\()p Fr(n)p Fu(\).)37 b(Prop)r(erties)26 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))p Fr(;)18 b Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))28 b(are)f(standard)f(consequences)h(of)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\).)381 b Fh(\004)p eop %%Page: 30 30 30 29 bop 0 226 a Fs(30)1695 b(Z.)29 b(AMMARI)1103 489 y @beginspecial @setspecial @endspecial Fu(6.)45 b Fv(Constr)n(uction) 31 b(of)h(the)g(w)-7 b(a)g(ve)31 b(opera)-6 b(tors)0 638 y @beginspecial @setspecial @endspecial Fu(6.1.)47 b Fz(Asymptotic)42 b(\014elds.)e Fu(In)e(this)g(subsection)f(w)n(e)g (pro)n(v)n(e)f(the)i(existence)f(of)h(asymptotic)f(\014elds)h(using)f (the)h(Co)r(ok)0 738 y(metho)r(d,)33 b(see)e(e.g.)p @beginspecial @setspecial @endspecial([19)o(].)49 b(W)-7 b(e)32 b(set)f @beginspecial @setspecial @endspecial Fr(h)1122 750 y Fq(t)1181 738 y Fu(:=)e(e)1335 708 y Fo(\000)p Fs(i)p Fq(t!)r Fs(\()p Fq(k)q Fs(\))1568 738 y Fr(h)p Fu(,)j(for)f Fr(h)f Fp(2)g Fm(h)i Fu(and)f(w)n(e)g(denote)h(b)n(y)f @beginspecial @setspecial @endspecial Fm(h)2765 750 y Fs(0)2834 738 y Fu(the)h(space)f Fr(C)3272 708 y Fo(1)3266 759 y Fs(0)3343 738 y Fu(\()p Fk(R)3429 708 y Fs(3)3493 738 y Fp(n)21 b(f)p Fu(0)p Fp(g)o Fu(\).)49 b(W)-7 b(e)0 838 y(in)n(tro)r(duce)27 b(Heisen)n(b)r(erg)g(deriv)-5 b(ativ)n(es:)1561 1048 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1602 980 a @beginspecial @setspecial @endspecial(I)-13 b(D)1682 992 y Fq(\024)1748 980 y Fu(:=)23 b Fr(@)1903 992 y Fq(t)1951 980 y Fu(+)18 b(i[)p Fr(H)2149 992 y Fq(\024)2192 980 y Fr(;)c(:)p Fu(])p Fr(;)1602 1114 y Fu(I)-13 b(D)23 b(:=)g Fr(@)1860 1126 y Fq(t)1908 1114 y Fu(+)18 b(i[)p Fr(H)r(;)c(:)p Fu(])p Fr(:)2339 1048 y @beginspecial @setspecial @endspecial 0 1200 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 157 x Fz(Theorem)30 b(6.1.)40 b @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b Fr(h)23 b Fp(2)g Fm(h)30 b Fn(the)g(str)l(ong)g(limits)1205 1508 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1295 1473 y Fs(+)1350 1508 y Fu(\()p Fr(h)p Fu(\))23 b(:=)g Fr(s)18 b Fp(\000)65 b Fu(lim)1736 1558 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1958 1508 y Fu(e)1995 1473 y Fs(i)p Fq(tH)2102 1508 y Fr(W)12 b Fu(\()p Fr(h)2272 1520 y Fq(t)2301 1508 y Fu(\)e)2370 1473 y Fo(\000)p Fs(i)p Fq(tH)2612 1508 y @beginspecial @setspecial @endspecial -2612 w Fu(\(6.1\))0 1686 y Fn(exist)29 b(and)i(ar)l(e)f(c)l(al)t(le)l (d)h Fu(asymptotic)c(W)-7 b(eyl)28 b(op)r(erators)p Fn(.)0 1786 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(urthermor)l(e)921 1937 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1011 1903 y Fs(+)1066 1937 y Fu(\()p Fr(h)p Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))1443 1903 y Fo(\000)p Fs(1)1555 1937 y Fu(=)69 b(lim)1643 1987 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1865 1937 y Fu(e)1902 1903 y Fs(i)p Fq(tH)2009 1937 y Fr(W)12 b Fu(\()p Fr(h)2179 1949 y Fq(t)2208 1937 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))2505 1903 y Fo(\000)p Fs(1)2594 1937 y Fu(e)2631 1903 y Fo(\000)p Fs(i)p Fq(tH)2790 1937 y Fr(:)83 b @beginspecial @setspecial @endspecial -2896 w Fu(\(6.2\))0 2120 y @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(The)h(map)1057 2336 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1098 2267 a @beginspecial @setspecial @endspecial Fm(h)24 b Fp(3)f Fr(h)g Fp(7!)g Fr(W)1510 2233 y Fs(+)1565 2267 y Fu(\()p Fr(h)p Fu(\))30 b Fn(is)g(str)l(ongly)g(c)l(ontinuous)o Fr(;)p @beginspecial @setspecial @endspecial 1098 2404 a Fm(h)24 b Fp(3)f Fr(h)g Fp(7!)g Fr(W)1510 2370 y Fs(+)1565 2404 y Fu(\()p Fr(h)p Fu(\)\()p Fr(H)j Fu(+)18 b(1\))1961 2370 y Fo(\000)p Fs(1)2080 2404 y Fn(is)30 b(norm)f(c)l(ontinuous)p Fr(:)2843 2336 y @beginspecial @setspecial @endspecial 4 2552 a Fu(_)-27 b Fr({v)s Fu(\))30 b Fn(The)h(Weyl)f(c)l(ommutation)g (r)l(elations)g(hold:)1223 2775 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1264 2714 a @beginspecial @setspecial @endspecial Fr(W)1354 2679 y Fs(+)1409 2714 y Fu(\()p Fr(h)p Fu(\))14 b Fr(W)1625 2679 y Fs(+)1681 2714 y Fu(\()p Fr(g)s Fu(\))23 b(=)f(e)1951 2657 y Fi(i)p 1945 2666 29 3 v 1945 2699 a(2)1984 2679 y Fs(Im)o(\()p Fq(h)p Fo(j)p Fq(g)r Fs(\))2211 2714 y Fr(W)2301 2679 y Fs(+)2356 2714 y Fu(\()p Fr(h)d Fu(+)f Fr(g)s Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1264 2849 a(W)1354 2814 y Fs(+)1409 2849 y Fu(\()p Fr(h)p Fu(\))1521 2814 y Fo(\003)1583 2849 y Fu(=)23 b Fr(W)1761 2814 y Fs(+)1816 2849 y Fu(\()p Fp(\000)p Fr(h)p Fu(\))p Fr(:)2677 2775 y @beginspecial @setspecial @endspecial 0 2997 a @beginspecial @setspecial @endspecial(v)s Fu(\))30 b Fn(The)h(Hamiltonian)g(pr)l(eserves)f(the)g(asymptotic)h(Weyl)f(op)l (er)l(ators:)1415 3148 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(e)1452 3114 y Fs(i)p Fq(tH)1559 3148 y Fr(W)1649 3114 y Fs(+)1704 3148 y Fu(\()p Fr(h)p Fu(\)e)1853 3114 y Fo(\000)p Fs(i)p Fq(tH)2035 3148 y Fu(=)23 b Fr(W)2213 3114 y Fs(+)2268 3148 y Fu(\()p Fr(h)2348 3160 y Fo(\000)p Fq(t)2429 3148 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 3305 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(relation)f(on)g @beginspecial @setspecial @endspecial Fp(H)1385 3452 y @beginspecial @setspecial @endspecial Fr(W)12 b Fu(\()p Fr(h)1555 3464 y Fq(t)1585 3452 y Fu(\))23 b(=)g(e)1765 3418 y Fo(\000)p Fs(i)p Fq(tH)1915 3426 y Fi(0)1951 3452 y Fr(W)12 b Fu(\()p Fr(h)p Fu(\)e)2190 3418 y Fs(i)p Fq(tH)2288 3426 y Fi(0)2325 3452 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 3599 a Fu(Hence)28 b(w)n(e)f(can)g(de\014ne)h @beginspecial @setspecial @endspecial Fr(@)805 3611 y Fq(t)834 3599 y Fr(W)12 b Fu(\()p Fr(h)1004 3611 y Fq(t)1034 3599 y Fu(\))28 b(as)f(quadratic)f(form)h(on)h Fp(D)r Fu(\()p Fr(H)2046 3611 y Fs(0)2084 3599 y Fu(\))1393 3747 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(@)1437 3759 y Fq(t)1467 3747 y Fr(W)12 b Fu(\()p Fr(h)1637 3759 y Fq(t)1666 3747 y Fu(\))23 b(=)g Fp(\000)p Fu(i[)p Fr(H)1989 3759 y Fs(0)2026 3747 y Fr(;)14 b(W)e Fu(\()p Fr(h)2233 3759 y Fq(t)2262 3747 y Fu(\)])p Fr(:)84 b @beginspecial @setspecial @endspecial -2424 w Fu(\(6.3\))0 3894 y(Using)27 b(\()p @beginspecial @setspecial @endspecial(6.3\))h(and)f(Thm.)37 b(2.9)27 b(w)n(e)g(ha)n(v)n(e,)g(since)g @beginspecial @setspecial @endspecial(Im\()p Fr(h)1681 3906 y Fq(t)1711 3894 y Fp(j)p Fr(v)1774 3906 y Fq(\024)1817 3894 y Fu(\))c Fp(2)h(B)s Fu(\()p Fp(K)q Fu(\),)k(the)g(follo)n(wing)e(iden)n(tit)n(y)i(on)f @beginspecial @setspecial @endspecial Fp(H)907 4045 y Fr(@)951 4057 y Fq(t)980 4045 y Fu(\(e)1049 4011 y Fs(i)p Fq(tH)1147 4019 y Fj(\024)1190 4045 y Fr(W)12 b Fu(\()p Fr(h)1360 4057 y Fq(t)1390 4045 y Fu(\)e)1459 4011 y Fo(\000)p Fs(i)p Fq(tH)1609 4019 y Fj(\024)1652 4045 y Fu(\))23 b(=)g(ie)1855 4011 y Fs(i)p Fq(tH)1953 4019 y Fj(\024)1996 4045 y Fu(Im\()p Fr(h)2175 4057 y Fq(t)2204 4045 y Fp(j)p Fr(v)2267 4057 y Fq(\024)2310 4045 y Fu(\))p Fr(W)12 b Fu(\()p Fr(h)2512 4057 y Fq(t)2542 4045 y Fu(\)e)2611 4011 y Fo(\000)p Fs(i)p Fq(tH)2761 4019 y Fj(\024)2804 4045 y Fr(:)83 b @beginspecial @setspecial @endspecial -2910 w Fu(\(6.4\))0 4192 y(W)-7 b(e)25 b(will)g(\014rst)g(pro)n(v)n(e) e(\()p @beginspecial @setspecial @endspecial(6.1\))i(and)f(\(6.2\))h (for)f Fr(h)f Fp(2)g Fm(h)1550 4204 y Fs(0)1613 4192 y Fu(then)i(w)n(e)g(extend)g(to)f Fr(h)f Fp(2)h Fm(h)p Fu(.)36 b(Let)25 b Fr(h)e Fp(2)g Fm(h)2874 4204 y Fs(0)2912 4192 y Fu(,)i(w)n(e)g(notice)g(in)g(this)g(case)f(that)0 4292 y(\()p Fr(h)80 4304 y Fq(s)116 4292 y Fp(j)p Fr(v)179 4304 y Fq(\024)222 4292 y Fu(\))j(=)f(\()p Fr(h)452 4304 y Fq(s)487 4292 y Fp(j)p Fr(v)550 4304 y Fq(\024)589 4312 y Fi(1)626 4292 y Fu(\),)31 b(for)e @beginspecial @setspecial @endspecial Fr(\024)d Fp(\025)g Fr(\024)1054 4304 y Fs(1)1091 4292 y Fu(.)44 b(Since)30 b(e)1414 4262 y Fs(i)p Fq(tH)1512 4270 y Fj(\024)1584 4292 y Fu(is)g(a)f(strongly)f (con)n(tin)n(uous)h(unitary)g(group)g(and)g(using)g(the)i(inequalit)n (y)0 4392 y(\()p @beginspecial @setspecial @endspecial(2.1\))j(w)n(e)f (see)h(that)g Fr(t)g Fp(7!)g Fu(e)878 4362 y Fs(i)p Fq(tH)976 4370 y Fj(\024)1019 4392 y Fu(Im\()p Fr(h)1198 4404 y Fq(t)1227 4392 y Fp(j)p Fr(v)1290 4404 y Fq(\024)1329 4412 y Fi(1)1366 4392 y Fu(\))p Fr(W)12 b Fu(\()p Fr(h)1568 4404 y Fq(t)1598 4392 y Fu(\)e)1667 4362 y Fo(\000)p Fs(i)p Fq(tH)1817 4370 y Fj(\024)1894 4392 y Fu(is)34 b(strongly)e(measurable.)55 b(Hence)34 b(b)n(y)g(in)n(tegrating)f(\()p @beginspecial @setspecial @endspecial(6.4\))g(w)n(e)0 4491 y(obtain)27 b(on)h @beginspecial @setspecial @endspecial Fp(H)g Fu(the)g(iden)n(tit)n(y:)532 4692 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(e)569 4657 y Fs(i)p Fq(tH)667 4665 y Fj(\024)710 4692 y Fr(W)12 b Fu(\()p Fr(h)880 4704 y Fq(t)910 4692 y Fu(\)e)979 4657 y Fo(\000)p Fs(i)p Fq(tH)1129 4665 y Fj(\024)1195 4692 y Fu(=)22 b Fr(W)12 b Fu(\()p Fr(h)p Fu(\))19 b(+)f(i)1623 4579 y Fl(Z)1706 4599 y Fq(t)1669 4767 y Fs(0)1763 4692 y Fu(e)1800 4657 y Fs(i)p Fq(sH)1904 4665 y Fj(\024)1947 4692 y Fu(Im\()p Fr(h)2126 4704 y Fq(s)2162 4692 y Fp(j)p Fr(v)2225 4704 y Fq(\024)2264 4712 y Fi(1)2300 4692 y Fu(\))p Fr(W)12 b Fu(\()p Fr(h)2502 4704 y Fq(s)2538 4692 y Fu(\)e)2607 4657 y Fo(\000)p Fs(i)p Fq(sH)2763 4665 y Fj(\024)2807 4692 y Fr(ds;)37 b(h)22 b Fp(2)i Fm(h)3141 4704 y Fs(0)3178 4692 y Fr(:)84 b @beginspecial @setspecial @endspecial -3285 w Fu(\(6.5\))0 4886 y(Using)27 b(Thm.)38 b @beginspecial @setspecial @endspecial(3.8)26 b(and)i(the)g(con)n(v)n(ergence)d(dominated)i(theorem,)h(letting)g @beginspecial @setspecial @endspecial Fr(\024)22 b Fp(!)h(1)28 b Fu(in)g(\(6.5\),)f(w)n(e)h(obtain)601 5091 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(e)638 5056 y Fs(i)p Fq(tH)744 5091 y Fr(W)12 b Fu(\()p Fr(h)914 5103 y Fq(t)944 5091 y Fu(\)e)1013 5056 y Fo(\000)p Fs(i)p Fq(tH)1195 5091 y Fu(=)22 b Fr(W)12 b Fu(\()p Fr(h)p Fu(\))19 b(+)f(i)1623 4978 y Fl(Z)1706 4998 y Fq(t)1669 5166 y Fs(0)1763 5091 y Fu(e)1800 5056 y Fs(i)p Fq(sH)1913 5091 y Fu(Im\()p Fr(h)2092 5103 y Fq(s)2128 5091 y Fp(j)p Fr(v)2191 5103 y Fq(\024)2230 5111 y Fi(1)2266 5091 y Fu(\))p Fr(W)12 b Fu(\()p Fr(h)2468 5103 y Fq(s)2504 5091 y Fu(\)e)2573 5056 y Fo(\000)p Fs(i)p Fq(sH)2738 5091 y Fr(ds;)37 b(h)23 b Fp(2)h Fm(h)3073 5103 y Fs(0)3110 5091 y Fr(:)83 b @beginspecial @setspecial @endspecial -3216 w Fu(\(6.6\))0 5279 y(Moreo)n(v)n(er)221 5464 y(e)258 5429 y Fs(i)p Fq(tH)365 5464 y Fr(W)12 b Fu(\()p Fr(h)535 5476 y Fq(t)564 5464 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))861 5429 y Fo(\000)p Fs(1)951 5464 y Fu(e)988 5429 y Fo(\000)p Fs(i)p Fq(tH)1169 5464 y Fu(=)23 b Fr(W)12 b Fu(\()p Fr(h)p Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))1724 5429 y Fo(\000)p Fs(1)1832 5464 y Fu(+)g(i)1952 5351 y Fl(Z)2035 5371 y Fq(t)1998 5539 y Fs(0)2078 5464 y Fu(e)2115 5429 y Fs(i)p Fq(sH)2228 5464 y Fu(Im\()p Fr(h)2407 5476 y Fq(s)2443 5464 y Fp(j)p Fr(v)2506 5476 y Fq(\024)2545 5484 y Fi(1)2581 5464 y Fu(\))p Fr(W)12 b Fu(\()p Fr(h)2783 5476 y Fq(s)2819 5464 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))3116 5429 y Fo(\000)p Fs(1)3205 5464 y Fu(e)3242 5429 y Fo(\000)p Fs(i)p Fq(sH)3407 5464 y Fr(ds:)84 b @beginspecial @setspecial @endspecial -3596 w Fu(\(6.7\))p eop %%Page: 31 31 31 30 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(31)100 489 y Fu(Clearly)20 b(there)i(exists)f @beginspecial @setspecial @endspecial Fr(\017)i(>)g Fu(0)e(suc)n(h)h (that)1376 427 y Fl(P)1463 514 y Fq(i)1491 489 y Fp(h)p Fu(x)1567 501 y Fq(i)1595 489 y Fp(i)1627 459 y Fs(1+)p Fq(\017)1743 489 y Fu(\()p Fr(H)14 b Fu(+)7 b(i\))1985 459 y Fo(\000)p Fs(1)2096 489 y Fu(is)21 b(b)r(ounded)i(since)e @beginspecial @setspecial @endspecial -62 x Fl(P)2792 514 y Fq(i)2820 489 y Fp(h)p Fu(x)2896 501 y Fq(i)2923 489 y Fp(i)2955 459 y Fs(1+)p Fq(\017)3072 489 y Fu(\()3126 468 y(^)3104 489 y Fr(H)3173 501 y Fo(1)3250 489 y Fu(+)7 b(i\))3377 459 y Fo(\000)p Fs(1)3488 489 y Fu(is)22 b(b)r(ounded.)0 590 y(Lemma)27 b @beginspecial @setspecial @endspecial(A.2)g(giv)n(es)g (that)833 527 y Fl(P)921 615 y Fq(i)949 590 y Fp(h)p Fu(x)1025 602 y Fq(i)1052 590 y Fp(i)1084 560 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)1253 590 y Fu(Im)q(\()p Fr(h)1433 602 y Fq(s)1468 590 y Fp(j)p Fr(v)1531 602 y Fq(\024)1570 610 y Fi(1)1607 590 y Fu(\))c Fp(2)h Fr(O)r Fu(\()p Fr(s)1877 560 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)2046 590 y Fu(\).)37 b(Then)28 b(the)g(existence)f(of)g(the)h(limits)g(\()p @beginspecial @setspecial @endspecial(6.2\))f(and)g(conse-)0 689 y(quen)n(tly)g(\(6.1\))h(for)f Fr(h)c Fp(2)g Fm(h)815 701 y Fs(0)880 689 y Fu(follo)n(ws.)0 789 y(Let)28 b @beginspecial @setspecial @endspecial Fr(h)23 b Fp(2)g Fm(h)28 b Fu(and)f Fr(h)578 801 y Fq(n)647 789 y Fp(2)c Fm(h)768 801 y Fs(0)833 789 y Fu(a)k(sequence)g(suc)n(h)h(that)g @beginspecial @setspecial @endspecial 29 w(lim)1614 839 y Fq(n)p Fo(!1)1801 789 y Fr(h)1849 801 y Fq(n)1917 789 y Fu(=)23 b Fr(h)k Fu(in)h @beginspecial @setspecial @endspecial Fm(h)p Fu(.)37 b(Corollary)25 b(4.11)i(and)g(inequalit)n(y) g(\()p @beginspecial @setspecial @endspecial(2.1\))g(giv)n(es)612 997 y @beginspecial @setspecial @endspecial 167 w Fp(jj)p Fu(\(e)894 963 y Fs(i)p Fq(sH)1007 997 y Fr(W)12 b Fu(\()p Fr(h)1177 1009 y Fq(s)1212 997 y Fu(\)e)1281 963 y Fo(\000)p Fs(i)p Fq(sH)1465 997 y Fp(\000)18 b Fu(e)1585 963 y Fs(i)p Fq(tH)1692 997 y Fr(W)12 b Fu(\()p Fr(h)1862 1009 y Fq(t)1891 997 y Fu(\)e)1960 963 y Fo(\000)p Fs(i)p Fq(tH)2119 997 y Fu(\))i(\()p Fr(H)26 b Fu(+)18 b(i\))2430 963 y Fo(\000)p Fs(1)2519 997 y Fp(jj)23 b(\024)0 1043 y @beginspecial @setspecial @endspecial 779 1192 a Fr(c)g Fu(\()p Fp(jj)p Fu(\(e)985 1158 y Fs(i)p Fq(sH)1098 1192 y Fr(W)12 b Fu(\()p Fr(h)1268 1204 y Fq(n;s)1365 1192 y Fu(\)e)1434 1158 y Fo(\000)p Fs(i)p Fq(sH)1617 1192 y Fp(\000)18 b Fu(e)1737 1158 y Fs(i)p Fq(tH)1844 1192 y Fr(W)12 b Fu(\()p Fr(h)2014 1204 y Fq(n;t)2104 1192 y Fu(\)e)2173 1158 y Fo(\000)p Fs(i)p Fq(tH)2332 1192 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))2629 1158 y Fo(\000)p Fs(1)2718 1192 y Fp(jj)h Fu(+)f Fp(jj)p Fr(h)2960 1204 y Fq(n)3023 1192 y Fp(\000)g Fr(h)p Fp(jj)3200 1158 y Fq(\017)3232 1192 y Fu(\))p Fr(:)100 1360 y Fu(This)37 b(inequalit)n(y)g(giv)n(es)g(the)h(existence)f(of)g(\()p @beginspecial @setspecial @endspecial(6.2\))h(and)f(\()p @beginspecial @setspecial @endspecial(6.1\))g(for)g Fr(h)j Fp(2)g Fm(h)p Fu(.)67 b(This)37 b(sho)n(ws)k(_)-27 b Fr({)p Fu(\),)40 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\).)67 b(Using)37 b(\(2.1\))g(and)0 1459 y(Corollary)25 b(4.11)h(w)n(e)i(ha)n(v)n(e)875 1626 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(e)958 1592 y Fs(i)p Fq(sH)1072 1626 y Fu(\()p Fr(W)12 b Fu(\()p Fr(h)1274 1638 y Fq(s)1309 1626 y Fu(\))19 b Fp(\000)f Fr(W)12 b Fu(\()p Fr(g)1605 1638 y Fq(s)1641 1626 y Fu(\)\)e)1742 1592 y Fo(\000)p Fs(i)p Fq(sH)1907 1626 y Fu(\()p Fr(H)25 b Fu(+)18 b(i\))2171 1592 y Fo(\000)p Fs(1)2261 1626 y Fp(jj)23 b(\024)g Fr(c)14 b Fp(jj)p Fr(h)k Fp(\000)g Fr(g)s Fp(jj)2752 1592 y Fo(\000)p Fq(\017)2835 1626 y Fr(:)83 b @beginspecial @setspecial @endspecial -2941 w Fu(\(6.8\))0 1793 y(T)-7 b(aking)27 b(the)h(limit)g @beginspecial @setspecial @endspecial Fr(s)23 b Fp(!)g(1)28 b Fu(in)g(\(6.8\))f(w)n(e)g(obtain)1104 1962 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(W)1272 1927 y Fs(+)1328 1962 y Fu(\()p Fr(h)p Fu(\))18 b Fp(\000)g Fr(W)1631 1927 y Fs(+)1687 1962 y Fu(\()p Fr(g)s Fu(\)\)\()p Fr(H)25 b Fu(+)18 b(1\))2109 1927 y Fo(\000)p Fs(1)2198 1962 y Fp(jj)24 b(\024)e Fr(c)14 b Fp(jj)p Fr(h)k Fp(\000)g Fr(g)s Fp(jj)2689 1927 y Fo(\000)p Fq(\017)2773 1962 y Fr(:)p @beginspecial @setspecial @endspecial 0 2129 a Fu(This)28 b(pro)n(v)n(es)d @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\).)37 b(The)27 b(rest)h(follo)n(ws)e(from)h(simple)h (computations.)p @beginspecial @setspecial @endspecial 1643 w Fh(\004)0 2150 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 147 x Fz(Theorem)i(6.2.)40 b Fn(The)31 b(\014ve)f(fol)t(lowing)i(assertions)e(hold:)4 2397 y Fu(_)-27 b Fr({)p Fu(\))30 b Fn(Ther)l(e)g(exist)g(selfadjoint)i (op)l(er)l(ators)f @beginspecial @setspecial @endspecial Fr(\036)1332 2366 y Fs(+)1387 2397 y Fu(\()p Fr(h)p Fu(\))p Fn(,)g(c)l(al)t(le)l(d)g Fu(asymptotic)c(\014elds)p Fn(,)j(such)g(that) 1393 2580 y @beginspecial @setspecial @endspecial Fr(W)1483 2546 y Fs(+)1538 2580 y Fu(\()p Fr(h)p Fu(\))23 b(=)g(e)1798 2546 y Fs(i)p Fq(\036)1857 2521 y Fi(+)1903 2546 y Fs(\()p Fq(h)p Fs(\))2012 2580 y Fu(for)13 b Fr(h)23 b Fp(2)h Fm(h)p Fr(:)83 b @beginspecial @setspecial @endspecial 0 2814 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b Fr(h)319 2826 y Fq(i)369 2814 y Fp(2)23 b Fm(h)p Fr(;)14 b(i)23 b Fu(=)g(1)p Fr(:::n)p Fn(.)38 b(We)30 b(have)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\(\()p Fr(H)26 b Fu(+)18 b(i\))1606 2761 y Fj(n)p 1606 2770 37 3 v 1610 2804 a Fi(2)1657 2814 y Fu(\))23 b Fp(\032)g(D)r Fu(\()1931 2710 y Fq(n)1898 2735 y Fl(Y)1935 2911 y Fs(1)2019 2814 y Fr(\036)2068 2780 y Fs(+)2123 2814 y Fu(\()p Fr(h)2203 2826 y Fq(i)2231 2814 y Fu(\)\))30 b Fn(and)852 3132 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 885 3028 a Fq(n)853 3053 y Fl(Y)852 3230 y Fq(i)p Fs(=1)973 3132 y Fr(\036)1022 3098 y Fs(+)1078 3132 y Fu(\()p Fr(h)1158 3144 y Fq(i)1186 3132 y Fu(\)\()p Fr(H)25 b Fu(+)19 b(i\))1483 3098 y Fo(\000)1545 3075 y Fj(n)p 1545 3084 V 1549 3118 a Fi(2)1618 3132 y Fu(=)70 b(lim)1706 3182 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1928 3132 y Fu(e)1965 3098 y Fs(i)p Fq(tH)2119 3028 y(n)2087 3053 y Fl(Y)2086 3230 y Fq(i)p Fs(=1)2207 3132 y Fr(\036)2256 3098 y Fs(+)2312 3132 y Fu(\()p Fr(h)2392 3144 y Fq(i)2420 3132 y Fu(\)e)2489 3098 y Fo(\000)p Fs(i)p Fq(tH)2648 3132 y Fu(\()p Fr(H)25 b Fu(+)18 b(i\))2912 3098 y Fo(\000)2974 3075 y Fj(n)p 2974 3084 V 2978 3118 a Fi(2)3025 3132 y Fr(:)p @beginspecial @setspecial @endspecial 0 3376 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(The)h(map)945 3610 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(h)1025 3622 y Fs(1)1062 3610 y Fr(;)14 b(:::;)g(h)1253 3622 y Fq(n)1298 3610 y Fu(\))24 b Fp(7!)1493 3506 y Fq(n)1461 3531 y Fl(Y)1460 3708 y Fq(i)p Fs(=1)1581 3610 y Fr(\036)1630 3576 y Fs(+)1686 3610 y Fu(\()p Fr(h)1766 3622 y Fq(i)1794 3610 y Fu(\)\()p Fr(H)h Fu(+)18 b(i\))2090 3576 y Fo(\000)2153 3553 y Fj(n)p 2153 3562 V 2157 3596 a Fi(2)2233 3610 y Fn(is)30 b(norm)g(c)l(ontinuous)o Fr(:)p @beginspecial @setspecial @endspecial 0 3858 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({v)s Fu(\))30 b Fn(The)h(c)l(ommutation)e(r)l (elations)i(hold)g(as)f(quadr)l(atic)h(forms)f(on)g @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(\036)2285 3828 y Fs(+)2341 3858 y Fu(\()p Fr(h)p Fu(\)\))19 b Fp(\\)g(D)r Fu(\()p Fr(\036)2725 3828 y Fs(+)2781 3858 y Fu(\()p Fr(g)s Fu(\)\))1477 4025 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial([)p Fr(\036)1549 3991 y Fs(+)1605 4025 y Fu(\()p Fr(h)p Fu(\))p Fr(;)14 b(\036)1803 3991 y Fs(+)1859 4025 y Fu(\()p Fr(g)s Fu(\)])23 b(=)g(iIm\()p Fr(h)p Fp(j)p Fr(g)s Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4193 a @beginspecial @setspecial @endspecial(v)s Fu(\))30 b Fn(We)g(have)1456 4363 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(e)1493 4329 y Fs(i)p Fq(tH)1600 4363 y Fr(\036)1649 4329 y Fs(+)1704 4363 y Fu(\()p Fr(h)p Fu(\)e)1853 4329 y Fo(\000)p Fs(i)p Fq(tH)2035 4363 y Fu(=)23 b Fr(\036)2172 4329 y Fs(+)2228 4363 y Fu(\()p Fr(h)2308 4375 y Fo(\000)p Fq(t)2389 4363 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 4531 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Since)27 b Fr(s)c Fp(!)g Fr(W)898 4501 y Fs(+)953 4531 y Fu(\()p Fr(sh)p Fu(\))28 b(is)g(strongly)e(con)n (tin)n(uous)h(using)g(Thm.)37 b @beginspecial @setspecial @endspecial(6.1)31 b(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)f Fr({)p Fu(\))28 b(,)k(_)-27 b Fr({)p Fu(\))28 b(follo)n(ws)e(from)i (Stone's)f(theorem.)0 4631 y(W)-7 b(e)28 b(in)n(tend)g(to)f(sho)n(w)g (the)h(existence)f(of)h(the)g(follo)n(wing)f(limit)h(for)f @beginspecial @setspecial @endspecial Fr(h)2164 4643 y Fq(i)2214 4631 y Fp(2)d Fm(h)p Fr(;)k(i)23 b Fu(=)f(1)p Fr(:::n)1192 4860 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 46 w Fu(lim)1192 4910 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1414 4860 y Fu(e)1451 4826 y Fs(i)p Fq(tH)1605 4756 y(n)1573 4781 y Fl(Y)1572 4958 y Fq(i)p Fs(=1)1693 4860 y Fr(\036)p Fu(\()p Fr(h)1822 4872 y Fq(i;t)1895 4860 y Fu(\)e)1964 4826 y Fo(\000)p Fs(i)p Fq(tH)2123 4860 y Fu(\()p Fr(H)k Fu(+)18 b(1\))2407 4826 y Fo(\000)2468 4803 y Fj(n)p 2468 4812 V 2472 4846 a Fi(2)2519 4860 y Fr(:)83 b @beginspecial @setspecial @endspecial -2625 w Fu(\(6.9\))100 5104 y(Let)23 b @beginspecial @setspecial @endspecial Fr(h)292 5116 y Fq(i)343 5104 y Fp(2)g Fm(h)464 5116 y Fs(0)502 5104 y Fr(;)28 b(i)22 b Fu(=)h(1)p Fr(:::n)p Fu(.)35 b(As)24 b(in)f(the)h(previous)f(pro)r (of)g(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing)e(iden)n(tit)n(y)i(as)f (quadratic)f(form)i(on)f @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)3830 5116 y Fs(0)3868 5104 y Fu(\))0 5204 y(whic)n(h)28 b(extends)f(as)g(an)g(op)r(erator)f(iden)n(tit)n(y)i (on)f Fp(H)q Fu(:)841 5433 y @beginspecial @setspecial @endspecial Fr(@)885 5445 y Fq(t)914 5433 y Fu(\()980 5329 y Fq(n)947 5354 y Fl(Y)946 5531 y Fq(i)p Fs(=1)1068 5433 y Fr(\036)p Fu(\()p Fr(h)1197 5445 y Fq(i;t)1270 5433 y Fu(\)\()p Fr(N)h Fu(+)18 b(1\))1586 5399 y Fo(\000)1648 5376 y Fj(n)p 1648 5385 V 1652 5419 a Fi(2)1698 5433 y Fu(\))24 b(=)e Fp(\000)p Fu(i[)p Fr(H)2021 5445 y Fs(0)2058 5433 y Fr(;)2128 5329 y Fq(n)2096 5354 y Fl(Y)2095 5531 y Fq(i)p Fs(=1)2217 5433 y Fr(\036)p Fu(\()p Fr(h)2346 5445 y Fq(i;t)2419 5433 y Fu(\)]\()p Fr(N)27 b Fu(+)18 b(1\))2757 5399 y Fo(\000)2819 5376 y Fj(n)p 2819 5385 V 2823 5419 a Fi(2)2870 5433 y Fr(:)83 b @beginspecial @setspecial @endspecial -2976 w Fu(\(6.10\))p eop %%Page: 32 32 32 31 bop 0 226 a Fs(32)1695 b(Z.)29 b(AMMARI)0 489 y Fu(No)n(w)e(w)n(e)g(compute)h(the)g(deriv)-5 b(ativ)n(e)995 1119 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1037 696 a @beginspecial @setspecial @endspecial Fr(@)1081 708 y Fq(t)1110 696 y Fu(\(e)1179 662 y Fs(i)p Fq(tH)1277 670 y Fj(\024)1367 593 y Fq(n)1335 618 y Fl(Y)1334 794 y Fq(i)p Fs(=1)1455 696 y Fr(\036)p Fu(\()p Fr(h)1584 708 y Fq(i;t)1657 696 y Fu(\)\()p Fr(H)1790 708 y Fq(\024)1852 696 y Fp(\000)19 b Fr(E)1997 708 y Fq(\024)2058 696 y Fu(+)f(i\))2196 662 y Fo(\000)2258 640 y Fj(n)p 2258 649 37 3 v 2262 682 a Fi(2)2309 696 y Fu(e)2346 662 y Fo(\000)p Fs(i)p Fq(tH)2496 670 y Fj(\024)2539 696 y Fr(u;)c(v)s Fu(\))23 b(=)p @beginspecial @setspecial @endspecial 1037 975 a(\()p Fr(@)1113 987 y Fq(t)1189 871 y(n)1157 896 y Fl(Y)1156 1073 y Fq(i)p Fs(=1)1278 975 y Fr(\036)p Fu(\()p Fr(h)1407 987 y Fq(i;t)1480 975 y Fu(\)\()p Fr(H)1613 987 y Fq(\024)1675 975 y Fp(\000)18 b Fr(E)1819 987 y Fq(\024)1880 975 y Fu(+)g(i\))2018 940 y Fo(\000)2081 918 y Fj(n)p 2081 927 V 2085 960 a Fi(2)2131 975 y Fu(e)2168 940 y Fo(\000)p Fs(i)p Fq(tH)2318 948 y Fj(\024)2361 975 y Fr(u;)c Fu(e)2483 940 y Fo(\000)p Fs(i)p Fq(tH)2633 948 y Fj(\024)2675 975 y Fr(v)s Fu(\))p @beginspecial @setspecial @endspecial 1037 1253 a(+\()1167 1149 y Fq(n)1135 1174 y Fl(Y)1134 1351 y Fq(i)p Fs(=1)1255 1253 y Fr(\036)p Fu(\()p Fr(h)1384 1265 y Fq(i;t)1457 1253 y Fu(\)\()p Fr(H)1590 1265 y Fq(\024)1652 1253 y Fp(\000)k Fr(E)1796 1265 y Fq(\024)1858 1253 y Fu(+)g(i\))1996 1219 y Fo(\000)2058 1196 y Fj(n)p 2058 1205 V 2062 1239 a Fi(2)2109 1253 y Fu(e)2146 1219 y Fo(\000)p Fs(i)p Fq(tH)2296 1227 y Fj(\024)2338 1253 y Fr(u;)c(H)2492 1265 y Fq(\024)2535 1253 y Fu(e)2572 1219 y Fo(\000)p Fs(i)p Fq(tH)2722 1227 y Fj(\024)2765 1253 y Fr(v)s Fu(\))p @beginspecial @setspecial @endspecial 1037 1531 a Fp(\000)p Fu(\()p Fr(H)1203 1543 y Fq(\024)1246 1531 y Fu(\()p Fr(H)1347 1543 y Fq(\024)1408 1531 y Fp(\000)k Fr(E)1552 1543 y Fq(\024)1614 1531 y Fu(+)g(i\))1752 1497 y Fo(\000)1814 1475 y Fj(n)p 1814 1484 V 1818 1517 a Fi(2)1865 1531 y Fu(e)1902 1497 y Fo(\000)p Fs(i)p Fq(tH)2052 1505 y Fj(\024)2095 1531 y Fr(u;)2212 1427 y Fq(n)2180 1452 y Fl(Y)2180 1629 y Fq(i)p Fs(=1)2301 1531 y Fr(\036)p Fu(\()p Fr(h)2430 1543 y Fq(i;t)2503 1531 y Fu(\)e)2572 1497 y Fo(\000)p Fs(i)p Fq(tH)2722 1505 y Fj(\024)2765 1531 y Fr(v)s Fu(\))p Fr(:)2905 1119 y @beginspecial @setspecial @endspecial 0 1754 a Fu(W)-7 b(e)28 b(ha)n(v)n(e)e @beginspecial @setspecial @endspecial(\()p Fr(h)414 1766 y Fq(i;t)487 1754 y Fp(j)p Fr(v)550 1766 y Fq(\024)593 1754 y Fu(\))e(=)e(\()p Fr(h)816 1766 y Fq(i;t)889 1754 y Fp(j)p Fr(v)952 1766 y Fq(\024)991 1774 y Fi(1)1028 1754 y Fu(\))28 b(for)f @beginspecial @setspecial @endspecial Fr(\024)22 b Fp(\025)h Fr(\024)1421 1766 y Fs(1)1458 1754 y Fu(.)37 b(Hence)28 b(w)n(e)f(ha)n(v)n(e)g(on)g Fp(H)1044 1969 y @beginspecial @setspecial @endspecial Fr(@)1088 1981 y Fq(t)1117 1969 y Fu(\(e)1186 1935 y Fs(i)p Fq(tH)1284 1943 y Fj(\024)1374 1865 y Fq(n)1342 1890 y Fl(Y)1341 2067 y Fq(i)p Fs(=1)1462 1969 y Fr(\036)p Fu(\()p Fr(h)1591 1981 y Fq(i;t)1665 1969 y Fu(\)\()p Fr(H)1798 1981 y Fq(\024)1860 1969 y Fp(\000)18 b Fr(E)2004 1981 y Fq(\024)2065 1969 y Fu(+)g(i\))2203 1935 y Fo(\000)2266 1912 y Fj(n)p 2266 1921 V 2270 1955 a Fi(2)2316 1969 y Fu(e)2353 1935 y Fo(\000)p Fs(i)p Fq(tH)2503 1943 y Fj(\024)2546 1969 y Fu(\))23 b(=)0 2100 y @beginspecial @setspecial @endspecial 1044 2291 a(ie)1104 2257 y Fs(i)p Fq(tH)1202 2265 y Fj(\024)1298 2188 y Fq(n)1259 2212 y Fl(X)1261 2389 y Fq(j)s Fs(=1)1425 2188 y Fq(n)1392 2212 y Fl(Y)1393 2391 y Fq(i)p Fo(6)p Fs(=)p Fq(j)1512 2291 y Fu(Im\()p Fr(h)1691 2303 y Fq(j;t)1768 2291 y Fp(j)p Fr(v)1831 2303 y Fq(\024)1870 2311 y Fi(1)1906 2291 y Fu(\))p Fr(\036)p Fu(\()p Fr(h)2067 2303 y Fq(i;t)2141 2291 y Fu(\)\()p Fr(H)2274 2303 y Fq(\024)2336 2291 y Fp(\000)18 b Fr(E)2480 2303 y Fq(\024)2542 2291 y Fu(+)g(i\))2680 2257 y Fo(\000)2742 2235 y Fj(n)p 2742 2244 V 2746 2277 a Fi(2)2792 2291 y Fu(e)2829 2257 y Fo(\000)p Fs(i)p Fq(tH)2979 2265 y Fj(\024)0 2291 y Fu(\(6.11\))0 2527 y(Letting)28 b @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(!)g(1)k Fu(in)h(\(6.11\))f(w)n(e)g(obtain)365 2739 y @beginspecial @setspecial @endspecial Fr(@)409 2751 y Fq(t)439 2739 y Fu(\(e)508 2705 y Fs(i)p Fq(tH)662 2636 y(n)629 2660 y Fl(Y)629 2837 y Fq(i)p Fs(=1)750 2739 y Fr(\036)p Fu(\()p Fr(h)879 2751 y Fq(i;t)952 2739 y Fu(\)\()p Fr(H)f Fu(+)18 b(i\))1249 2705 y Fo(\000)1311 2683 y Fj(n)p 1311 2692 V 1315 2725 a Fi(2)1362 2739 y Fu(e)1399 2705 y Fo(\000)p Fs(i)p Fq(tH)1557 2739 y Fu(\))24 b(=)e(ie)1760 2705 y Fs(i)p Fq(tH)1920 2636 y(n)1881 2660 y Fl(X)1883 2837 y Fq(j)s Fs(=1)2047 2636 y Fq(n)2015 2660 y Fl(Y)2015 2839 y Fq(i)p Fo(6)p Fs(=)p Fq(j)2135 2739 y Fu(Im\()p Fr(h)2314 2751 y Fq(j;t)2390 2739 y Fp(j)p Fr(v)2453 2751 y Fq(\024)2492 2759 y Fi(1)2529 2739 y Fu(\))p Fr(\036)p Fu(\()p Fr(h)2690 2751 y Fq(i;t)2763 2739 y Fu(\)\()p Fr(H)k Fu(+)18 b(i\))3060 2705 y Fo(\000)3122 2683 y Fj(n)p 3122 2692 V 3126 2725 a Fi(2)3173 2739 y Fu(e)3210 2705 y Fo(\000)p Fs(i)p Fq(tH)3452 2739 y @beginspecial @setspecial @endspecial -3452 w Fu(\(6.12\))0 2981 y(Since)32 b @beginspecial @setspecial @endspecial -63 x Fl(P)309 3005 y Fq(i)336 2981 y Fp(h)p Fu(x)412 2993 y Fq(i)440 2981 y Fp(i)472 2950 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)641 2981 y Fu(Im\()p Fr(h)820 2993 y Fq(i;t)893 2981 y Fp(j)p Fr(v)956 2993 y Fq(\024)995 3001 y Fi(1)1032 2981 y Fu(\))e Fp(2)h Fr(O)r Fu(\()p Fr(t)1307 2950 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)1476 2981 y Fu(\),)j(the)e(dominated)g(con)n(v)n(ergence)d(theorem)j(giv)n (es)f(the)h(existence)f(of)h(\()p @beginspecial @setspecial @endspecial(6.9\))0 3080 y(for)27 b @beginspecial @setspecial @endspecial Fr(h)c Fp(2)g Fm(h)319 3092 y Fs(0)357 3080 y Fu(.)0 3180 y(Let)j @beginspecial @setspecial @endspecial Fr(h)195 3192 y Fq(i)245 3180 y Fp(2)e Fm(h)p Fr(;)14 b(i)22 b Fu(=)h(1)p Fr(:::n)i Fu(and)g Fr(h)936 3192 y Fq(i;`)1035 3180 y Fp(2)e Fm(h)1156 3192 y Fs(0)1219 3180 y Fu(sequences)i(suc)n(h)g(that)h @beginspecial @setspecial @endspecial(lim)2072 3192 y Fq(`)2118 3180 y Fr(h)2166 3192 y Fq(i;`)2264 3180 y Fu(=)c Fr(h)2399 3192 y Fq(i)2427 3180 y Fu(,)k(in)g @beginspecial @setspecial @endspecial Fm(h)p Fu(.)36 b(Using)26 b(Corollary)d(2.4)h(and)i (Corollary)0 3279 y @beginspecial @setspecial @endspecial(4.8,)h(w)n(e) g(obtain)g(the)h(inequalit)n(y)433 3631 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 475 3487 a @beginspecial @setspecial @endspecial Fp(jj)p Fu(\(e)590 3453 y Fs(i)p Fq(sH)750 3383 y(n)718 3408 y Fl(Y)717 3585 y Fq(i)p Fs(=1)838 3487 y Fr(\036)p Fu(\()p Fr(h)967 3499 y Fq(i;s)1047 3487 y Fu(\)e)1116 3453 y Fo(\000)p Fs(i)p Fq(sH)1299 3487 y Fp(\000)18 b Fu(e)1419 3453 y Fs(i)p Fq(tH)1573 3383 y(n)1541 3408 y Fl(Y)1540 3585 y Fq(i)p Fs(=1)1661 3487 y Fr(\036)p Fu(\()p Fr(h)1790 3499 y Fq(i;t)1863 3487 y Fu(\)e)1932 3453 y Fo(\000)p Fs(i)p Fq(tH)2091 3487 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(1\))2407 3453 y Fo(\000)2469 3430 y Fj(n)p 2469 3439 V 2473 3473 a Fi(2)2519 3487 y Fp(jj)24 b(\024)p @beginspecial @setspecial @endspecial 475 3765 a Fr(c)14 b Fu(\()596 3661 y Fq(n)557 3686 y Fl(X)563 3863 y Fq(i)p Fs(=1)691 3765 y Fp(jj)p Fr(h)785 3777 y Fq(i)831 3765 y Fp(\000)k Fr(h)962 3777 y Fq(i;`)1037 3765 y Fp(jj)g Fu(+)g Fp(jj)p Fu(\(e)1299 3731 y Fs(i)p Fq(sH)1460 3661 y(n)1427 3686 y Fl(Y)1426 3863 y Fq(i)p Fs(=1)1548 3765 y Fr(\036)p Fu(\()p Fr(h)1677 3777 y Fq(i;`;s)1804 3765 y Fu(\)e)1873 3731 y Fo(\000)p Fs(i)p Fq(sH)2056 3765 y Fp(\000)g Fu(e)2176 3731 y Fs(i)p Fq(tH)2330 3661 y(n)2298 3686 y Fl(Y)2297 3863 y Fq(i)p Fs(=1)2418 3765 y Fr(\036)p Fu(\()p Fr(h)2547 3777 y Fq(i;`;t)2668 3765 y Fu(\)e)2737 3731 y Fo(\000)p Fs(i)p Fq(tH)2896 3765 y Fu(\)\()p Fr(H)25 b Fu(+)18 b(1\))3211 3731 y Fo(\000)3273 3709 y Fj(n)p 3273 3718 V 3277 3751 a Fi(2)3324 3765 y Fp(jj)p Fu(\))p Fr(:)3467 3631 y @beginspecial @setspecial @endspecial 0 3988 a Fu(Cauc)n(h)n(y)26 b(criterion)h(for)g(the)h(con)n(v)n(ergence,)d(pro)n (v)n(es)h(the)i(existence)f(of)h(the)g(limit)g(\()p @beginspecial @setspecial @endspecial(6.9\).)0 4087 y(T)-7 b(o)27 b(complete)32 b(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))28 b(it)g(su\016ces)g(to)f(sho)n (w)g(b)n(y)g(induction)h(in)g @beginspecial @setspecial @endspecial Fr(n)f Fu(the)h(existence)g(of)f(follo)n(wing)g(limit)304 4300 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 8 w(lim)304 4351 y Fq(s)p Fo(!)p Fs(0)495 4300 y Fu(lim)449 4349 y Fq(t)p Fo(!)p Fs(+)p Fo(1)671 4300 y Fu(e)708 4265 y Fs(i)p Fq(tH)815 4300 y Fu(\()857 4243 y(1)p 857 4280 42 4 v 858 4356 a Fr(s)909 4300 y Fu(\()p Fr(W)12 b Fu(\()p Fr(sh)1150 4312 y Fs(1)p Fq(;t)1232 4300 y Fu(\))19 b Fp(\000)f Fu(1\))1487 4196 y Fq(n)1454 4221 y Fl(Y)1454 4398 y Fq(i)p Fs(=2)1575 4300 y Fr(\036)p Fu(\()p Fr(h)1704 4312 y Fq(i;t)1777 4300 y Fu(\))g Fp(\000)g Fu(i)1981 4196 y Fq(n)1948 4221 y Fl(Y)1947 4398 y Fq(i)p Fs(=1)2069 4300 y Fr(\036)p Fu(\()p Fr(h)2198 4312 y Fq(i;t)2271 4300 y Fu(\)\)\()p Fr(H)26 b Fu(+)18 b(1\))2619 4265 y Fo(\000)2681 4243 y Fj(n)p 2681 4252 37 3 v 2685 4285 a Fi(2)2731 4300 y Fu(e)2768 4265 y Fo(\000)p Fs(i)p Fq(tH)2927 4300 y Fr(u)k Fu(=)h(0)p Fr(;)37 b(u)22 b Fp(2)i(H)q Fr(:)83 b @beginspecial @setspecial @endspecial -3513 w Fu(\(6.13\))0 4526 y(W)-7 b(e)24 b(\014rst)e(pro)n(v)n(e)g(\()p @beginspecial @setspecial @endspecial(6.13\))g(for)h Fr(u)f Fp(2)i(D)r Fu(\(\()p Fr(H)17 b Fu(+)10 b(1\))1397 4496 y Fo(\000)p Fq(\017)1480 4526 y Fu(\))p Fr(;)k(\017)23 b(>)f Fu(0,)i(then)g(b)n(y)e (an)h(argumen)n(t)f(of)h(densit)n(y)g(w)n(e)g(obtain)g(\()p @beginspecial @setspecial @endspecial(6.13\))f(for)h Fr(u)g Fp(2)g(H)q Fu(.)0 4626 y(W)-7 b(e)28 b(recall)f(\()p @beginspecial @setspecial @endspecial(2.2\):)968 4784 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 8 w(lim)968 4835 y Fq(s)p Fo(!)p Fs(0)1150 4784 y Fu(sup)1113 4858 y Fo(jj)p Fq(h)p Fo(jj\024)p Fq(c)1326 4784 y Fr(s)1365 4750 y Fo(\000)p Fs(1)1454 4784 y Fp(jj)p Fu(\()p Fr(W)12 b Fu(\()p Fr(sh)p Fu(\))19 b Fp(\000)f Fu(1)-15 b(l)18 b Fp(\000)g Fu(i)p Fr(\036)p Fu(\()p Fr(h)p Fu(\)\)\()p Fr(N)29 b Fu(+)18 b(1\))2527 4750 y Fo(\000)2588 4727 y Fi(1)p 2588 4736 29 3 v 2588 4770 a(2)2627 4750 y Fo(\000)p Fq(\017)2710 4784 y Fp(jj)23 b Fu(=)g(0)p Fr(:)p @beginspecial @setspecial @endspecial 0 4992 a Fu(W)-7 b(e)28 b(see)f(that)677 5184 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 8 w(lim)677 5235 y Fq(s)p Fo(!)p Fs(0)821 5184 y Fu(sup)828 5254 y Fq(t)p Fo(2)p Fe(R)960 5184 y Fp(jj)p Fu(\()1048 5127 y(1)p 1048 5164 42 4 v 1049 5241 a Fr(s)1100 5184 y Fu(\()p Fr(W)12 b Fu(\()p Fr(sh)1341 5196 y Fs(1)p Fq(;t)1423 5184 y Fu(\))19 b Fp(\000)f Fu(1\))1678 5080 y Fq(n)1645 5105 y Fl(Y)1645 5282 y Fq(i)p Fs(=2)1766 5184 y Fr(\036)p Fu(\()p Fr(h)1895 5196 y Fq(i;t)1968 5184 y Fu(\))h Fp(\000)f Fu(i)2172 5080 y Fq(n)2139 5105 y Fl(Y)2139 5282 y Fq(i)p Fs(=1)2260 5184 y Fr(\036)p Fu(\()p Fr(h)2389 5196 y Fq(i;t)2462 5184 y Fu(\)\)\()p Fr(H)26 b Fu(+)18 b(1\))2810 5149 y Fo(\000)2872 5127 y Fj(n)p 2872 5136 37 3 v 2876 5169 a Fi(2)2918 5149 y Fo(\000)p Fq(\017)3002 5184 y Fp(jj)23 b Fu(=)g(0)p Fr(:)p @beginspecial @setspecial @endspecial 0 5406 a Fu(This)k(completes)g(the)h(pro)r(of)f(of)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\).)41 b(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\))27 b(follo)n(ws)g(from)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))28 b(and)f(Corollary)e(2.4)31 b(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\).)42 b(_)-27 b Fr({)o(v)s Fu(\))28 b(follo)n(ws)f(from)g(the)h (prop)r(erties)e(of)0 5506 y(CCR)i(represen)n(tations.)35 b @beginspecial @setspecial @endspecial 3018 w Fh(\004)0 5522 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial eop %%Page: 33 33 33 32 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(33)0 489 y Fz(De\014nition)31 b(6.3.)40 b Fn(We)30 b(de\014ne)g(the)f Fu(asymptotic)f(creation)g Fn(and)i Fu(annihilation)e(op)r(erators)f Fn(on)j @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(\036)3172 459 y Fs(+)3228 489 y Fu(\()p Fr(h)p Fu(\)\))19 b Fp(\\)g(D)r Fu(\()p Fr(\036)3612 459 y Fs(+)3668 489 y Fu(\(i)p Fr(h)p Fu(\)\))1371 670 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)1415 635 y Fs(+)1470 670 y Fu(\()p Fr(h)p Fu(\))k(:=)1760 613 y(1)p 1726 650 111 4 v 1726 667 a Fp(p)p 1795 667 42 4 v 69 x Fu(2)1860 670 y(\()p Fr(\036)1941 635 y Fs(+)1997 670 y Fu(\()p Fr(h)p Fu(\))c Fp(\000)f Fu(i)p Fr(\036)2283 635 y Fs(+)2339 670 y Fu(\(i)p Fr(h)p Fu(\)\))p Fr(;)p @beginspecial @setspecial @endspecial 1354 948 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(a)1398 914 y Fs(+)p Fo(\003)1487 948 y Fu(\()p Fr(h)p Fu(\))23 b(:=)1777 892 y(1)p 1743 929 111 4 v 1743 946 a Fp(p)p 1812 946 42 4 v 69 x Fu(2)1877 948 y(\()p Fr(\036)1958 914 y Fs(+)2014 948 y Fu(\()p Fr(h)p Fu(\))c(+)f(i)p Fr(\036)2300 914 y Fs(+)2356 948 y Fu(\(i)p Fr(h)p Fu(\)\))p Fr(;)p @beginspecial @setspecial @endspecial 0 1146 a Fn(We)30 b(denote)g(by)g @beginspecial @setspecial @endspecial Fr(a)568 1116 y Fs(+)p Fq(])650 1146 y Fu(\()p Fr(h)p Fu(\))g Fn(the)g(op)l(er)l(ator)h Fr(a)1301 1116 y Fs(+)p Fo(\003)1390 1146 y Fu(\()p Fr(h)p Fu(\))f Fn(or)g @beginspecial @setspecial @endspecial Fr(a)1683 1116 y Fs(+)1738 1146 y Fu(\()p Fr(h)p Fu(\))p Fn(.)0 1167 y @beginspecial @setspecial @endspecial 100 1300 a Fu(W)-7 b(e)28 b(form)n(ulate)e(no)n (w)h(a)h(theorem)f(whic)n(h)g(follo)n(ws)g(from)g(the)h(previous)f (one.)0 1316 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 137 x Fz(Theorem)j(6.4.)40 b @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))24 b Fr(a)722 1423 y Fs(+)p Fo(\003)811 1453 y Fu(\()p Fr(h)p Fu(\))30 b Fn(and)g Fr(a)1158 1423 y Fs(+)1213 1453 y Fu(\()p Fr(h)p Fu(\))g Fn(ar)l(e)g(close)l(d)h(op)l (er)l(ators.)0 1619 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b Fr(h)319 1631 y Fq(i)369 1619 y Fp(2)23 b Fm(h)p Fr(;)14 b(i)23 b Fu(=)g(1)p Fr(:::n)p Fn(.)38 b(We)30 b(have)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\(\()p Fr(H)26 b Fu(+)18 b(i\))1606 1563 y Fj(n)p 1606 1572 37 3 v 1610 1605 a Fi(2)1657 1619 y Fu(\))23 b Fp(\032)g(D)r Fu(\()1932 1515 y Fq(n)1899 1540 y Fl(Y)1898 1717 y Fq(i)p Fs(=1)2020 1619 y Fr(a)2064 1585 y Fs(+)p Fq(])2146 1619 y Fu(\()p Fr(h)2226 1631 y Fq(i)2254 1619 y Fu(\)\))30 b Fn(and)869 1913 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 902 1809 a Fq(n)870 1834 y Fl(Y)869 2011 y Fq(i)p Fs(=1)991 1913 y Fr(a)1035 1878 y Fs(+)p Fq(])1116 1913 y Fu(\()p Fr(h)p Fu(\)\()p Fr(H)c Fu(+)18 b(i\))1493 1878 y Fo(\000)1556 1856 y Fj(n)p 1556 1865 V 1560 1898 a Fi(2)1629 1913 y Fu(=)70 b(lim)1717 1963 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1939 1913 y Fu(e)1976 1878 y Fs(i)p Fq(tH)2130 1809 y(n)2098 1834 y Fl(Y)2097 2011 y Fq(i)p Fs(=1)2218 1913 y Fr(a)2262 1878 y Fq(])2293 1913 y Fu(\()p Fr(h)2373 1925 y Fq(t)2402 1913 y Fu(\)e)2471 1878 y Fo(\000)p Fs(i)p Fq(tH)2630 1913 y Fu(\()p Fr(H)26 b Fu(+)18 b(i\))2895 1878 y Fo(\000)2957 1856 y Fj(n)p 2957 1865 V 2961 1898 a Fi(2)3008 1913 y Fr(:)p @beginspecial @setspecial @endspecial 0 2131 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(The)h(map)948 2339 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(h)1028 2351 y Fs(1)1066 2339 y Fr(;)14 b(:::;)g(h)1257 2351 y Fq(n)1302 2339 y Fu(\))23 b Fp(7!)1496 2236 y Fq(n)1464 2261 y Fl(Y)1463 2437 y Fq(i)p Fs(=1)1584 2339 y Fr(a)1628 2305 y Fs(+)p Fq(])1710 2339 y Fu(\()p Fr(h)p Fu(\)\()p Fr(H)j Fu(+)18 b(i\))2087 2305 y Fo(\000)2149 2283 y Fj(n)p 2149 2292 V 2153 2325 a Fi(2)2230 2339 y Fn(is)30 b(norm)g(c)l(ontinuous)o Fr(:)p @beginspecial @setspecial @endspecial 0 2562 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({v)s Fu(\))30 b Fn(The)h(c)l(ommutation)e(r)l(elations)i(hold)g(as)f (quadr)l(atic)h(form)f(on)g @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(a)2246 2532 y Fs(+)2302 2562 y Fu(\()p Fr(h)p Fu(\)\))19 b Fp(\\)g(D)r Fu(\()p Fr(a)2681 2532 y Fs(+)2736 2562 y Fu(\()p Fr(g)s Fu(\)\))1501 2703 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial([)p Fr(a)1568 2669 y Fs(+)1623 2703 y Fu(\()p Fr(h)p Fu(\))p Fr(;)14 b(a)1816 2669 y Fs(+)p Fo(\003)1905 2703 y Fu(\()p Fr(g)s Fu(\)])24 b(=)e(\()p Fr(h)p Fp(j)p Fr(g)s Fu(\)1)-13 b(l)p Fr(;)p @beginspecial @setspecial @endspecial 1272 2887 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu([)p Fr(a)1339 2852 y Fs(+)1394 2887 y Fu(\()p Fr(h)p Fu(\))p Fr(;)14 b(a)1587 2852 y Fs(+)1643 2887 y Fu(\()p Fr(g)s Fu(\)])23 b(=)g([)p Fr(a)1951 2852 y Fs(+)p Fo(\003)2040 2887 y Fu(\()p Fr(h)p Fu(\))p Fr(;)14 b(a)2233 2852 y Fs(+)p Fo(\003)2322 2887 y Fu(\()p Fr(g)s Fu(\)])23 b(=)g(0)p Fr(:)p @beginspecial @setspecial @endspecial 0 3028 a @beginspecial @setspecial @endspecial(v)s Fu(\))30 b Fn(We)g(have)1435 3174 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(e)1472 3139 y Fs(i)p Fq(tH)1578 3174 y Fr(a)1622 3139 y Fs(+)p Fq(])1704 3174 y Fu(\()p Fr(h)p Fu(\)e)1853 3139 y Fo(\000)p Fs(i)p Fq(tH)2035 3174 y Fu(=)23 b Fr(a)2167 3139 y Fs(+)p Fq(])2249 3174 y Fu(\()p Fr(h)2329 3186 y Fo(\000)p Fq(t)2410 3174 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 3335 a Fu(Similar)k(results)g(hold)g(for)g(the)h(mo)r(di\014ed)g (Hamiltonian)g @beginspecial @setspecial @endspecial 1943 3314 a(^)1921 3335 y Fr(H)1990 3347 y Fo(1)2060 3335 y Fu(.)37 b(W)-7 b(e)28 b(form)n(ulate)f(this)h(in)g(the)g(follo)n (wing)e(theorem.)0 3351 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 138 x Fz(Theorem)k(6.5.)45 b Fu(_)-28 b Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b Fr(h)23 b Fp(2)g Fm(h)30 b Fn(the)g(fol)t(lowing)j(limit)d(exists)1148 3638 y @beginspecial @setspecial @endspecial 1173 3617 a Fu(^)1148 3638 y Fr(W)1238 3604 y Fs(+)1293 3638 y Fu(\()p Fr(h)p Fu(\))24 b(:)83 b(=)g Fr(U)1749 3604 y Fo(\003)1740 3659 y(1)1810 3638 y Fr(W)1900 3604 y Fs(+)1955 3638 y Fu(\()p Fr(h)p Fu(\))p Fr(U)2124 3650 y Fo(1)1535 3823 y Fu(=)g @beginspecial @setspecial @endspecial Fr(s)18 b Fp(\000)65 b Fu(lim)1823 3873 y Fq(t)p Fo(!)p Fs(+)p Fo(1)2045 3823 y Fu(e)2082 3789 y Fs(i)p Fq(t)2142 3774 y Fs(^)2126 3789 y Fq(H)2180 3797 y Fg(1)2245 3823 y Fr(W)12 b Fu(\()p Fr(h)2415 3835 y Fq(t)2444 3823 y Fu(\)e)2513 3789 y Fo(\000)p Fs(i)p Fq(t)2626 3774 y Fs(^)2609 3789 y Fq(H)2663 3797 y Fg(1)2729 3823 y Fr(:)0 4017 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))23 b Fr(h)f Fp(!)310 3996 y Fu(^)286 4017 y Fr(W)376 3987 y Fs(+)431 4017 y Fu(\()p Fr(h)p Fu(\))h Fn(is)g(a)g(CCR)g(r)l(epr)l(esentation.)37 b(We)23 b(denote)g(by)g @beginspecial @setspecial @endspecial 1 w Fu(^)-43 b Fr(a)2035 3987 y Fo(\003)p Fs(+)2124 4017 y Fu(\()p Fr(h)p Fu(\))p Fr(;)16 b Fu(^)-44 b Fr(a)2317 3987 y Fs(+)2372 4017 y Fu(\()p Fr(h)p Fu(\))24 b Fn(the)f Fu(creation)e Fn(and)i Fu(annihilation)d(op)r(erators)0 4117 y Fn(asso)l(ciate)l(d)31 b(to)f(this)g(r)l(epr)l(esentation.)0 4278 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(F)-6 b(or)30 b Fr(h)345 4290 y Fq(i)396 4278 y Fp(2)23 b Fm(h)p Fr(;)14 b(i)23 b Fu(=)g(1)p Fr(:::n)p Fn(.)38 b(We)30 b(have)h @beginspecial @setspecial @endspecial Fp(D)r Fu(\(\()1412 4257 y(^)1390 4278 y Fr(H)1459 4290 y Fo(1)1548 4278 y Fu(+)18 b(i\))1696 4222 y Fj(n)p 1697 4231 V 1701 4264 a Fi(2)1747 4278 y Fu(\))24 b Fp(\032)e(D)r Fu(\()2022 4175 y Fq(n)1989 4199 y Fl(Y)1988 4376 y Fq(i)p Fs(=1)2112 4278 y Fu(^)-44 b Fr(a)2154 4244 y Fs(+)p Fq(])2236 4278 y Fu(\()p Fr(h)2316 4290 y Fq(i)2344 4278 y Fu(\)\))30 b Fn(and)749 4572 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 783 4468 a Fq(n)750 4493 y Fl(Y)749 4670 y Fq(i)p Fs(=1)872 4572 y Fu(^)-43 b Fr(a)915 4538 y Fs(+)p Fq(])997 4572 y Fu(\()p Fr(h)p Fu(\)\()1163 4551 y(^)1141 4572 y Fr(H)1210 4584 y Fo(1)1299 4572 y Fu(+)18 b(i\))1437 4538 y Fo(\000)1499 4515 y Fj(n)p 1499 4524 V 1503 4558 a Fi(2)1573 4572 y Fu(=)69 b(lim)1661 4622 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1883 4572 y Fu(e)1920 4538 y Fs(i)p Fq(t)1980 4523 y Fs(^)1964 4538 y Fq(H)2018 4546 y Fg(1)2130 4468 y Fq(n)2098 4493 y Fl(Y)2097 4670 y Fq(i)p Fs(=1)2218 4572 y Fr(a)2262 4538 y Fq(])2293 4572 y Fu(\()p Fr(h)2373 4584 y Fq(t)2402 4572 y Fu(\)e)2471 4538 y Fo(\000)p Fs(i)p Fq(t)2584 4523 y Fs(^)2567 4538 y Fq(H)2621 4546 y Fg(1)2687 4572 y Fu(\()2741 4551 y(^)2719 4572 y Fr(H)2788 4584 y Fo(1)2877 4572 y Fu(+)18 b(i\))3015 4538 y Fo(\000)3077 4515 y Fj(n)p 3077 4524 V 3081 4558 a Fi(2)3128 4572 y Fr(;)p @beginspecial @setspecial @endspecial 0 4799 a Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial 1 w Fu(^)-43 b Fr(a)278 4769 y Fs(+)p Fq(])360 4799 y Fu(\()p Fr(h)p Fu(\))30 b Fn(denote)h(either)g Fu(^)-43 b Fr(a)1048 4769 y Fs(+)p Fo(\003)1137 4799 y Fu(\()p Fr(h)p Fu(\))30 b Fn(or)g @beginspecial @setspecial @endspecial 1 w Fu(^)-43 b Fr(a)1430 4769 y Fs(+)1485 4799 y Fu(\()p Fr(h)p Fu(\))p Fn(.)0 4898 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({v)s Fu(\))30 b Fn(The)h(map)916 5106 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(h)996 5118 y Fs(1)1034 5106 y Fr(;)14 b(:::;)g(h)1225 5118 y Fq(n)1270 5106 y Fu(\))23 b Fp(7!)1464 5003 y Fq(n)1432 5027 y Fl(Y)1431 5204 y Fq(i)p Fs(=1)1554 5106 y Fu(^)-43 b Fr(a)1597 5072 y Fs(+)p Fq(])1678 5106 y Fu(\()p Fr(h)p Fu(\)\()1845 5085 y(^)1822 5106 y Fr(H)1891 5118 y Fo(1)1981 5106 y Fu(+)18 b(i\))2119 5072 y Fo(\000)2181 5050 y Fj(n)p 2181 5059 V 2185 5092 a Fi(2)2262 5106 y Fn(is)30 b(norm)f(c)l (ontinuous)p Fr(:)p @beginspecial @setspecial @endspecial 0 5325 a @beginspecial @setspecial @endspecial(v)s Fu(\))h Fn(We)g(have)1378 5485 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(e)1415 5451 y Fs(i)p Fq(t)1476 5436 y Fs(^)1459 5451 y Fq(H)1513 5459 y Fg(1)1580 5485 y Fu(^)-44 b Fr(a)1622 5451 y Fs(+)p Fq(])1704 5485 y Fu(\()p Fr(h)p Fu(\)e)1853 5451 y Fo(\000)p Fs(i)p Fq(t)1966 5436 y Fs(^)1949 5451 y Fq(H)2003 5459 y Fg(1)2092 5485 y Fu(=)23 b(^)-43 b Fr(a)2223 5451 y Fs(+)p Fq(])2305 5485 y Fu(\()p Fr(h)2385 5497 y Fo(\000)p Fq(t)2466 5485 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial eop %%Page: 34 34 34 33 bop 0 226 a Fs(34)1695 b(Z.)29 b(AMMARI)100 489 y Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(The)27 b(existence)g(of)h(the)g(strong)e(limit)j(follo)n(ws)d (from)i(\()p @beginspecial @setspecial @endspecial(6.1\))f(and)g(the)h (fact)g(that)1373 636 y Fr(U)1439 602 y Fo(\003)1430 657 y(1)1500 636 y Fr(W)12 b Fu(\()p Fr(h)1670 648 y Fq(t)1699 636 y Fu(\))p Fr(U)1788 648 y Fo(1)1882 636 y Fu(=)22 b(e)2006 602 y Fo(\000)p Fs(i)p Fq(I)5 b(m)p Fs(\()p Fq(G)2248 610 y Fg(1)2308 602 y Fo(j)p Fq(h)2367 610 y Fj(t)2394 602 y Fs(\))2424 636 y Fr(W)12 b Fu(\()p Fr(h)2594 648 y Fq(t)2624 636 y Fu(\))p Fr(;)0 682 y @beginspecial @setspecial @endspecial 1373 801 a(w)21 b Fp(\000)65 b Fu(lim)1536 851 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1759 801 y Fr(h)1807 813 y Fq(t)1859 801 y Fu(=)22 b(0)p Fr(:)0 977 y Fu(This)28 b(pro)n(v)n(e)d @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\).)38 b(Thm.)f(6.1)31 b(_)-27 b Fr({)o(v)s Fu(\))28 b(and)k(_)-27 b Fr({)p Fu(\))28 b(giv)n(e)1174 1131 y @beginspecial @setspecial @endspecial 1365 1110 a(^)1340 1131 y Fr(W)1430 1096 y Fs(+)1485 1131 y Fu(\()p Fr(h)p Fu(\))1636 1110 y(^)1611 1131 y Fr(W)1701 1096 y Fs(+)1757 1131 y Fu(\()p Fr(g)s Fu(\))23 b(=)g(e)2027 1074 y Fi(i)p 2021 1083 29 3 v 2021 1116 a(2)2060 1096 y Fs(Im)o(\()p Fq(h)p Fo(j)p Fq(g)r Fs(\))2311 1110 y Fu(^)2287 1131 y Fr(W)2377 1096 y Fs(+)2432 1131 y Fu(\()p Fr(h)c Fu(+)f Fr(g)s Fu(\))p Fr(;)0 1176 y @beginspecial @setspecial @endspecial 1365 1282 a Fu(^)1340 1303 y Fr(W)1430 1269 y Fs(+)1485 1303 y Fu(\()p Fr(h)p Fu(\))1597 1269 y Fo(\003)1659 1303 y Fu(=)1771 1282 y(^)1747 1303 y Fr(W)1837 1269 y Fs(+)1892 1303 y Fu(\()p Fp(\000)p Fr(h)p Fu(\))p Fr(:)0 1443 y Fu(This)28 b(pro)n(v)n(es)d(CCR)j(represen)n(tation.)35 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))28 b(is)g(a)f(consequence)g(of)g (Thm.)37 b @beginspecial @setspecial @endspecial(6.2)31 b(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(and)f(the)h(fact)g(that)1231 1583 y @beginspecial @setspecial @endspecial Fr(U)1297 1549 y Fo(\003)1288 1604 y(1)1359 1583 y Fr(\036)p Fu(\()p Fr(h)1488 1595 y Fq(t)1517 1583 y Fu(\))p Fr(U)1606 1595 y Fo(1)1700 1583 y Fu(=)22 b Fr(\036)p Fu(\()p Fr(h)1916 1595 y Fq(t)1946 1583 y Fu(\))d(+)f(Im\()p Fr(G)2276 1595 y Fo(1)2347 1583 y Fp(j)p Fr(h)2418 1595 y Fq(t)2447 1583 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 1723 a Fu(The)28 b(rest)f(follo)n(ws)f(from)i(Thm.)37 b @beginspecial @setspecial @endspecial(6.4.)2668 b Fh(\004)0 1883 y @beginspecial @setspecial @endspecial Fu(6.2.)47 b Fz(W)-8 b(a)m(v)m(e)33 b(op)s(erators.)41 b Fu(W)-7 b(e)29 b(recall)e(the)h(construction)f(of)h(the)h(F)-7 b(o)r(c)n(k)27 b(subrepresen)n(tation)g(of)h(a)g(CCR)g(represen)n (tation.)0 1983 y(Details)g(can)f(b)r(e)i(found)f(in)g @beginspecial @setspecial @endspecial([6],)g([12)o(].)38 b(Let)28 b Fm(g)g Fu(b)r(e)g(pre-Hilb)r(ert)g(space,)f(and)h(denote)f (b)n(y)h @beginspecial @setspecial @endspecial -4 x(\026)2758 1983 y Fm(g)g Fu(its)g(completion.)37 b(W)-7 b(e)29 b(de\014ne)f(the)0 2082 y(space)f(of)g Fn(vacua)1326 2222 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(K)1389 2234 y Fq(\031)1457 2222 y Fu(:=)c Fp(f)p Fr(u)f Fp(2)i(H)14 b(j)p Fr(a)1910 2234 y Fq(\031)1955 2222 y Fu(\()p Fr(h)p Fu(\))p Fr(u)23 b Fu(=)g(0)p Fr(;)k(h)c Fp(2)h Fm(g)p Fp(g)p Fr(:)p @beginspecial @setspecial @endspecial 0 2375 a @beginspecial @setspecial @endspecial Fz(Prop)s(osition)30 b(6.6.)44 b Fu(_)-27 b Fr({)p Fu(\))14 b Fp(K)842 2387 y Fq(\031)917 2375 y Fn(is)30 b(a)g(close)l(d)h(sp)l (ac)l(e.)4 2474 y Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))14 b Fp(K)163 2486 y Fq(\031)238 2474 y Fn(is)30 b(c)l(ontaine)l(d)g(in)g (the)g(set)f(of)i(analytic)g(ve)l(ctors)f(of)g @beginspecial @setspecial @endspecial Fr(\036)1904 2486 y Fq(\031)1950 2474 y Fu(\()p Fr(h)p Fu(\))p Fr(;)14 b(h)23 b Fp(2)h Fm(g)p Fn(.)0 2495 y @beginspecial @setspecial @endspecial 100 2627 a Fu(Let)j @beginspecial @setspecial @endspecial Fp(H)318 2639 y Fq(\031)387 2627 y Fu(:=)22 b Fp(K)560 2639 y Fq(\031)624 2627 y Fp(\012)c Fu(\000\()791 2623 y(\026)791 2627 y Fm(g)p Fu(\).)37 b(W)-7 b(e)28 b(de\014ne)1093 2834 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1135 2766 a @beginspecial @setspecial @endspecial(\012)1195 2778 y Fq(\031)1263 2766 y Fu(:)23 b Fp(K)1372 2778 y Fq(\031)1436 2766 y Fp(\012)18 b Fu(\000)1571 2778 y Fs(\014n)1649 2766 y Fu(\()1681 2762 y(\026)1681 2766 y Fm(g)p Fu(\))24 b Fp(!)f(H)q @beginspecial @setspecial @endspecial 1135 2900 a Fu(\012)1195 2912 y Fq(\031)1240 2900 y Fr( )f Fp(\012)c Fr(\036)p Fu(\()p Fr(h)p Fu(\))1560 2866 y Fq(p)1599 2900 y Fu(\012)23 b(:=)g Fr(\036)1842 2912 y Fq(\031)1887 2900 y Fu(\()p Fr(h)p Fu(\))1999 2866 y Fq(p)2038 2900 y Fr( )s(;)83 b(h)23 b Fp(2)g Fm(g)p Fr(;)60 b( )26 b Fp(2)e(K)2697 2912 y Fq(\031)2742 2900 y Fr(:)2807 2834 y @beginspecial @setspecial @endspecial 0 2978 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 153 x Fz(Prop)s(osition)30 b(6.7.)40 b Fn(The)31 b(map)f @beginspecial @setspecial @endspecial Fu(\012)1118 3143 y Fq(\031)1193 3131 y Fn(extends)f(to)h(an)g (isometric)h(map)1694 3271 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\012)1754 3283 y Fq(\031)1822 3271 y Fu(:)23 b Fp(H)1938 3283 y Fq(\031)2006 3271 y Fp(!)g(H)q Fr(;)p @beginspecial @setspecial @endspecial 0 3411 a Fn(satis\014es)30 b @beginspecial @setspecial @endspecial Fu(\012)372 3423 y Fq(\031)417 3411 y Fu(1)-13 b(l)18 b Fp(\012)g Fr(a)614 3380 y Fq(])645 3411 y Fu(\()p Fr(h)p Fu(\))23 b(=)g Fr(a)912 3380 y Fq(])912 3431 y(\031)957 3411 y Fu(\()p Fr(h)p Fu(\))p Fr(;)60 b(h)23 b Fp(2)g Fm(g)p Fr(:)0 3431 y @beginspecial @setspecial @endspecial 100 3563 a Fu(Thm.)37 b @beginspecial @setspecial @endspecial(6.1)27 b(sho)n(ws)f(that)i(asymptotic)f(W)-7 b(eyl)28 b(op)r(erators)d(de\014ne)j(a)f(CCR)h(represen)n(tation.)35 b(Then)28 b(w)n(e)f(de\014ne)h(the)g(space)0 3663 y(of)g Fn(vacua)g Fu(in)g(our)f(case)1257 3803 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1299 3802 a @beginspecial @setspecial @endspecial Fp(K)1363 3772 y Fs(+)1441 3802 y Fu(:=)c Fp(f)p Fr(u)f Fp(2)h(H)q(j)46 b Fr(a)1926 3772 y Fs(+)1981 3802 y Fu(\()p Fr(h)p Fu(\))p Fr(u)23 b Fu(=)g(0)p Fr(;)k(h)c Fp(2)h Fm(h)p Fp(g)p Fr(:)2643 3803 y @beginspecial @setspecial @endspecial 100 3946 a Fu(W)-7 b(e)28 b(denote)f(b)n(y)g @beginspecial @setspecial @endspecial Fp(H)696 3916 y Fs(+)779 3946 y Fu(the)h(space)f Fp(K)1208 3916 y Fs(+)1282 3946 y Fp(\012)18 b Fu(\000\()p Fm(h)p Fu(\).)0 3967 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 132 x Fz(Prop)s(osition)30 b(6.8.)40 b Fn(The)31 b(fol)t(lowing)h(thr)l (e)l(e)e(assertions)g(hold:)4 4198 y Fu(_)-27 b Fr({)p Fu(\))14 b Fp(K)137 4168 y Fs(+)222 4198 y Fn(is)30 b(close)l(d)h @beginspecial @setspecial @endspecial Fr(H)7 b Fn(-invariant)30 b(sp)l(ac)l(e.)4 4298 y Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b @beginspecial @setspecial @endspecial Fr(h)319 4310 y Fq(i)369 4298 y Fp(2)23 b Fm(h)p Fr(;)14 b(i)23 b Fu(=)g(1)p Fr(:::n)p Fn(.)38 b(One)29 b(has)1561 4504 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(K)1625 4470 y Fs(+)1703 4504 y Fp(\032)23 b(D)r Fu(\()1923 4400 y Fq(n)1890 4425 y Fl(Y)1889 4602 y Fq(i)p Fs(=1)2011 4504 y Fr(a)2055 4470 y Fs(+)p Fo(\003)2144 4504 y Fu(\()p Fr(h)2224 4516 y Fq(i)2252 4504 y Fu(\)\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4759 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))1689 4898 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(1)-13 b(l)1740 4910 y Fs(pp)1818 4898 y Fu(\()p Fr(H)7 b Fu(\))23 b Fp(\032)g(K)2133 4864 y Fs(+)2188 4898 y Fr(:)p @beginspecial @setspecial @endspecial 100 5051 a Fz(Pro)s(of.)45 b @beginspecial @setspecial @endspecial 24 w Fu(The)29 b(fact)g(that)f Fp(K)1009 5021 y Fs(+)1093 5051 y Fu(is)h Fr(H)7 b Fp(\000)p Fu(in)n(v)-5 b(arian)n(t)27 b(follo)n(ws)g(from)h (Thm.)40 b @beginspecial @setspecial @endspecial(6.4)28 b Fr(v)s Fu(\).)44 b(_)-27 b Fr({)p Fu(\))29 b(and)j(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))29 b(follo)n(w)e(b)n(y)i(Prop.)38 b @beginspecial @setspecial @endspecial(6.7.)h(Let)0 5151 y(us)28 b(pro)n(v)n(e)d(no)n(w)32 b(_)-28 b Fr({)t Fu(_)h Fr({)t Fu(_)g Fr({)p Fu(\).)37 b(Let)28 b Fr(u)22 b Fp(2)i(H)k Fu(suc)n(h)g(that)f @beginspecial @setspecial @endspecial Fr(H)7 b(u)23 b Fu(=)f Fr(E)5 b(u)p Fu(,)28 b(one)f(has)1465 5294 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 47 w(lim)1465 5344 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1687 5294 y Fu(e)1724 5260 y Fs(i)p Fq(tH)1831 5294 y Fr(a)p Fu(\()p Fr(h)1955 5306 y Fq(t)1984 5294 y Fu(\)e)2053 5260 y Fo(\000)p Fs(i)p Fq(tH)2212 5294 y Fr(u)c Fu(=)f(0)p Fr(;)p @beginspecial @setspecial @endspecial 0 5481 a Fu(since)27 b @beginspecial @setspecial @endspecial Fr(s)19 b Fp(\000)39 b Fu(lim)344 5531 y Fq(t)p Fo(!1)515 5481 y Fr(a)p Fu(\()p Fr(h)639 5493 y Fq(t)668 5481 y Fu(\))24 b(=)e(0)27 b(and)h @beginspecial @setspecial @endspecial(e)1079 5451 y Fs(i)p Fq(tH)1186 5481 y Fr(a)p Fu(\()p Fr(h)1310 5493 y Fq(t)1339 5481 y Fu(\)e)1408 5451 y Fo(\000)p Fs(i)p Fq(tH)1567 5481 y Fr(u)22 b Fu(=)h(\()p Fr(E)h Fu(+)18 b(i\))c(e)2031 5451 y Fs(i)p Fq(t)p Fs(\()p Fq(H)t Fo(\000)p Fq(E)s Fs(\))2294 5481 y Fr(a)p Fu(\()p Fr(h)2418 5493 y Fq(t)2447 5481 y Fu(\)\()p Fr(H)26 b Fu(+)18 b(i\))2744 5451 y Fo(\000)p Fs(1)2833 5481 y Fr(u)p Fu(.)37 b(This)27 b(means)g @beginspecial @setspecial @endspecial Fr(a)3428 5451 y Fs(+)3483 5481 y Fu(\()p Fr(h)p Fu(\))p Fr(u)c Fu(=)g(0.)16 b Fh(\004)p eop %%Page: 35 35 35 34 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(35)100 489 y Fu(W)-7 b(e)28 b(de\014ne)1401 640 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(H)1477 606 y Fs(+)1555 640 y Fu(:=)22 b Fr(H)1734 655 y Fo(jK)1805 639 y Fi(+)1874 640 y Fp(\012)c Fu(1)-15 b(l)18 b(+)g(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fr(!)s Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 100 791 a Fu(and)27 b(the)h Fn(wave)j(op)l(er)l (ator)718 1076 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 759 936 a @beginspecial @setspecial @endspecial Fu(\012)819 902 y Fs(+)898 936 y Fu(:)23 b Fp(H)1015 902 y Fs(+)1093 936 y Fp(!)g(H)q Fr(;)p @beginspecial @setspecial @endspecial 759 1138 a Fu(\012)819 1104 y Fs(+)874 1138 y Fr( )f Fp(\012)1066 1035 y Fq(n)1034 1059 y Fl(Y)1033 1236 y Fq(i)p Fs(=1)1154 1138 y Fr(a)1198 1104 y Fo(\003)1236 1138 y Fu(\()p Fr(h)1316 1150 y Fq(i)1344 1138 y Fu(\)\012)h(:=)1603 1035 y Fq(n)1571 1059 y Fl(Y)1570 1236 y Fq(i)p Fs(=1)1692 1138 y Fr(a)1736 1104 y Fs(+)p Fo(\003)1825 1138 y Fu(\()p Fr(h)1905 1150 y Fq(i)1932 1138 y Fu(\))p Fr( )s(;)28 b Fu(for)14 b Fr( )26 b Fp(2)d(K)2408 1104 y Fs(+)2464 1138 y Fr(;)60 b(h)2595 1150 y Fq(i)2645 1138 y Fp(2)24 b Fm(h)p Fr(;)k(i)22 b Fu(=)h(1)p Fr(:::n:)3182 1076 y @beginspecial @setspecial @endspecial 0 1304 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 159 x Fz(Theorem)30 b(6.9.)40 b Fu(\012)655 1433 y Fs(+)740 1463 y Fn(is)30 b(a)g(unitary)g(map)h(satisfying)1294 1614 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(a)1338 1579 y Fs(+)p Fq(])1420 1614 y Fu(\()p Fr(h)p Fu(\)\012)1592 1579 y Fs(+)1670 1614 y Fu(=)23 b(\012)1818 1579 y Fs(+)1887 1614 y Fu(1)-13 b(l)17 b Fp(\012)h Fr(a)2083 1579 y Fq(])2114 1614 y Fu(\()p Fr(h)p Fu(\))p Fr(;)28 b Fu(for)14 b Fr(h)23 b Fp(2)g Fm(h)p Fr(;)p @beginspecial @setspecial @endspecial 1665 1816 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(H)7 b Fu(\012)1801 1781 y Fs(+)1879 1816 y Fu(=)22 b(\012)2026 1781 y Fs(+)2081 1816 y Fr(H)2157 1781 y Fs(+)2212 1816 y Fr(:)p @beginspecial @setspecial @endspecial 100 1974 a Fz(Pro)s(of.)64 b @beginspecial @setspecial @endspecial 34 w Fu(Prop.)55 b(6.8)34 b(giv)n(es)f(that)h(\012)1310 1944 y Fs(+)1399 1974 y Fu(is)g(isometric)g(and)g(satis\014es)f(prop)r (erties)g(announced)h(in)g(the)h(theorem.)56 b(Let)0 2074 y(pro)n(v)n(e)24 b(that)h @beginspecial @setspecial @endspecial(\012)459 2044 y Fs(+)540 2074 y Fu(is)g(unitary)-7 b(.)36 b(Using)25 b([12)o(,)h(Thm.)36 b(3.3],)25 b(it)h(su\016ces)f(to) g(sho)n(w)f(that)i(the)g(CCR)f(represen)n(tation)f @beginspecial @setspecial @endspecial Fr(h)e Fp(!)i Fr(W)3733 2044 y Fs(+)3788 2074 y Fu(\()p Fr(h)p Fu(\))0 2174 y(admits)30 b(a)f(densely)g(de\014ned)h(n)n(um)n(b)r(er)g(op)r(erator.)41 b(F)-7 b(or)29 b(eac)n(h)g(\014nite)h(dimensional)g(space)f @beginspecial @setspecial @endspecial Fm(f)d Fp(\032)g Fm(h)p Fu(,)31 b(w)n(e)e(de\014ne)h(as)f(quadratic)0 2273 y(form)e(the)h(follo)n(wing)f(expression)732 2527 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(n)782 2491 y Fs(+)782 2569 y Fm(f)837 2527 y Fu(\()p Fr(u)p Fu(\))c(:=)1083 2415 y Fs(dim)o Fm(f)1091 2448 y Fl(X)1097 2625 y Fq(i)p Fs(=1)1234 2527 y Fp(jj)p Fr(a)1324 2492 y Fs(+)1379 2527 y Fu(\()p Fr(h)1459 2539 y Fq(i)1486 2527 y Fu(\))p Fr(u)p Fp(jj)1612 2492 y Fs(2)1650 2527 y Fr(;)60 b Fp(f)p Fr(h)1823 2539 y Fq(i)1850 2527 y Fp(g)14 b Fu(is)g(an)g(orthonormal)g(basis)g(of)47 b Fm(f)p Fr(;)28 b(u)23 b Fp(2)g(H)q Fr(:)p @beginspecial @setspecial @endspecial 0 2764 a Fu(No)n(w)k(w)n(e)g(sho)n(w)g(that)h @beginspecial @setspecial @endspecial Fr(n)747 2776 y Fs(+)802 2764 y Fu(\()p Fr(u)p Fu(\))23 b(:=)g(sup)1097 2851 y Fm(f)1210 2764 y Fr(n)1260 2728 y Fs(+)1260 2806 y Fm(f)1315 2764 y Fu(\()p Fr(u)p Fu(\))k(is)h(densely)f(de\014ned:) 1133 3185 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1174 3113 a @beginspecial @setspecial @endspecial Fp(jj)p Fr(n)1270 3077 y Fs(+)1270 3155 y Fm(f)1325 3113 y Fu(\()p Fr(u)p Fu(\))p Fp(jj)1483 3078 y Fs(2)1604 3113 y Fp(\024)129 b Fu(lim)1751 3162 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1974 3001 y Fs(dim)o Fm(f)1982 3034 y Fl(X)1988 3211 y Fq(i)p Fs(=1)2124 3113 y Fp(jj)p Fr(a)p Fu(\()p Fr(h)2294 3125 y Fq(i;t)2367 3113 y Fu(\)e)2436 3078 y Fo(\000)p Fs(i)p Fq(tH)2595 3113 y Fr(u)p Fp(jj)2689 3078 y Fs(2)2726 3113 y @beginspecial @setspecial @endspecial 1604 3328 a Fp(\024)g Fu(lim)1751 3378 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1960 3328 y Fu(\(e)2029 3293 y Fo(\000)p Fs(i)p Fq(tH)2188 3328 y Fr(u;)14 b(N)9 b Fu(e)2386 3293 y Fo(\000)p Fs(i)p Fq(tH)2543 3328 y Fr(u)p Fu(\))p Fr(:)2767 3185 y @beginspecial @setspecial @endspecial 0 3531 a Fu(W)-7 b(e)28 b(conclude)f(using)h(Corollary)d @beginspecial @setspecial @endspecial(4.11)h(that)i Fr(n)1474 3501 y Fs(+)1529 3531 y Fu(\()p Fr(u)p Fu(\))23 b Fp(\024)g Fr(c)14 b Fp(jj)p Fu(\()p Fr(H)25 b Fu(+)18 b Fr(b)p Fu(\))2135 3479 y Fi(1)p 2135 3488 29 3 v 2135 3521 a(2)2177 3531 y Fr(u)p Fp(jj)p Fu(.)37 b(Th)n(us)27 b @beginspecial @setspecial @endspecial Fp(D)r Fu(\()p Fr(H)2726 3479 y Fi(1)p 2726 3488 V 2726 3521 a(2)2768 3531 y Fu(\))c Fp(\032)g(D)r Fu(\()p Fr(n)3059 3501 y Fs(+)3114 3531 y Fu(\))28 b(and)g @beginspecial @setspecial @endspecial(Ran\012)3545 3501 y Fs(+)3623 3531 y Fu(=)22 b Fp(H)q Fu(.)31 b Fh(\004)100 3731 y Fu(W)-7 b(e)28 b(de\014ne)f(an)h Fn(extende)l(d)h(wave)i(op)l (er)l(ator)821 4087 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 862 3943 a @beginspecial @setspecial @endspecial 871 3922 a Fu(^)862 3943 y(\012)922 3909 y Fs(ext)p Fq(;)p Fs(+)1110 3943 y Fu(:)1186 3840 y Fo(1)1157 3865 y Fl(M)1157 4040 y Fq(n)p Fs(=0)1296 3943 y Fp(D)r Fu(\(\()1449 3922 y(^)1426 3943 y Fr(H)1495 3955 y Fo(1)1585 3943 y Fu(+)18 b(1\))1752 3887 y Fj(n)p 1751 3896 37 3 v 1755 3929 a Fi(2)1802 3943 y Fu(\))h Fp(\012)f(\012)2001 3909 y Fq(n)2001 3964 y(s)2046 3943 y Fm(h)23 b Fp(!)g(H)q Fr(;)p @beginspecial @setspecial @endspecial 871 4200 a Fu(^)862 4221 y(\012)922 4186 y Fs(ext)p Fq(;)p Fs(+)1087 4221 y Fr( )f Fp(\012)1279 4117 y Fq(n)1247 4142 y Fl(Y)1246 4319 y Fq(i)p Fs(=1)1367 4221 y Fr(a)1411 4186 y Fo(\003)1449 4221 y Fu(\()p Fr(h)1529 4233 y Fq(i)1557 4221 y Fu(\)\012)h(:=)1816 4117 y Fq(n)1784 4142 y Fl(Y)1783 4319 y Fq(i)p Fs(=1)1906 4221 y Fu(^)-43 b Fr(a)1949 4186 y Fs(+)p Fo(\003)2038 4221 y Fu(\()p Fr(h)2118 4233 y Fq(i)2145 4221 y Fu(\))p Fr( )s(;)61 b( )26 b Fp(2)d(D)r Fu(\(\()2629 4200 y(^)2606 4221 y Fr(H)2675 4233 y Fo(1)2765 4221 y Fu(+)18 b(1\))2932 4164 y Fj(n)p 2932 4173 V 2936 4206 a Fi(2)2982 4221 y Fu(\))p Fr(:)3079 4087 y @beginspecial @setspecial @endspecial 0 4443 a Fu(W)-7 b(e)28 b(set)1221 4594 y @beginspecial @setspecial @endspecial 1237 4573 a(^)1221 4594 y Fp(K)1285 4560 y Fs(+)1363 4594 y Fu(:=)23 b Fr(U)1540 4560 y Fo(\003)1531 4615 y(1)1601 4594 y Fp(K)1665 4560 y Fs(+)1721 4594 y Fr(;)1874 4573 y Fu(^)1850 4594 y Fp(H)1921 4560 y Fs(+)1999 4594 y Fu(:=)2126 4573 y(^)2109 4594 y Fp(K)2173 4560 y Fs(+)2248 4594 y Fp(\012)18 b Fu(\000\()p Fm(h)p Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 4745 a Fu(Then)28 b(w)n(e)f(ha)n(v)n(e)f(a)i(w)n(a)n(v)n (e)e(op)r(erator)f(of)j(the)g(mo)r(di\014ed)g(Hamiltonian:)676 4899 y @beginspecial @setspecial @endspecial 852 4878 a(^)842 4899 y(\012)902 4865 y Fs(+)981 4899 y Fu(:)1051 4878 y(^)1027 4899 y Fp(H)1098 4865 y Fs(+)1176 4899 y Fp(!)23 b(H)q Fr(;)0 4940 y @beginspecial @setspecial @endspecial 852 5116 a Fu(^)842 5137 y(\012)902 5102 y Fs(+)957 5137 y Fr( )f Fp(\012)1149 5033 y Fq(n)1117 5058 y Fl(Y)1116 5235 y Fq(i)p Fs(=1)1237 5137 y Fr(a)1281 5102 y Fo(\003)1319 5137 y Fu(\()p Fr(h)1399 5149 y Fq(i)1427 5137 y Fu(\)\012)i(:=)1686 5033 y Fq(n)1654 5058 y Fl(Y)1653 5235 y Fq(i)p Fs(=1)1776 5137 y Fu(^)-43 b Fr(a)1819 5102 y Fs(+)p Fo(\003)1908 5137 y Fu(\()p Fr(h)1988 5149 y Fq(i)2015 5137 y Fu(\))p Fr( )s(;)28 b Fu(for)14 b Fr( )26 b Fp(2)2443 5116 y Fu(^)2427 5137 y Fp(K)2491 5102 y Fs(+)2547 5137 y Fr(;)60 b(h)2678 5149 y Fq(i)2728 5137 y Fp(2)24 b Fm(h)p Fr(;)k(i)22 b Fu(=)h(1)p Fr(:::n:)0 5381 y Fu(W)-7 b(e)40 b(notice)f(that)g @beginspecial @setspecial @endspecial 611 5360 a(^)602 5381 y(\012)662 5341 y Fs(ext)p Fq(;)p Fs(+)662 5421 y Fo(j)701 5406 y Fs(^)682 5421 y Fo(H)739 5404 y Fi(+)869 5381 y Fu(=)986 5360 y(^)976 5381 y(\012)1036 5351 y Fs(+)1091 5381 y Fu(.)72 b(This)39 b(suggests)f(to)h(treat)g(sometimes)g @beginspecial @setspecial @endspecial 2474 5360 a(^)2465 5381 y(\012)2525 5351 y Fs(ext)p Fq(;)p Fs(+)2729 5381 y Fu(as)f(a)h(partial)g(isometry)-7 b(.)70 b(Another)0 5506 y(construction)27 b(of)g(the)h(extended)g(w)n(a)n(v)n(e)e(op)r (erator)g(is)h(giv)n(en)g(b)n(y)h(the)f(follo)n(wing)g(theorem,)g(see)g @beginspecial @setspecial @endspecial([11)o(,)h(Thm.)37 b(5.7]:)0 5527 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial eop %%Page: 36 36 36 35 bop 0 226 a Fs(36)1695 b(Z.)29 b(AMMARI)0 489 y Fz(Theorem)h(6.10.)40 b @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))30 b Fn(L)l(et)f Fr(u)23 b Fp(2)g(D)r Fu(\()1131 468 y(^)1122 489 y(\012)1182 459 y Fs(ext)p Fq(;)p Fs(+)1348 489 y Fu(\))p Fn(.)39 b(Then)30 b(one)g(has)1352 644 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 46 w Fu(lim)1352 694 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1574 644 y Fu(e)1611 610 y Fs(i)p Fq(t)1671 595 y Fs(^)1655 610 y Fq(H)1709 618 y Fg(1)1774 644 y Fr(I)7 b Fu(e)1854 610 y Fo(\000)p Fs(i)p Fq(t)1966 595 y Fs(^)1950 610 y Fq(H)2008 585 y Fi(ext)2004 626 y Fg(1)2094 644 y Fr(u)23 b Fu(=)2262 623 y(^)2253 644 y(\012)2313 610 y Fs(ext)p Fq(;)p Fs(+)2478 644 y Fr(u;)p @beginspecial @setspecial @endspecial 0 812 a Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial Fr(I)37 b Fn(is)30 b(the)g(sc)l(attering)g(identi\014c)l (ation)h(op)l(er)l(ator)f(de\014ne)l(d)g(in)g(the)g(Subse)l(ct.)38 b @beginspecial @setspecial @endspecial(2.1.)4 915 y Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))24 b Fn(L)l(et)h Fr(\037)e Fp(2)g Fr(C)467 885 y Fo(1)461 936 y Fs(0)538 915 y Fu(\()p Fk(R)p Fu(\))p Fn(.)43 b @beginspecial @setspecial @endspecial Fu(Ran)14 b Fr(\037)p Fu(\()993 894 y(^)971 915 y Fr(H)1047 885 y Fs(ext)1040 935 y Fo(1)1141 915 y Fu(\))24 b Fp(\032)e(D)r Fu(\()1391 894 y(^)1382 915 y(\012)1442 885 y Fs(ext)p Fq(;)p Fs(+)1608 915 y Fu(\))j Fn(and)g @beginspecial @setspecial @endspecial Fr(I)7 b(\037)p Fu(\()1970 894 y(^)1948 915 y Fr(H)2024 885 y Fs(ext)2017 935 y Fo(1)2119 915 y Fu(\))p Fr(;)2197 894 y Fu(^)2188 915 y(\012)2248 885 y Fs(ext)p Fq(;)p Fs(+)2413 915 y Fr(\037)p Fu(\()2519 894 y(^)2497 915 y Fr(H)2573 885 y Fs(ext)2566 935 y Fo(1)2668 915 y Fu(\))25 b Fn(ar)l(e)g(b)l(ounde)l(d)g(op)l(er)l(ators.)38 b(Mor)l(e)l(over)1113 1070 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 46 w Fu(lim)1113 1120 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1335 1070 y Fu(e)1372 1036 y Fs(i)p Fq(t)1432 1021 y Fs(^)1416 1036 y Fq(H)1470 1044 y Fg(1)1535 1070 y Fr(I)7 b Fu(e)1615 1036 y Fo(\000)p Fs(i)p Fq(t)1727 1021 y Fs(^)1711 1036 y Fq(H)1769 1011 y Fi(ext)1765 1052 y Fg(1)1855 1070 y Fr(\037)p Fu(\()1961 1049 y(^)1939 1070 y Fr(H)2015 1036 y Fs(ext)2008 1091 y Fo(1)2110 1070 y Fu(\))23 b(=)2262 1049 y(^)2253 1070 y(\012)2313 1036 y Fs(ext)p Fq(;)p Fs(+)2478 1070 y Fr(\037)p Fu(\()2584 1049 y(^)2562 1070 y Fr(H)2638 1036 y Fs(ext)2631 1091 y Fo(1)2732 1070 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 1415 1275 a @beginspecial @setspecial @endspecial Fu(7.)46 b Fv(Pr)n(op)-6 b(a)n(ga)g(tion)32 b(estima)-6 b(tes)100 1425 y Fu(W)f(e)28 b(mak)n(e)e(the)i(follo)n(wing)f (notations)g(for)g(the)h(Heisen)n(b)r(erg)f(deriv)-5 b(ativ)n(es)1511 1695 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1552 1556 a @beginspecial @setspecial @endspecial(d)-12 b(l)1610 1568 y Fs(0)1670 1556 y Fu(:=)23 b Fr(@)1825 1568 y Fq(t)1873 1556 y Fu(+)18 b(i[)p Fr(!)s Fu(\()p Fr(k)s Fu(\))p Fr(;)c(:)p Fu(])p Fr(;)1552 1690 y Fu(I)-13 b(D)1632 1702 y Fs(0)1693 1690 y Fu(:=)22 b Fr(@)1847 1702 y Fq(t)1895 1690 y Fu(+)c(i[d\000\()p Fr(!)s Fu(\))p Fr(;)c(:)p Fu(])p Fr(;)p @beginspecial @setspecial @endspecial 1572 1813 a Fu(^)1552 1834 y(I)-13 b(D)1632 1846 y Fq(\024)1698 1834 y Fu(:=)23 b Fr(@)1853 1846 y Fq(t)1901 1834 y Fu(+)18 b(i[)2052 1813 y(^)2030 1834 y Fr(H)2099 1846 y Fq(\024)2142 1834 y Fr(;)c(:)p Fu(])p Fr(:)2389 1695 y @beginspecial @setspecial @endspecial 0 1985 a @beginspecial @setspecial @endspecial Fz(Prop)s(osition)30 b(7.1.)40 b Fn(L)l(et)32 b Fr(\037)c Fp(2)h Fr(C)1081 1955 y Fo(1)1075 2006 y Fs(0)1151 1985 y Fu(\()p Fk(R)q Fu(\))p Fn(.)53 b(F)-6 b(or)33 b @beginspecial @setspecial @endspecial Fr(R)1570 1955 y Fo(0)1621 1985 y Fr(>)27 b(R)i(>)f Fu(1)p Fn(,)33 b(ther)l(e)g(exists)f Fr(c)g Fn(indep)l(endent)h(fr)l(om)g Fr(\024)g Fn(such)f(that)h(we)g(have)0 2085 y(for)e @beginspecial @setspecial @endspecial Fr(\024)22 b Fp(\024)h(1)989 2264 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)1072 2172 y Fo(1)1035 2340 y Fs(1)1170 2264 y Fp(jj)p Fu(d\000\(1)-13 b(l)1398 2279 y Fs([)p Fq(R;R)1537 2263 y Fg(0)1559 2279 y Fs(])1582 2264 y Fu(\()1624 2208 y Fp(j)p Fr(x)p Fp(j)p 1624 2245 94 4 v 1656 2321 a Fr(t)1728 2264 y Fu(\)\))1802 2208 y Fi(1)p 1803 2217 29 3 v 1803 2250 a(2)1845 2264 y Fr(\037)p Fu(\()1951 2243 y(^)1929 2264 y Fr(H)1998 2276 y Fq(\024)2041 2264 y Fu(\))14 b(e)2124 2230 y Fo(\000)p Fs(i)p Fq(t)2237 2215 y Fs(^)2220 2230 y Fq(H)2274 2238 y Fj(\024)2317 2264 y Fr(u)p Fp(jj)2411 2230 y Fs(2)2458 2208 y Fr(dt)p 2458 2245 74 4 v 2480 2321 a(t)2564 2264 y Fp(\024)23 b Fr(c)g Fp(jj)p Fr(u)p Fp(jj)2851 2230 y Fs(2)2888 2264 y Fr(:)p @beginspecial @setspecial @endspecial 100 2461 a Fz(Pro)s(of.)47 b @beginspecial @setspecial @endspecial 26 w Fu(W)-7 b(e)29 b(use)g(a)g(standard)f(metho)r(d)i(in)g (scattering)e(theory)g(of)h(the)h(N-b)r(o)r(dy)f(problem)g @beginspecial @setspecial @endspecial([15)o(],[24)o(].)42 b(It)30 b(is)f(based)f(on)0 2561 y(a)f(tec)n(hnical)g(lemma,)h(see)f (e.g.)37 b @beginspecial @setspecial @endspecial([10)o(,)27 b(lemma)h(B.4.1].)0 2661 y(Let)j @beginspecial @setspecial @endspecial Fr(F)40 b Fp(2)29 b Fr(C)394 2631 y Fo(1)465 2661 y Fu(\()p Fk(R)p Fu(\))37 b(b)r(e)31 b(a)g(cuto\013)g(function)g (equal)g(to)f(1)h(near)f @beginspecial @setspecial @endspecial Fp(1)p Fu(,)h(to)g(0)g(near)e(the)j(origin,)e(with)i Fr(F)3130 2631 y Fo(0)3153 2661 y Fu(\()p Fr(s)p Fu(\))d Fp(\025)f Fu(1)-15 b(l)3427 2676 y Fs([)p Fq(R;R)3566 2659 y Fg(0)3589 2676 y Fs(])3612 2661 y Fu(\()p Fr(s)p Fu(\).)47 b(W)-7 b(e)0 2760 y(consider)27 b(the)h(observ)-5 b(able)1363 2924 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\010\()p Fr(t)p Fu(\))24 b(:=)e Fr(\037)p Fu(\()1757 2903 y(^)1735 2924 y Fr(H)1804 2936 y Fq(\024)1848 2924 y Fu(\)d\000\()p Fr(F)12 b Fu(\()2117 2868 y Fp(j)p Fr(x)p Fp(j)p 2117 2905 94 4 v 2149 2981 a Fr(t)2221 2924 y Fu(\)\))p Fr(\037)p Fu(\()2391 2903 y(^)2369 2924 y Fr(H)2438 2936 y Fq(\024)2482 2924 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3083 a Fu(By)27 b(Lemma)h @beginspecial @setspecial @endspecial(A.3,)f(it)h(is)g(enough)f(to)g(sho)n(w)g(that)871 3263 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 890 3242 a(^)871 3263 y(I)-13 b(D)951 3275 y Fq(\024)994 3263 y Fu(\010\()p Fr(t)p Fu(\))23 b Fp(\025)g Fr(t)1289 3229 y Fo(\000)p Fs(1)1378 3263 y Fr(C)1437 3275 y Fs(0)1521 3263 y Fr(\037)p Fu(\()1627 3242 y(^)1605 3263 y Fr(H)1674 3275 y Fq(\024)1717 3263 y Fu(\)d\000\()p Fr(F)1944 3229 y Fo(0)1968 3263 y Fu(\()2010 3207 y Fp(j)p Fr(x)p Fp(j)p 2010 3244 V 2042 3320 a Fr(t)2113 3263 y Fu(\)\))14 b Fr(\037)p Fu(\()2298 3242 y(^)2275 3263 y Fr(H)2344 3275 y Fq(\024)2388 3263 y Fu(\))19 b(+)f Fr(O)r Fu(\()p Fr(t)2649 3229 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2831 3263 y Fu(\))83 b @beginspecial @setspecial @endspecial -2946 w(\(7.1\))0 3423 y(uniformly)27 b(in)h @beginspecial @setspecial @endspecial Fr(\024)g Fu(to)f(ha)n(v)n(e)g(the)h(inequalit)n(y)-7 b(.)0 3522 y(One)27 b(has)661 3686 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 680 3665 a(^)661 3686 y(I)-13 b(D)741 3698 y Fq(\024)784 3686 y Fu(\010\()p Fr(t)p Fu(\))23 b(=)g Fr(\037)p Fu(\()1155 3665 y(^)1133 3686 y Fr(H)1202 3698 y Fq(\024)1245 3686 y Fu(\))h(d\000\(d)-12 b(l)1488 3698 y Fs(0)1526 3686 y Fr(F)12 b Fu(\()1633 3630 y Fp(j)p Fr(x)p Fp(j)p 1633 3667 V 1665 3743 a Fr(t)1736 3686 y Fu(\)\))24 b Fr(\037)p Fu(\()1930 3665 y(^)1908 3686 y Fr(H)1977 3698 y Fq(\024)2020 3686 y Fu(\))19 b(+)f Fr(\037)p Fu(\()2260 3665 y(^)2238 3686 y Fr(H)2307 3698 y Fq(\024)2350 3686 y Fu(\)[)2415 3665 y(^)2405 3686 y Fr(I)2441 3698 y Fq(\024)2485 3686 y Fr(;)c Fu(i)g(d\000\()p Fr(F)e Fu(\()2796 3630 y Fp(j)p Fr(x)p Fp(j)p 2796 3667 V 2828 3743 a Fr(t)2900 3686 y Fu(\)\)])p Fr(\037)p Fu(\()3094 3665 y(^)3071 3686 y Fr(H)3140 3698 y Fq(\024)3184 3686 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3872 a Fu(Using)35 b(the)h(fact)f(that)h @beginspecial @setspecial @endspecial(d)-12 b(l)809 3884 y Fs(0)846 3872 y Fr(F)12 b Fu(\()953 3831 y Fo(j)p Fq(x)p Fo(j)p 953 3853 78 4 v 979 3900 a Fq(t)1040 3872 y Fu(\))36 b Fp(\025)1219 3838 y Fq(c)1249 3846 y Fi(0)p 1219 3853 63 4 v 1237 3900 a Fq(t)1327 3872 y Fr(F)1392 3842 y Fo(0)1415 3872 y Fu(\()1457 3831 y Fo(j)p Fq(x)p Fo(j)p 1457 3853 78 4 v 1483 3900 a Fq(t)1544 3872 y Fu(\))24 b(+)g Fr(O)r Fu(\()p Fr(t)1816 3842 y Fo(\000)p Fs(2)1906 3872 y Fu(\),)37 b(it)f(is)f(su\016cien)n(t)h(to)f(sho)n(w)g(that)g (the)h(second)f(term)g(in)h(the)0 3976 y(previous)26 b(iden)n(tit)n(y)i(is)g @beginspecial @setspecial @endspecial Fr(O)r Fu(\()p Fr(t)848 3946 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)1030 3976 y Fu(\))p Fr(;)14 b(\026)23 b(>)g Fu(0)k(uniformly)h (in)f @beginspecial @setspecial @endspecial Fr(\024)p Fu(,)h(to)f(ha)n(v)n(e)g(\(7.1\).)0 4076 y(By)g(simple)h(comm)n (utation)f(relations)f(w)n(e)i(obtain:)840 4255 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial([)p Fr(\036)p Fu(\()p Fr(v)984 4267 y Fq(\024)1023 4275 y Fi(0)1061 4255 y Fu(\))p Fr(;)14 b Fu(d\000\()p Fr(F)e Fu(\()1367 4199 y Fp(j)p Fr(x)p Fp(j)p 1367 4236 94 4 v 1399 4312 a Fr(t)1471 4255 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e(i)p Fr(\036)p Fu(\(i)p Fr(F)12 b Fu(\()2023 4199 y Fp(j)p Fr(x)p Fp(j)p 2023 4236 V 2056 4312 a Fr(t)2128 4255 y Fu(\))p Fr(v)2200 4267 y Fq(\024)2239 4275 y Fi(0)2276 4255 y Fu(\))p Fr(;)0 4337 y @beginspecial @setspecial @endspecial 811 4501 a Fu([)p Fr(a)878 4467 y Fo(\003)912 4441 y Fi(2)948 4501 y Fu(\()p Fr(r)1019 4467 y Fq(j)1017 4521 y(\024)1061 4501 y Fu(\))p Fr(;)i Fu(d\000\()p Fr(F)e Fu(\()1367 4445 y Fp(j)p Fr(x)p Fp(j)p 1367 4482 V 1399 4558 a Fr(t)1471 4501 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e(2)14 b Fr(a)1889 4467 y Fo(\003)1927 4501 y Fu(\()p Fr(r)1998 4467 y Fq(j)1996 4521 y(\024)2040 4501 y Fu(\))p Fr(a)2116 4467 y Fo(\003)2154 4501 y Fu(\()p Fr(F)e Fu(\()2293 4445 y Fp(j)p Fr(x)p Fp(j)p 2293 4482 V 2325 4558 a Fr(t)2397 4501 y Fu(\))p Fr(r)2468 4467 y Fq(j)2466 4521 y(\024)2510 4501 y Fu(\))p Fr(;)0 4583 y @beginspecial @setspecial @endspecial 844 4746 a Fu([)p Fr(a)911 4712 y Fs(2)948 4746 y Fu(\()p Fr(r)1019 4712 y Fq(j)1017 4767 y(\024)1061 4746 y Fu(\))p Fr(;)i Fu(d\000\()p Fr(F)e Fu(\()1367 4690 y Fp(j)p Fr(x)p Fp(j)p 1367 4727 V 1399 4803 a Fr(t)1471 4746 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e Fp(\000)p Fu(2)14 b Fr(a)p Fu(\()p Fr(r)2025 4712 y Fq(j)2023 4767 y(\024)2066 4746 y Fu(\))p Fr(a)p Fu(\()p Fr(F)e Fu(\()2281 4690 y Fp(j)p Fr(x)p Fp(j)p 2281 4727 V 2313 4803 a Fr(t)2385 4746 y Fu(\))p Fr(r)2456 4712 y Fq(j)2454 4767 y(\024)2498 4746 y Fu(\))p Fr(;)0 4828 y @beginspecial @setspecial @endspecial 654 4992 a Fu([)p Fr(a)721 4957 y Fo(\003)759 4992 y Fu(\()p Fr(r)830 4957 y Fq(j)828 5012 y(\024)872 4992 y Fu(\))p Fr(a)p Fu(\()p Fr(r)1019 4957 y Fq(j)1017 5012 y(\024)1061 4992 y Fu(\))p Fr(;)i Fu(d\000\()p Fr(F)e Fu(\()1367 4935 y Fp(j)p Fr(x)p Fp(j)p 1367 4972 V 1399 5048 a Fr(t)1471 4992 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e Fr(a)1833 4957 y Fo(\003)1871 4992 y Fu(\()p Fr(r)1942 4957 y Fq(j)1940 5012 y(\024)1984 4992 y Fu(\))p Fr(a)p Fu(\()p Fr(F)12 b Fu(\()2199 4935 y Fp(j)p Fr(x)p Fp(j)p 2199 4972 V 2232 5048 a Fr(t)2303 4992 y Fu(\))p Fr(r)2374 4957 y Fq(j)2372 5012 y(\024)2416 4992 y Fu(\))19 b(+)f Fr(a)2594 4957 y Fo(\003)2632 4992 y Fu(\()p Fr(F)12 b Fu(\()2771 4935 y Fp(j)p Fr(x)p Fp(j)p 2771 4972 V 2803 5048 a Fr(t)2875 4992 y Fu(\))p Fr(r)2946 4957 y Fq(j)2944 5012 y(\024)2988 4992 y Fu(\))p Fr(a)p Fu(\()p Fr(r)3135 4957 y Fq(j)3133 5012 y(\024)3177 4992 y Fu(\))p Fr(;)-3232 b Fu(\(7.2\))0 5073 y @beginspecial @setspecial @endspecial 764 5237 a([)p Fr(D)856 5249 y Fq(j)904 5237 y Fr(a)p Fu(\()p Fr(r)1019 5203 y Fq(j)1017 5257 y(\024)1061 5237 y Fu(\))p Fr(;)14 b Fu(d\000\()p Fr(F)e Fu(\()1367 5181 y Fp(j)p Fr(x)p Fp(j)p 1367 5218 V 1399 5294 a Fr(t)1471 5237 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e Fr(D)1858 5249 y Fq(j)1907 5237 y Fr(a)p Fu(\()p Fr(F)12 b Fu(\()2090 5181 y Fp(j)p Fr(x)p Fp(j)p 2090 5218 V 2122 5294 a Fr(t)2194 5237 y Fu(\))p Fr(r)2265 5203 y Fq(j)2263 5257 y(\024)2307 5237 y Fu(\))p Fr(;)0 5319 y @beginspecial @setspecial @endspecial 726 5482 a Fu([)p Fr(a)793 5448 y Fo(\003)831 5482 y Fu(\()p Fr(r)902 5448 y Fq(j)900 5503 y(\024)943 5482 y Fu(\))i Fr(D)1058 5494 y Fq(j)1093 5482 y Fr(;)g Fu(d\000\()p Fr(F)e Fu(\()1367 5426 y Fp(j)p Fr(x)p Fp(j)p 1367 5463 V 1399 5539 a Fr(t)1471 5482 y Fu(\)\)])84 b @beginspecial @setspecial @endspecial(=)e Fr(a)1833 5448 y Fo(\003)1871 5482 y Fu(\()p Fr(F)12 b Fu(\()2010 5426 y Fp(j)p Fr(x)p Fp(j)p 2010 5463 V 2043 5539 a Fr(t)2114 5482 y Fu(\))p Fr(r)2185 5448 y Fq(j)2183 5503 y(\024)2227 5482 y Fu(\))i Fr(D)2342 5494 y Fq(j)2377 5482 y Fr(:)p eop %%Page: 37 37 37 36 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(37)0 489 y Fu(Using)34 b(the)g(functional)g (calculus)f(form)n(ula)g(\()p @beginspecial @setspecial @endspecial(4.18\))g(and)h(Corollary)d(4.8,)k(it)f(is)g(enough)f(to)h (estimate)g @beginspecial @setspecial @endspecial(\()p Fr(N)d Fu(+)22 b(1\))3577 459 y Fo(\000)p Fq(n)3674 489 y Fu(\()p Fr(H)3775 501 y Fs(0)3835 489 y Fu(+)0 602 y Fr(c)p Fu(\))68 572 y Fo(\000)130 549 y Fi(1)p 130 558 29 3 v 130 592 a(2)172 602 y Fu([)205 581 y(^)195 602 y Fr(I)231 614 y Fq(\024)275 602 y Fr(;)14 b Fu(i)g(d\000\()p Fr(F)e Fu(\()586 562 y Fo(j)p Fq(x)p Fo(j)p 586 583 78 4 v 612 631 a Fq(t)673 602 y Fu(\)\)]\()p Fr(H)861 614 y Fs(0)918 602 y Fu(+)18 b Fr(c)p Fu(\))1069 572 y Fo(\000)1131 549 y Fi(1)p 1131 558 29 3 v 1131 592 a(2)1173 602 y Fr(;)c(c)23 b(>)g Fu(0.)36 b(Using)28 b(\()p @beginspecial @setspecial @endspecial(7.2\))f(and)h(Lemma)f(3.4,)g(w)n(e)g(obtain:) 693 812 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 166 w Fp(jj)p Fu(\()p Fr(N)g Fu(+)18 b(1\))1188 778 y Fo(\000)p Fq(n)1285 812 y Fu(\()p Fr(H)1386 824 y Fs(0)1442 812 y Fu(+)g Fr(c)p Fu(\))1593 778 y Fo(\000)1655 756 y Fi(1)p 1655 765 V 1655 798 a(2)1698 812 y Fu([)1731 791 y(^)1721 812 y Fr(I)1757 824 y Fq(\024)1800 812 y Fr(;)c Fu(i)g(d\000\()p Fr(F)e Fu(\()2111 756 y Fp(j)p Fr(x)p Fp(j)p 2111 793 94 4 v 2143 869 a Fr(t)2215 812 y Fu(\)\)]\()p Fr(H)2403 824 y Fs(0)2460 812 y Fu(+)18 b Fr(c)p Fu(\))2611 778 y Fo(\000)2673 756 y Fi(1)p 2673 765 29 3 v 2673 798 a(2)2715 812 y Fp(jj)23 b(\024)0 894 y @beginspecial @setspecial @endspecial 859 1076 a Fr(c)p Fu(\()p Fp(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))1248 1042 y Fo(\000)1309 1020 y Fi(1)p 1309 1029 V 1309 1062 a(2)1352 1076 y Fr(F)12 b Fu(\()1459 1020 y Fp(j)p Fr(x)p Fp(j)p 1459 1057 94 4 v 1491 1133 a Fr(t)1562 1076 y Fu(\))p Fr(v)1634 1088 y Fq(\024)1673 1096 y Fi(0)1710 1076 y Fp(jj)19 b Fu(+)f Fp(jj)p Fr(!)1969 1016 y Fj(s)p Fg(\000)p Fi(1)p 1969 1029 102 3 v 2005 1062 a(4)2084 1076 y Fr(r)2123 1042 y Fq(j)2121 1097 y(\024)2164 1076 y Fp(jj)23 b(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))2554 1042 y Fo(\000)2616 1020 y Fj(s)p 2616 1029 29 3 v 2616 1062 a Fi(2)2658 1076 y Fr(!)2723 1016 y Fj(s)p Fg(\000)p Fi(1)p 2723 1029 102 3 v 2759 1062 a(4)2838 1076 y Fr(F)12 b Fu(\()2945 1020 y Fp(j)p Fr(x)p Fp(j)p 2945 1057 94 4 v 2977 1133 a Fr(t)3049 1076 y Fu(\))p Fr(r)3120 1042 y Fq(j)3118 1097 y(\024)3161 1076 y Fp(jj)0 1158 y @beginspecial @setspecial @endspecial 859 1340 a Fu(+)p Fp(jj)p Fu(\()p Fr(V)37 b Fu(+)18 b(1\))1254 1280 y Fj(s)p Fg(\000)p Fi(1)p 1254 1293 102 3 v 1290 1326 a(2)1369 1340 y Fr(!)1424 1306 y Fo(\000)1486 1283 y Fj(s)p 1485 1292 29 3 v 1485 1326 a Fi(2)1528 1340 y Fr(r)1567 1306 y Fq(j)1565 1360 y(\024)1608 1340 y Fp(jj)23 b(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))1998 1306 y Fo(\000)2060 1283 y Fj(s)p 2060 1292 V 2060 1326 a Fi(2)2102 1340 y Fr(!)2167 1280 y Fj(s)p Fg(\000)p Fi(1)p 2167 1293 102 3 v 2203 1326 a(2)2282 1340 y Fr(F)12 b Fu(\()2389 1284 y Fp(j)p Fr(x)p Fp(j)p 2389 1321 94 4 v 2421 1397 a Fr(t)2493 1340 y Fu(\))p Fr(r)2564 1306 y Fq(j)2562 1360 y(\024)2605 1340 y Fp(jj)-2651 b Fu(\(7.3\))0 1422 y @beginspecial @setspecial @endspecial 859 1604 a(+)p Fp(jj)p Fr(D)15 b Fu(\()p Fr(K)24 b Fu(+)18 b Fr(c)p Fu(\))1332 1569 y Fo(\000)1394 1547 y Fi(1)p 1394 1556 29 3 v 1394 1589 a(2)1437 1604 y Fp(jj)23 b(jj)p Fu(\()p Fr(V)38 b Fu(+)18 b(1\))1827 1569 y Fo(\000)1889 1547 y Fj(s)p 1888 1556 V 1888 1589 a Fi(2)1931 1604 y Fr(!)1996 1544 y Fj(s)p Fg(\000)p Fi(1)p 1995 1557 102 3 v 2032 1589 a(2)2111 1604 y Fr(F)12 b Fu(\()2218 1547 y Fp(j)p Fr(x)p Fp(j)p 2218 1585 94 4 v 2249 1661 a Fr(t)2321 1604 y Fu(\))p Fr(r)2392 1569 y Fq(j)2390 1624 y(\024)2434 1604 y Fp(jj)p Fu(\))p Fr(:)0 1808 y Fu(It)24 b(remains)g(to)g(see)f(that)i(the)f(terms)g @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(V)31 b Fu(+)11 b(1\))1472 1778 y Fo(\000)1533 1755 y Fi(1)p 1533 1764 29 3 v 1533 1798 a(2)1576 1808 y Fr(F)h Fu(\()1683 1768 y Fo(j)p Fq(x)p Fo(j)p 1683 1789 78 4 v 1709 1837 a Fq(t)1770 1808 y Fu(\))p Fr(v)1842 1820 y Fq(\024)1881 1828 y Fi(0)1918 1808 y Fp(jj)p Fr(;)24 b Fp(jj)p Fu(\()p Fr(V)31 b Fu(+)11 b(1\))2318 1778 y Fo(\000)2380 1755 y Fj(s)p 2379 1764 29 3 v 2379 1798 a Fi(2)2445 1808 y Fr(!)2500 1778 y Fo(\000)2561 1752 y Fj(s)p Fg(\000)p Fi(1)p 2561 1765 102 3 v 2598 1798 a(2)2699 1808 y Fr(F)h Fu(\()2806 1768 y Fo(j)p Fq(x)p Fo(j)p 2807 1789 78 4 v 2833 1837 a Fq(t)2894 1808 y Fu(\))p Fr(r)2963 1820 y Fq(\024)3006 1808 y Fp(jj)25 b Fu(and)f @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(V)30 b Fu(+)11 b(1\))3541 1778 y Fo(\000)3603 1755 y Fj(s)p 3603 1764 29 3 v 3603 1798 a Fi(2)3668 1808 y Fr(!)3723 1778 y Fo(\000)3785 1752 y Fj(s)p Fg(\000)p Fi(1)p 3785 1765 102 3 v 3821 1798 a(4)0 1929 y Fr(F)h Fu(\()107 1888 y Fo(j)p Fq(x)p Fo(j)p 107 1910 78 4 v 133 1957 a Fq(t)194 1929 y Fu(\))p Fr(r)263 1941 y Fq(\024)307 1929 y Fp(jj)26 b Fu(are)f(in)n(tegrable)g(for)g @beginspecial @setspecial @endspecial(\(1)15 b Fp(\000)g Fr(s)p Fu(\))26 b(small)g(enough.)35 b(They)26 b(are)f Fr(O)r Fu(\()p Fr(t)2301 1899 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2484 1929 y Fu(\))h(b)n(y)g(\()p @beginspecial @setspecial @endspecial(4.24\).)35 b(Then)26 b(using)g(Lemma)g(A.3)f(w)n(e)0 2028 y(\014nish)j(the)g(pro)r(of)f(of)g(the)h(estimate)g(announced)f (in)h(the)g(prop)r(osition.)p @beginspecial @setspecial @endspecial 1595 w Fh(\004)0 2044 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 145 x Fz(Prop)s(osition)i(7.2.)40 b Fn(L)l(et)30 b @beginspecial @setspecial @endspecial Fr(\037)23 b Fp(2)g Fr(C)1068 2159 y Fo(1)1062 2210 y Fs(0)1139 2189 y Fu(\()p Fk(R)p Fu(\))p Fr(;)34 b Fu(0)22 b Fr(<)h(c)1502 2201 y Fs(0)1562 2189 y Fr(<)g(c)1686 2201 y Fs(1)1723 2189 y Fn(,)30 b(and)947 2365 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\002)1012 2380 y Fs([)p Fq(c)1061 2388 y Fi(0)1093 2380 y Fq(;c)1143 2388 y Fi(1)1175 2380 y Fs(])1198 2365 y Fu(\()p Fr(t)p Fu(\))23 b(:=)g(d\000\()p Fp(h)1598 2309 y Fr(x)p 1598 2346 48 4 v 1607 2422 a(t)1674 2365 y Fp(\000)18 b(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\))p Fr(;)c Fu(1)-13 b(l)2080 2380 y Fs([)p Fq(c)2129 2388 y Fi(0)2161 2380 y Fq(;c)2211 2388 y Fi(1)2242 2380 y Fs(])2265 2365 y Fu(\()2307 2309 y Fr(x)p 2307 2346 V 2316 2422 a(t)2365 2365 y Fu(\)\()2439 2309 y Fr(x)p 2439 2346 V 2448 2422 a(t)2515 2365 y Fp(\000)18 b(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\)\))p Fp(i)p Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 2544 a Fn(One)29 b(has)i(uniformly)f(in)g @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(\024)g(1)1123 2741 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)1206 2648 y Fo(1)1170 2816 y Fs(1)1304 2741 y Fp(jj)p Fu(\002)1415 2756 y Fs([)p Fq(c)1464 2764 y Fi(0)1496 2756 y Fq(;c)1546 2764 y Fi(1)1578 2756 y Fs(])1601 2741 y Fu(\()p Fr(t)p Fu(\))1705 2684 y Fi(1)p 1705 2693 29 3 v 1705 2726 a(2)1748 2741 y Fr(\037)p Fu(\()1854 2720 y(^)1832 2741 y Fr(H)1901 2753 y Fq(\024)1944 2741 y Fu(\)e)2013 2706 y Fo(\000)p Fs(i)p Fq(t)2125 2691 y Fs(^)2109 2706 y Fq(H)2163 2714 y Fj(\024)2206 2741 y Fr(u)p Fp(jj)2300 2706 y Fs(2)2347 2684 y Fr(dt)p 2347 2722 74 4 v 2368 2798 a(t)2453 2741 y Fp(\024)g Fr(c)p Fp(jj)p Fr(u)p Fp(jj)2717 2706 y Fs(2)2754 2741 y Fr(:)p @beginspecial @setspecial @endspecial 100 2948 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Let)27 b Fr(R)635 2960 y Fs(0)673 2948 y Fu(\()p Fr(x)p Fu(\))d Fp(2)f Fr(C)951 2918 y Fo(1)1049 2948 y Fu(b)r(e)28 b(a)g(function)g (suc)n(h)f(that:)1343 3297 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1385 3124 a @beginspecial @setspecial @endspecial Fr(R)1448 3136 y Fs(0)1485 3124 y Fu(\()p Fr(x)p Fu(\))d(=)f(0)p Fr(;)36 b Fu(for)22 b Fp(j)p Fr(x)p Fp(j)i(\024)2146 3068 y Fr(c)2182 3080 y Fs(0)p 2146 3105 V 2162 3181 a Fu(2)2229 3124 y Fr(;)p @beginspecial @setspecial @endspecial 1385 3326 a @beginspecial @setspecial @endspecial(R)1448 3338 y Fs(0)1485 3326 y Fu(\()p Fr(x)p Fu(\))g(=)1717 3270 y(1)p 1717 3307 42 4 v 1717 3383 a(2)1769 3326 y Fr(x)1816 3292 y Fs(2)1872 3326 y Fu(+)18 b Fr(c;)37 b Fu(for)22 b Fp(j)p Fr(x)p Fp(j)i(\025)f Fu(2)p Fr(c)2456 3338 y Fs(1)2492 3326 y Fr(;)p @beginspecial @setspecial @endspecial 1385 3489 a @beginspecial @setspecial @endspecial Fp(r)1454 3455 y Fs(2)1454 3510 y Fq(x)1496 3489 y Fr(R)1559 3501 y Fs(0)1619 3489 y Fp(\025)f Fu(1)-15 b(l)1756 3504 y Fs([)p Fq(c)1805 3512 y Fi(0)1837 3504 y Fq(;c)1887 3512 y Fi(1)1918 3504 y Fs(])1941 3489 y Fu(\()p Fp(j)p Fr(x)p Fp(j)p Fu(\))p Fr(:)2557 3297 y @beginspecial @setspecial @endspecial 0 3645 a Fu(W)-7 b(e)33 b(c)n(ho)r(ose)f @beginspecial @setspecial @endspecial Fr(c)453 3657 y Fs(1)522 3645 y Fr(>)g Fu(2)p Fr(;)45 b(c)765 3657 y Fs(2)834 3645 y Fr(>)32 b(c)967 3657 y Fs(1)1026 3645 y Fu(+)22 b(1)32 b(and)h(w)n(e)g(de\014ne)g(the)g(function)h @beginspecial @setspecial @endspecial Fr(R)q Fu(\()p Fr(x)p Fu(\))e(:=)g Fr(F)12 b Fu(\()p Fp(j)p Fr(x)p Fp(j)p Fu(\))p Fr(R)2818 3657 y Fs(0)2856 3645 y Fu(\()p Fr(x)p Fu(\))35 b(where)d Fr(F)12 b Fu(\()p Fr(s)p Fu(\))32 b(=)g(1)p Fr(;)45 b(if)c(s)31 b Fp(\024)0 3745 y Fr(c)36 3757 y Fs(1)73 3745 y Fr(;)14 b(F)e Fu(\()p Fr(s)p Fu(\))24 b(=)e(0)p Fr(;)37 b(if)31 b(s)23 b Fp(\025)g Fr(c)778 3757 y Fs(2)815 3745 y Fu(.)0 3844 y(W)-7 b(e)28 b(set)1118 4017 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(b)p Fu(\()p Fr(t)p Fu(\))23 b(:=)g Fr(R)q Fu(\()1488 3961 y Fr(x)p 1488 3998 48 4 v 1497 4074 a(t)1545 4017 y Fu(\))c Fp(\000)1689 3961 y Fu(1)p 1689 3998 42 4 v 1689 4074 a(2)1740 4017 y(\()p Fp(hr)p Fr(R)q Fu(\()1979 3961 y Fr(x)p 1979 3998 48 4 v 1988 4074 a(t)2037 4017 y Fu(\))p Fr(;)2116 3961 y(x)p 2116 3998 V 2125 4074 a(t)2192 4017 y Fp(\000)f(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\))p Fp(i)h Fu(+)f Fr(hc)p Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4195 a Fu(W)-7 b(e)28 b(consider)f(the)h(observ)-5 b(able)1419 4350 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\010\()p Fr(t)p Fu(\))24 b(:=)f Fr(\037)p Fu(\()1814 4329 y(^)1792 4350 y Fr(H)1861 4362 y Fq(\024)1904 4350 y Fu(\)d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()2335 4329 y(^)2312 4350 y Fr(H)2381 4362 y Fq(\024)2425 4350 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4505 a Fu(Pseudo)r(di\013eren)n(tial)k (calculus)g(giv)n(es)312 4693 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 353 4703 a @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()459 4682 y(^)437 4703 y Fr(H)506 4715 y Fq(\024)550 4703 y Fu(\)I)-13 b(D)662 4715 y Fs(0)699 4703 y Fu(d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()1098 4682 y(^)1075 4703 y Fr(H)1144 4715 y Fq(\024)1188 4703 y Fu(\))23 b Fp(\025)g Fr(\037)p Fu(\()1437 4682 y(^)1415 4703 y Fr(H)1484 4715 y Fq(\024)1527 4703 y Fu(\)\()1601 4647 y(1)p 1601 4684 42 4 v 1607 4760 a Fr(t)1653 4703 y Fu(\002)1718 4718 y Fs([)p Fq(c)1767 4726 y Fi(0)1799 4718 y Fq(;c)1849 4726 y Fi(2)1881 4718 y Fs(])1904 4703 y Fu(\()p Fr(t)p Fu(\))c Fp(\000)2110 4647 y Fu(1)p 2110 4684 V 2116 4760 a Fr(t)2161 4703 y Fu(d\000\(1)-15 b(l)2341 4718 y Fs([)p Fq(v)2393 4726 y Fj(max)2513 4718 y Fs(+1)p Fq(;c)2647 4726 y Fi(2)2679 4718 y Fs(])2702 4703 y Fu(\()2744 4647 y Fp(j)p Fr(x)p Fp(j)p 2744 4684 94 4 v 2776 4760 a Fr(t)2847 4703 y Fu(\)\)\))p Fr(\037)p Fu(\()3050 4682 y(^)3027 4703 y Fr(H)3096 4715 y Fq(\024)3141 4703 y Fu(\))18 b(+)g Fr(O)r Fu(\()p Fr(t)3401 4669 y Fo(\000)p Fs(2)3491 4703 y Fu(\))p Fr(:)3588 4693 y @beginspecial @setspecial @endspecial 0 4881 a Fu(The)28 b(\014rst)g(term)g(will)h (serv)n(es)d(in)i(the)h(application)e(of)i(Lemma)e @beginspecial @setspecial @endspecial(A.4)i(and)e(the)i(second)f(is)g(in)n(tegrable)e (along)h(the)i(ev)n(olution)0 4981 y(using)e(Prop.)36 b @beginspecial @setspecial @endspecial(7.1.)g(T)-7 b(o)27 b(complete)h(the)g(pro)r(of)f(of)g(the)h(prop)r(osition,)f(it)h (su\016ces)f(to)h(sho)n(w)e(uniformly)i(in)g @beginspecial @setspecial @endspecial Fr(\024)f Fu(that:)1040 5139 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1146 5118 y(^)1124 5139 y Fr(H)1193 5151 y Fq(\024)1236 5139 y Fu(\)[)1301 5118 y(^)1291 5139 y Fr(I)1327 5151 y Fq(\024)1371 5139 y Fr(;)14 b Fu(i)g(d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\)])p Fr(\037)p Fu(\()1866 5118 y(^)1844 5139 y Fr(H)1913 5151 y Fq(\024)1957 5139 y Fu(\))23 b Fp(2)g Fr(O)r Fu(\()p Fr(t)2217 5105 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2400 5139 y Fu(\))p Fr(;)14 b(\026)23 b(>)g Fu(0)p Fr(:)82 b @beginspecial @setspecial @endspecial -2777 w Fu(\(7.4\))0 5294 y(As)44 b(in)g(the)g(Prop.)84 b @beginspecial @setspecial @endspecial(7.1,)47 b(using)c(\(7.2\))h(and)f(\()p @beginspecial @setspecial @endspecial(7.3\),)48 b(w)n(e)43 b(see)h(that)g(\(7.4\))f (is)h(b)r(ounded)g(b)n(y)g(a)f(sum)h(of)f(terms)h @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(V)k Fu(+)0 5406 y(1\))74 5376 y Fo(\000)136 5354 y Fi(1)p 136 5363 29 3 v 136 5396 a(2)178 5406 y Fr(b)p Fu(\()p Fr(t)p Fu(\))p Fr(v)348 5418 y Fq(\024)387 5426 y Fi(0)424 5406 y Fp(jj)p Fr(;)14 b Fp(jj)p Fu(\()p Fr(V)49 b Fu(+)28 b(1\))849 5376 y Fo(\000)911 5354 y Fj(s)p 911 5363 V 911 5396 a Fi(2)1004 5406 y Fr(!)1059 5376 y Fo(\000)1120 5351 y Fi(1)p Fg(\000)p Fj(s)p 1120 5364 102 3 v 1157 5396 a Fi(2)1286 5406 y Fr(b)p Fu(\()p Fr(t)p Fu(\))p Fr(r)1453 5418 y Fq(\024)1497 5406 y Fp(jj)44 b Fu(and)g @beginspecial @setspecial @endspecial Fp(jj)p Fu(\()p Fr(V)k Fu(+)29 b(1\))2107 5376 y Fo(\000)2169 5354 y Fj(s)p 2169 5363 29 3 v 2169 5396 a Fi(2)2262 5406 y Fr(!)2317 5376 y Fo(\000)2378 5351 y Fi(1)p Fg(\000)p Fj(s)p 2378 5364 102 3 v 2414 5396 a Fi(4)2544 5406 y Fr(b)p Fu(\()p Fr(t)p Fu(\))p Fr(r)2711 5418 y Fq(\024)2754 5406 y Fp(jj)p Fu(.)86 b(By)44 b(\()p @beginspecial @setspecial @endspecial(4.24\))f(these)h(terms)g(are)0 5506 y Fr(O)r Fu(\()p Fr(t)127 5476 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)309 5506 y Fu(\))p Fr(;)14 b(\026)24 b(>)e Fu(0,)27 b(for)h @beginspecial @setspecial @endspecial(\(1)18 b Fp(\000)g Fr(s)p Fu(\))28 b(small)f(enough.)36 b(W)-7 b(e)28 b(end)g(the)g(pro)r (of)f(b)n(y)g(using)g(lemma)h @beginspecial @setspecial @endspecial(A.3.)857 b Fh(\004)0 5527 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial eop %%Page: 38 38 38 37 bop 0 226 a Fs(38)1695 b(Z.)29 b(AMMARI)0 489 y Fz(Prop)s(osition)h(7.3.)40 b Fn(L)l(et)f Fu(0)h Fr(<)g(c)1082 501 y Fs(0)1159 489 y Fr(<)g(c)1300 501 y Fs(1)1337 489 y Fr(;)28 b(J)48 b Fp(2)41 b Fr(C)1643 459 y Fo(1)1637 510 y Fs(0)1713 489 y Fu(\()p Fp(f)p Fr(c)1823 501 y Fs(0)1900 489 y Fr(<)f Fp(j)p Fr(x)p Fp(j)h Fr(<)f(c)2280 501 y Fs(1)2317 489 y Fp(g)p Fu(\))p Fr(;)27 b(\037)41 b Fp(2)f Fr(C)2694 459 y Fo(1)2688 510 y Fs(0)2765 489 y Fu(\()p Fk(R)p Fu(\))p Fn(.)74 b(F)-6 b(or)39 b @beginspecial @setspecial @endspecial Fu(1)h Fp(\024)g Fr(i)f Fp(\024)h Fu(3)p Fn(,)i(one)d(has)0 589 y(uniformly)31 b(in)f @beginspecial @setspecial @endspecial Fr(\024)22 b Fp(\024)h(1)727 767 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)810 674 y Fo(1)773 842 y Fs(1)908 767 y Fp(jj)p Fu(d\000\()p Fp(j)h Fr(J)8 b Fu(\()1227 710 y Fr(x)p 1227 747 48 4 v 1236 823 a(t)1284 767 y Fu(\)\()1358 710 y Fr(x)1405 722 y Fq(i)p 1359 747 76 4 v 1381 823 a Fr(t)1462 767 y Fp(\000)18 b Fr(@)1589 779 y Fq(i)1617 767 y Fr(!)s Fu(\()p Fr(k)s Fu(\)\))h(+)f Fr(hc)23 b Fp(j)p Fu(\))2088 710 y Fi(1)p 2088 719 29 3 v 2088 752 a(2)2144 767 y Fr(\037)p Fu(\()2250 746 y(^)2228 767 y Fr(H)2297 779 y Fq(\024)2340 767 y Fu(\)e)2409 732 y Fo(\000)p Fs(i)p Fq(t)2522 717 y Fs(^)2505 732 y Fq(H)2559 740 y Fj(\024)2602 767 y Fr(u)p Fp(jj)2696 732 y Fs(2)2743 710 y Fr(dt)p 2743 747 74 4 v 2765 823 a(t)2849 767 y Fp(\024)g Fr(c)p Fp(jj)p Fr(u)p Fp(jj)3113 732 y Fs(2)3150 767 y Fr(:)p @beginspecial @setspecial @endspecial 100 959 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(set)1437 1190 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1478 1095 a @beginspecial @setspecial @endspecial Fr(A)23 b Fu(:=)g(\()1716 1039 y Fr(x)p 1716 1076 48 4 v 1725 1152 a(t)1792 1095 y Fp(\000)18 b(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\)\))2141 1061 y Fs(2)2197 1095 y Fu(+)h Fr(t)2311 1061 y Fo(\000)p Fq(\016)2399 1095 y Fr(;)1478 1279 y(b)p Fu(\()p Fr(t)p Fu(\))k(:=)g Fr(J)8 b Fu(\()1838 1222 y Fr(x)p 1838 1259 V 1847 1335 a(t)1896 1279 y Fu(\))p Fr(A)2000 1222 y Fi(1)p 2000 1231 29 3 v 2000 1264 a(2)2043 1279 y Fr(J)g Fu(\()2139 1222 y Fr(x)p 2139 1259 48 4 v 2148 1335 a(t)2196 1279 y Fu(\))p Fr(:)2463 1190 y @beginspecial @setspecial @endspecial 0 1442 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(J)195 1454 y Fs(1)255 1442 y Fp(2)23 b Fr(C)398 1412 y Fo(1)392 1463 y Fs(0)469 1442 y Fu(\()p Fp(f)p Fr(c)579 1454 y Fs(0)639 1442 y Fr(<)g Fp(j)p Fr(x)p Fp(j)g Fr(<)g(c)967 1454 y Fs(1)1004 1442 y Fp(g)p Fu(\))p Fr(;)28 b Fu(0)22 b Fp(\024)h Fr(J)31 b Fp(\024)23 b Fu(1)p Fr(;)36 b(J)31 b Fu(=)23 b(1)k(near)g(the)h(supp)r(ort)f(of)h @beginspecial @setspecial @endspecial Fr(J)2557 1454 y Fs(1)2594 1442 y Fu(.)37 b(W)-7 b(e)28 b(consider)f(the)g(observ)-5 b(able)1387 1586 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\010\()p Fr(t)p Fu(\))24 b(:=)e Fp(\000)p Fr(\037)p Fu(\()1846 1565 y(^)1824 1586 y Fr(H)1893 1598 y Fq(\024)1936 1586 y Fu(\)d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()2367 1565 y(^)2344 1586 y Fr(H)2413 1598 y Fq(\024)2458 1586 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 1721 a Fu(One)27 b(has)995 1857 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1101 1836 y(^)1079 1857 y Fr(H)1148 1869 y Fq(\024)1191 1857 y Fu(\)I)-13 b(D)1303 1869 y Fs(0)1341 1857 y Fu(d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()1739 1836 y(^)1717 1857 y Fr(H)1786 1869 y Fq(\024)1830 1857 y Fu(\))23 b(=)g Fp(\000)p Fr(\037)p Fu(\()2143 1836 y(^)2122 1857 y Fr(H)2191 1869 y Fq(\024)2234 1857 y Fu(\)d\000\(d)-12 b(l)2454 1869 y Fs(0)2491 1857 y Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()2759 1836 y(^)2737 1857 y Fr(H)2806 1869 y Fq(\024)2850 1857 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 0 1993 a Fu(and)27 b(w)n(e)h(ha)n(v)n(e)e(using)h @beginspecial @setspecial @endspecial([11)o(,)h(Lemma)f(6.4])568 2207 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 609 2150 a @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()715 2129 y(^)693 2150 y Fr(H)762 2162 y Fq(\024)806 2150 y Fu(\)I)-13 b(D)918 2162 y Fs(0)955 2150 y Fu(d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()1354 2129 y(^)1331 2150 y Fr(H)1400 2162 y Fq(\024)1444 2150 y Fu(\))24 b Fp(\025)1597 2093 y Fr(c)1633 2105 y Fs(0)p 1597 2131 74 4 v 1619 2207 a Fr(t)1680 2150 y(\037)p Fu(\()1786 2129 y(^)1764 2150 y Fr(H)1833 2162 y Fq(\024)1877 2150 y Fu(\)d\000\()p Fp(j)f Fr(J)2131 2162 y Fs(1)2169 2150 y Fu(\()2211 2093 y Fr(x)p 2211 2131 48 4 v 2220 2207 a(t)2268 2150 y Fu(\)\()2342 2093 y Fr(x)2389 2105 y Fq(i)p 2343 2131 76 4 v 2365 2207 a Fr(t)2446 2150 y Fp(\000)18 b Fr(@)2573 2162 y Fq(i)2601 2150 y Fr(!)s Fu(\()p Fr(k)s Fu(\)\))h(+)f Fr(hc)23 b Fp(j)p Fu(\))p Fr(\037)p Fu(\()3168 2129 y(^)3146 2150 y Fr(H)3215 2162 y Fq(\024)3258 2150 y Fu(\))p @beginspecial @setspecial @endspecial 609 2285 a Fp(\000)684 2252 y Fq(c)p 684 2266 30 4 v 686 2314 a(t)724 2285 y Fr(\037)p Fu(\()830 2264 y(^)808 2285 y Fr(H)877 2297 y Fq(\024)920 2285 y Fu(\)d\000\()p Fp(h)1124 2252 y Fq(x)p 1125 2266 38 4 v 1131 2314 a(t)1191 2285 y Fp(\000)18 b(r)p Fr(!)s(;)c(J)1481 2297 y Fs(2)1518 2285 y Fu(\()1560 2252 y Fq(x)p 1560 2266 V 1566 2314 a(t)1608 2285 y Fu(\)\()1682 2252 y Fq(x)p 1682 2266 V 1688 2314 a(t)1748 2285 y Fp(\000)k(r)p Fr(!)s Fu(\))p Fp(i)p Fu(\))p Fr(\037)p Fu(\()2158 2264 y(^)2135 2285 y Fr(H)2204 2297 y Fq(\024)2248 2285 y Fu(\))h(+)f Fr(O)r Fu(\()p Fr(t)2509 2255 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2691 2285 y Fu(\))p Fr(:)3332 2207 y @beginspecial @setspecial @endspecial 0 2417 a Fu(The)28 b(second)f(term)g(is)h(in)n(tegrable)e(along)g(the)i(ev)n (olution)f(b)n(y)g(Prop.)36 b @beginspecial @setspecial @endspecial(7.2.)g(It's)28 b(enough)f(to)g(sho)n(w)g(that)847 2556 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()953 2535 y(^)931 2556 y Fr(H)1000 2568 y Fq(\024)1043 2556 y Fu(\)[)1108 2535 y(^)1098 2556 y Fr(I)1134 2568 y Fq(\024)1178 2556 y Fr(;)14 b Fu(i)g(d\000\()p Fr(b)p Fu(\()p Fr(t)p Fu(\)\)])p Fr(\037)p Fu(\()1673 2535 y(^)1651 2556 y Fr(H)1720 2568 y Fq(\024)1764 2556 y Fu(\))23 b Fp(2)h Fr(O)r Fu(\()p Fr(t)2025 2522 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2207 2556 y Fu(\))p Fr(;)14 b(\026)23 b(>)g Fu(0)p Fr(;)36 b Fu(uniformly)24 b(in)f Fr(\024)p @beginspecial @setspecial @endspecial 0 2700 a Fu(This)31 b(follo)n(ws)f(b)n(y)h(using)g(\()p @beginspecial @setspecial @endspecial(7.3\),)h(the)g(fact)f(that)h @beginspecial @setspecial @endspecial Fr(J)8 b Fu(\()1629 2668 y Fq(x)p 1629 2682 V 1635 2729 a(t)1677 2700 y Fu(\))p Fr(A)1781 2648 y Fi(1)p 1781 2657 29 3 v 1781 2690 a(2)1853 2700 y Fp(2)29 b Fr(O)r Fu(\(1\))j(and)f(Lemma)g @beginspecial @setspecial @endspecial(A.2.)48 b(Using)31 b(Lemma)g(A.3)g(w)n(e)g(end) h(the)0 2800 y(pro)r(of.)p @beginspecial @setspecial @endspecial 3622 w Fh(\004)0 2816 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 138 x Fz(Prop)s(osition)e(7.4.)40 b Fn(L)l(et)30 b Fr(\037)23 b Fp(2)g Fr(C)1068 2923 y Fo(1)1062 2974 y Fs(0)1139 2954 y Fu(\()p Fk(R)p Fu(\))q Fn(,)36 b(supp)l(orte)l(d)30 b(in)g @beginspecial @setspecial @endspecial Fk(R)25 b Fp(n)18 b Fu(\()p Fr(\034)28 b Fp([)19 b Fr(\033)2147 2966 y Fs(pp)2225 2954 y Fu(\()2279 2933 y(^)2257 2954 y Fr(H)2326 2966 y Fo(1)2397 2954 y Fu(\)\))p Fn(.)39 b(Ther)l(e)31 b(exist)f @beginspecial @setspecial @endspecial Fr(\017)23 b(>)f Fu(0)p Fr(;)28 b(C)36 b Fn(indep)l(endent)30 b(in)g Fr(\024)0 3064 y Fn(and)g(a)g(se)l(quenc)l(e)g(of)690 3043 y Fu(^)668 3064 y Fr(H)737 3076 y Fq(\024)810 3064 y Fn(such)g(that)g(for)g @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(\024)g(1)p Fn(,)30 b(we)g(have)1066 3243 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)1149 3151 y Fo(1)1112 3319 y Fs(1)1247 3243 y Fp(jj)p Fu(\000\(1)-13 b(l)1429 3258 y Fs([0)p Fq(;\017)p Fs(])1551 3243 y Fu(\()1593 3187 y Fp(j)p Fr(x)p Fp(j)p 1593 3224 94 4 v 1625 3300 a Fr(t)1697 3243 y Fu(\)\))p Fr(\037)p Fu(\()1868 3222 y(^)1845 3243 y Fr(H)1914 3255 y Fq(\024)1958 3243 y Fu(\))14 b(e)2041 3209 y Fo(\000)p Fs(i)p Fq(t)2153 3194 y Fs(^)2137 3209 y Fq(H)2191 3217 y Fj(\024)2234 3243 y Fr(u)p Fp(jj)2328 3209 y Fs(2)2375 3187 y Fr(dt)p 2375 3224 74 4 v 2396 3300 a(t)2481 3243 y Fp(\024)22 b Fr(C)6 b Fp(jj)p Fr(u)p Fp(jj)2773 3209 y Fs(2)2811 3243 y Fr(:)p @beginspecial @setspecial @endspecial 100 3456 a Fz(Pro)s(of.)39 b @beginspecial @setspecial @endspecial 23 w Fu(W)-7 b(e)23 b(notice)f(that)g(the)g(Prop.)34 b(7.4)21 b(is)h(a)g(minimal)g (v)n(elo)r(cit)n(y)f(estimate)h(for)g(a)g(sequence)f(of)h @beginspecial @setspecial @endspecial 3194 3435 a(^)3172 3456 y Fr(H)3241 3468 y Fq(\024)3306 3456 y Fu(whic)n(h)g(is)g(uniform) 0 3559 y(in)29 b @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)38 b(Let)29 b @beginspecial @setspecial @endspecial Fr(\037)f Fu(supp)r(orted)g(near)g Fr(\025)g Fu(suc)n(h)g(that)h Fr(\025)24 b Fp(2)h Fk(R)g Fp(n)18 b Fu(\()p Fr(\034)29 b Fp([)19 b Fr(\033)1968 3571 y Fs(pp)2046 3559 y Fu(\()2100 3538 y(^)2078 3559 y Fr(H)2147 3571 y Fo(1)2218 3559 y Fu(\)\).)39 b(Then)29 b(there)f(exists)g(a)g (sequence)f @beginspecial @setspecial @endspecial 3441 3538 a(^)3419 3559 y Fr(H)3488 3571 y Fq(\024)3560 3559 y Fu(suc)n(h)h(that)0 3670 y Fr(\025)23 b Fp(2)h Fk(R)g Fp(n)18 b Fu(\()p Fr(\034)29 b Fp([)18 b Fr(\033)505 3682 y Fs(pp)583 3670 y Fu(\()637 3649 y(^)615 3670 y Fr(H)684 3682 y Fq(\024)728 3670 y Fu(\)\).)37 b(Lemma)28 b @beginspecial @setspecial @endspecial(5.5)e(in)i(Mourre)f(estimate)g (section)g(giv)n(es)1340 3817 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1446 3796 y(^)1424 3817 y Fr(H)1493 3829 y Fq(\024)1536 3817 y Fu(\)[)1613 3796 y(^)1591 3817 y Fr(H)1660 3829 y Fq(\024)1703 3817 y Fr(;)14 b Fu(i)p Fr(B)t Fu(])p Fr(\037)p Fu(\()1959 3796 y(^)1937 3817 y Fr(H)2006 3829 y Fq(\024)2050 3817 y Fu(\))23 b Fp(\025)g Fr(c)2229 3829 y Fq(\024)2272 3817 y Fr(\037)2324 3782 y Fs(2)2361 3817 y Fu(\()2415 3796 y(^)2393 3817 y Fr(H)2462 3829 y Fq(\024)2505 3817 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 3952 a Fu(Let)28 b @beginspecial @setspecial @endspecial Fr(\017)23 b(>)f Fu(0.)37 b(Let)27 b Fr(q)f Fp(2)e Fr(C)750 3922 y Fo(1)744 3973 y Fs(0)820 3952 y Fu(\()p Fp(j)p Fr(x)p Fp(j)g(\024)f Fu(2)p Fr(\017)p Fu(\))k(suc)n(h)g(that)h @beginspecial @setspecial @endspecial(0)23 b Fp(\024)g Fr(q)j Fp(\024)c Fu(1)p Fr(;)14 b(q)s Fu(\()p Fr(x)p Fu(\))24 b(=)e(1)p Fr(;)28 b Fu(if)g Fp(j)p Fr(x)p Fp(j)23 b(\024)g Fr(\017)p Fu(.)0 4052 y(W)-7 b(e)28 b(set)1315 4209 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\010)1375 4221 y Fq(\024)1418 4209 y Fu(\()p Fr(t)p Fu(\))c(:=)e Fr(\037)p Fu(\()1752 4188 y(^)1730 4209 y Fr(H)1799 4221 y Fq(\024)1843 4209 y Fu(\)\000\()p Fr(q)1999 4175 y Fq(t)2028 4209 y Fu(\))2070 4153 y Fr(B)p 2070 4190 68 4 v 2089 4266 a(t)2148 4209 y Fu(\000\()p Fr(q)2272 4175 y Fq(t)2301 4209 y Fu(\))p Fr(\037)p Fu(\()2439 4188 y(^)2417 4209 y Fr(H)2486 4221 y Fq(\024)2530 4209 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 4373 a Fu(The)28 b(Heisen)n(b)r(erg)e(deriv)-5 b(ativ)n(e)27 b(of)h @beginspecial @setspecial @endspecial(\010)1129 4385 y Fq(\024)1172 4373 y Fu(\()p Fr(t)p Fu(\))g(is)939 4874 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -436 x @beginspecial @setspecial @endspecial 1000 4530 a(^)980 4551 y(I)-13 b(D)1060 4563 y Fq(\024)1103 4551 y Fu(\010)1163 4563 y Fq(\024)1206 4551 y Fu(\()p Fr(t)p Fu(\))84 b(=)106 b Fr(\037)p Fu(\()1661 4530 y(^)1639 4551 y Fr(H)1708 4563 y Fq(\024)1751 4551 y Fu(\)d\000\()p Fr(q)1953 4521 y Fq(t)1983 4551 y Fr(;)14 b Fu(d)-12 b(l)2077 4563 y Fs(0)2114 4551 y Fr(q)2154 4521 y Fq(t)2184 4551 y Fu(\))2240 4518 y Fq(B)p 2240 4532 53 4 v 2254 4580 a(t)2303 4551 y Fu(\000\()p Fr(q)2427 4521 y Fq(t)2456 4551 y Fu(\))p Fr(\037)p Fu(\()2594 4530 y(^)2572 4551 y Fr(H)2641 4563 y Fq(\024)2685 4551 y Fu(\))19 b(+)f Fr(hc)p @beginspecial @setspecial @endspecial 1555 4729 a Fu(+)p Fr(\037)p Fu(\()1725 4708 y(^)1704 4729 y Fr(H)1773 4741 y Fq(\024)1816 4729 y Fu(\)[)1881 4708 y(^)1871 4729 y Fr(I)1907 4741 y Fq(\024)1950 4729 y Fr(;)c Fu(i\000\()p Fr(q)2134 4695 y Fq(t)2164 4729 y Fu(\)])2243 4673 y Fr(B)p 2243 4710 68 4 v 2261 4786 a(t)2320 4729 y Fu(\000\()p Fr(q)2444 4695 y Fq(t)2473 4729 y Fu(\))p Fr(\037)p Fu(\()2612 4708 y(^)2589 4729 y Fr(H)2658 4741 y Fq(\024)2702 4729 y Fu(\))19 b(+)f Fr(hc)p @beginspecial @setspecial @endspecial 980 4899 a @beginspecial @setspecial @endspecial 575 w Fu(+)p Fr(t)1650 4865 y Fo(\000)p Fs(1)1738 4899 y Fr(\037)p Fu(\()1844 4878 y(^)1822 4899 y Fr(H)1891 4911 y Fq(\024)1935 4899 y Fu(\)\000\()p Fr(q)2091 4865 y Fq(t)2120 4899 y Fu(\)[)2197 4878 y(^)2175 4899 y Fr(H)2244 4911 y Fq(\024)2288 4899 y Fr(;)c Fu(i)p Fr(B)t Fu(]\000\()p Fr(q)2562 4865 y Fq(t)2591 4899 y Fu(\))p Fr(\037)p Fu(\()2730 4878 y(^)2707 4899 y Fr(H)2776 4911 y Fq(\024)2820 4899 y Fu(\))p @beginspecial @setspecial @endspecial 980 5077 a @beginspecial @setspecial @endspecial 575 w Fp(\000)p Fr(t)1650 5043 y Fo(\000)p Fs(1)1738 5077 y Fr(\037)p Fu(\()1844 5056 y(^)1822 5077 y Fr(H)1891 5089 y Fq(\024)1935 5077 y Fu(\)\000\()p Fr(q)2091 5043 y Fq(t)2120 5077 y Fu(\))2162 5021 y Fr(B)p 2162 5058 V 2181 5134 a(t)2240 5077 y Fu(\000\()p Fr(q)2364 5043 y Fq(t)2393 5077 y Fu(\)\000\()p Fr(q)2549 5043 y Fq(t)2579 5077 y Fu(\))p Fr(\037)p Fu(\()2717 5056 y(^)2695 5077 y Fr(H)2764 5089 y Fq(\024)2807 5077 y Fu(\))p @beginspecial @setspecial @endspecial 980 5238 a @beginspecial @setspecial @endspecial 404 w(=:)83 b Fr(R)1619 5208 y Fs(1)1618 5259 y Fq(\024)1679 5238 y Fu(+)18 b Fr(R)1826 5208 y Fs(2)1825 5259 y Fq(\024)1887 5238 y Fu(+)g Fr(R)2034 5208 y Fs(3)2033 5259 y Fq(\024)2094 5238 y Fu(+)g Fr(R)2241 5208 y Fs(4)2240 5259 y Fq(\024)2283 5238 y Fr(:)2961 4874 y @beginspecial @setspecial @endspecial 0 5370 a Fu(W)-7 b(e)28 b(claim)f(that)1454 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(R)1518 5472 y Fs(2)1517 5526 y Fq(\024)1584 5506 y Fp(2)c Fr(O)r Fu(\()p Fr(t)1789 5472 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)1971 5506 y Fu(\))p Fr(;)28 b(\026)23 b(>)g Fu(0)p Fr(:)83 b @beginspecial @setspecial @endspecial -2363 w Fu(\(7.5\))p eop %%Page: 39 39 39 38 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(39)0 489 y Fu(T)-7 b(o)27 b(pro)n(v)n(e)f(this,) i(w)n(e)f(use)h(the)g(estimates)f(\()p @beginspecial @setspecial @endspecial(4.22\))g(in)h(the)g(pro)r(of)f(of)g(Lemma)h @beginspecial @setspecial @endspecial(4.14)e(to)h(obtain)950 641 y Fr(\037)p Fu(\()1056 620 y(^)1034 641 y Fr(H)1103 653 y Fq(\024)1146 641 y Fu(\)[)1211 620 y(^)1201 641 y Fr(I)1237 653 y Fq(\024)1281 641 y Fr(;)14 b Fu(i\000\()p Fr(q)1465 607 y Fq(t)1494 641 y Fu(\)]\(d\000\()p Fr(!)s Fu(\))20 b(+)e(1\))1975 607 y Fo(\000)2036 584 y Fi(1)p 2036 593 29 3 v 2036 627 a(2)2102 641 y Fp(2)23 b Fr(O)r Fu(\()p Fr(t)2307 607 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2490 641 y Fu(\))p Fr(;)14 b(\026)23 b(>)f Fu(0)p Fr(:)83 b @beginspecial @setspecial @endspecial -2867 w Fu(\(7.6\))0 780 y(T)-7 b(o)27 b(pro)n(v)n(e)f(\(7.5\))h(it)i(su\016ces)e(b)n(y)g (Corollary)e(4.8)i(to)g(sho)n(w)g(that)893 961 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial (\(d\000\()p Fr(!)s Fu(\))19 b(+)f(1\))1328 905 y Fi(1)p 1328 914 V 1328 947 a(2)1380 905 y Fr(B)p 1380 942 68 4 v 1399 1018 a(t)1457 961 y Fu(\000\()p Fr(q)1581 927 y Fq(t)1611 961 y Fu(\)\(d\000\()p Fr(!)s Fu(\))h(+)f(1\))2068 927 y Fo(\000)2130 905 y Fi(1)p 2130 914 29 3 v 2130 947 a(2)2172 961 y Fu(\()p Fr(N)28 b Fu(+)18 b(1\))2456 927 y Fo(\000)p Fs(2)2568 961 y Fp(2)23 b Fr(O)r Fu(\(1\))p Fr(:)84 b @beginspecial @setspecial @endspecial -2924 w Fu(\(7.7\))100 1124 y(A)28 b(simple)f(explicit)h(calculus)f(giv)n(es) 618 1300 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial(\(d\000\()p Fr(!)s Fu(\))19 b(+)f(1\))1053 1244 y Fi(1)p 1052 1253 V 1052 1286 a(2)1105 1244 y Fr(B)p 1105 1281 68 4 v 1123 1357 a(t)1182 1300 y Fu(\000\()p Fr(q)1306 1266 y Fq(t)1335 1300 y Fu(\))84 b @beginspecial @setspecial @endspecial(=)1608 1244 y Fr(B)p 1608 1281 V 1627 1357 a(t)1685 1300 y Fu(\000\()p Fr(q)1809 1266 y Fq(t)1839 1300 y Fu(\)\(d\000\()p Fr(!)s Fu(\))19 b(+)f(1\))2306 1244 y Fi(1)p 2306 1253 29 3 v 2306 1286 a(2)2366 1300 y Fu(+)g([\(d\000\()p Fr(!)s Fu(\))i(+)e(1\))2908 1244 y Fi(1)p 2907 1253 V 2907 1286 a(2)2950 1300 y Fr(;)2997 1244 y(B)p 2997 1281 68 4 v 3015 1357 a(t)3074 1300 y Fu(]\000\()p Fr(q)3221 1266 y Fq(t)3250 1300 y Fu(\))0 1382 y @beginspecial @setspecial @endspecial 1598 1543 a(+)1673 1487 y Fr(B)p 1673 1524 V 1691 1600 a(t)1750 1543 y Fu([\(d\000\()p Fr(!)s Fu(\))h(+)f(1\))2208 1486 y Fi(1)p 2208 1495 29 3 v 2208 1529 a(2)2250 1543 y Fr(;)c Fu(\000\()p Fr(q)2411 1509 y Fq(t)2440 1543 y Fu(\)])p Fr(:)-2518 b Fu(\(7.8\))0 1720 y(Clearly)27 b @beginspecial @setspecial @endspecial 299 1687 a Fq(B)p 298 1701 53 4 v 312 1749 a(t)361 1720 y Fu(\000\()p Fr(q)485 1690 y Fq(t)515 1720 y Fu(\)\()p Fr(N)h Fu(+)18 b(1\))831 1690 y Fo(\000)p Fs(2)943 1720 y Fp(2)23 b Fr(O)r Fu(\(1\))28 b(and)698 1909 y @beginspecial @setspecial @endspecial([\(d\000\()p Fr(!)s Fu(\))19 b(+)f(1\))1156 1853 y Fi(1)p 1156 1862 29 3 v 1156 1895 a(2)1198 1909 y Fr(;)1245 1853 y(B)p 1245 1890 68 4 v 1263 1966 a(t)1322 1909 y Fu(]\000\()p Fr(q)1469 1875 y Fq(t)1499 1909 y Fu(\))23 b(=)f(\(d\000\()p Fr(!)s Fu(\))d(+)f(1\))2066 1875 y Fo(\000)2128 1853 y Fi(1)p 2128 1862 29 3 v 2128 1895 a(2)2170 1909 y Fu(d\000\()p Fp(jr)p Fr(!)s Fp(j)2470 1875 y Fs(2)2508 1909 y Fu(\)\000\()p Fr(q)2664 1875 y Fq(t)2694 1909 y Fu(\))37 b Fp(2)23 b Fr(O)r Fu(\(1\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 2076 a Fu(T)-7 b(o)27 b(estimate)h(the)g(last)f(term)h (in)f(\()p @beginspecial @setspecial @endspecial(7.8\),)h(w)n(e)f (write)g(on)h(the)g(n-particle)f(sector:)468 2299 y @beginspecial @setspecial @endspecial([\(d\000\()p Fr(!)s Fu(\))19 b(+)f(1\))926 2243 y Fi(1)p 926 2252 V 926 2285 a(2)968 2299 y Fr(;)c Fu(\000\()p Fr(q)1129 2265 y Fq(t)1159 2299 y Fu(\)])1214 2328 y Fo(jK\012)1337 2312 y Fj(s)1337 2345 y(n)1377 2328 y Fm(h)1519 2299 y @beginspecial @setspecial @endspecial Fu(=)1718 2196 y Fq(n)1678 2220 y Fl(X)1681 2397 y Fq(j)s Fs(=1)1812 2192 y Fq(j)s Fo(\000)p Fs(1)1817 2220 y Fl(Y)1816 2397 y Fq(i)p Fs(=1)1942 2299 y Fr(q)s Fu(\()2024 2243 y Fr(x)2071 2255 y Fq(i)p 2024 2280 76 4 v 2047 2356 a Fr(t)2109 2299 y Fu(\)[\()2236 2196 y Fq(n)2196 2220 y Fl(X)2202 2397 y Fq(i)p Fs(=1)2330 2299 y Fr(!)s Fu(\()p Fr(k)2460 2311 y Fq(i)2488 2299 y Fu(\))19 b(+)f(1\))2706 2243 y Fi(1)p 2706 2252 29 3 v 2706 2285 a(2)2748 2299 y Fr(;)c(q)s Fu(\()2867 2243 y Fr(x)2914 2255 y Fq(j)p 2867 2280 83 4 v 2894 2356 a Fr(t)2960 2299 y Fu(\)])3103 2196 y Fq(n)3071 2220 y Fl(Y)3029 2397 y Fq(i)p Fs(=)p Fq(j)s Fs(+1)3232 2299 y Fr(q)s Fu(\()3314 2243 y Fr(x)3361 2255 y Fq(i)p 3314 2280 76 4 v 3337 2356 a Fr(t)3399 2299 y Fu(\))0 2442 y @beginspecial @setspecial @endspecial 1508 2627 a(=:)1718 2523 y Fq(n)1678 2548 y Fl(X)1681 2725 y Fq(j)s Fs(=1)1812 2627 y Fr(R)1875 2639 y Fq(j)1910 2627 y Fu(\()p Fr(t)p Fu(\))p Fr(:)0 2850 y Fu(Pseudo)r(di\013eren)n(tial)27 b(calculus)g(giv)n(es)f(that) 1277 2989 y @beginspecial @setspecial @endspecial Fr(x)1324 3001 y Fq(k)1365 2989 y Fr(R)1428 3001 y Fq(j)1463 2989 y Fu(\()p Fr(t)p Fu(\))e Fp(2)f Fr(O)r Fu(\(1\))p Fr(;)14 b Fu(uniformly)i(in)e Fr(k)s(;)g(j:)83 b @beginspecial @setspecial @endspecial 0 3128 a Fu(This)28 b(pro)n(v)n(es)d(\(7.7\).) 37 b(No)n(w)27 b(\()p @beginspecial @setspecial @endspecial(7.6\))g (and)h(\(7.7\))f(imply)h(that)g Fr(R)1901 3098 y Fs(2)1900 3148 y Fq(\024)1966 3128 y Fp(2)23 b Fr(O)r Fu(\()p Fr(t)2171 3098 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\027)2350 3128 y Fu(\),)28 b(for)g @beginspecial @setspecial @endspecial Fr(\024)22 b(<)h Fp(1)p Fu(.)0 3227 y(W)-7 b(e)28 b(consider)f(no)n(w)g @beginspecial @setspecial @endspecial Fr(R)704 3197 y Fs(1)703 3248 y Fq(\024)746 3227 y Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e:)871 3405 y(d)-12 b(l)928 3417 y Fs(0)965 3405 y Fr(q)1005 3371 y Fq(t)1058 3405 y Fu(=)22 b Fp(\000)1235 3349 y Fu(1)p 1220 3386 72 4 v 1220 3462 a(2)p Fr(t)1301 3405 y Fp(h)1343 3349 y Fr(x)p 1343 3386 48 4 v 1352 3462 a(t)1419 3405 y Fp(\000)c(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\))p Fr(;)c Fp(r)p Fr(q)s Fu(\()1924 3349 y Fr(x)p 1924 3386 V 1933 3462 a(t)1983 3405 y Fu(\))p Fp(i)19 b Fu(+)f Fr(hc)g Fu(+)g Fr(r)2373 3371 y Fq(t)2426 3405 y Fu(=:)2547 3349 y(1)p 2547 3386 42 4 v 2553 3462 a Fr(t)2598 3405 y(g)2641 3371 y Fq(t)2688 3405 y Fu(+)g Fr(r)2810 3371 y Fq(t)2840 3405 y Fr(;)83 b @beginspecial @setspecial @endspecial 0 3578 a Fu(where)27 b @beginspecial @setspecial @endspecial Fr(r)279 3548 y Fq(t)332 3578 y Fp(2)d Fr(O)r Fu(\()p Fr(t)538 3548 y Fo(\000)p Fs(2)628 3578 y Fu(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(using)h(\(2.3\))h(and)f (Corollary)e @beginspecial @setspecial @endspecial(4.8:)1100 3759 y Fp(jj)p Fr(\037)p Fu(\()1252 3738 y(^)1230 3759 y Fr(H)1299 3771 y Fq(\024)1342 3759 y Fu(\)d\000\()p Fr(q)1544 3725 y Fq(t)1574 3759 y Fr(;)14 b(r)1650 3725 y Fq(t)1680 3759 y Fu(\))1722 3703 y Fr(B)p 1722 3740 68 4 v 1741 3816 a(t)1800 3759 y Fu(\000\()p Fr(q)1924 3725 y Fq(t)1953 3759 y Fu(\))p Fr(\037)p Fu(\()2091 3738 y(^)2069 3759 y Fr(H)2138 3771 y Fq(\024)2182 3759 y Fu(\))p Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(t)2488 3725 y Fo(\000)p Fs(2)2579 3759 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 3921 a Fu(W)-7 b(e)28 b(set)695 4081 y @beginspecial @setspecial @endspecial Fr(B)758 4093 y Fs(1)818 4081 y Fu(:=)23 b Fr(\037)p Fu(\()1035 4060 y(^)1013 4081 y Fr(H)1082 4093 y Fq(\024)1125 4081 y Fu(\)d\000\()p Fr(q)1327 4047 y Fq(t)1357 4081 y Fr(;)14 b(g)1437 4047 y Fq(t)1465 4081 y Fu(\)\()p Fr(N)28 b Fu(+)18 b(1\))1781 4047 y Fo(\000)1843 4025 y Fi(1)p 1843 4034 29 3 v 1843 4067 a(2)1885 4081 y Fr(;)37 b(B)2008 4093 y Fs(2)2069 4081 y Fu(:=)22 b(\()p Fr(N)28 b Fu(+)18 b(1\))2473 4025 y Fi(1)p 2473 4034 V 2473 4067 a(2)2525 4025 y Fr(B)p 2525 4062 68 4 v 2543 4138 a(t)2602 4081 y Fu(\000\()p Fr(q)2726 4047 y Fq(t)2755 4081 y Fu(\))p Fr(\037)p Fu(\()2894 4060 y(^)2871 4081 y Fr(H)2940 4093 y Fq(\024)2984 4081 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 4244 a Fu(So)27 b(w)n(e)h(obtain)f(the)h (inequalit)n(y)1243 4383 y Fr(R)1307 4348 y Fs(1)1306 4403 y Fq(\024)1372 4383 y Fp(\025)22 b(\000)p Fr(\017)1558 4347 y Fo(\000)p Fs(1)1558 4405 y(0)1647 4383 y Fr(t)1677 4348 y Fo(\000)p Fs(1)1766 4383 y Fr(B)1829 4395 y Fs(1)1866 4383 y Fr(B)1933 4348 y Fo(\003)1929 4403 y Fs(1)1990 4383 y Fp(\000)c Fr(\017)2107 4395 y Fs(0)2144 4383 y Fr(t)2174 4348 y Fo(\000)p Fs(1)2263 4383 y Fr(B)2326 4395 y Fs(2)2363 4383 y Fr(B)2430 4348 y Fo(\003)2426 4403 y Fs(2)2468 4383 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4522 a Fu(Using)27 b(argumen)n(ts)g(in)h([11)o(,)f(Prop.) 36 b(6.5],)27 b(w)n(e)g(obtain)1095 4669 y @beginspecial @setspecial @endspecial Fp(\000)p Fr(B)1223 4681 y Fs(2)1259 4669 y Fr(B)1326 4635 y Fo(\003)1322 4689 y Fs(2)1388 4669 y Fp(\025)22 b(\000)p Fr(C)1599 4681 y Fs(1)1637 4669 y Fr(\037)p Fu(\()1742 4648 y(^)1721 4669 y Fr(H)1790 4681 y Fq(\024)1833 4669 y Fu(\)\000\()p Fr(q)1989 4635 y Fq(t)2019 4669 y Fu(\))2051 4635 y Fs(2)2088 4669 y Fr(\037)p Fu(\()2194 4648 y(^)2172 4669 y Fr(H)2241 4681 y Fq(\024)2284 4669 y Fu(\))d Fp(\000)f Fr(C)6 b(t)2513 4635 y Fo(\000)p Fs(1)2602 4669 y Fr(;)97 b @beginspecial @setspecial @endspecial 1626 4767 a Fl(Z)1709 4787 y Fo(1)1673 4955 y Fs(1)1794 4880 y Fp(jj)p Fr(B)1903 4892 y Fs(1)1940 4880 y Fu(e)1977 4846 y Fo(\000)p Fs(i)p Fq(t)2089 4831 y Fs(^)2073 4846 y Fq(H)2127 4854 y Fj(\024)2170 4880 y Fr(u)p Fp(jj)22 b(\024)h Fr(C)6 b Fp(jj)p Fr(u)p Fp(jj)2579 4846 y Fs(2)2616 4880 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 5061 a Fu(Using)27 b(Lemma)h(4.14)e(and)h (Thm.)38 b @beginspecial @setspecial @endspecial(5.7,)26 b(w)n(e)i(ha)n(v)n(e)1156 5203 y Fr(R)1220 5169 y Fs(3)1219 5224 y Fq(\024)1285 5203 y Fp(\025)23 b Fr(C)1432 5215 y Fs(0)1470 5203 y Fr(t)1500 5169 y Fo(\000)p Fs(1)1589 5203 y Fr(\037)p Fu(\()1695 5182 y(^)1673 5203 y Fr(H)1742 5215 y Fq(\024)1785 5203 y Fu(\)\000\()p Fr(q)1941 5169 y Fq(t)1971 5203 y Fu(\))2003 5169 y Fs(2)2040 5203 y Fr(\037)p Fu(\()2146 5182 y(^)2124 5203 y Fr(H)2193 5215 y Fq(\024)2236 5203 y Fu(\))c Fp(\000)f Fr(C)6 b(t)2465 5169 y Fo(\000)p Fs(2)2555 5203 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 5343 a Fu(W)-7 b(e)28 b(ha)n(v)n(e)1157 5482 y @beginspecial @setspecial @endspecial Fp(\000)p Fr(R)1286 5448 y Fs(4)1285 5503 y Fq(\024)1350 5482 y Fp(\024)23 b Fr(C)1497 5494 y Fs(2)1545 5426 y Fr(\017)p 1545 5463 34 4 v 1547 5539 a(t)1588 5482 y(\037)p Fu(\()1694 5461 y(^)1672 5482 y Fr(H)1741 5494 y Fq(\024)1785 5482 y Fu(\)\000\()p Fr(q)1941 5448 y Fq(t)1970 5482 y Fu(\))2002 5448 y Fs(2)2040 5482 y Fr(\037)p Fu(\()2146 5461 y(^)2124 5482 y Fr(H)2193 5494 y Fq(\024)2236 5482 y Fu(\))c(+)f Fr(C)6 b(t)2465 5448 y Fo(\000)p Fs(2)2554 5482 y Fr(:)83 b @beginspecial @setspecial @endspecial eop %%Page: 40 40 40 39 bop 0 226 a Fs(40)1695 b(Z.)29 b(AMMARI)0 489 y Fu(Collecting)e(the)h(four)f(terms,)h(w)n(e)f(obtain)873 635 y @beginspecial @setspecial @endspecial 892 614 a(^)873 635 y(I)-13 b(D)953 647 y Fq(\024)996 635 y Fr(\036)1045 647 y Fq(\024)1088 635 y Fu(\()p Fr(t)p Fu(\))84 b Fp(\025)f(\000)p Fr(\017)1513 647 y Fs(0)1549 635 y Fr(t)1579 601 y Fo(\000)p Fs(1)1668 635 y Fr(B)1731 647 y Fs(2)1768 635 y Fr(B)1835 601 y Fo(\003)1831 656 y Fs(2)1892 635 y Fu(+)18 b Fr(R)2039 601 y Fs(2)2038 656 y Fq(\024)2100 635 y Fu(+)g Fr(R)2247 601 y Fs(3)2246 656 y Fq(\024)2307 635 y Fu(+)g Fr(R)2454 601 y Fs(4)2453 656 y Fq(\024)0 681 y @beginspecial @setspecial @endspecial 1266 811 a @beginspecial @setspecial @endspecial Fp(\025)83 b Fu(\()p Fp(\000)p Fr(\017)1545 823 y Fs(0)1581 811 y Fr(C)1640 823 y Fs(1)1696 811 y Fp(\000)18 b Fr(\017C)1872 823 y Fs(2)1910 811 y Fu(\))p Fr(t)1972 777 y Fo(\000)p Fs(1)2061 811 y Fr(\037)p Fu(\()2167 790 y(^)2145 811 y Fr(H)2214 823 y Fq(\024)2257 811 y Fu(\)\000\()p Fr(q)2413 777 y Fq(t)2443 811 y Fu(\))2475 777 y Fs(2)2513 811 y Fr(\037)p Fu(\()2619 790 y(^)2597 811 y Fr(H)2666 823 y Fq(\024)2709 811 y Fu(\))h(+)f Fr(C)6 b(t)2938 777 y Fo(\000)p Fs(2)0 857 y @beginspecial @setspecial @endspecial 1266 987 a Fp(\025)1432 966 y Fu(~)1414 987 y Fr(C)1479 953 y Fq(\024)1473 1008 y Fs(0)1522 987 y Fr(\037)p Fu(\()1628 966 y(^)1606 987 y Fr(H)1675 999 y Fq(\024)1718 987 y Fu(\)\000\()p Fr(q)1874 953 y Fq(t)1904 987 y Fu(\))1936 953 y Fs(2)1973 987 y Fr(\037)p Fu(\()2079 966 y(^)2057 987 y Fr(H)2126 999 y Fq(\024)2170 987 y Fu(\))18 b Fp(\000)g Fr(R)q Fu(\()p Fr(t)p Fu(\))p Fr(;)0 1130 y Fu(where)39 b @beginspecial @setspecial @endspecial Fr(R)q Fu(\()p Fr(t)p Fu(\))h(is)f(in)n(tegrable.)71 b(By)39 b(Lemma)g(A.3)h(w)n(e)f (obtain)g(the)g(inequalit)n(y)g(announced)g(in)h(the)g(prop)r(osition)e (for)h @beginspecial @setspecial @endspecial Fr(\037)0 1230 y Fu(supp)r(orted)f(near)f(a)h(one)g(energy)f(lev)n(el)g @beginspecial @setspecial @endspecial Fr(\025)p Fu(.)69 b(Then)38 b(w)n(e)g(complete)g(the)h(pro)r(of)e(for)h(an)f(arbitrary)f @beginspecial @setspecial @endspecial Fr(\037)j Fu(using)e(a)h (standard)0 1330 y(argumen)n(t,)27 b(see)g(e.g.[10)n(,)h(Prop.)36 b(4.4.7].)p @beginspecial @setspecial @endspecial 2570 w Fh(\004)1350 1523 y @beginspecial @setspecial @endspecial Fu(8.)46 b Fv(Asymptotic)31 b(completeness)100 1673 y Fu(In)23 b(this)h(section)f(w)n(e)f(pro)n(v)n(e)g(the)i(main)f(result)g (of)g(this)h(pap)r(er)f(,)h(whic)n(h)f(is)g(the)h(asymptotic)f (completeness)f(of)i(the)f(Nelson)0 1772 y(Hamiltonian.)37 b(This)27 b(is)h(the)g(sub)5 b(ject)27 b(of)h(Thm.)37 b @beginspecial @setspecial @endspecial(8.5,)27 b(where)g(w)n(e)g(pro)n (v)n(e)1606 1915 y(1)-15 b(l)1656 1927 y Fs(pp)1733 1915 y Fu(\()p Fr(H)7 b Fu(\))23 b(=)g Fp(K)2048 1881 y Fo(\006)2105 1915 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 2070 a @beginspecial @setspecial @endspecial Fz(Theorem)30 b(8.1.)40 b Fn(L)l(et)29 b @beginspecial @setspecial @endspecial Fr(q)s(;)20 b Fu(~)-48 b Fr(q)26 b Fp(2)d Fr(C)1021 2040 y Fo(1)1015 2090 y Fs(0)1092 2070 y Fu(\()p Fk(R)1178 2040 y Fs(3)1221 2070 y Fu(\))29 b Fn(such)g(that)g Fu(0)22 b Fp(\024)h Fr(q)s(;)d Fu(~)-48 b Fr(q)26 b Fp(\024)d Fu(1)p Fr(;)41 b(q)s(;)20 b Fu(~)-48 b Fr(q)26 b Fu(=)c(1)29 b Fn(on)g(a)g(neighb)l(orho)l(o)l(d)i(of)f(zer)l(o)f(and)g @beginspecial @setspecial @endspecial Fr(q)3575 2040 y Fq(t)3627 2070 y Fu(:=)23 b Fr(q)s Fu(\()3820 2037 y Fq(x)p 3820 2051 38 4 v 3826 2098 a(t)3868 2070 y Fu(\))0 2169 y @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))30 b Fn(The)g(fol)t(lowing)j(limits)d(exist)1092 2434 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1133 2331 a @beginspecial @setspecial @endspecial Fu(\000)1185 2297 y Fs(+)1240 2331 y Fu(\()p Fr(q)s Fu(\))84 b(:=)e Fr(s)19 b Fp(\000)64 b Fu(lim)1739 2381 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1975 2331 y Fu(e)2012 2297 y Fs(i)p Fq(t)2072 2282 y Fs(^)2056 2297 y Fq(H)2110 2305 y Fg(1)2189 2331 y Fu(\000\()p Fr(q)2313 2297 y Fq(t)2342 2331 y Fu(\)e)2411 2297 y Fo(\000)p Fs(i)p Fq(t)2523 2282 y Fs(^)2507 2297 y Fq(H)2561 2305 y Fg(1)2626 2331 y @beginspecial @setspecial @endspecial 1428 2520 a Fu(=)133 b(lim)1598 2570 y Fq(\024)p Fo(!1)1783 2520 y Fr(s)19 b Fp(\000)64 b Fu(lim)1924 2570 y Fq(t)p Fo(!)p Fs(+)p Fo(1)2160 2520 y Fu(e)2197 2486 y Fs(i)p Fq(t)2257 2471 y Fs(^)2241 2486 y Fq(H)2295 2494 y Fj(\024)2352 2520 y Fu(\000\()p Fr(q)2476 2486 y Fq(t)2505 2520 y Fu(\)e)2574 2486 y Fo(\000)p Fs(i)p Fq(t)2686 2471 y Fs(^)2670 2486 y Fq(H)2724 2494 y Fj(\024)2808 2434 y @beginspecial @setspecial @endspecial 0 2699 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(We)f(have)1232 2982 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1274 2842 a @beginspecial @setspecial @endspecial Fu(\000)1326 2808 y Fs(+)1381 2842 y Fu(\()p Fr(q)9 b Fu(~)-48 b Fr(q)s Fu(\))24 b(=)e(\000)1688 2808 y Fs(+)1743 2842 y Fu(\()p Fr(q)s Fu(\)\000)1899 2808 y Fs(+)1955 2842 y Fu(\()6 b(~)-48 b Fr(q)s Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 1274 2977 a Fu(0)22 b Fp(\024)h Fu(\000)1478 2943 y Fs(+)1533 2977 y Fu(\()p Fr(q)s Fu(\))h Fp(\024)e Fu(\000)1800 2943 y Fs(+)1855 2977 y Fu(\()6 b(~)-48 b Fr(q)t Fu(\))23 b Fp(\024)g Fu(1)-13 b(l)o Fr(;)37 b Fu(if)30 b(0)23 b Fp(\024)f Fr(q)k Fp(\024)j Fu(~)-48 b Fr(q)s(:)p @beginspecial @setspecial @endspecial 1274 3120 a Fu([)1319 3099 y(^)1297 3120 y Fr(H)1366 3132 y Fo(1)1436 3120 y Fr(;)14 b Fu(\000)1525 3086 y Fs(+)1580 3120 y Fu(\()p Fr(q)s Fu(\)])24 b(=)f(0)p Fr(:)2668 2982 y @beginspecial @setspecial @endspecial 0 3264 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))30 b Fn(We)g(have)1560 3415 y @beginspecial @setspecial @endspecial Fu(Ran\000)1761 3381 y Fs(+)1816 3415 y Fu(\()p Fr(q)s Fu(\))24 b Fp(\032)2047 3394 y Fu(^)2031 3415 y Fp(K)2095 3381 y Fs(+)2151 3415 y Fr(:)83 b @beginspecial @setspecial @endspecial 100 3570 a Fz(Pro)s(of.)46 b @beginspecial @setspecial @endspecial 26 w Fu(It)29 b(is)g(su\016cien)n(t)g(using)g(a)f(densit)n(y)h(argumen) n(t)f(and)h(Lemma)g @beginspecial @setspecial @endspecial(4.14)e(to)i (sho)n(w)f(for)h Fr(\037)c Fp(2)h Fr(C)3206 3539 y Fo(1)3200 3590 y Fs(0)3276 3570 y Fu(\()p Fk(R)q Fu(\))35 b(the)29 b(existence)0 3669 y(of)f(the)g(limit)1093 3812 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(s)18 b Fp(\000)65 b Fu(lim)1233 3862 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1455 3812 y Fu(e)1492 3778 y Fs(i)p Fq(t)1552 3763 y Fs(^)1536 3778 y Fq(H)1590 3786 y Fg(1)1655 3812 y Fr(\037)p Fu(\()1761 3791 y(^)1739 3812 y Fr(H)1808 3824 y Fo(1)1879 3812 y Fu(\))14 b(\000\()p Fr(q)2049 3778 y Fq(t)2078 3812 y Fu(\))p Fr(\037)p Fu(\()2217 3791 y(^)2194 3812 y Fr(H)2263 3824 y Fo(1)2334 3812 y Fu(\)e)2403 3778 y Fo(\000)p Fs(i)p Fq(t)2516 3763 y Fs(^)2499 3778 y Fq(H)2553 3786 y Fg(1)2618 3812 y Fr(:)83 b @beginspecial @setspecial @endspecial -2724 w Fu(\(8.1\))0 3987 y(Using)30 b(Lemma)g(A.4,)g(w)n(e)g(see)g(that)g (as)g(for)f(all)h(asymptotic)g(limits)g(these)g(amoun)n(ts)g(to)g(b)r (ound)g(Heisen)n(b)r(erg)f(deriv)-5 b(ativ)n(es)0 4086 y(uniformly)27 b(in)h @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(on)i Fp(H)q Fu(:)572 4243 y @beginspecial @setspecial @endspecial Fr(@)616 4255 y Fq(t)645 4243 y Fu(\(e)714 4209 y Fs(i)p Fq(t)775 4194 y Fs(^)758 4209 y Fq(H)812 4217 y Fj(\024)855 4243 y Fr(\037)p Fu(\()961 4222 y(^)939 4243 y Fr(H)1008 4255 y Fq(\024)1052 4243 y Fu(\))14 b(\000\()p Fr(q)1222 4209 y Fq(t)1251 4243 y Fu(\))p Fr(\037)p Fu(\()1389 4222 y(^)1367 4243 y Fr(H)1436 4255 y Fq(\024)1480 4243 y Fu(\)e)1549 4209 y Fo(\000)p Fs(i)p Fq(t)1661 4194 y Fs(^)1645 4209 y Fq(H)1699 4217 y Fj(\024)1742 4243 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)g @beginspecial @setspecial @endspecial(e)2042 4209 y Fs(i)p Fq(t)2102 4194 y Fs(^)2086 4209 y Fq(H)2140 4217 y Fj(\024)2182 4243 y Fr(\037)p Fu(\()2288 4222 y(^)2266 4243 y Fr(H)2335 4255 y Fq(\024)2379 4243 y Fu(\)d\000\()p Fr(q)2581 4209 y Fq(t)2610 4243 y Fr(;)14 b Fu(d)-12 b(l)2705 4255 y Fs(0)2742 4243 y Fr(q)2782 4209 y Fq(t)2811 4243 y Fu(\))p Fr(\037)p Fu(\()2950 4222 y(^)2927 4243 y Fr(H)2996 4255 y Fq(\024)3040 4243 y Fu(\)e)3109 4209 y Fo(\000)p Fs(i)p Fq(t)3221 4194 y Fs(^)3205 4209 y Fq(H)3259 4217 y Fj(\024)0 4289 y @beginspecial @setspecial @endspecial 2005 4430 a Fu(+e)2107 4396 y Fs(i)p Fq(t)2166 4381 y Fs(^)2151 4396 y Fq(H)2205 4404 y Fj(\024)2247 4430 y Fr(\037)p Fu(\()2353 4409 y(^)2331 4430 y Fr(H)2400 4442 y Fq(\024)2443 4430 y Fu(\)[)2508 4409 y(^)2498 4430 y Fr(I)2534 4442 y Fq(\024)2578 4430 y Fr(;)14 b Fu(i\000\()p Fr(q)2762 4396 y Fq(t)2791 4430 y Fu(\)])p Fr(\037)p Fu(\()2953 4409 y(^)2930 4430 y Fr(H)2999 4442 y Fq(\024)3043 4430 y Fu(\)e)3112 4396 y Fo(\000)p Fs(i)p Fq(t)3224 4381 y Fs(^)3208 4396 y Fq(H)3262 4404 y Fj(\024)3305 4430 y Fr(:)0 4573 y Fu(By)27 b(\()p @beginspecial @setspecial @endspecial(4.22\))g(w)n(e)g(ha)n(v)n(e)g(uniformly)g(in)h Fr(\024)p Fu(:)1226 4724 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1332 4703 y(^)1310 4724 y Fr(H)1379 4736 y Fq(\024)1422 4724 y Fu(\)[)1487 4703 y(^)1477 4724 y Fr(I)1513 4736 y Fq(\024)1557 4724 y Fr(;)14 b Fu(i\000\()p Fr(q)1741 4690 y Fq(t)1771 4724 y Fu(\)])p Fr(\037)p Fu(\()1932 4703 y(^)1910 4724 y Fr(H)1979 4736 y Fq(\024)2022 4724 y Fu(\))24 b Fp(2)f Fr(O)r Fu(\()p Fr(t)2283 4690 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)2453 4724 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial -2591 w Fu(\(8.2\))0 4867 y(W)-7 b(e)28 b(use)f(no)n(w)g(an)h(argumen)n(t)e(in)n(tro)r (duced)i(in)f @beginspecial @setspecial @endspecial([11]:)825 5111 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 866 5010 a @beginspecial @setspecial @endspecial(d)-12 b(l)924 5022 y Fs(0)961 5010 y Fr(q)1001 4979 y Fq(t)1053 5010 y Fu(=)1151 4977 y Fs(1)p 1151 4991 34 4 v 1155 5038 a Fq(t)1194 5010 y Fr(g)1237 4979 y Fq(t)1284 5010 y Fu(+)18 b Fr(r)1406 4979 y Fq(t)1436 5010 y Fr(;)889 5184 y Fu(where)23 b Fr(g)1168 5150 y Fq(t)1220 5184 y Fu(=)f Fp(\000)1382 5128 y Fu(1)p 1382 5165 42 4 v 1382 5241 a(2)1433 5184 y(\(\()1507 5128 y Fr(x)p 1507 5165 48 4 v 1516 5241 a(t)1584 5184 y Fp(\000)c Fr(@)5 b(!)s Fu(\()p Fr(k)s Fu(\)\))23 b Fr(@)5 b(q)s Fu(\()2067 5128 y Fr(x)p 2067 5165 V 2076 5241 a(t)2124 5184 y Fu(\))19 b(+)f Fr(hc)p Fu(\))p Fr(;)37 b Fu(and)23 b Fr(r)2630 5150 y Fq(t)2683 5184 y Fp(2)g Fr(O)r Fu(\()p Fr(t)2888 5150 y Fo(\000)p Fs(2)2978 5184 y Fu(\))p Fr(:)3075 5111 y @beginspecial @setspecial @endspecial 0 5355 a Fu(The)28 b(estimate)f(\()p @beginspecial @setspecial @endspecial(2.3\))h(and)f(Corollary)e(4.8)i(giv)n(e)g(uniformly)g(in)h @beginspecial @setspecial @endspecial Fr(\024)1189 5506 y Fp(jj)p Fr(\037)p Fu(\()1341 5485 y(^)1319 5506 y Fr(H)1388 5518 y Fq(\024)1431 5506 y Fu(\)d\000\()p Fr(q)1633 5472 y Fq(t)1663 5506 y Fr(;)14 b Fu(d)-12 b(l)1758 5518 y Fs(0)1795 5506 y Fr(r)1834 5472 y Fq(t)1864 5506 y Fu(\))p Fr(\037)p Fu(\()2002 5485 y(^)1980 5506 y Fr(H)2049 5518 y Fq(\024)2093 5506 y Fu(\))p Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(t)2399 5472 y Fo(\000)p Fs(2)2489 5506 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial -2628 w Fu(\(8.3\))p eop %%Page: 41 41 41 40 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(41)0 489 y Fu(The)28 b(other)f(term)g(will)h(b)r (e)g(estimated:)438 631 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(j)p Fu(\(e)530 597 y Fo(\000)p Fs(i)p Fq(t)642 582 y Fs(^)626 597 y Fq(H)680 605 y Fj(\024)723 631 y Fr(u;)14 b(\037)p Fu(\()913 610 y(^)892 631 y Fr(H)961 643 y Fq(\024)1004 631 y Fu(\)d\000\()p Fr(q)1206 597 y Fq(t)1235 631 y Fr(;)g(g)1315 597 y Fq(t)1344 631 y Fu(\))p Fr(\037)p Fu(\()1482 610 y(^)1460 631 y Fr(H)1529 643 y Fq(\024)1573 631 y Fu(\)e)1642 597 y Fo(\000)p Fs(i)p Fq(t)1754 582 y Fs(^)1738 597 y Fq(H)1792 605 y Fj(\024)1835 631 y Fr(u)p Fu(\))p Fp(j)23 b(\024)f(jj)p Fu(d\000\()p Fp(j)p Fr(g)2290 597 y Fq(t)2319 631 y Fp(j)p Fu(\))2384 575 y Fi(1)p 2385 584 29 3 v 2385 617 a(2)2427 631 y Fr(\037)p Fu(\()2533 610 y(^)2511 631 y Fr(H)2580 643 y Fq(\024)2623 631 y Fu(\)e)2692 597 y Fo(\000)p Fs(i)p Fq(t)2805 582 y Fs(^)2788 597 y Fq(H)2842 605 y Fj(\024)2885 631 y Fr(u)p Fp(jj)2979 597 y Fs(2)3016 631 y Fr(;)14 b(u)23 b Fp(2)g(H)q Fr(:)83 b @beginspecial @setspecial @endspecial -3379 w Fu(\(8.4\))0 773 y(Using)27 b(\()p @beginspecial @setspecial @endspecial(8.2\)-\(8.4\),)g(Prop.)36 b(7.3)26 b(and)i(Lemma)f @beginspecial @setspecial @endspecial(A.4,)h (w)n(e)f(obtain)g(the)h(existence)g(of)f(the)h(limit)h(\()p @beginspecial @setspecial @endspecial(8.1\).)100 873 y(The)e(\014rst)h(statemen)n(t)f(in)h @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(follo)n(ws)e(from)h(the)h(fact)g(that)1487 1015 y @beginspecial @setspecial @endspecial(\000\()p Fr(q)1611 981 y Fq(t)1646 1015 y Fu(~)-48 b Fr(q)1680 981 y Fq(t)1710 1015 y Fu(\))23 b(=)g(\000\()p Fr(q)1977 981 y Fq(t)2006 1015 y Fu(\)\000\()6 b(~)-48 b Fr(q)2162 981 y Fq(t)2192 1015 y Fu(\))p Fr(:)0 1157 y Fu(The)28 b(second)f(statemen)n(t)g(follo)n(ws)g(b)n(y)1229 1299 y @beginspecial @setspecial @endspecial(0)22 b Fp(\024)h Fu(\000\()p Fr(q)1505 1265 y Fq(t)1534 1299 y Fu(\))h Fp(\024)e Fu(\000\()6 b(~)-48 b Fr(q)1801 1265 y Fq(t)1831 1299 y Fu(\))23 b Fp(\024)g Fu(1)f Fr(if)31 b Fu(0)23 b Fp(\024)g Fr(q)j Fp(\024)j Fu(~)-49 b Fr(q)s(:)83 b @beginspecial @setspecial @endspecial 0 1441 a Fu(The)28 b(last)f(statemen)n(t)h(is)f(a)g(consequence)g(of)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)p Fu(\))28 b(and)g(Lemma)f @beginspecial @setspecial @endspecial(4.14.)100 1549 y(Let)e(us)h(pro)n(v)n(e)e @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)f Fr({)t Fu(_)h Fr({)p Fu(\).)36 b(Since)981 1528 y(^)959 1549 y Fr(H)1028 1561 y Fo(1)1099 1549 y Fr(;)14 b Fu(\000)1188 1519 y Fs(+)1242 1549 y Fu(\()p Fr(q)s Fu(\))27 b(comm)n(ute,)f @beginspecial @setspecial @endspecial(\000)1804 1519 y Fs(+)1859 1549 y Fu(\()p Fr(q)s Fu(\))g(preserv)n(es)d Fp(D)r Fu(\()2464 1528 y(^)2441 1549 y Fr(H)2510 1561 y Fo(1)2581 1549 y Fu(\).)37 b(Consequen)n(tly)24 b @beginspecial @setspecial @endspecial Fp(D)r Fu(\()3306 1528 y(^)3283 1549 y Fr(H)3352 1561 y Fo(1)3423 1549 y Fu(\))15 b Fp(\\)g Fu(Ran)o(\000)3740 1519 y Fs(+)3795 1549 y Fu(\()p Fr(q)s Fu(\))0 1657 y(is)26 b(dense)h(in)f @beginspecial @setspecial @endspecial(Ran\000)604 1627 y Fs(+)659 1657 y Fu(\()p Fr(q)s Fu(\).)37 b(Since)1055 1636 y(^)1039 1657 y Fp(K)1103 1627 y Fs(+)1185 1657 y Fu(is)26 b(closed,)g(it)h(is)f(enough)g(to)g(sho)n(w)g(that)g @beginspecial @setspecial @endspecial Fp(D)r Fu(\()2587 1636 y(^)2564 1657 y Fr(H)2633 1669 y Fo(1)2704 1657 y Fu(\))17 b Fp(\\)f Fu(Ran\000)3025 1627 y Fs(+)3080 1657 y Fu(\()p Fr(q)s Fu(\))24 b Fp(\032)3311 1636 y Fu(^)3295 1657 y Fp(K)3359 1627 y Fs(+)3441 1657 y Fu(to)j(pro)n(v)n(e) d @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\).)0 1765 y(Let)28 b Fr(v)e Fp(2)d(D)r Fu(\()414 1744 y(^)391 1765 y Fr(H)460 1777 y Fo(1)531 1765 y Fu(\))c Fp(\\)g Fu(Ran\000)857 1734 y Fs(+)912 1765 y Fu(\()p Fr(q)s Fu(\),)28 b @beginspecial @setspecial @endspecial Fr(v)e Fu(=)d(\000)1273 1734 y Fs(+)1328 1765 y Fu(\()p Fr(q)s Fu(\))p Fr(u)p Fu(.)535 2123 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 577 1925 a @beginspecial @setspecial @endspecial(\()631 1904 y(^)609 1925 y Fr(H)678 1937 y Fo(1)767 1925 y Fu(+)18 b Fr(b)p Fu(\))918 1895 y Fo(\000)p Fs(1)1008 1925 y Fu(^)-43 b Fr(a)1051 1895 y Fs(+)1106 1925 y Fu(\()p Fr(h)p Fu(\)\000)1270 1895 y Fs(+)1325 1925 y Fu(\()p Fr(q)s Fu(\))p Fr(u)83 b @beginspecial @setspecial @endspecial Fu(=)130 b(lim)1708 1975 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1930 1925 y Fu(e)1967 1891 y Fs(i)p Fq(t)2027 1876 y Fs(^)2011 1891 y Fq(H)2065 1899 y Fg(1)2130 1925 y Fu(\()2184 1904 y(^)2162 1925 y Fr(H)2231 1937 y Fo(1)2320 1925 y Fu(+)18 b Fr(b)p Fu(\))2471 1891 y Fo(\000)p Fs(1)2560 1925 y Fr(a)p Fu(\()p Fr(h)2684 1937 y Fq(t)2714 1925 y Fu(\)e)2783 1891 y Fo(\000)p Fs(i)p Fq(t)2895 1876 y Fs(^)2879 1891 y Fq(H)2933 1899 y Fg(1)2998 1925 y Fu(\000)3050 1891 y Fs(+)3105 1925 y Fu(\()p Fr(q)s Fu(\))p Fr(u)p @beginspecial @setspecial @endspecial 1560 2115 a Fu(=)130 b(lim)1708 2164 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1930 2115 y Fu(e)1967 2080 y Fs(i)p Fq(t)2027 2065 y Fs(^)2011 2080 y Fq(H)2065 2088 y Fg(1)2130 2115 y Fu(\()2184 2094 y(^)2162 2115 y Fr(H)2231 2127 y Fo(1)2320 2115 y Fu(+)18 b Fr(b)p Fu(\))2471 2080 y Fo(\000)p Fs(1)2560 2115 y Fr(a)p Fu(\()p Fr(h)2684 2127 y Fq(t)2714 2115 y Fu(\)\000\()p Fr(q)2870 2080 y Fq(t)2899 2115 y Fu(\)e)2968 2080 y Fo(\000)p Fs(i)p Fq(t)3081 2065 y Fs(^)3064 2080 y Fq(H)3118 2088 y Fg(1)3183 2115 y Fr(u)p @beginspecial @setspecial @endspecial 577 2304 a @beginspecial @setspecial @endspecial 983 w Fu(=)130 b(lim)1708 2354 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1930 2304 y Fu(e)1967 2270 y Fs(i)p Fq(t)2027 2255 y Fs(^)2011 2270 y Fq(H)2065 2278 y Fg(1)2130 2304 y Fu(\()2184 2283 y(^)2162 2304 y Fr(H)2231 2316 y Fo(1)2320 2304 y Fu(+)18 b Fr(b)p Fu(\))2471 2270 y Fo(\000)p Fs(1)2560 2304 y Fu(\000\()p Fr(q)2684 2270 y Fq(t)2714 2304 y Fu(\))p Fr(a)p Fu(\()p Fr(q)2862 2270 y Fq(t)2892 2304 y Fr(h)2940 2316 y Fq(t)2969 2304 y Fu(\)e)3038 2270 y Fo(\000)p Fs(i)p Fq(t)3150 2255 y Fs(^)3134 2270 y Fq(H)3188 2278 y Fg(1)3253 2304 y Fr(u:)3365 2123 y @beginspecial @setspecial @endspecial 0 2486 a Fu(If)41 b @beginspecial @setspecial @endspecial Fr(h)k Fp(2)h Fm(h)333 2498 y Fs(0)370 2486 y Fu(,)e(b)n(y)d(a)f(stationary)g(phase)g(argumen)n(t)g (w)n(e)g(ha)n(v)n(e)g @beginspecial @setspecial @endspecial Fr(q)2066 2455 y Fq(t)2095 2486 y Fr(h)2143 2498 y Fq(t)2217 2486 y Fp(2)46 b Fr(o)p Fu(\(1\))p Fr(;)14 b(t)45 b Fp(!)g Fu(+)p Fp(1)p Fu(.)76 b(Using)41 b(the)g(fact)g(that)g @beginspecial @setspecial @endspecial Fr(h)k Fp(!)0 2593 y Fu(\()54 2572 y(^)32 2593 y Fr(H)101 2605 y Fo(1)190 2593 y Fu(+)18 b(i\))328 2563 y Fo(\000)p Fs(1)418 2593 y Fr(a)p Fu(\()p Fr(h)542 2605 y Fq(t)571 2593 y Fu(\))28 b(is)f(con)n(tin)n(uous,)g(w)n(e)g(obtain)g @beginspecial @setspecial @endspecial 1 w(^)-43 b Fr(a)1577 2563 y Fs(+)1632 2593 y Fu(\()p Fr(h)p Fu(\))p Fr(v)27 b Fu(=)c(0)k(for)g(all) g Fr(h)c Fp(2)h Fm(h)p Fu(.)37 b(This)27 b(ends)h(the)g(pro)r(of.)p @beginspecial @setspecial @endspecial 637 w Fh(\004)0 2614 y @beginspecial @setspecial @endspecial 133 x Fz(Corollary)33 b(8.2.)40 b Fn(L)l(et)32 b Fp(f)p Fr(q)841 2759 y Fq(n)886 2747 y Fp(g)26 b(2)i Fr(C)1102 2717 y Fo(1)1096 2768 y Fs(0)1173 2747 y Fu(\()p Fk(R)1259 2717 y Fs(3)1302 2747 y Fu(\))33 b Fn(b)l(e)f(a)g(de)l(cr)l(e)l(asing)i(se)l(quenc)l(e)d (of)i(functions)f(such)h(that)f @beginspecial @setspecial @endspecial Fu(0)27 b Fp(\024)f Fr(q)3307 2759 y Fq(n)3380 2747 y Fp(\024)h Fu(1)p Fr(;)41 b(q)30 b Fu(=)d(1)32 b Fn(on)0 2847 y(a)i(neighb)l(orho)l(o)l(d)i(of)e @beginspecial @setspecial @endspecial 14 w Fu(0)f Fn(and)h Fp(\\)983 2812 y Fo(1)983 2867 y Fq(n)p Fs(=1)1112 2847 y Fu(supp)14 b Fr(q)1334 2859 y Fq(n)1409 2847 y Fu(=)30 b Fp(f)p Fu(0)p Fp(g)p Fn(.)48 b(Then)34 b(the)g(fol)t(lowing)i(limit)e(exist)f (and)h(it)f(do)l(es)i(not)e(dep)l(end)h(in)g(the)0 2946 y(cho)l(ose)d(of)g(the)f(se)l(quenc)l(e)1465 3088 y @beginspecial @setspecial @endspecial 170 w Fu(_)-27 b Fr({)p Fu(\))14 b Fr(P)1769 3053 y Fs(+)1757 3111 y(0)1847 3088 y Fu(:=)51 b(lim)1957 3138 y Fq(n)p Fo(!1)2145 3088 y Fu(\000)2197 3054 y Fs(+)2252 3088 y Fu(\()p Fr(q)2321 3100 y Fq(n)2366 3088 y Fu(\))p Fr(;)0 3172 y @beginspecial @setspecial @endspecial 1548 3301 a @beginspecial @setspecial @endspecial 87 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))14 b(Ran)p Fr(P)1944 3265 y Fs(+)1932 3323 y(0)2022 3301 y Fp(\032)2126 3280 y Fu(^)2110 3301 y Fp(K)2174 3267 y Fs(+)2229 3301 y Fr(:)0 3445 y Fn(Mor)l(e)l(over)31 b @beginspecial @setspecial @endspecial Fr(P)430 3410 y Fs(+)418 3468 y(0)515 3445 y Fn(is)f(an)g(ortho)l(gonal)h(pr)l(oje)l(ction.)0 3468 y @beginspecial @setspecial @endspecial 100 3599 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 23 w Fu(The)26 b(existence)g(of)h(the)f(limit)32 b(_)-28 b Fr({)p Fu(\))27 b(follo)n(ws)e(from)h(Thm.)37 b @beginspecial @setspecial @endspecial(8.1)29 b(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))27 b(and)f(Lemma)g(A.5.)36 b(The)26 b(indep)r(endence)h (from)0 3699 y(the)34 b(c)n(ho)r(ose)d(of)j(the)f(sequence)g(follo)n (ws)f(from)h(the)g(fact)h(that)f(there)g(exists)g(an)g(index)g @beginspecial @setspecial @endspecial Fr(m)2875 3711 y Fq(n)2953 3699 y Fu(suc)n(h)g(that)h Fr(q)3369 3711 y Fq(n)3446 3699 y Fp(\025)k Fu(~)-48 b Fr(q)3580 3711 y Fq(m)3639 3719 y Fj(n)3684 3699 y Fr(;)20 b Fu(~)-48 b Fr(q)3758 3711 y Fq(n)3835 3699 y Fp(\025)0 3798 y Fr(q)37 3810 y Fq(m)96 3818 y Fj(n)141 3798 y Fu(;)14 b(lim)293 3810 y Fq(n)p Fo(!1)484 3798 y Fr(m)557 3810 y Fq(n)626 3798 y Fu(=)22 b(+)p Fp(1)28 b Fu(and)f(Thm.)37 b @beginspecial @setspecial @endspecial(8.1)31 b(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\).)38 b @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)f Fr({)p Fu(\))28 b(is)g(a)f(consequence)f(of)i(Thm.)37 b @beginspecial @setspecial @endspecial(8.1)31 b(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))28 b(and)k(_)-27 b Fr({)p Fu(\).)688 b Fh(\004)100 3998 y Fu(Let)29 b @beginspecial @setspecial @endspecial Fr(j)284 4010 y Fs(0)347 3998 y Fp(2)d Fr(C)493 3968 y Fo(1)487 4018 y Fs(0)564 3998 y Fu(\()p Fk(R)650 3968 y Fs(3)693 3998 y Fu(\))p Fr(;)14 b Fu(0)26 b Fp(\024)f Fr(j)954 4010 y Fs(0)992 3998 y Fr(;)14 b Fu(0)25 b Fp(\024)g Fr(j)1220 4010 y Fo(1)1291 3998 y Fr(;)14 b(j)1367 3968 y Fs(2)1362 4018 y(0)1423 3998 y Fu(+)20 b Fr(j)1547 3968 y Fs(2)1542 4018 y Fo(1)1638 3998 y Fp(\024)25 b Fu(1)p Fr(;)14 b(j)1841 4010 y Fs(0)1904 3998 y Fu(=)25 b(1)k(near)f @beginspecial @setspecial @endspecial(0.)42 b(Set)29 b @beginspecial @setspecial @endspecial Fr(j)i Fu(:=)26 b(\()p Fr(j)2747 4010 y Fs(0)2784 3998 y Fr(;)14 b(j)2855 4010 y Fo(1)2926 3998 y Fu(\))29 b(and)g @beginspecial @setspecial @endspecial Fr(j)3189 3968 y Fq(t)3244 3998 y Fu(:=)d(\()p Fr(j)3429 3968 y Fq(t)3424 4018 y Fs(0)3462 3998 y Fr(;)14 b(j)3538 3968 y Fq(t)3533 4018 y Fo(1)3603 3998 y Fu(\))p Fr(;)30 b Fu(where)0 4097 y @beginspecial @setspecial @endspecial Fr(j)39 4067 y Fq(t)34 4118 y Fs(0)94 4097 y Fu(:=)23 b Fr(j)239 4109 y Fs(0)277 4097 y Fu(\()319 4065 y Fq(x)p 319 4079 38 4 v 325 4126 a(t)366 4097 y Fu(\))p Fr(;)14 b(j)474 4067 y Fq(t)469 4118 y Fo(1)563 4097 y Fu(:=)23 b Fr(j)708 4109 y Fo(1)778 4097 y Fu(\()820 4065 y Fq(x)p 821 4079 V 827 4126 a(t)868 4097 y Fu(\).)38 b(W)-7 b(e)28 b(recall)e(that)i @beginspecial @setspecial @endspecial Fr(I)7 b Fu(\()p Fr(j)1619 4067 y Fq(t)1649 4097 y Fu(\))28 b(is)f(the)h(op)r(erator)e(in)n(tro)r(duced)h(in)h(Subsect.)37 b @beginspecial @setspecial @endspecial(2.1.)0 4126 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 125 x Fz(Theorem)30 b(8.3.)40 b @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))30 b Fn(The)h(fol)t(lowing)h(limits)e(exist)1179 4515 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1220 4412 a Fr(s)19 b Fp(\000)64 b Fu(lim)1361 4462 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1583 4412 y Fu(e)1620 4378 y Fs(i)p Fq(t)1680 4363 y Fs(^)1664 4378 y Fq(H)1722 4353 y Fi(ext)1718 4394 y Fg(1)1808 4412 y Fr(I)1851 4378 y Fo(\003)1890 4412 y Fu(\()p Fr(j)1961 4378 y Fq(t)1990 4412 y Fu(\)e)2059 4378 y Fo(\000)p Fs(i)p Fq(t)2171 4363 y Fs(^)2155 4378 y Fq(H)2209 4386 y Fg(1)2297 4412 y Fu(=:)23 b Fr(W)2498 4378 y Fs(+)2553 4412 y Fu(\()p Fr(j)5 b Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1220 4601 a(s)19 b Fp(\000)64 b Fu(lim)1361 4651 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1583 4601 y Fu(e)1620 4567 y Fs(i)p Fq(t)1680 4552 y Fs(^)1664 4567 y Fq(H)1718 4575 y Fg(1)1783 4601 y Fr(I)7 b Fu(\()p Fr(j)1897 4567 y Fq(t)1926 4601 y Fu(\)e)1995 4567 y Fo(\000)p Fs(i)p Fq(t)2108 4552 y Fs(^)2091 4567 y Fq(H)2149 4542 y Fi(ext)2145 4584 y Fg(1)2259 4601 y Fu(=)23 b Fr(W)2437 4567 y Fs(+)2492 4601 y Fu(\()p Fr(j)5 b Fu(\))2595 4567 y Fo(\003)2633 4601 y Fr(:)2721 4515 y @beginspecial @setspecial @endspecial 0 4779 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))30 b Fn(F)-6 b(or)30 b(a)g(b)l(ounde)l(d)g (Bor)l(el)h(function)e(F,)i(we)f(have)1339 4930 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1429 4895 y Fs(+)1484 4930 y Fu(\()p Fr(j)5 b Fu(\))23 b Fr(F)12 b Fu(\()1729 4909 y(^)1707 4930 y Fr(H)1776 4942 y Fo(1)1847 4930 y Fu(\))23 b(=)g Fr(F)12 b Fu(\()2109 4909 y(^)2087 4930 y Fr(H)2163 4895 y Fs(ext)2156 4950 y Fo(1)2257 4930 y Fu(\))p Fr(W)2379 4895 y Fs(+)2435 4930 y Fu(\()p Fr(j)5 b Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 5076 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))27 b Fn(L)l(et)e Fr(q)315 5088 y Fs(0)352 5076 y Fr(;)14 b(q)426 5088 y Fo(1)520 5076 y Fp(2)23 b Fr(C)663 5046 y Fo(1)734 5076 y Fu(\()p Fk(R)820 5046 y Fs(3)863 5076 y Fu(\))p Fr(;)28 b Fp(r)p Fr(q)1052 5088 y Fs(0)1090 5076 y Fr(;)f Fp(r)p Fr(q)1246 5088 y Fo(1)1340 5076 y Fp(2)d Fr(C)1484 5046 y Fo(1)1478 5096 y Fs(0)1554 5076 y Fu(\()p Fk(R)1640 5046 y Fs(3)1684 5076 y Fu(\))p Fr(;)k Fu(0)22 b Fp(\024)h Fr(q)1956 5088 y Fs(0)1993 5076 y Fr(;)14 b(q)2067 5088 y Fo(1)2161 5076 y Fp(\024)22 b Fu(1)p Fr(;)27 b(q)2377 5088 y Fs(0)2438 5076 y Fu(=)22 b(1)k Fn(ne)l(ar)g @beginspecial @setspecial @endspecial Fu(0)p Fn(.)37 b(Set)3018 5057 y Fu(~)3019 5076 y Fr(j)28 b Fu(:=)23 b(\()p Fr(q)3261 5088 y Fs(0)3298 5076 y Fr(j)3332 5088 y Fs(0)3370 5076 y Fr(;)14 b(q)3444 5088 y Fo(1)3514 5076 y Fr(j)3548 5088 y Fo(1)3619 5076 y Fu(\))p Fn(.)37 b(Then)1200 5222 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\000)1252 5187 y Fs(+)1307 5222 y Fu(\()p Fr(q)1376 5234 y Fs(0)1413 5222 y Fu(\))19 b Fp(\012)f Fu(\000\()p Fr(q)1668 5234 y Fo(1)1738 5222 y Fu(\()p Fp(r)p Fr(!)s Fu(\()p Fr(k)s Fu(\)\)\))p Fr(W)2158 5187 y Fs(+)2215 5222 y Fu(\()p Fr(j)5 b Fu(\))23 b(=)g Fr(W)2519 5187 y Fs(+)2574 5222 y Fu(\()2605 5203 y(~)2606 5222 y Fr(j)5 b Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 0 5364 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({v)s Fu(\))30 b Fn(L)l(et)f Fr(q)d Fp(2)e Fr(C)482 5334 y Fo(1)552 5364 y Fu(\()p Fk(R)639 5334 y Fs(3)682 5364 y Fu(\))p Fr(;)k Fp(r)p Fr(q)e Fp(2)e Fr(C)1041 5334 y Fo(1)1035 5384 y Fs(0)1111 5364 y Fu(\()p Fk(R)1197 5334 y Fs(3)1241 5364 y Fu(\))p Fr(;)k Fu(0)22 b Fp(\024)h Fr(q)j Fp(\024)d Fu(1)p Fr(;)k(q)f Fu(=)d(1)29 b Fn(ne)l(ar)h @beginspecial @setspecial @endspecial Fu(0)p Fn(.)38 b(Then)1076 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1166 5472 y Fs(+)1221 5506 y Fu(\()p Fr(j)5 b Fu(\)\000)1376 5472 y Fs(+)1431 5506 y Fu(\()p Fr(q)s Fu(\))24 b(=)f Fr(W)1737 5472 y Fs(+)1792 5506 y Fu(\()p Fr(q)s(j)5 b Fu(\))p Fr(;)60 b Fu(where)23 b Fr(q)s(j)28 b Fu(=)23 b(\()p Fr(q)s(j)2550 5518 y Fs(0)2587 5506 y Fr(;)14 b(q)s(j)2698 5518 y Fo(1)2769 5506 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial eop %%Page: 42 42 42 41 bop 0 226 a Fs(42)1695 b(Z.)29 b(AMMARI)0 489 y @beginspecial @setspecial @endspecial Fr(v)s Fu(\))h Fn(L)l(et)247 470 y Fu(~)249 489 y Fr(j)e Fu(=)22 b(\()429 470 y(~)430 489 y Fr(j)464 501 y Fs(0)502 489 y Fr(;)538 470 y Fu(~)539 489 y Fr(j)573 501 y Fo(1)643 489 y Fu(\))30 b Fn(b)l(e)g(another)h(p)l(air)f(satisfying)i(the)d(c)l(onditions)i (state)l(d)f(b)l(efor)l(e)h(the)e(the)l(or)l(em.)39 b(Then)1331 628 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(W)1421 593 y Fs(+)1476 628 y Fu(\()1507 609 y(~)1508 628 y Fr(j)5 b Fu(\))1579 593 y Fo(\003)1617 628 y Fr(W)1707 593 y Fs(+)1762 628 y Fu(\()p Fr(j)g Fu(\))24 b(=)f(\000\()2060 609 y(~)2061 628 y Fr(j)2095 640 y Fs(0)2132 628 y Fr(j)2166 640 y Fs(0)2222 628 y Fu(+)2304 609 y(~)2305 628 y Fr(j)2339 640 y Fo(1)2410 628 y Fr(j)2444 640 y Fo(1)2514 628 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 0 762 a Fn(in)30 b(p)l(articular)h(if)f @beginspecial @setspecial @endspecial Fr(j)601 732 y Fs(2)596 783 y(0)657 762 y Fu(+)18 b Fr(j)779 732 y Fs(2)774 783 y Fo(1)867 762 y Fu(=)23 b(1)p Fr(;)29 b Fn(then)h Fr(W)1324 732 y Fs(+)1379 762 y Fu(\()p Fr(j)5 b Fu(\))30 b Fn(is)g(isometric.)0 862 y @beginspecial @setspecial @endspecial Fr(v)7 b Fu(_)-27 b Fr({)p Fu(\))30 b Fn(L)l(et)f Fr(j)309 874 y Fs(0)365 862 y Fu(+)18 b Fr(j)482 874 y Fo(1)576 862 y Fu(=)k(1)p Fn(.)39 b(If)30 b Fr(\037)23 b Fp(2)g Fr(C)1074 832 y Fo(1)1068 883 y Fs(0)1145 862 y Fu(\()p Fk(R)p Fu(\))p Fn(,)36 b(then)1375 1005 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1384 984 a Fu(^)1375 1005 y(\012)1435 970 y Fs(ext)p Fq(;)p Fs(+)1600 1005 y Fr(\037)p Fu(\()1706 984 y(^)1684 1005 y Fr(H)1760 970 y Fs(ext)1753 1025 y Fo(1)1855 1005 y Fu(\))p Fr(W)1977 970 y Fs(+)2032 1005 y Fu(\()p Fr(j)5 b Fu(\))23 b(=)g Fr(\037)p Fu(\()2352 984 y(^)2330 1005 y Fr(H)2399 1017 y Fo(1)2470 1005 y Fu(\))p Fr(:)p @beginspecial @setspecial @endspecial 100 1154 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 23 w Fu(T)-7 b(o)25 b(pro)n(v)n(e)j(_)-27 b Fr({)p Fu(\))26 b(w)n(e)f(use)h(the)g(same)f (argumen)n(ts)f(as)h(in)h(Thm.)36 b @beginspecial @setspecial @endspecial(8.1.)g(Using)25 b(Lemma)h(A.4,)g(it)g(is)f(enough)g(to)h (pro)n(v)n(e)0 1254 y(the)i(existence)f(of)h(the)g(limit)1057 1389 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(s)19 b Fp(\000)64 b Fu(lim)1198 1438 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1420 1389 y Fu(e)1457 1354 y Fs(i)p Fq(t)1517 1339 y Fs(^)1501 1354 y Fq(H)1559 1329 y Fi(ext)1555 1371 y Fg(1)1645 1389 y Fr(\037)p Fu(\()1751 1368 y(^)1729 1389 y Fr(H)1805 1354 y Fs(ext)1798 1409 y Fo(1)1900 1389 y Fu(\))p Fr(I)1975 1354 y Fo(\003)2013 1389 y Fu(\()p Fr(j)2084 1354 y Fq(t)2114 1389 y Fu(\)e)2183 1354 y Fo(\000)p Fs(i)p Fq(t)2295 1339 y Fs(^)2279 1354 y Fq(H)2333 1362 y Fg(1)2398 1389 y Fr(\037)p Fu(\()2504 1368 y(^)2482 1389 y Fr(H)2551 1401 y Fo(1)2621 1389 y Fu(\))p Fr(;)84 b @beginspecial @setspecial @endspecial -2760 w Fu(\(8.5\))0 1559 y(for)27 b(some)g @beginspecial @setspecial @endspecial Fr(\037)c Fp(2)h Fr(C)554 1529 y Fo(1)548 1580 y Fs(0)624 1559 y Fu(\()p Fk(R)p Fu(\))q(.)43 b(W)-7 b(e)28 b(compute)278 1713 y @beginspecial @setspecial @endspecial Fr(@)322 1725 y Fq(t)351 1713 y Fu(\(e)420 1678 y Fs(i)p Fq(t)480 1663 y Fs(^)464 1678 y Fq(H)522 1653 y Fi(ext)518 1695 y Fj(\024)609 1713 y Fr(\037)p Fu(\()715 1692 y(^)693 1713 y Fr(H)769 1678 y Fs(ext)762 1733 y Fq(\024)863 1713 y Fu(\))p Fr(I)938 1678 y Fo(\003)977 1713 y Fu(\()p Fr(j)1048 1678 y Fq(t)1077 1713 y Fu(\))p Fr(\037)p Fu(\()1215 1692 y(^)1193 1713 y Fr(H)1262 1725 y Fq(\024)1306 1713 y Fu(\)e)1375 1678 y Fo(\000)p Fs(i)p Fq(t)1487 1663 y Fs(^)1471 1678 y Fq(H)1525 1686 y Fj(\024)1568 1713 y Fu(\))83 b @beginspecial @setspecial @endspecial(=)g(e)1868 1678 y Fs(i)p Fq(t)1928 1663 y Fs(^)1912 1678 y Fq(H)1970 1653 y Fi(ext)1966 1695 y Fj(\024)2056 1713 y Fu(\()p Fr(\037)p Fu(\()2194 1692 y(^)2172 1713 y Fr(H)2248 1678 y Fs(ext)2241 1733 y Fq(\024)2343 1713 y Fu(\))p Fm(D)2444 1725 y Fs(0)2481 1713 y Fr(I)2524 1678 y Fo(\003)2563 1713 y Fu(\()p Fr(j)2634 1678 y Fq(t)2663 1713 y Fu(\))p Fr(\037)p Fu(\()2801 1692 y(^)2779 1713 y Fr(H)2848 1725 y Fq(\024)2892 1713 y Fu(\))0 1758 y @beginspecial @setspecial @endspecial 1715 1891 a @beginspecial @setspecial @endspecial 116 w(+i)p Fr(\037)p Fu(\()2024 1870 y(^)2003 1891 y Fr(H)2079 1857 y Fs(ext)2072 1912 y Fq(\024)2173 1891 y Fu(\)\()2247 1870 y(^)2237 1891 y Fr(I)2273 1903 y Fq(\024)2335 1891 y Fp(\012)18 b Fu(1)-15 b(l)p Fr(I)2511 1857 y Fo(\003)2549 1891 y Fu(\()p Fr(j)2620 1857 y Fq(t)2649 1891 y Fu(\))19 b Fp(\000)f Fr(I)2826 1857 y Fo(\003)2864 1891 y Fu(\()p Fr(j)2935 1857 y Fq(t)2965 1891 y Fu(\))3007 1870 y(^)2997 1891 y Fr(I)3033 1903 y Fq(\024)3076 1891 y Fu(\))p Fr(\037)p Fu(\()3215 1870 y(^)3192 1891 y Fr(H)3261 1903 y Fq(\024)3305 1891 y Fu(\)\)e)3406 1857 y Fo(\000)p Fs(i)p Fq(t)3519 1842 y Fs(^)3502 1857 y Fq(H)3556 1865 y Fj(\024)3599 1891 y Fr(;)0 2026 y Fu(where)27 b @beginspecial @setspecial @endspecial Fm(D)309 2038 y Fs(0)374 2026 y Fu(is)h(the)f(asymmetric)g(Heisen)n(b)r(erg)g(deriv)-5 b(ativ)n(e)27 b @beginspecial @setspecial @endspecial Fr(@)1894 2038 y Fq(t)1942 2026 y Fu(+)18 b(i)p Fr(H)2124 1996 y Fs(ext)2117 2046 y(0)2218 2026 y Fr(:)g Fp(\000)g Fr(:)p Fu(i)p Fr(H)2457 2038 y Fs(0)2495 2026 y Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e Fm(D)2958 2038 y Fs(0)2995 2026 y Fr(I)3038 1996 y Fo(\003)3077 2026 y Fu(\()p Fr(j)3148 1996 y Fq(t)3177 2026 y Fu(\))d(=)g(d)p Fr(I)3409 1996 y Fo(\003)3447 2026 y Fu(\()p Fr(j)3518 1996 y Fq(t)3548 2026 y Fr(;)14 b Fm(d)3626 2038 y Fs(0)3663 2026 y Fr(j)3702 1996 y Fq(t)3731 2026 y Fu(\).)0 2125 y(Pseudo)r(di\013eren)n(tial)27 b(calculus)g(giv)n(es)1294 2295 y @beginspecial @setspecial @endspecial Fm(d)1335 2307 y Fs(0)1373 2295 y Fr(j)1412 2260 y Fq(t)1524 2295 y @beginspecial @setspecial @endspecial Fu(=)1682 2238 y(1)p 1682 2276 42 4 v 1688 2352 a Fr(t)1733 2295 y(g)1776 2260 y Fq(t)1824 2295 y Fu(+)18 b Fr(r)1946 2260 y Fq(t)1976 2295 y Fr(;)816 2530 y(g)859 2495 y Fq(t)911 2530 y Fu(=)k(\()p Fr(g)1073 2495 y Fq(t)1070 2550 y Fs(0)1107 2530 y Fr(;)14 b(g)1187 2495 y Fq(t)1184 2550 y Fo(1)1254 2530 y Fu(\))p Fr(;)60 b(g)1412 2495 y Fq(t)1409 2550 y(\017)1524 2530 y @beginspecial @setspecial @endspecial Fu(=)83 b Fp(\000)1747 2473 y Fu(1)p 1747 2510 V 1747 2586 a(2)1798 2530 y(\(\()1872 2473 y Fr(x)p 1872 2510 48 4 v 1881 2586 a(t)1948 2530 y Fp(\000)18 b Fr(@)5 b(!)s Fu(\()p Fr(k)s Fu(\)\))p Fr(@)g(j)2360 2542 y Fq(\017)2392 2530 y Fu(\()2434 2473 y Fr(x)p 2434 2510 V 2443 2586 a(t)2492 2530 y Fu(\))18 b(+)h Fr(hc)p Fu(\))p Fr(;)14 b(\017)22 b Fu(=)h(0)p Fr(;)14 b Fp(1)0 2697 y Fu(with)28 b @beginspecial @setspecial @endspecial Fr(r)228 2667 y Fq(t)281 2697 y Fp(2)c Fr(O)r Fu(\()p Fr(t)487 2667 y Fo(\000)p Fs(2)577 2697 y Fu(\).)37 b(Using)27 b(Corollary)e(4.8)i(and)h(\()p @beginspecial @setspecial @endspecial(2.4\))f(w)n(e)g(obtain)1193 2840 y Fp(jj)p Fr(\037)p Fu(\()1345 2819 y(^)1323 2840 y Fr(H)1399 2806 y Fs(ext)1392 2861 y Fq(\024)1494 2840 y Fu(\)d)p Fr(I)1615 2806 y Fo(\003)1653 2840 y Fu(\()p Fr(j)1724 2806 y Fq(t)1754 2840 y Fr(;)14 b(r)1830 2806 y Fq(t)1860 2840 y Fu(\))p Fr(\037)p Fu(\()1998 2819 y(^)1976 2840 y Fr(H)2045 2852 y Fq(\024)2088 2840 y Fu(\))p Fp(jj)24 b(2)f Fr(O)r Fu(\()p Fr(t)2395 2806 y Fo(\000)p Fs(2)2485 2840 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial -2624 w Fu(\(8.6\))0 2990 y(Using)27 b(no)n(w)g(\(2.5\))h(with)g @beginspecial @setspecial @endspecial Fr(u)842 2959 y Fq(t)842 3011 y(i)894 2990 y Fu(:=)22 b(e)1041 2959 y Fs(i)p Fq(t)1102 2944 y Fs(^)1085 2959 y Fq(H)1139 2967 y Fj(\024)1182 2990 y Fr(u)1230 3002 y Fq(i)1257 2990 y Fu(,)28 b(one)f(obtain)940 3278 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 981 3133 a @beginspecial @setspecial @endspecial Fp(j)p Fu(\()p Fr(u)1084 3103 y Fq(t)1084 3154 y Fs(1)1121 3133 y Fp(j)p Fr(\037)p Fu(\()1250 3112 y(^)1228 3133 y Fr(H)1304 3103 y Fs(ext)1297 3154 y Fq(\024)1399 3133 y Fu(\)d)p Fr(I)1520 3103 y Fo(\003)1559 3133 y Fu(\()p Fr(j)1630 3103 y Fq(t)1659 3133 y Fr(;)14 b(g)1739 3103 y Fq(t)1768 3133 y Fu(\))p Fr(\037)p Fu(\()1906 3112 y(^)1884 3133 y Fr(H)1953 3145 y Fq(\024)1996 3133 y Fu(\))p Fr(u)2076 3103 y Fq(t)2076 3154 y Fs(2)2113 3133 y Fu(\))p Fp(j)p @beginspecial @setspecial @endspecial 981 3281 a(\024)23 b(jj)p Fu(d\000\()p Fp(j)p Fr(g)1311 3247 y Fq(t)1308 3302 y Fs(0)1345 3281 y Fp(j)p Fu(\))1410 3225 y Fi(1)p 1411 3234 29 3 v 1411 3267 a(2)1471 3281 y Fp(\012)18 b Fu(1)-15 b(l)p Fr(\037)p Fu(\()1710 3260 y(^)1688 3281 y Fr(H)1764 3247 y Fs(ext)1757 3302 y Fq(\024)1858 3281 y Fu(\))p Fr(u)1938 3247 y Fq(t)1938 3302 y Fs(2)1975 3281 y Fp(jj)23 b(jj)p Fu(d\000\()p Fp(j)p Fr(g)2286 3247 y Fq(t)2283 3302 y Fs(0)2321 3281 y Fp(j)p Fu(\))2386 3225 y Fi(1)p 2386 3234 V 2386 3267 a(2)2428 3281 y Fr(\037)p Fu(\()2534 3260 y(^)2512 3281 y Fr(H)2581 3293 y Fq(\024)2625 3281 y Fu(\))p Fr(u)2705 3247 y Fq(t)2705 3302 y Fs(1)2742 3281 y Fp(jj)p @beginspecial @setspecial @endspecial 981 3429 a Fu(+)p Fp(jj)p Fu(\(1)-15 b(l)18 b Fp(\012)g Fu(d\000\()p Fp(j)p Fr(g)1471 3395 y Fq(t)1468 3450 y Fo(1)1538 3429 y Fp(j)p Fu(\))1603 3373 y Fi(1)p 1604 3382 V 1604 3415 a(2)1646 3429 y Fu(\))p Fr(\037)p Fu(\()1784 3408 y(^)1762 3429 y Fr(H)1838 3395 y Fs(ext)1831 3450 y Fq(\024)1933 3429 y Fu(\))p Fr(u)2013 3395 y Fq(t)2013 3450 y Fs(2)2050 3429 y Fp(jj)23 b(jj)p Fu(d\000\()p Fp(j)p Fr(g)2361 3395 y Fq(t)2358 3450 y Fo(1)2429 3429 y Fp(j)p Fu(\))2494 3373 y Fi(1)p 2494 3382 V 2494 3415 a(2)2536 3429 y Fr(\037)p Fu(\()2642 3408 y(^)2620 3429 y Fr(H)2689 3441 y Fq(\024)2732 3429 y Fu(\))p Fr(u)2812 3395 y Fq(t)2812 3450 y Fs(1)2850 3429 y Fp(jj)p Fr(:)2960 3278 y @beginspecial @setspecial @endspecial 0 3581 a Fu(Then)28 b(the)g @beginspecial @setspecial @endspecial Fr(\024)p Fp(\000)p Fu(uniform)f(in)n(tegrabilit)n(y)f(of)i(the)g(term) f Fr(\037)p Fu(\()1794 3560 y(^)1772 3581 y Fr(H)1848 3551 y Fs(ext)1841 3602 y Fq(\024)1943 3581 y Fu(\))p Fm(D)2044 3593 y Fs(0)2081 3581 y Fr(I)2124 3551 y Fo(\003)2162 3581 y Fu(\()p Fr(j)2233 3551 y Fq(t)2263 3581 y Fu(\))p Fr(\037)p Fu(\()2401 3560 y(^)2379 3581 y Fr(H)2448 3593 y Fq(\024)2491 3581 y Fu(\))h(follo)n(ws)f(using)g(Prop.)36 b @beginspecial @setspecial @endspecial(7.1.)0 3681 y(Using)27 b(\(4.25\))g(w)n(e)g(obtain)h(uniformly)f(in)h @beginspecial @setspecial @endspecial Fr(\024)924 3824 y(\037)p Fu(\()1030 3803 y(^)1008 3824 y Fr(H)1084 3789 y Fs(ext)1077 3844 y Fq(\024)1179 3824 y Fu(\)\()1253 3803 y(^)1243 3824 y Fr(I)1279 3836 y Fq(\024)1341 3824 y Fp(\012)18 b Fu(1)-15 b(l)p Fr(I)1517 3789 y Fo(\003)1555 3824 y Fu(\()p Fr(j)1626 3789 y Fq(t)1655 3824 y Fu(\))19 b Fp(\000)f Fr(I)1832 3789 y Fo(\003)1870 3824 y Fu(\()p Fr(j)1941 3789 y Fq(t)1971 3824 y Fu(\))2013 3803 y(^)2003 3824 y Fr(I)2039 3836 y Fq(\024)2083 3824 y Fu(\))p Fr(\037)p Fu(\()2221 3803 y(^)2199 3824 y Fr(H)2268 3836 y Fq(\024)2311 3824 y Fu(\))24 b Fp(2)f Fr(O)r Fu(\()p Fr(t)2572 3789 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\026)2754 3824 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 3958 a Fu(Then)28 b(the)g(existence)f(of)h(the)g(limit)g(in)k(_)-27 b Fr({)p Fu(\))28 b(follo)n(ws.)0 4058 y @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(follo)n(ws)e(b)n(y)h(Lemma)h(4.15.)39 b(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\))28 b(follo)n(ws)e(using)h(the)h(fact) g(that)1132 4200 y @beginspecial @setspecial @endspecial 47 w(lim)1132 4250 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1355 4200 y Fu(e)1392 4166 y Fs(i)p Fq(t)p Fs(d\000\()p Fq(!)r Fs(\))1613 4200 y Fu(\000\()p Fr(q)1737 4166 y Fq(t)1766 4200 y Fu(\)e)1835 4166 y Fo(\000)p Fs(i)p Fq(t)p Fs(d\000\()p Fq(!)r Fs(\))2132 4200 y Fu(=)23 b(\000\()p Fr(q)s Fu(\()p Fp(r)p Fr(!)s Fu(\)\))p Fr(;)98 b @beginspecial @setspecial @endspecial 1524 4408 a Fu(\000\()p Fr(q)1648 4374 y Fq(t)1645 4428 y Fs(0)1682 4408 y Fu(\))19 b Fp(\012)f Fu(\000\()p Fr(q)1940 4374 y Fq(t)1937 4428 y Fo(1)2008 4408 y Fu(\))p Fr(I)2083 4374 y Fo(\003)2121 4408 y Fu(\()p Fr(j)2192 4374 y Fq(t)2222 4408 y Fu(\))23 b(=)g Fr(I)2408 4374 y Fo(\003)2446 4408 y Fu(\()2477 4389 y(~)2478 4408 y Fr(j)2517 4374 y Fq(t)2546 4408 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 4 4543 a Fu(_)-27 b Fr({v)s Fu(\))28 b(is)f(true)h(since)1441 4677 y @beginspecial @setspecial @endspecial Fr(I)1484 4643 y Fo(\003)1522 4677 y Fu(\()p Fr(j)1593 4643 y Fq(t)1623 4677 y Fu(\)\000\()p Fr(q)1779 4643 y Fq(t)1808 4677 y Fu(\))c(=)e Fr(I)1994 4643 y Fo(\003)2033 4677 y Fu(\(\()p Fr(j)5 b(q)s Fu(\))2208 4643 y Fq(t)2238 4677 y Fu(\))p Fr(:)0 4812 y @beginspecial @setspecial @endspecial(v)s Fu(\))28 b(is)g(a)f(consequence)f(of)i(the) g(fact)1376 4951 y Fr(I)7 b Fu(\()1450 4932 y(~)1451 4951 y Fr(j)1490 4916 y Fq(t)1519 4951 y Fu(\))p Fr(I)1594 4916 y Fo(\003)1633 4951 y Fu(\()p Fr(j)1704 4916 y Fq(t)1733 4951 y Fu(\)\000\()1848 4932 y(~)1849 4951 y Fr(j)1888 4916 y Fq(t)1883 4971 y Fs(0)1921 4951 y Fr(j)1960 4916 y Fq(t)1955 4971 y Fs(0)2011 4951 y Fu(+)2093 4932 y(~)2094 4951 y Fr(j)2133 4916 y Fq(t)2128 4971 y Fo(1)2198 4951 y Fr(j)2237 4916 y Fq(t)2232 4971 y Fo(1)2303 4951 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 5085 a(v)7 b Fu(_)-27 b Fr({)p Fu(\))28 b(One)f(has)1370 5207 y(^)1348 5228 y Fr(H)1424 5194 y Fs(ext)1417 5249 y Fo(1)1518 5228 y Fu(1)-15 b(l)1568 5243 y Fs([)p Fq(k)q(;)p Fo(1)p Fs([)1732 5228 y Fu(\()p Fr(N)1831 5240 y Fo(1)1901 5228 y Fu(\))24 b Fp(\025)e Fr(mk)f Fu(+)d Fr(E)2325 5240 y Fs(0)2363 5228 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 5363 a Fu(Let)28 b Fr(\037)23 b Fp(2)g Fr(C)367 5333 y Fo(1)361 5384 y Fs(0)438 5363 y Fu(\()p Fk(R)p Fu(\).)43 b(There)27 b(exists)h @beginspecial @setspecial @endspecial Fr(n)1142 5375 y Fs(0)1202 5363 y Fp(2)23 b Fk(N)38 b Fu(suc)n(h)27 b(that)h(for)f Fr(n)c Fp(\025)f Fr(n)2072 5375 y Fs(0)1426 5506 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(\037)p Fu(\()1532 5485 y(^)1510 5506 y Fr(H)1586 5472 y Fs(ext)1579 5526 y Fo(1)1681 5506 y Fu(\)1)-15 b(l)1762 5521 y Fs(])p Fq(n;)p Fo(1)p Fs([)1931 5506 y Fu(\()p Fr(N)2030 5518 y Fo(1)2100 5506 y Fu(\))24 b(=)e(0)p Fr(:)83 b @beginspecial @setspecial @endspecial -2391 w Fu(\(8.7\))p eop %%Page: 43 43 43 42 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(43)0 489 y Fu(W)-7 b(e)28 b(ha)n(v)n(e)617 611 y(^)608 632 y(\012)668 598 y Fs(ext)p Fq(;)p Fs(+)833 632 y Fr(\037)p Fu(\()939 611 y(^)917 632 y Fr(H)993 598 y Fs(ext)986 653 y Fo(1)1087 632 y Fu(\))p Fr(W)1209 598 y Fs(+)1265 632 y Fu(\()p Fr(j)5 b Fu(\))83 b @beginspecial @setspecial @endspecial(=)1608 611 y(^)1599 632 y(\012)1659 598 y Fs(ext)p Fq(;)p Fs(+)1824 632 y Fu(1)-15 b(l)1873 647 y Fs([0)p Fq(;n)p Fs(])2009 632 y Fu(\()p Fr(N)2108 644 y Fo(1)2178 632 y Fu(\))p Fr(\037)p Fu(\()2316 611 y(^)2294 632 y Fr(H)2370 598 y Fs(ext)2363 653 y Fo(1)2465 632 y Fu(\))p Fr(W)2587 598 y Fs(+)2642 632 y Fu(\()p Fr(j)5 b Fu(\))-2745 b(\(8.8\))0 687 y @beginspecial @setspecial @endspecial 1368 828 a @beginspecial @setspecial @endspecial 83 w(=)83 b Fr(s)18 b Fp(\000)39 b Fu(lim)1739 878 y Fq(t)p Fo(!1)1910 828 y Fu(e)1947 794 y Fs(i)p Fq(t)2008 779 y Fs(^)1991 794 y Fq(H)2045 802 y Fg(1)2110 828 y Fr(I)7 b Fu(1)-15 b(l)2203 843 y Fs([0)p Fq(;n)p Fs(])2338 828 y Fu(\()p Fr(N)2437 840 y Fo(1)2508 828 y Fu(\))p Fr(\037)p Fu(\()2646 807 y(^)2624 828 y Fr(H)2700 794 y Fs(ext)2693 849 y Fo(1)2794 828 y Fu(\))p Fr(I)2869 794 y Fo(\003)2908 828 y Fu(\()p Fr(j)2979 794 y Fq(t)3008 828 y Fu(\)e)3077 794 y Fo(\000)p Fs(i)p Fq(t)3190 779 y Fs(^)3173 794 y Fq(H)3227 802 y Fg(1)0 828 y Fu(\(8.9\))0 911 y @beginspecial @setspecial @endspecial 1451 1052 a(=)83 b Fr(s)18 b Fp(\000)39 b Fu(lim)1739 1102 y Fq(t)p Fo(!1)1910 1052 y Fu(e)1947 1018 y Fs(i)p Fq(t)2008 1003 y Fs(^)1991 1018 y Fq(H)2045 1026 y Fg(1)2110 1052 y Fr(I)7 b Fu(1)-15 b(l)2203 1067 y Fs([0)p Fq(;n)p Fs(])2338 1052 y Fu(\()p Fr(N)2437 1064 y Fo(1)2508 1052 y Fu(\))p Fr(I)2583 1018 y Fo(\003)2621 1052 y Fu(\()p Fr(j)2692 1018 y Fq(t)2722 1052 y Fu(\)e)2791 1018 y Fo(\000)p Fs(i)p Fq(t)2903 1003 y Fs(^)2887 1018 y Fq(H)2941 1026 y Fg(1)3006 1052 y Fr(\037)p Fu(\()3112 1031 y(^)3090 1052 y Fr(H)3159 1064 y Fo(1)3229 1052 y Fu(\))-3261 b(\(8.10\))0 1225 y(\()p @beginspecial @setspecial @endspecial(8.8\))28 b(follo)n(ws)f(from)h(\(8.7\).)39 b(\()p @beginspecial @setspecial @endspecial(8.9\))29 b(follo)n(ws)e(from)h(the)h(limit)k(_) -27 b Fr({)p Fu(\))28 b(and)g(Thm.)40 b @beginspecial @setspecial @endspecial(6.10.)e(Lemma)28 b(4.15)f(and)h(the)h(b)r (oundeness)f(of)0 1324 y(the)g(op)r(erator)e @beginspecial @setspecial @endspecial Fr(I)7 b Fu(1)-15 b(l)570 1339 y Fs([0)p Fq(;n)p Fs(])706 1324 y Fu(\()p Fr(N)805 1336 y Fo(1)875 1324 y Fu(\)\()p Fr(N)1006 1336 y Fs(0)1062 1324 y Fu(+)18 b(1\))1219 1294 y Fo(\000)1281 1272 y Fj(n)p 1281 1281 37 3 v 1285 1314 a Fi(2)1359 1324 y Fu(giv)n(es)27 b(\()p @beginspecial @setspecial @endspecial(8.10\).)36 b(W)-7 b(e)28 b(use)f(no)n(w)g(an)g(estimate)h(pro)n(v)n(ed)e(in)i([12) o(]:)1056 1478 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(jj)p Fr(I)7 b Fu(1)-15 b(l)1195 1493 y Fs(])p Fq(n;)p Fo(1)p Fs([)1363 1478 y Fu(\()p Fr(N)1462 1490 y Fo(1)1533 1478 y Fu(\))p Fr(I)1608 1443 y Fo(\003)1646 1478 y Fu(\()p Fr(j)1717 1443 y Fq(t)1747 1478 y Fu(\)\()p Fr(N)27 b Fu(+)18 b(1\))2062 1443 y Fo(\000)p Fs(1)2151 1478 y Fp(jj)24 b(\024)e Fu(\()p Fr(n)d Fu(+)f(1\))2566 1443 y Fo(\000)p Fs(1)2655 1478 y Fr(:)83 b @beginspecial @setspecial @endspecial -2761 w Fu(\(8.11\))0 1638 y(Since)28 b Fr(I)7 b(I)303 1608 y Fo(\003)341 1638 y Fu(\()p Fr(j)412 1608 y Fq(t)441 1638 y Fu(\))24 b(=)e(1)-15 b(l,)27 b(letting)h Fr(n)23 b Fp(!)g(1)28 b Fu(w)n(e)f(obtain)h @beginspecial @setspecial @endspecial 1627 1617 a(^)1618 1638 y(\012)1678 1608 y Fs(ext)p Fq(;)p Fs(+)1843 1638 y Fr(\037)p Fu(\()1949 1617 y(^)1927 1638 y Fr(H)2003 1608 y Fs(ext)1996 1658 y Fo(1)2097 1638 y Fu(\))p Fr(W)2219 1608 y Fs(+)2275 1638 y Fu(\()p Fr(j)5 b Fu(\))23 b(=)g Fr(\037)p Fu(\()2595 1617 y(^)2573 1638 y Fr(H)2642 1650 y Fo(1)2712 1638 y Fu(\).)38 b(This)27 b(completes)g(the)h(pro)r(of.)p @beginspecial @setspecial @endspecial 102 w Fh(\004)0 1658 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 134 x Fz(Theorem)i(8.4.)40 b Fn(L)l(et)29 b @beginspecial @setspecial @endspecial Fr(j)772 1804 y Fq(n)841 1792 y Fu(=)22 b(\()p Fr(j)994 1804 y Fs(0)p Fq(;n)1093 1792 y Fr(;)14 b(j)1164 1804 y Fo(1)p Fq(;n)1295 1792 y Fu(\))30 b Fn(b)l(e)f(a)h(se)l(quenc)l(e)f (satisfying)h(the)g(hyp)l(othesis)h(state)l(d)e(in)h(the)f(b)l(e)l (ginning)h(of)g(Thm.)0 1892 y @beginspecial @setspecial @endspecial(8.3)h(such)f(that)g Fr(j)533 1904 y Fs(0)588 1892 y Fu(+)18 b Fr(j)705 1904 y Fo(1)799 1892 y Fu(=)23 b(1)29 b Fn(and)h(for)h(any)f @beginspecial @setspecial @endspecial Fr(\017)23 b(>)f Fu(0)30 b Fn(ther)l(e)f(exists)h Fr(m;)14 b Fp(8)p Fr(n)22 b(>)g(m;)37 b Fu(supp)p Fr(j)2715 1904 y Fs(0)p Fq(;n)2836 1892 y Fp(\032)23 b Fu([)p Fp(\000)p Fr(\017;)14 b(\017)p Fu(])p Fn(.)38 b(Then)1448 2128 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1490 2046 a @beginspecial @setspecial @endspecial 1499 2025 a Fu(^)1490 2046 y(\012)1550 2012 y Fs(+)p Fo(\003)1662 2046 y Fu(=)23 b Fr(w)e Fp(\000)46 b Fu(lim)1913 2096 y Fq(\024)p Fo(!1)2098 2046 y Fr(W)2188 2012 y Fs(+)2243 2046 y Fu(\()p Fr(j)2309 2058 y Fq(n)2355 2046 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1506 2197 a Fu(^)1490 2218 y Fp(K)1554 2183 y Fs(+)1633 2218 y Fu(=)22 b(Ran)p Fr(P)1934 2182 y Fs(+)1922 2240 y(0)1989 2218 y Fr(;)2452 2128 y @beginspecial @setspecial @endspecial 100 2373 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 23 w Fu(Let)27 b Fr(q)f Fp(2)e Fr(C)778 2343 y Fo(1)772 2394 y Fs(0)848 2373 y Fu(\()p Fk(R)q Fu(\))p Fr(;)14 b Fu(0)28 b Fp(\024)23 b Fr(q)j Fp(\024)d Fu(1)j(and)h Fr(q)f Fu(=)c(1)27 b(in)g(a)f(neigh)n(b)r(orho)r (o)r(d)g(of)g(zero)g(suc)n(h)h(that)g @beginspecial @setspecial @endspecial Fr(q)s(j)3154 2385 y Fs(0)p Fq(;n)3275 2373 y Fu(=)c Fr(j)3397 2385 y Fs(0)p Fq(;n)3522 2373 y Fu(for)j Fr(n)g Fu(large)0 2473 y(enough.)36 b(Using)28 b(Thm.)37 b @beginspecial @setspecial @endspecial(8.3)31 b(_)-27 b Fr({)t Fu(_)f Fr({)t Fu(_)h Fr({)p Fu(\))28 b(and)f(Corollary)e @beginspecial @setspecial @endspecial(8.2)i(w)n(e)g(obtain)1318 2616 y(\000)1370 2582 y Fs(+)1425 2616 y Fu(\()p Fr(q)s Fu(\))19 b Fp(\012)f Fu(1)-15 b(l)23 b Fr(W)1794 2582 y Fs(+)1849 2616 y Fu(\()p Fr(j)1915 2628 y Fq(n)1961 2616 y Fu(\))g(=)g Fr(W)2194 2582 y Fs(+)2249 2616 y Fu(\()p Fr(j)2315 2628 y Fq(n)2360 2616 y Fu(\))p Fr(;)84 b @beginspecial @setspecial @endspecial 1117 2804 a(w)22 b Fp(\000)47 b Fu(lim)1281 2854 y Fq(n)p Fo(!1)1468 2804 y Fr(P)1533 2768 y Fs(+)1521 2826 y(0)1606 2804 y Fp(\012)18 b Fu(1)-15 b(l)23 b Fr(W)1852 2770 y Fs(+)1907 2804 y Fu(\()p Fr(j)1973 2816 y Fq(n)2018 2804 y Fu(\))c Fp(\000)f Fr(W)2242 2770 y Fs(+)2297 2804 y Fu(\()p Fr(j)2363 2816 y Fq(n)2409 2804 y Fu(\))23 b(=)g(0)p Fr(:)82 b @beginspecial @setspecial @endspecial -2699 w Fu(\(8.12\))100 2976 y(Let)27 b @beginspecial @setspecial @endspecial Fr(\037)c Fp(2)h Fr(C)467 2946 y Fo(1)461 2997 y Fs(0)537 2976 y Fu(\()p Fk(R)q Fu(\).)43 b(W)-7 b(e)28 b(ha)n(v)n(e)835 3128 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 844 3107 a(^)835 3128 y(\012)895 3093 y Fs(+)p Fo(\003)984 3128 y Fr(\037)p Fu(\()1090 3107 y(^)1068 3128 y Fr(H)1137 3140 y Fo(1)1208 3128 y Fu(\))83 b(=)g @beginspecial @setspecial @endspecial 1480 3107 a(^)1471 3128 y(\012)1531 3093 y Fs(+)p Fo(\003)1629 3107 y Fu(^)1620 3128 y(\012)1680 3093 y Fs(ext)p Fq(;)p Fs(+)1845 3128 y Fr(\037)p Fu(\()1951 3107 y(^)1929 3128 y Fr(H)2005 3093 y Fs(ext)1998 3148 y Fo(1)2099 3128 y Fu(\))p Fr(W)2221 3093 y Fs(+)2277 3128 y Fu(\()p Fr(j)2343 3140 y Fq(n)2388 3128 y Fu(\))-2420 b(\(8.13\))0 3173 y @beginspecial @setspecial @endspecial 1240 3304 a @beginspecial @setspecial @endspecial 83 w(=)83 b Fr(w)21 b Fp(\000)47 b Fu(lim)1634 3354 y Fq(n)p Fo(!1)1830 3283 y Fu(^)1821 3304 y(\012)1881 3270 y Fs(+)p Fo(\003)1979 3283 y Fu(^)1970 3304 y(\012)2030 3270 y Fs(ext)p Fq(;)p Fs(+)2195 3304 y Fr(\037)p Fu(\()2301 3283 y(^)2279 3304 y Fr(H)2355 3270 y Fs(ext)2348 3324 y Fo(1)2450 3304 y Fu(\))p Fr(P)2547 3268 y Fs(+)2535 3326 y(0)2620 3304 y Fp(\012)18 b Fu(1)-15 b(l)23 b Fr(W)2866 3270 y Fs(+)2921 3304 y Fu(\()p Fr(j)2987 3316 y Fq(n)3033 3304 y Fu(\))-3065 b(\(8.14\))0 3387 y @beginspecial @setspecial @endspecial 1323 3518 a(=)83 b @beginspecial @setspecial @endspecial Fr(w)21 b Fp(\000)47 b Fu(lim)1634 3567 y Fq(n)p Fo(!1)1821 3518 y Fr(P)1886 3482 y Fs(+)1874 3540 y(0)1959 3518 y Fp(\012)18 b Fu(1)-15 b(l)p Fr(\037)p Fu(\()2198 3497 y(^)2176 3518 y Fr(H)2252 3483 y Fs(ext)2245 3538 y Fo(1)2346 3518 y Fu(\))p Fr(W)2468 3483 y Fs(+)2524 3518 y Fu(\()p Fr(j)2590 3530 y Fq(n)2635 3518 y Fu(\))-2667 b(\(8.15\))0 3601 y @beginspecial @setspecial @endspecial 1323 3731 a(=)83 b Fr(w)21 b Fp(\000)47 b Fu(lim)1634 3781 y Fq(n)p Fo(!1)1821 3731 y Fr(\037)p Fu(\()1927 3710 y(^)1905 3731 y Fr(H)1981 3697 y Fs(ext)1974 3752 y Fo(1)2076 3731 y Fu(\))p Fr(W)2198 3697 y Fs(+)2253 3731 y Fu(\()p Fr(j)2319 3743 y Fq(n)2365 3731 y Fu(\))-2397 b(\(8.16\))0 3814 y @beginspecial @setspecial @endspecial 1240 3945 a @beginspecial @setspecial @endspecial 83 w(=)83 b Fr(w)21 b Fp(\000)47 b Fu(lim)1634 3995 y Fq(n)p Fo(!1)1821 3945 y Fr(W)1911 3911 y Fs(+)1966 3945 y Fu(\()p Fr(j)2032 3957 y Fq(n)2078 3945 y Fu(\))p Fr(\037)p Fu(\()2216 3924 y(^)2194 3945 y Fr(H)2263 3957 y Fo(1)2334 3945 y Fu(\))p Fr(:)-2389 b Fu(\(8.17\))0 4134 y(\()p @beginspecial @setspecial @endspecial(8.13\))34 b(follo)n(ws)f(from)i(Thm.)58 b @beginspecial @setspecial @endspecial(8.3)38 b(_)-27 b Fr({)o(v)s Fu(\).)59 b(\(8.14\))34 b(follo)n(ws)f(b)n(y)i(\()p @beginspecial @setspecial @endspecial(8.12\).)57 b(\(8.15\))33 b(is)i(true)f(since)h Fr(P)3047 4098 y Fs(+)3035 4156 y(0)3136 4134 y Fu(comm)n(utes)g(with)g @beginspecial @setspecial @endspecial 3751 4113 a(^)3730 4134 y Fr(H)3806 4103 y Fs(ext)3799 4154 y Fo(1)0 4243 y Fu(and)29 b(that)g(Ran)o Fr(P)557 4207 y Fs(+)545 4265 y(0)637 4243 y Fp(\032)743 4222 y Fu(^)727 4243 y Fp(K)791 4212 y Fs(+)846 4243 y Fr(;)892 4222 y Fu(^)883 4243 y(\012)943 4212 y Fs(ext)p Fq(;)p Fs(+)1108 4243 y Fu(1)-15 b(l)1170 4255 y Fs(^)1158 4270 y Fo(K)1209 4252 y Fi(+)1279 4243 y Fp(\012)18 b Fu(1)-15 b(l)25 b(=)1535 4222 y(^)1526 4243 y(\012)1586 4212 y Fs(+)1670 4243 y Fu(and)k @beginspecial @setspecial @endspecial 1842 4222 a(^)1833 4243 y(\012)1893 4212 y Fs(+)p Fo(\003)1991 4222 y Fu(^)1982 4243 y(\012)2042 4212 y Fs(+)2122 4243 y Fu(=)24 b(1)-15 b(l)2273 4255 y Fs(^)2261 4270 y Fo(K)2312 4252 y Fi(+)2382 4243 y Fp(\012)18 b Fu(1)-15 b(l.)40 b(\()p @beginspecial @setspecial @endspecial(8.16\))28 b(follo)n(ws)g(from)g(the)h(fact)g(that)g Fr(P)3845 4207 y Fs(+)3833 4265 y(0)0 4356 y Fu(comm)n(utes)e(with)h @beginspecial @setspecial @endspecial 601 4335 a(^)579 4356 y Fr(H)655 4326 y Fs(ext)648 4377 y Fo(1)777 4356 y Fu(and)g(\(8.12\).)36 b(\()p @beginspecial @setspecial @endspecial(8.17\))27 b(is)g(Thm.)37 b(8.3)31 b(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\).)37 b(So)27 b(w)n(e)g(conclude)h(b)n(y)f (a)g(densit)n(y)h(argumen)n(t)e(that)1406 4507 y @beginspecial @setspecial @endspecial 1415 4486 a(^)1406 4507 y(\012)1466 4473 y Fs(+)p Fo(\003)1578 4507 y Fu(=)d Fr(w)e Fp(\000)47 b Fu(lim)1829 4557 y Fq(n)p Fo(!1)2016 4507 y Fr(W)2106 4473 y Fs(+)2161 4507 y Fu(\()p Fr(j)2227 4519 y Fq(n)2273 4507 y Fu(\))p Fr(;)83 b @beginspecial @setspecial @endspecial 1504 4740 a(P)1569 4704 y Fs(+)1557 4762 y(0)1642 4740 y Fp(\012)19 b Fu(1)-15 b(l)1807 4719 y(^)1798 4740 y(\012)1858 4705 y Fs(+)p Fo(\003)1970 4740 y Fu(=)2067 4719 y(^)2058 4740 y(\012)2118 4705 y Fs(+)p Fo(\003)2207 4740 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4883 a Fu(So)27 b(w)n(e)h(obtain)895 5026 y @beginspecial @setspecial @endspecial(Ran)1053 5005 y(^)1043 5026 y(\012)1103 4992 y Fs(+)p Fo(\003)1216 5026 y Fu(=)1319 5005 y(^)1303 5026 y Fp(K)1367 4992 y Fs(+)1441 5026 y Fp(\012)18 b Fu(\000\()p Fm(h)p Fu(\))24 b Fp(\032)f Fu(Ran)p Fr(P)2009 4991 y Fs(+)1997 5049 y(0)2082 5026 y Fp(\012)18 b Fu(\000\()p Fm(h)p Fu(\))24 b Fp(\032)2451 5005 y Fu(^)2435 5026 y Fp(K)2499 4992 y Fs(+)2573 5026 y Fp(\012)18 b Fu(\000\()p Fm(h)p Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 5179 a Fu(Hence)28 b(w)n(e)f(pro)n(v)n(e)f(that)789 5158 y(^)773 5179 y Fp(K)837 5149 y Fs(+)915 5179 y Fu(=)d(Ran)p Fr(P)1217 5144 y Fs(+)1205 5201 y(0)1272 5179 y Fu(.)p @beginspecial @setspecial @endspecial 2540 w Fh(\004)0 5201 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 133 x Fz(Theorem)30 b(8.5.)40 b Fn(We)30 b(have)1647 5477 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1688 5476 a @beginspecial @setspecial @endspecial Fu(1)-13 b(l)1740 5488 y Fs(pp)1817 5476 y Fu(\()p Fr(H)7 b Fu(\))24 b(=)e Fp(K)2132 5446 y Fo(\006)2189 5476 y Fr(:)2253 5477 y @beginspecial @setspecial @endspecial eop %%Page: 44 44 44 43 bop 0 226 a Fs(44)1695 b(Z.)29 b(AMMARI)100 489 y Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(By)27 b(Prop.)35 b(6.8)c(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)p Fu(\))27 b(w)n(e)h(ha)n(v)n(e)1575 645 y @beginspecial @setspecial @endspecial(1)-15 b(l)1624 657 y Fs(pp)1702 645 y Fu(\()1756 624 y(^)1734 645 y Fr(H)1803 657 y Fo(1)1873 645 y Fu(\))24 b Fp(\032)2032 624 y Fu(^)2016 645 y Fp(K)2080 611 y Fs(+)2136 645 y Fr(:)0 805 y Fu(Then)h(it)h(su\016ces)e(to)h(sho)n(w)g(that)g @beginspecial @setspecial @endspecial 1070 784 a(^)1054 805 y Fp(K)1118 774 y Fs(+)1197 805 y Fp(\032)d Fu(1)-15 b(l)1334 817 y Fs(pp)1411 805 y Fu(\()1465 784 y(^)1443 805 y Fr(H)1512 817 y Fo(1)1583 805 y Fu(\).)36 b(Prop.)f @beginspecial @setspecial @endspecial(7.4)24 b(giv)n(es)g(the)h (existence)g(of)g Fr(\017)e(>)g Fu(0)h(and)h(a)g(sequence)f @beginspecial @setspecial @endspecial 3625 784 a(^)3603 805 y Fr(H)3672 817 y Fq(\024)3740 805 y Fu(suc)n(h)0 904 y(that)1111 977 y Fl(Z)1194 997 y Fs(+)p Fo(1)1157 1165 y Fs(1)1330 1090 y Fp(jj)p Fu(\000\()p Fr(q)1500 1055 y Fq(t)1529 1090 y Fu(\))p Fr(\037)p Fu(\()1667 1069 y(^)1645 1090 y Fr(H)1714 1102 y Fq(\024)1758 1090 y Fu(\)e)1827 1055 y Fo(\000)p Fs(i)p Fq(t)1939 1040 y Fs(^)1923 1055 y Fq(H)1977 1063 y Fj(\024)2020 1090 y Fr(u)p Fp(jj)2114 1055 y Fs(2)2160 1033 y Fu(d)p Fr(t)p 2160 1070 77 4 v 2184 1146 a(t)2270 1090 y Fp(\024)e Fr(C)6 b Fp(jj)p Fr(u)p Fp(jj)2562 1055 y Fs(2)2600 1090 y Fr(;)83 b @beginspecial @setspecial @endspecial 0 1301 a Fu(where)27 b Fr(\037)c Fp(2)g Fr(C)458 1271 y Fo(1)452 1321 y Fs(0)529 1301 y Fu(\()p Fk(R)i Fp(n)18 b Fu(\()p Fr(\034)28 b Fp(\\)19 b Fr(\033)917 1313 y Fs(pp)995 1301 y Fu(\()1049 1280 y(^)1027 1301 y Fr(H)1096 1313 y Fo(1)1167 1301 y Fu(\)\)\))28 b(and)g @beginspecial @setspecial @endspecial Fr(q)e Fp(2)d Fr(C)1659 1271 y Fo(1)1653 1321 y Fs(0)1730 1301 y Fu(\([)p Fp(\000)p Fr(\017;)14 b(\017)p Fu(]\))p Fr(;)36 b(q)26 b Fu(=)d(1)k(for)g @beginspecial @setspecial @endspecial Fp(j)p Fr(x)p Fp(j)d Fr(<)2633 1268 y Fq(\017)p 2630 1282 34 4 v 2630 1329 a Fs(2)2673 1301 y Fu(.)37 b(Thm.)h(8.1)26 b(giv)n(es)h(that)1019 1476 y @beginspecial @setspecial @endspecial Fp(jj)p Fu(\000\()p Fr(q)1189 1441 y Fq(t)1219 1476 y Fu(\))p Fr(\037)p Fu(\()1357 1455 y(^)1335 1476 y Fr(H)1404 1488 y Fq(\024)1447 1476 y Fu(\)e)1516 1441 y Fo(\000)p Fs(i)p Fq(t)1629 1426 y Fs(^)1612 1441 y Fq(H)1666 1449 y Fj(\024)1709 1476 y Fr(u)p Fp(jj)c(!)g(jj)p Fu(\000)2030 1441 y Fs(+)2085 1476 y Fu(\()p Fr(q)s Fu(\))p Fr(\037)p Fu(\()2296 1455 y(^)2273 1476 y Fr(H)2342 1488 y Fo(1)2413 1476 y Fu(\))p Fr(u)p Fp(jj)g Fu(=)g(0)p Fr(;)83 b @beginspecial @setspecial @endspecial 0 1632 a Fu(then)26 b @beginspecial @setspecial @endspecial(\000)239 1602 y Fs(+)294 1632 y Fu(\()p Fr(q)s Fu(\))p Fr(\037)p Fu(\()505 1611 y(^)482 1632 y Fr(H)551 1644 y Fo(1)623 1632 y Fu(\))d(=)g(0.)36 b(So)25 b(w)n(e)h(ha)n(v)n(e)f (Ran)o Fr(P)1504 1596 y Fs(+)1492 1654 y(0)1582 1632 y Fp(\032)e Fu(1)-15 b(l)1719 1658 y Fq(\034)7 b Fo(\\)p Fq(\033)1840 1666 y Fi(pp)1906 1658 y Fs(\()1949 1643 y(^)1932 1658 y Fq(H)1986 1666 y Fg(1)2047 1658 y Fs(\))2077 1632 y Fu(\()2131 1611 y(^)2109 1632 y Fr(H)2178 1644 y Fo(1)2249 1632 y Fu(\).)37 b(Thm.)f @beginspecial @setspecial @endspecial(5.7)25 b(giv)n(es)g(that)h Fr(\034)36 b Fu(is)26 b(a)g(closed)f(coun)n(table)0 1762 y(set)k(and)h @beginspecial @setspecial @endspecial Fr(\033)342 1774 y Fs(pp)420 1762 y Fu(\()474 1741 y(^)452 1762 y Fr(H)521 1774 y Fo(1)591 1762 y Fu(\))g(can)f(accum)n(ulate)g(only)g(at)g @beginspecial @setspecial @endspecial Fr(\034)9 b Fu(,)31 b(so)e(1)-15 b(l)1784 1774 y Fs(pp)1861 1762 y Fu(\()1915 1741 y(^)1893 1762 y Fr(H)1962 1774 y Fo(1)2033 1762 y Fu(\))26 b(=)g(1)-15 b(l)2232 1788 y Fq(\034)7 b Fo(\\)p Fq(\033)2353 1796 y Fi(pp)2419 1788 y Fs(\()2461 1773 y(^)2445 1788 y Fq(H)2499 1796 y Fg(1)2559 1788 y Fs(\))2590 1762 y Fu(\()2644 1741 y(^)2622 1762 y Fr(H)2691 1774 y Fo(1)2761 1762 y Fu(\).)43 b(This)29 b(pro)n(v)n(es)f @beginspecial @setspecial @endspecial(1)-15 b(l)3358 1774 y Fs(pp)3436 1762 y Fu(\()3490 1741 y(^)3468 1762 y Fr(H)3537 1774 y Fo(1)3607 1762 y Fu(\))27 b(=)3773 1741 y(^)3756 1762 y Fp(K)3820 1732 y Fo(\006)3877 1762 y Fu(.)0 1870 y(Then)h(w)n(e)f(pro)n(v)n(e)f(the)i(theorem.)36 b @beginspecial @setspecial @endspecial 2774 w Fh(\004)1676 2179 y Fv(Appendix)c @beginspecial @setspecial @endspecial Fu(A.)100 2328 y(The)27 b(follo)n(wing)g(theorem)g(follo)n(ws)g(from)g (the)h(KLMN)f(theorem)g(and)h @beginspecial @setspecial @endspecial([23)o(,)g(Thm.)37 b(VI)r(I)r(I.25].)0 2349 y @beginspecial @setspecial @endspecial 136 x Fz(Theorem)30 b(A.1.)41 b Fn(L)l(et)32 b @beginspecial @setspecial @endspecial Fr(H)835 2497 y Fs(0)904 2485 y Fn(b)l(e)g(a)g(p)l(ositive) i(selfadjoint)g(op)l(er)l(ator)f(on)g @beginspecial @setspecial @endspecial Fp(H)q Fn(.)45 b(L)l(et)32 b(for)h Fr(\024)27 b Fp(\024)g(1)p Fn(,)33 b @beginspecial @setspecial @endspecial Fr(B)3031 2497 y Fq(\024)3106 2485 y Fn(b)l(e)f(quadr)l(atic)h(forms)g (on)0 2607 y Fr(D)r Fu(\()p Fr(H)189 2537 y Fi(1)p 189 2546 29 3 v 189 2579 a(2)172 2630 y Fs(0)231 2607 y Fu(\))d Fn(such)g(that)1344 2784 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fp(j)p Fr(B)1430 2796 y Fq(\024)1473 2784 y Fu(\()p Fr( )s(;)14 b( )s Fu(\))p Fp(j)24 b(\024)f Fr(a)14 b Fp(jj)p Fr(H)2012 2713 y Fi(1)p 2012 2722 V 2012 2756 a(2)1996 2806 y Fs(0)2055 2784 y Fr( )s Fp(jj)2158 2750 y Fs(2)2214 2784 y Fu(+)k Fr(b)c Fp(jj)p Fr( )s Fp(jj)2496 2750 y Fs(2)2533 2784 y Fr(;)p @beginspecial @setspecial @endspecial 0 2961 a Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial Fr(a)23 b(<)g Fu(1)29 b Fn(uniformly)i(in)f Fr(\024)f Fn(and)i Fr(B)1243 2973 y Fq(\024)1309 2961 y Fp(!)23 b Fr(B)1478 2973 y Fo(1)1578 2961 y Fn(on)30 b @beginspecial @setspecial @endspecial Fr(D)r Fu(\()p Fr(H)1886 2890 y Fi(1)p 1886 2899 V 1886 2932 a(2)1869 2983 y Fs(0)1928 2961 y Fu(\))p Fn(.)39 b(Then)0 3089 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))30 b Fn(Ther)l(e)g(exist)g(for)g Fr(\024)23 b Fp(\024)g(1)30 b Fn(selfadjoint)i(op)l(er)l(ators)e @beginspecial @setspecial @endspecial Fr(H)1755 3101 y Fq(\024)1828 3089 y Fn(with)g Fr(D)r Fu(\()p Fr(H)2180 3101 y Fq(\024)2224 3089 y Fu(\))23 b Fp(\032)g Fr(D)r Fu(\()p Fr(H)2556 3018 y Fi(1)p 2556 3027 V 2556 3060 a(2)2539 3111 y Fs(0)2598 3089 y Fu(\))30 b Fn(and)1039 3266 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fu(\()p Fr(H)1140 3278 y Fq(\024)1183 3266 y Fr( )s(;)14 b( )s Fu(\))24 b(=)e Fr(B)1540 3278 y Fq(\024)1583 3266 y Fu(\()p Fr( )s(;)14 b( )s Fu(\))19 b(+)f(\()p Fr(H)2018 3195 y Fi(1)p 2018 3204 V 2018 3237 a(2)2001 3288 y Fs(0)2061 3266 y Fr( )s(;)c(H)2240 3195 y Fi(1)p 2240 3204 V 2240 3237 a(2)2224 3288 y Fs(0)2283 3266 y Fr( )s Fu(\))p Fr(;)37 b( )26 b Fp(2)d Fr(D)r Fu(\()p Fr(H)2762 3278 y Fq(\024)2806 3266 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 0 3414 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))1402 3563 y(lim)1374 3613 y Fq(\024)p Fo(!1)1546 3563 y Fu(\()p Fr(z)22 b Fp(\000)c Fr(H)1791 3575 y Fq(\024)1834 3563 y Fu(\))1866 3529 y Fo(\000)p Fs(1)1978 3563 y Fu(=)23 b(\()p Fr(z)f Fp(\000)c Fr(H)2311 3575 y Fo(1)2381 3563 y Fu(\))2413 3529 y Fo(\000)p Fs(1)2503 3563 y Fr(;)p @beginspecial @setspecial @endspecial 4 3741 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)g Fr({)o Fu(\))1480 3892 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial Fr(s)18 b Fp(\000)46 b Fu(lim)1620 3942 y Fq(\024)p Fo(!1)1805 3892 y Fu(e)1842 3858 y Fo(\000)p Fs(i)p Fq(tH)1992 3866 y Fj(\024)2058 3892 y Fu(=)22 b(e)2182 3858 y Fo(\000)p Fs(i)p Fq(tH)2332 3866 y Fg(1)2397 3892 y Fr(:)p @beginspecial @setspecial @endspecial 0 4086 a @beginspecial @setspecial @endspecial Fz(Lemma)30 b(A.2.)40 b Fn(L)l(et)f Fr(F)52 b Fp(2)41 b Fr(C)972 4056 y Fo(1)1043 4086 y Fu(\()p Fk(R)p Fu(\))p Fr(;)46 b Fn(e)l(qual)39 b(to)h(0)g(ne)l(ar)f(the)h(origin)g(and)g(b)l (ounde)l(d)g(ne)l(ar)f @beginspecial @setspecial @endspecial Fp(1)p Fn(.)68 b(we)39 b(denote)h(by)g Fr(F)3698 4056 y Fq(R)3792 4086 y Fn(the)0 4199 y(derivative)31 b(op)l(er)l(ator)f @beginspecial @setspecial @endspecial Fr(F)12 b Fu(\()813 4159 y Fo(j)p Fq(x)p Fo(j)p 813 4180 78 4 v 826 4228 a Fq(R)900 4199 y Fu(\))p Fn(.)39 b(We)29 b(r)l(e)l(c)l(al)t(l)g(that)g Fu(x)g Fn(denote)g(the)g(nucle)l(on)g(p)l(osition)h(observable.)40 b(One)29 b(has)g(uniformly)h(in)f @beginspecial @setspecial @endspecial Fr(\024)1194 4425 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 1236 4354 a @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)p Fu(\))14 b Fp(jjh)p Fu(x)p Fp(i)1463 4324 y Fo(\000)p Fq(s)1551 4354 y Fr(F)1616 4324 y Fq(R)1670 4354 y Fr(v)1710 4366 y Fq(\024)1749 4374 y Fi(0)1786 4354 y Fp(jj)23 b(2)g Fr(O)r Fu(\()p Fr(R)2094 4324 y Fo(\000)p Fq(s)2182 4354 y Fu(\))p Fr(;)p @beginspecial @setspecial @endspecial 1240 4493 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))p Fp(jjh)p Fu(x)p Fp(i)1476 4459 y Fo(\000)p Fq(s)1564 4493 y Fr(!)s Fu(\()p Fr(k)s Fu(\))1729 4459 y Fo(\000)p Fq(\017)1812 4493 y Fr(F)1877 4459 y Fq(R)1932 4493 y Fr(r)1969 4505 y Fq(\024)2012 4493 y Fp(jj)24 b(2)f Fr(O)r Fu(\()p Fr(R)2321 4459 y Fo(\000)p Fq(s)2409 4493 y Fu(\))p Fr(;)14 b(\017)23 b(>)f Fu(0)2706 4425 y @beginspecial @setspecial @endspecial 100 4657 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(ha)n(v)n(e)e Fp(h)p Fu(x)p Fp(i)866 4626 y Fo(\000)p Fq(s)954 4657 y Fr(v)994 4669 y Fq(\024)1033 4677 y Fi(0)1092 4657 y Fp(2)e Fr(H)1247 4626 y Fq(s)1282 4657 y Fu(\()p Fk(R)1368 4626 y Fs(3)1412 4657 y Fr(;)14 b Fp(B)s Fu(\()p Fp(K)q Fu(\)\).)37 b(Since)28 b(w)n(e)f(ha)n(v)n(e) 1092 4853 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(R)1202 4818 y Fq(s)1238 4853 y Fr(F)12 b Fu(\()1345 4797 y Fp(j)p Fr(D)1437 4809 y Fq(k)1477 4797 y Fp(j)p 1345 4834 156 4 v 1391 4910 a Fr(R)1510 4853 y Fu(\))p Fp(h)p Fu(x)p Fp(i)1650 4818 y Fo(\000)p Fq(s)1738 4853 y Fr(v)1778 4865 y Fq(\024)1817 4873 y Fi(0)1854 4853 y Fp(jj)23 b(\024)g Fr(c)p Fp(jjj)p Fr(D)2185 4865 y Fq(k)2226 4853 y Fp(j)2249 4818 y Fq(s)2284 4853 y Fp(h)p Fu(x)p Fp(i)2392 4818 y Fo(\000)p Fq(s)2480 4853 y Fr(v)2520 4865 y Fq(\024)2559 4873 y Fi(0)2596 4853 y Fp(jj)83 b @beginspecial @setspecial @endspecial 0 5029 a Fu(w)n(e)27 b(obtain)32 b(_)-27 b Fr({)o Fu(\).)38 b(Let)27 b(us)h(pro)n(v)n(e)e @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\).)38 b(Pseudo)r(di\013eren)n(tial)26 b(calculus)h(giv)n(es)571 5225 y @beginspecial @setspecial @endspecial Fp(jj)p Fr(!)s Fu(\()p Fr(k)s Fu(\))782 5191 y Fo(\000)p Fq(\017)866 5225 y Fr(F)12 b Fu(\()973 5169 y Fp(j)p Fr(D)1065 5181 y Fq(k)1106 5169 y Fp(j)p 973 5206 V 1020 5282 a Fr(R)1162 5225 y Fp(\025)23 b Fu(1\))p Fr(r)1361 5237 y Fq(\024)1404 5225 y Fp(jj)g(\024)g Fr(c)p Fp(jj)p Fr(F)12 b Fu(\()1750 5169 y Fp(j)p Fr(D)1842 5181 y Fq(k)1883 5169 y Fp(j)p 1750 5206 V 1796 5282 a Fr(R)1939 5225 y Fp(\025)2037 5169 y Fu(1)p 2037 5206 42 4 v 2037 5282 a(2)2088 5225 y(\))p Fr(!)s Fu(\()p Fr(k)s Fu(\))2285 5191 y Fo(\000)p Fq(\017)2369 5225 y Fr(r)2406 5237 y Fq(\024)2450 5225 y Fp(jj)18 b Fu(+)2640 5169 y Fr(c)p 2607 5206 101 4 v 2607 5282 a(R)2671 5258 y Fs(2)2718 5225 y Fp(jj)p Fr(!)s Fu(\()p Fr(k)s Fu(\))2929 5191 y Fo(\000)p Fq(\017)3013 5225 y Fr(r)3050 5237 y Fq(\024)3093 5225 y Fp(jj)p Fr(:)84 b @beginspecial @setspecial @endspecial 0 5406 a Fu(Then)28 b(w)n(e)f(obtain)g @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)t Fu(_)g Fr({)p Fu(\))28 b(using)j(_)-27 b Fr({)p Fu(\))28 b(for)f Fr(!)1197 5376 y Fo(\000)p Fq(\017)1280 5406 y Fr(r)1317 5418 y Fq(\024)1361 5406 y Fu(,)g(whic)n(h)h(is)f @beginspecial @setspecial @endspecial Fr(L)1789 5376 y Fs(2)1826 5406 y Fu(\()p Fk(R)1912 5376 y Fs(3)1956 5406 y Fr(;)14 b(dk)s Fu(\))28 b(uniformly)f(in)h @beginspecial @setspecial @endspecial Fr(\024)p Fu(.)1147 b Fh(\004)p eop %%Page: 45 45 45 44 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(45)100 489 y Fu(Let)32 b @beginspecial @setspecial @endspecial Fp(H)h Fu(a)f(Hilb)r(ert)h(space.)49 b(Let)33 b Fp(f)p Fr(H)1253 501 y Fq(\024)1295 489 y Fp(g)f Fu(b)r(e)g(a)g(sequence)g(of)g(selfadjoin)n(t)g(op)r(erators)e (on)i(a)f(common)h(domain)g @beginspecial @setspecial @endspecial Fp(D)h(\032)d(H)q Fu(.)0 592 y(W)-7 b(e)31 b(supp)r(ose)f(that)h Fr(s)21 b Fp(\000)48 b Fu(lim)789 642 y Fq(\024)p Fo(!1)974 592 y Fu(e)1011 558 y Fo(\000)p Fs(i)p Fq(tH)1161 566 y Fj(\024)1232 592 y Fu(=)28 b(e)1362 558 y Fo(\000)p Fs(i)p Fq(tH)1512 566 y Fg(1)1577 592 y Fu(,)j(where)f @beginspecial @setspecial @endspecial Fr(H)1943 604 y Fo(1)2044 592 y Fu(is)h(a)f(selfadjoin)n(t)h(op)r (erator.)44 b(W)-7 b(e)31 b(ha)n(v)n(e)f(for)g @beginspecial @setspecial @endspecial Fr(\037)e Fp(2)h Fr(C)3682 562 y Fo(1)3676 613 y Fs(0)3752 592 y Fu(\()p Fk(R)q Fu(\),)0 721 y(using)e([23)o(,)h(Thm.)37 b(VI)r(I)r(I.20])28 b(and)f @beginspecial @setspecial @endspecial([23)o(,)h(Thm.)37 b(VI)r(I)r(I.21])1395 863 y Fr(s)19 b Fp(\000)46 b Fu(lim)1536 913 y Fq(\024)p Fo(!1)1721 863 y Fr(\037)p Fu(\()p Fr(H)1874 875 y Fq(\024)1917 863 y Fu(\))23 b(=)g Fr(\037)p Fu(\()p Fr(H)2213 875 y Fo(1)2283 863 y Fu(\))p Fr(:)84 b @beginspecial @setspecial @endspecial 0 1005 a @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 154 x Fz(Lemma)30 b(A.3.)40 b Fn(L)l(et)29 b Fr(t)23 b Fp(!)g Fr(B)918 1171 y Fq(t)971 1159 y Fp(2)g(B)s Fu(\()p Fp(H)q Fu(\))29 b Fn(and)i Fr(\037)23 b Fp(2)g Fr(C)1651 1129 y Fo(1)1645 1179 y Fs(0)1722 1159 y Fu(\()p Fk(R)p Fu(\))36 b Fn(such)30 b(that)f(for)i @beginspecial @setspecial @endspecial Fr(\024)23 b Fp(\024)g(1)1501 1301 y @beginspecial @setspecial @endspecial(jj)p Fr(B)1614 1267 y Fo(\003)1610 1322 y Fq(t)1653 1301 y Fr(B)1716 1313 y Fq(t)1759 1301 y Fr(\037)p Fu(\()p Fr(H)1912 1313 y Fq(\024)1955 1301 y Fu(\))p Fp(jj)h(\024)e Fr(c)2180 1313 y Fq(t)2209 1301 y Fr(;)84 b @beginspecial @setspecial @endspecial 0 1443 a Fn(wher)l(e)31 b @beginspecial @setspecial @endspecial Fr(c)271 1455 y Fq(t)330 1443 y Fn(is)g Fr(\024)p Fp(\000)p Fn(indep)l(endent)f(lo)l(c)l(al)t(ly)i(inte)l(gr)l(able)f(function)f (in)g Fr(t)p Fn(.)40 b(If)31 b(ther)l(e)f(exist)g(a)g(c)l(onstant)g(c)g (indep)l(endent)h(of)g @beginspecial @setspecial @endspecial Fr(\024)f Fn(such)0 1543 y(that)g(for)g @beginspecial @setspecial @endspecial Fr(\024)23 b(<)g Fp(1)1056 1727 y @beginspecial @setspecial @endspecial -113 x Fl(Z)1139 1635 y Fo(1)1102 1803 y Fs(1)1223 1727 y Fp(jj)p Fr(B)1332 1739 y Fq(t)1375 1727 y Fr(\037)p Fu(\()p Fr(H)1528 1739 y Fq(\024)1572 1727 y Fu(\)e)1641 1693 y Fo(\000)p Fs(i)p Fq(tH)1791 1701 y Fj(\024)1834 1727 y Fr(u)p Fp(jj)1928 1693 y Fs(2)1964 1727 y Fr(dt)h Fp(\024)e Fr(c)14 b Fp(jj)p Fr(u)p Fp(jj)2338 1693 y Fs(2)2375 1727 y Fr(;)37 b(u)23 b Fp(2)g(H)q Fr(;)83 b @beginspecial @setspecial @endspecial 0 1911 a Fn(then:)1171 2080 y @beginspecial @setspecial @endspecial -113 x Fl(Z)1254 1987 y Fo(1)1217 2155 y Fs(1)1338 2080 y Fp(jj)p Fr(B)1447 2092 y Fq(t)1490 2080 y Fr(\037)p Fu(\()p Fr(H)1643 2092 y Fo(1)1714 2080 y Fu(\)e)1783 2045 y Fo(\000)p Fs(i)p Fq(tH)1933 2053 y Fg(1)1998 2080 y Fr(u)p Fp(jj)2092 2045 y Fs(2)2129 2080 y Fr(dt)23 b Fp(\024)g Fr(c)14 b Fp(jj)p Fr(u)p Fp(jj)2503 2045 y Fs(2)2540 2080 y Fr(:)83 b @beginspecial @setspecial @endspecial 100 2276 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(W)-7 b(e)28 b(ha)n(v)n(e)e(to)h(pro)n (v)n(e,)f(uniformly)i(in)g Fr(T)12 b Fu(,)26 b(that:)1180 2475 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial -113 x Fl(Z)1263 2382 y Fq(T)1226 2550 y Fs(1)1329 2475 y Fp(jj)p Fr(B)1438 2487 y Fq(t)1481 2475 y Fr(\037)p Fu(\()p Fr(H)1634 2487 y Fo(1)1705 2475 y Fu(\)e)1774 2440 y Fo(\000)p Fs(i)p Fq(tH)1924 2448 y Fg(1)1989 2475 y Fr(u)p Fp(jj)2083 2440 y Fs(2)2120 2475 y Fr(dt)d Fp(\024)g Fr(c)14 b Fp(jj)p Fr(u)p Fp(jj)2494 2440 y Fs(2)2531 2475 y Fr(:)83 b @beginspecial @setspecial @endspecial -2637 w Fu(\(A.1\))0 2659 y(W)-7 b(e)28 b(apply)f(the)h (dominated)g(con)n(v)n(ergence)d(theorem.)36 b(It)28 b(is)g(enough)f(to)g(sho)n(w)g(that)942 2801 y @beginspecial @setspecial @endspecial 27 w(lim)942 2851 y Fq(\024)p Fo(!1)1127 2801 y Fp(jj)p Fr(B)1236 2813 y Fq(t)1279 2801 y Fr(\037)p Fu(\()p Fr(H)1432 2813 y Fq(\024)1475 2801 y Fu(\)e)1544 2767 y Fo(\000)p Fs(i)p Fq(tH)1694 2775 y Fj(\024)1737 2801 y Fr(u)p Fp(jj)1831 2767 y Fs(2)1891 2801 y Fu(=)c Fp(jj)p Fr(B)2088 2813 y Fq(t)2131 2801 y Fr(\037)p Fu(\()p Fr(H)2284 2813 y Fo(1)2354 2801 y Fu(\)e)2423 2767 y Fo(\000)p Fs(i)p Fq(tH)2573 2775 y Fg(1)2638 2801 y Fr(u)p Fp(jj)2732 2767 y Fs(2)2769 2801 y Fr(;)83 b @beginspecial @setspecial @endspecial 0 2980 a Fu(since)28 b(w)n(e)g(ha)n(v)n(e)f Fp(jj)p Fr(B)628 2992 y Fq(t)671 2980 y Fr(\037)p Fu(\()p Fr(H)824 2992 y Fq(\024)867 2980 y Fu(\)e)936 2950 y Fo(\000)p Fs(i)p Fq(tH)1086 2958 y Fj(\024)1129 2980 y Fr(u)p Fp(jj)1223 2950 y Fs(2)1284 2980 y Fp(\024)d Fr(c)1409 2950 y Fo(0)1446 2980 y Fp(jj)p Fr(B)1559 2950 y Fo(\003)1555 3000 y Fq(t)1597 2980 y Fr(B)1660 2992 y Fq(t)1703 2980 y Fr(\037)p Fu(\()p Fr(H)1856 2992 y Fq(\024)1899 2980 y Fu(\))p Fp(jj)h(jj)p Fr(u)p Fp(jj)2142 2950 y Fs(2)2179 2980 y Fu(.)39 b(It)28 b(is)g(easy)f(to)h(see)g(that)h(to)f(pro)n(v)n(e)e(\()p @beginspecial @setspecial @endspecial(A.1\))j(it)f(su\016ces)0 3079 y(to)f(sho)n(w)944 3222 y @beginspecial @setspecial @endspecial Fr(w)21 b Fp(\000)46 b Fu(lim)1107 3271 y Fq(\024)p Fo(!1)1292 3222 y Fr(\037)p Fu(\()p Fr(H)1445 3234 y Fq(\024)1489 3222 y Fu(\))p Fr(B)1588 3187 y Fo(\003)1584 3242 y Fq(t)1626 3222 y Fr(B)1689 3234 y Fq(t)1718 3222 y Fr(\037)p Fu(\()p Fr(H)1871 3234 y Fq(\024)1915 3222 y Fu(\))23 b(=)g Fr(\037)p Fu(\()p Fr(H)2211 3234 y Fo(1)2281 3222 y Fu(\))p Fr(B)2380 3187 y Fo(\003)2376 3242 y Fq(t)2419 3222 y Fr(B)2482 3234 y Fq(t)2511 3222 y Fr(\037)p Fu(\()p Fr(H)2664 3234 y Fo(1)2735 3222 y Fu(\))p Fr(:)83 b @beginspecial @setspecial @endspecial 0 3393 a Fu(This)28 b(follo)n(ws)e(from)h(the)h (h)n(yp)r(othesis)g Fp(jj)p Fr(B)1324 3363 y Fo(\003)1320 3413 y Fq(t)1362 3393 y Fr(B)1425 3405 y Fq(t)1454 3393 y Fr(\037)p Fu(\()p Fr(H)1607 3405 y Fq(\024)1650 3393 y Fu(\))p Fp(jj)c(\024)f Fr(c)1876 3405 y Fq(t)1932 3393 y Fu(for)k @beginspecial @setspecial @endspecial Fr(\024)c Fp(\024)g(1)p Fu(.)37 b @beginspecial @setspecial @endspecial 1474 w Fh(\004)100 3592 y Fu(Let)28 b @beginspecial @setspecial @endspecial Fp(H)319 3604 y Fq(i)347 3592 y Fr(;)14 b(i)23 b Fu(=)h(1)p Fr(;)14 b Fu(2)27 b(b)r(e)i(t)n(w)n(o)e(Hilb)r(ert)i (spaces.)38 b(Let)28 b Fr(H)1738 3604 y Fq(i;\024)1825 3592 y Fr(;)14 b(i)23 b Fu(=)h(1)p Fr(;)14 b Fu(2)27 b(b)r(e)i(t)n(w)n(o)e(sequences)h(of)g(selfadjoin)n(t)g(op)r(erators)e (on)i @beginspecial @setspecial @endspecial Fp(H)3849 3604 y Fq(i)3877 3592 y Fu(,)0 3692 y(suc)n(h)f(that)h @beginspecial @setspecial @endspecial Fr(H)436 3704 y Fq(i;\024)550 3692 y Fu(con)n(v)n(erge)d(in)j(the)g(strong)f(resolv)n (en)n(t)e(sense)j(to)f @beginspecial @setspecial @endspecial Fr(H)2116 3704 y Fq(i;)p Fo(1)2230 3692 y Fu(.)0 3791 y(The)h(Lemma)f(A.4)h(follo)n(ws)e(from)h(the)h(pro)r(of)f(of)h(Lemma)f @beginspecial @setspecial @endspecial(A.3)h(and)f([10)o(,)h(Lemma)f (B.4.2])0 3812 y @beginspecial @setspecial @endspecial 133 x Fz(Lemma)j(A.4.)40 b Fn(L)l(et)c @beginspecial @setspecial @endspecial Fr(t)f Fp(!)h Fr(C)6 b Fu(\()p Fr(t)p Fu(\))36 b Fp(2)g(B)s Fu(\()p Fp(H)1333 3957 y Fs(1)1369 3945 y Fr(;)14 b Fp(H)1476 3957 y Fs(2)1514 3945 y Fu(\))36 b Fn(and)h Fr(\037)f Fp(2)f Fr(C)1993 3915 y Fo(1)1987 3966 y Fs(0)2064 3945 y Fu(\()p Fk(R)p Fu(\))p Fn(.)65 b(We)37 b(supp)l(ose)g(that)g(the)f(asymmetric)i (Heisenb)l(er)l(g)0 4045 y(derivatives:)1066 4187 y @beginspecial @setspecial @endspecial Fr(D)1135 4199 y Fq(\024)1178 4187 y Fr(C)6 b Fu(\()p Fr(t)p Fu(\))24 b(:=)f Fr(@)1516 4199 y Fq(t)1545 4187 y Fr(C)6 b Fu(\()p Fr(t)p Fu(\))19 b(+)f(i\()p Fr(H)1930 4199 y Fs(1)p Fq(;\024)2026 4187 y Fr(C)6 b Fu(\()p Fr(t)p Fu(\))20 b Fp(\000)e Fr(C)6 b Fu(\()p Fr(t)p Fu(\))p Fr(H)2516 4199 y Fs(2)p Fq(;\024)2612 4187 y Fu(\))p Fr(;)84 b @beginspecial @setspecial @endspecial 0 4329 a Fn(satis\014es)30 b(for)g Fr(\024)23 b(<)g Fp(1)p Fn(,)30 b(the)g(fol)t(lowing)i(hyp)l(othesis:)634 4471 y @beginspecial @setspecial @endspecial 170 w Fu(_)-27 b Fr({)p Fu(\))14 b Fr(D)942 4483 y Fq(\024)985 4471 y Fr(C)6 b Fu(\()p Fr(t)p Fu(\))23 b(=)g Fr(B)t Fu(\()p Fr(t)p Fu(\))c(+)f Fr(R)1581 4483 y Fq(\024)1624 4471 y Fu(\()p Fr(t)p Fu(\))p Fr(:)800 4639 y @beginspecial @setspecial @endspecial 4 w Fu(_)-27 b Fr({)t Fu(_)g Fr({)o Fu(\))14 b Fr(\037)p Fu(\()p Fr(H)1052 4651 y Fs(2)p Fq(;\024)1149 4639 y Fu(\))g Fr(R)1258 4651 y Fq(\024)1301 4639 y Fu(\()p Fr(t)p Fu(\))g Fr(\037)p Fu(\()p Fr(H)1562 4651 y Fs(1)p Fq(;)1615 4660 y Fj(k)1652 4651 y Fq(k)1692 4639 y Fu(\))p Fp(jj)24 b(\024)f Fr(c)14 b(t)1962 4605 y Fo(\000)p Fs(1)p Fo(\000)p Fq(\017)2130 4639 y Fr(;)37 b(\017)23 b(>)f Fu(0)p Fr(;)51 b Fn(uniformly)j(for)24 b Fr(\024)f(<)f Fp(1)0 4688 y @beginspecial @setspecial @endspecial 804 4806 a Fu(_)-27 b Fr({)t Fu(_)g Fr({)t Fu(_)f Fr({)p Fu(\))14 b Fp(jj)p Fr(\037)p Fu(\()p Fr(H)1125 4818 y Fs(2)p Fq(;\024)1222 4806 y Fu(\))p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fp(jj)24 b(\024)e Fr(c)1608 4818 y Fq(t)1638 4806 y Fr(;)27 b Fp(jj)p Fr(B)t Fu(\()p Fr(t)p Fu(\))14 b Fr(\037)p Fu(\()p Fr(H)2062 4818 y Fs(1)p Fq(;\024)2159 4806 y Fu(\))p Fp(jj)23 b(\024)g Fr(c)2384 4818 y Fq(t)2413 4806 y Fr(;)37 b Fn(uniformly)55 b(for)23 b Fr(\024)g Fp(\024)g(1)p Fr(;)0 4949 y Fn(wher)l(e)30 b @beginspecial @setspecial @endspecial Fr(c)270 4961 y Fq(t)329 4949 y Fn(is)g Fr(\024)p Fp(\000)p Fn(indep)l(endent)g(lo)l(c)l(al)t(ly)i (inte)l(gr)l(able)e(function)g(in)g @beginspecial @setspecial @endspecial Fr(t)p Fn(.)808 5153 y Fu(_)-27 b Fr({)o(v)s Fu(\))14 b Fp(j)p Fu(\()p Fr(u)1022 5165 y Fs(2)1060 5153 y Fp(j)p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(u)1292 5165 y Fs(1)1329 5153 y Fu(\))p Fp(j)24 b(\024)e Fr(c)1598 5049 y Fq(n)1559 5074 y Fl(X)1561 5251 y Fq(j)s Fs(=1)1693 5153 y Fp(jj)p Fr(B)1802 5165 y Fs(2)p Fq(;j)1890 5153 y Fu(\()p Fr(t)p Fu(\))p Fr(u)2032 5165 y Fs(2)2069 5153 y Fp(jj)h(jj)p Fr(B)2247 5165 y Fs(1)p Fq(;j)2335 5153 y Fu(\()p Fr(t)p Fu(\))p Fr(u)2477 5165 y Fs(1)2514 5153 y Fp(jj)p Fr(;)14 b(w)r(her)r(e)0 5295 y @beginspecial @setspecial @endspecial 804 5351 a Fl(Z)887 5371 y Fo(1)850 5539 y Fs(1)971 5464 y Fp(jj)p Fr(B)1080 5476 y Fq(i;j)1158 5464 y Fu(\()p Fr(t)p Fu(\))g Fr(\037)p Fu(\()p Fr(H)1419 5476 y Fq(i;)p Fo(1)1533 5464 y Fu(\)e)1602 5429 y Fo(\000)p Fs(i)p Fq(tH)1752 5437 y Fj(i;)p Fg(1)1859 5464 y Fr(u)p Fp(jj)1953 5429 y Fs(2)1989 5464 y Fr(dt)24 b Fp(\024)e Fr(c)14 b Fp(jj)p Fr(u)p Fp(jj)2363 5429 y Fs(2)2400 5464 y Fr(;)g(i)23 b Fu(=)f(1)p Fr(;)14 b Fu(2)43 b Fn(and)58 b Fr(j)28 b Fu(=)23 b(1)p Fr(:::n:)p eop %%Page: 46 46 46 45 bop 0 226 a Fs(46)1695 b(Z.)29 b(AMMARI)0 489 y Fn(Then)h(the)g(fol)t(lowing)j(limit)d(exists)999 625 y @beginspecial @setspecial @endspecial Fr(s)18 b Fp(\000)64 b Fu(lim)1139 675 y Fq(t)p Fo(!)p Fs(+)p Fo(1)1361 625 y Fu(e)1398 591 y Fs(i)p Fq(tH)1496 599 y Fi(2)p Fj(;)p Fg(1)1608 625 y Fr(\037)p Fu(\()p Fr(H)1761 637 y Fs(2)p Fq(;)p Fo(1)1885 625 y Fu(\))14 b Fr(C)6 b Fu(\()p Fr(t)p Fu(\))14 b Fr(\037)p Fu(\()p Fr(H)2257 637 y Fs(1)p Fq(;)p Fo(1)2381 625 y Fu(\)e)2450 591 y Fo(\000)p Fs(i)p Fq(tH)2600 599 y Fi(1)p Fj(;)p Fg(1)2712 625 y Fr(:)83 b @beginspecial @setspecial @endspecial 100 807 a Fz(Pro)s(of.)41 b @beginspecial @setspecial @endspecial 24 w Fu(Let)27 b Fr(u)c Fp(2)g(H)791 819 y Fs(1)829 807 y Fr(;)14 b(v)26 b Fp(2)d(H)1080 819 y Fs(2)1118 807 y Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e:)348 943 y(\()p Fr(u)p Fp(j)p Fu(e)488 909 y Fs(i)p Fq(t)532 917 y Fi(2)564 909 y Fq(H)618 917 y Fi(2)p Fj(;)p Fg(1)730 943 y Fr(\037)p Fu(\()p Fr(H)883 955 y Fs(2)p Fq(;)p Fo(1)1007 943 y Fu(\))p Fr(C)6 b Fu(\()p Fr(t)1166 955 y Fs(2)1204 943 y Fu(\))p Fr(\037)p Fu(\()p Fr(H)1389 955 y Fs(1)p Fq(;)p Fo(1)1512 943 y Fu(\)e)1581 909 y Fs(i)p Fq(t)1625 917 y Fi(2)1658 909 y Fq(H)1712 917 y Fi(1)p Fj(;)p Fg(1)1824 943 y Fu(\))19 b Fp(\000)f Fu(\()p Fr(u)p Fp(j)p Fu(e)2098 909 y Fs(i)p Fq(t)2142 917 y Fi(1)2174 909 y Fq(H)2228 917 y Fi(2)p Fj(;)p Fg(1)2341 943 y Fr(\037)p Fu(\()p Fr(H)2494 955 y Fs(2)p Fq(;)p Fo(1)2617 943 y Fu(\))p Fr(C)6 b Fu(\()p Fr(t)2776 955 y Fs(1)2814 943 y Fu(\))p Fr(\037)p Fu(\()p Fr(H)2999 955 y Fs(1)p Fq(;)p Fo(1)3123 943 y Fu(\)e)3192 909 y Fo(\000)p Fs(i)p Fq(t)3288 917 y Fi(1)3320 909 y Fq(H)3374 917 y Fi(1)p Fj(;)p Fg(1)3487 943 y Fr(v)s Fu(\))0 992 y @beginspecial @setspecial @endspecial 348 1111 a(=)50 b(lim)435 1161 y Fq(\024)p Fo(!1)607 1111 y Fu(\()p Fr(u)p Fp(j)p Fu(e)747 1077 y Fs(i)p Fq(t)791 1085 y Fi(2)823 1077 y Fq(H)877 1085 y Fi(2)p Fj(;\024)967 1111 y Fr(\037)p Fu(\()p Fr(H)1120 1123 y Fs(2)p Fq(;\024)1216 1111 y Fu(\))p Fr(C)6 b Fu(\()p Fr(t)1375 1123 y Fs(2)1413 1111 y Fu(\))p Fr(\037)p Fu(\()p Fr(H)1598 1123 y Fs(1)p Fq(;\024)1695 1111 y Fu(\)e)1764 1077 y Fo(\000)p Fs(i)p Fq(t)1860 1085 y Fi(2)1892 1077 y Fq(H)1946 1085 y Fi(1)p Fj(;\024)2036 1111 y Fr(v)s Fu(\))19 b Fp(\000)f Fu(\()p Fr(u)p Fp(j)p Fu(e)2353 1077 y Fs(i)p Fq(t)2397 1085 y Fi(1)2429 1077 y Fq(H)2483 1085 y Fi(2)p Fj(;\024)2574 1111 y Fr(\037)p Fu(\()p Fr(H)2727 1123 y Fs(2)p Fq(;\024)2823 1111 y Fu(\))p Fr(C)6 b Fu(\()p Fr(t)2982 1123 y Fs(1)3020 1111 y Fu(\))p Fr(\037)p Fu(\()p Fr(H)3205 1123 y Fs(1)p Fq(;\024)3301 1111 y Fu(\)e)3370 1077 y Fo(\000)p Fs(i)p Fq(t)3466 1085 y Fi(1)3498 1077 y Fq(H)3552 1085 y Fi(1)p Fj(;\024)3643 1111 y Fr(v)s Fu(\))0 1194 y @beginspecial @setspecial @endspecial 348 1367 a(=)50 b(lim)435 1417 y Fq(\024)p Fo(!1)621 1254 y Fl(Z)704 1275 y Fq(t)729 1283 y Fi(2)667 1443 y Fq(t)692 1451 y Fi(1)765 1367 y Fu(\(e)834 1333 y Fo(\000)p Fs(i)p Fq(t)930 1341 y Fi(1)963 1333 y Fq(H)1017 1341 y Fi(2)p Fj(;\024)1107 1367 y Fr(u)p Fp(j)p Fr(\037)p Fu(\()p Fr(H)1331 1379 y Fs(2)p Fq(;\024)1427 1367 y Fu(\)I)-13 b(D)1539 1379 y Fq(\024)1582 1367 y Fr(C)6 b Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)1894 1379 y Fs(1)p Fq(;\024)1991 1367 y Fu(\)e)2060 1333 y Fo(\000)p Fs(i)p Fq(t)2156 1341 y Fi(1)2188 1333 y Fq(H)2242 1341 y Fi(1)p Fj(;\024)2332 1367 y Fr(v)s Fu(\))p Fr(dt)0 1476 y @beginspecial @setspecial @endspecial 348 1649 a Fu(=)50 b(lim)435 1699 y Fq(\024)p Fo(!1)621 1536 y Fl(Z)704 1557 y Fq(t)729 1565 y Fi(2)667 1725 y Fq(t)692 1733 y Fi(1)765 1649 y Fu(\(e)834 1615 y Fo(\000)p Fs(i)p Fq(t)930 1623 y Fi(1)963 1615 y Fq(H)1017 1623 y Fi(2)p Fj(;\024)1107 1649 y Fr(u)p Fp(j)p Fu(\()p Fr(B)t Fu(\()p Fr(t)p Fu(\))19 b(+)f Fr(R)q Fu(\()p Fr(t)p Fu(\)\))p Fr(\037)p Fu(\()p Fr(H)1816 1661 y Fs(1)p Fq(;\024)1912 1649 y Fu(\)e)1981 1615 y Fo(\000)p Fs(i)p Fq(t)2077 1623 y Fi(1)2110 1615 y Fq(H)2164 1623 y Fi(1)p Fj(;\024)2254 1649 y Fr(v)s Fu(\))p Fr(dt:)0 1836 y Fu(Using)27 b(Leb)r(esgue)g (dominated)h(con)n(v)n(ergence)d(theorem)i(and,)h(as)f(in)g(the)h(pro)r (of)f(of)h(Lemma)f @beginspecial @setspecial @endspecial(A.3,)h(the)g (fact)g(that)758 1972 y(_)-28 b Fr({)t Fu(_)h Fr({)t Fu(_)g Fr({)p Fu(\))23 b Fp(\))g Fr(w)e Fp(\000)46 b Fu(lim)1158 2022 y Fq(\024)p Fo(!1)1343 1972 y Fr(\037)p Fu(\()p Fr(H)1496 1984 y Fs(2)p Fq(;\024)1592 1972 y Fu(\))p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)1938 1984 y Fs(1)p Fq(;\024)2035 1972 y Fu(\))24 b(=)e Fr(\037)p Fu(\()p Fr(H)2331 1984 y Fs(2)p Fq(;)p Fo(1)2455 1972 y Fu(\))p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)2801 1984 y Fs(1)p Fq(;)p Fo(1)2925 1972 y Fu(\))p Fr(;)84 b @beginspecial @setspecial @endspecial 0 2133 a Fu(w)n(e)27 b(obtain)255 2306 y(lim)227 2355 y Fq(\024)p Fo(!1)412 2193 y Fl(Z)495 2213 y Fq(t)520 2221 y Fi(2)459 2381 y Fq(t)484 2389 y Fi(1)557 2306 y Fu(\(e)626 2271 y Fo(\000)p Fs(i)p Fq(t)722 2279 y Fi(1)754 2271 y Fq(H)808 2279 y Fi(2)p Fj(;\024)899 2306 y Fr(u)p Fp(j)p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)1284 2318 y Fs(1)p Fq(;\024)1380 2306 y Fu(\)e)1449 2271 y Fo(\000)p Fs(i)p Fq(t)1545 2279 y Fi(1)1577 2271 y Fq(H)1631 2279 y Fi(1)p Fj(;\024)1722 2306 y Fr(v)s Fu(\))p Fr(dt)d Fu(=)1981 2193 y Fl(Z)2064 2213 y Fq(t)2089 2221 y Fi(2)2027 2381 y Fq(t)2052 2389 y Fi(1)2126 2306 y Fu(\(e)2195 2271 y Fo(\000)p Fs(i)p Fq(t)2291 2279 y Fi(1)2323 2271 y Fq(H)2377 2279 y Fi(2)p Fj(;)p Fg(1)2490 2306 y Fr(u)p Fp(j)p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)2875 2318 y Fs(1)p Fq(;\024)2971 2306 y Fu(\)e)3040 2271 y Fo(\000)p Fs(i)p Fq(t)3136 2279 y Fi(1)3168 2271 y Fq(H)3222 2279 y Fi(1)p Fj(;)p Fg(1)3335 2306 y Fr(v)s Fu(\))p Fr(dt:)84 b @beginspecial @setspecial @endspecial -3590 w Fu(\(A.2\))0 2497 y(W)-7 b(e)28 b(ha)n(v)n(e)e(b)n(y)i @beginspecial @setspecial @endspecial 4 w(_)-27 b Fr({)o Fu(\):)769 2691 y(lim)741 2741 y Fq(\024)p Fo(!1)926 2578 y Fl(Z)1009 2598 y Fq(t)1034 2606 y Fi(2)972 2766 y Fq(t)997 2774 y Fi(1)1070 2691 y Fu(\(e)1139 2656 y Fo(\000)p Fs(i)p Fq(t)1235 2664 y Fi(1)1268 2656 y Fq(H)1322 2664 y Fi(2)p Fj(;\024)1412 2691 y Fr(u)p Fp(j)p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)1797 2703 y Fs(1)p Fq(;\024)1893 2691 y Fu(\)e)1962 2656 y Fo(\000)p Fs(i)p Fq(t)2058 2664 y Fi(1)2091 2656 y Fq(H)2145 2664 y Fi(1)p Fj(;\024)2235 2691 y Fr(v)s Fu(\))p Fr(dt)24 b Fp(\024)f Fr(c)14 b(t)2575 2655 y Fo(\000)p Fq(\017)2575 2713 y Fs(2)2672 2691 y Fp(jj)p Fr(v)s Fp(jj)23 b(jj)p Fr(u)p Fp(jj)p Fr(:)83 b @beginspecial @setspecial @endspecial -3076 w Fu(\(A.3\))0 2882 y(Using)32 b(_)-28 b Fr({v)s Fu(\))28 b(w)n(e)g(obtain:)698 3076 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 194 w(lim)864 3126 y Fq(\024)p Fo(!1)1049 2963 y Fl(Z)1132 2983 y Fq(t)1157 2991 y Fi(2)1095 3151 y Fq(t)1120 3159 y Fi(1)1193 3076 y Fu(\(e)1262 3041 y Fo(\000)p Fs(i)p Fq(t)1358 3049 y Fi(1)1391 3041 y Fq(H)1445 3049 y Fi(2)p Fj(;)p Fg(1)1557 3076 y Fr(u)p Fp(j)p Fr(B)t Fu(\()p Fr(t)p Fu(\))p Fr(\037)p Fu(\()p Fr(H)1942 3088 y Fs(1)p Fq(;)p Fo(1)2066 3076 y Fu(\)e)2135 3041 y Fo(\000)p Fs(i)p Fq(t)2231 3049 y Fi(1)2263 3041 y Fq(H)2317 3049 y Fi(1)p Fj(;)p Fg(1)2430 3076 y Fr(v)s Fu(\))p Fr(dt)0 3185 y @beginspecial @setspecial @endspecial 864 3367 a Fp(\024)22 b Fr(c)1054 3263 y Fq(n)1015 3288 y Fl(X)1017 3465 y Fq(j)s Fs(=1)1149 3254 y Fl(Z)1232 3274 y Fq(t)1257 3282 y Fi(2)1195 3442 y Fq(t)1220 3450 y Fi(1)1307 3367 y Fp(jj)p Fr(B)1416 3379 y Fs(2)p Fq(;j)1504 3367 y Fu(\()p Fr(t)p Fu(\)e)1635 3332 y Fo(\000)p Fs(i)p Fq(tH)1785 3340 y Fi(2)p Fj(;)p Fg(1)1898 3367 y Fr(u)p Fp(jj)1992 3332 y Fs(2)2052 3367 y Fr(dt)c Fp(\002)2226 3254 y Fl(Z)2309 3274 y Fq(t)2334 3282 y Fi(2)2272 3442 y Fq(t)2297 3450 y Fi(1)2385 3367 y Fp(jj)p Fr(B)2494 3379 y Fs(1)p Fq(;j)2582 3367 y Fu(\()p Fr(t)p Fu(\)e)2713 3332 y Fo(\000)p Fs(i)p Fq(tH)2863 3340 y Fi(1)p Fj(;)p Fg(1)2975 3367 y Fr(u)p Fp(jj)3069 3332 y Fs(2)3129 3367 y Fr(dt)-3202 b Fu(\(A.4\))0 3509 y @beginspecial @setspecial @endspecial 864 3616 a Fp(\024)22 b Fr(c)14 b Fp(jj)p Fr(u)p Fp(jj)g(jj)p Fr(v)s Fp(jj)p Fr(:)100 3753 y Fu(\()p @beginspecial @setspecial @endspecial(A.2\),)28 b(\(A.3\))g(and)f (\(A.4\))h(giv)n(e)f(the)h(existence)f(of)h(the)g(claimed)f(limit.)38 b @beginspecial @setspecial @endspecial 1454 w Fh(\004)100 3952 y Fu(W)-7 b(e)28 b(recall)e(here)h(a)h(con)n(v)n(ergence)d(lemma)i (of)h(p)r(ositiv)n(e)f(op)r(erators,)f(see)h(e.g.)36 b @beginspecial @setspecial @endspecial([11)o(,)28 b(Lemma)f(A.3])0 3973 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 129 x Fz(Lemma)j(A.5.)40 b Fn(L)l(et)29 b Fr(Q)762 4114 y Fq(n)837 4102 y Fn(b)l(e)h(a)g(se)l(quenc)l(e)f(of)i (c)l(ommuting)e(selfadjoint)j(op)l(er)l(ators.)40 b(If)1028 4239 y @beginspecial @setspecial @endspecial Fu(0)22 b Fp(\024)h Fr(Q)1246 4251 y Fq(n)1314 4239 y Fp(\024)g Fu(1)p Fr(;)82 b(Q)1615 4251 y Fq(n)p Fs(+1)1767 4239 y Fp(\024)23 b Fr(Q)1921 4251 y Fq(n)1966 4239 y Fr(;)83 b(Q)2138 4251 y Fq(n)p Fs(+1)2267 4239 y Fr(Q)2333 4251 y Fq(n)2400 4239 y Fu(=)23 b Fr(Q)2554 4251 y Fq(n)p Fs(+1)2683 4239 y Fr(:)83 b @beginspecial @setspecial @endspecial 0 4375 a Fn(Then)30 b(ther)l(e)g(exist)g @beginspecial @setspecial @endspecial Fr(Q)f Fn(a)h(pr)l(oje)l(ction) 1577 4511 y Fr(Q)23 b Fu(=)f Fr(s)d Fp(\000)f Fu(lim)1931 4561 y Fq(n)2023 4511 y Fr(Q)2089 4523 y Fq(n)2134 4511 y Fr(:)0 4644 y @beginspecial @setspecial @endspecial 1708 4709 a Fv(References)35 4842 y Fy([1])35 b(W.O.)25 b(Amrein,)g(A.)g(Boutet)i(de)g(Mon)n(v)n(el,)f(V.)f(Georgescu,)j(Comm)n (utator)d(Metho)r(ds)h(and)h(Sp)r(ectral)f(Theory)h(of)e(N-Bo)r(dy)h (Hamiltonian,)145 4925 y(Birkh\177)-35 b(auser,)23 b(1996.)35 5008 y([2])35 b(A.)20 b(Arai,)f(M.)h(Hirok)l(a)n(w)n(a,)g Ft(On)i(the)g(existenc)l(e)g(and)h(uniqueness)g(of)f(gr)l(ound)i (states)e(of)h(a)g(gener)l(alize)l(d)g(spin-b)l(oson)h(mo)l(del)p Fy(,)d(J.)f(F)-6 b(unct.)21 b(Anal.)145 5091 y Fd(151)i Fy(\(1997\))i(455-503.)35 5174 y([3])35 b(F.A.)23 b(Berezin,)g(The)h (Metho)r(d)h(of)e(Second)i(Quan)n(tization,)g(Academic)f(Press,)e (1966.)35 5257 y([4])35 b(V.)24 b(Bac)n(h,)i(J.)e(F)-6 b(r\177)-35 b(ohlic)n(h,)25 b(I.)g(M.)f(Sigal,)g Ft(Quantum)k(ele)l (ctr)l(o)l(dynamics)g(of)f(c)l(on\014ne)l(d)h(non-r)l(elativistic)e(p)l (articles)p Fy(,)g(Adv.)e(Math.)h Fd(137)f Fy(\(1998\))145 5340 y(299-395.)35 5423 y([5])35 b(V.)23 b(Bac)n(h,)h(J.)f(F)-6 b(r\177)-35 b(ohlic)n(h,)23 b(I.)h(M.)f(Sigal,)g(A.)g(So\013er,)g Ft(Positive)j(c)l(ommutators)h(and)g(sp)l(e)l(ctrum)f(of)g(nonr)l (elativistic)g(QED)p Fy(,)d(preprin)n(t)g(1997.)35 5506 y([6])35 b(O.)23 b(Brattelli,)g(D.)g(Robinson,)h(Op)r(erator)g (Algebras)f(and)h(Quan)n(tum)g(Statistical)h(Mec)n(hanics,)f(Springer,) f(Berlin,)f(1981.)p eop %%Page: 47 47 47 46 bop 992 226 a Fs(ASYMPTOTIC)30 b(COMPLETENESS)d(F)n(OR)i(A)h (RENORMALIZED...)925 b(47)35 489 y Fy([7])35 b(J.)26 b(C.)g(Baez,)h(I.)g(E.)f(Segal,)h(Z.)f(Zhou,)h(In)n(tro)r(duction)i(to) e(Algebraic)f(and)h(Constructiv)n(e)h(Quan)n(tum)e(Field)g(Theory)-6 b(,)27 b(Princeton)g(series)f(in)145 572 y(Ph)n(ysics,)d(1992.)35 655 y([8])35 b(J.)29 b(T.)f(Cannon,)i Ft(Quantum)h(\014eld)g(the)l(or)l (etic)g(pr)l(op)l(erties)g(of)g(a)g(mo)l(del)h(of)e(Nelson:)43 b(domain)32 b(and)f(eigenve)l(ctor)e(stability)h(for)h(p)l(erturb)l(e)l (d)145 738 y(line)l(ar)26 b(op)l(er)l(ators)p Fy(,)g(J.)d(F)-6 b(unct.)25 b(Anal.)e Fd(8)g Fy(\(1971\))i(101-152.)35 821 y([9])35 b(H.)19 b(L.)h(Cycon,)h(R.)f(F)-6 b(ro)r(ese,)21 b(W.)f(Kirsc)n(h,)f(B.)h(Simon,)f(Sc)n(hr\177)-35 b(odinger)21 b(Op)r(erators)f(with)g(Applications)h(to)f(Quan)n(tum)g(Mec)n(hanics)h (and)g(Global)145 904 y(Geometry)-6 b(,)24 b(Springer,)f(1987.)0 987 y([10])35 b(J.)20 b(Derezi)r(\023)-37 b(nski,)20 b(C.)g(G)n(\023)-33 b(erard,)20 b(Scattering)i(Theory)e(of)g(Classical) g(and)g(Quan)n(tum)g(N-particle)g(Systems,)g(T)-6 b(exts)21 b(and)g(Monographs)f(in)g(Ph)n(ys.)145 1070 y(Springer,)j(1997.)0 1153 y([11])35 b(J.)18 b(Derezi)r(\023)-37 b(nski,)19 b(C.)e(G)n(\023)-33 b(erard,)19 b Ft(Asymptotic)h(Completeness)h(in)g (Quantum)g(Field)g(The)l(ory.)g(Massive)f(Pauli-Fierz)i(Hamiltonians)p Fy(,)e(Reviews)145 1236 y(in)j(Math.)h(Ph)n(ys.)g Fd(11)e Fy(\(1999\))k(383-450.)0 1319 y([12])35 b(J.)20 b(Derezi)r(\023)-37 b(nski,)21 b(C.)e(G)n(\023)-33 b(erard,)21 b Ft(Sp)l(e)l(ctr)l(al)j (and)g(Sc)l(attering)e(The)l(ory)h(of)g(Sp)l(atial)t(ly)g(cut-o\013)f @beginspecial @setspecial @endspecial Fc(P)10 b Fy(\()p Fc(\036)p Fy(\))2642 1328 y Fw(2)2700 1319 y Ft(Hamiltonians)p Fy(,)22 b(preprin)n(t)e(Ecole)h(P)n(olytec)n(h-)145 1402 y(nique,)j(1998.)0 1485 y([13])35 b(J,)23 b(Derezi)r(\023)-37 b(nski,)24 b(V.)e(Jaksic,)i Ft(Sp)l(e)l(crtal)j(the)l(ory)f(of)g (Pauli-Fierz)g(Hamiltonians)h(I)p Fy(,)d(preprin)n(t.)0 1568 y([14])35 b(J.)e(F)-6 b(r\177)-35 b(ohlic)n(h,)34 b Ft(Existenc)l(e)g(of)g(Dr)l(esse)l(d)h(Ele)l(ctr)l(on)g(States)f(in)g (a)g(Class)h(of)f(Persistent)g(Mo)l(dels)p Fy(,)i(F)-6 b(ortsc)n(hritte)34 b(der)e(Ph)n(ysik,)j Fd(22)d Fy(\(1974\))145 1651 y(159-198.)0 1734 y([15])j(G.)29 b(M.)f(Graf,)h Ft(Asymptotic)h(Completeness)i(for)e(N-Bo)l(dy)h(Short-R)l(ange)g (Quantum)g(Systems:)42 b(A)31 b(New)f(Pr)l(o)l(of)p Fy(,)i(Comm.)26 b(Math.)j(ph)n(ys.)145 1817 y Fd(123)23 b Fy(\(1990\))i(73-101.)0 1900 y([16])35 b(J.)29 b(Glimm,)e(A.)h(Ja\013e,)j(Collected)f(P)n(ap)r (ers,)g(Quan)n(tum)f(Field)f(Theory)i(and)f(Statistical)h(Mec)n(hanics) g(Exp)r(ostions,)g(v)n(ol.)f(I,)g(Birkh\177)-35 b(auser,)145 1983 y(1985.)0 2066 y([17])35 b(E.)23 b(P)-6 b(.)23 b(Gross,)g(Ann.)h (Ph)n(ys.)f Fd(19)g Fy(\(1962\))j(219.)0 2149 y([18])35 b(O.)23 b(W.)h(Green)n(b)r(erg,)g(S.)f(S.)h(Sc)n(h)n(w)n(eb)r(er,)g (Nuo)n(v)n(o)g(Cimen)n(to)g(s,)f Fd(378)f Fy(\()i(1958\).)0 2232 y([19])35 b(R.)23 b(H\034egh-Krohn,)h Ft(Asymptotic)h(limits)h(in) f(some)h(mo)l(dels)i(of)d(quantum)i(\014eld)f(the)l(ory)g Fy(,)d(J.)g(Math.)h(Ph)n(ys.)f Fd(9)g Fy(\(1968\))j(2075-2079.)0 2315 y([20])35 b(E.)e(Mourre,)h Ft(A)n(bsenc)l(e)g(of)g(singular)h(c)l (ontinuous)h(sp)l(e)l(ctrum)f(for)g(c)l(ertain)f(selfadjoint)h(op)l(er) l(ators)p Fy(,)i(Comm.)31 b(Math.)i(ph)n(ys.)g Fd(78)f Fy(\(1981\))145 2398 y(519-567.)0 2481 y([21])j(E.)23 b(Nelson,)g Ft(Inter)l(action)k(of)e(Nonr)l(elativistic)h(p)l(articles) g(with)g(a)g(Quantize)l(d)g(Sc)l(alar)h(Field,)d Fy(J.)g(Math.)f(Ph)n (ys.)h Fd(5)f Fy(\(1964\))j(1190-1197.)0 2565 y([22])35 b(L.)23 b(Rosen,)48 b Ft(The)25 b @beginspecial @setspecial @endspecial Fy(\(\010)708 2541 y Fw(2)p Fb(n)782 2565 y Fy(\))809 2574 y Fw(2)869 2565 y Ft(Quantum)h(Field)h(The)l(ory:)33 b(Higher)25 b(Or)l(der)h(Estimates,)e Fy(Comm.)d(Pure)j(Appl.)f(Math.)g Fd(24)g Fy(\(1971\))j(417-457.)0 2648 y([23])35 b(M.)23 b(Reed,)i(B.)e(Simon,)g(Metho)r(ds)h(of)g(Mo)r(dern)g(Mathematical)g (Ph)n(ysics,)g(v)n(ol.)g(I)g(and)h(I)r(I,)f(1976,)h(v)n(ol.)e(I)r(I)r (I,)i(1979,)g(v)n(ol)f(IV,)f(1978,)i(Academic)145 2731 y(Press,)e(London.)0 2814 y([24])35 b(I.)26 b(M.)e(Sigal,)i(A.)f (So\013er,)h Ft(The)i(N-p)l(article)g(sc)l(attering)f(pr)l(oblem:)38 b(asymptotic)28 b(c)l(ompleteness)g(for)g(the)f(short-r)l(ange)h (quantum)h(systems)p Fy(,)145 2897 y(Ann.)23 b(of)h(Math.)g Fd(125)e Fy(\(1987\))k(35-108.)0 2913 y @beginspecial @setspecial @endspecial 73 x @beginspecial @setspecial @endspecial 100 3057 a Fx(Centre)f(de)h(Ma)l(th)658 3051 y(\023)658 3057 y(ema)l(tiques,)e(UMR)h(7640)e(CNRS,)1658 3040 y(\023)1652 3057 y(Ecole)j(Pol)l(ytechnique,)f(91128)f(P)-6 b(alaisea)o(u)24 b(Cedex,)i(France)100 3140 y Ft(E-mail)g(addr)l(ess)5 b Fy(:)33 b Fa(ammari@math.polytechniqu)q(e.f)q(r)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0001261006432--