Content-Type: multipart/mixed; boundary="-------------0104181729718" This is a multi-part message in MIME format. ---------------0104181729718 Content-Type: text/plain; name="01-146.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-146.keywords" absolutely continuous spectrum, localization, quasiperiodic ---------------0104181729718 Content-Type: application/postscript; name="longrn.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="longrn.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: longrn.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips longrn %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 2001.04.10:1540 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul 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Fl(2)3390 2167 y Fn(;)17 b Fj(k)p Fn(f)11 b Fj(k)3593 2182 y Fh(1)3666 2167 y Fn(:)474 2370 y Fo(Previous)37 b(pro)s(ofs)f(of)g(absolutely)g(con)m(tin)m(uous) h(sp)s(ectrum)g(for)f(quasip)s(erio)s(dic)e(op-)328 2491 y(erators)e(w)m(ere)h(mostly)e(KAM-t)m(yp)s(e)i(p)s(erturbativ)m(e)f (sc)m(hemes,)i(\014rst)e(dev)m(elop)s(ed)h(in)e(this)328 2611 y(con)m(text)g(b)m(y)g(Dinaburg-Sinai)26 b([6].)42 b(Early)30 b(results)g([1,)f(5])h(w)m(ere)h(restricted)f(to)g(the)g (case)328 2731 y(of)38 b(almost-Mathieu)d(t)m(yp)s(e)k(\()p Fn(f)48 b Fo(=)37 b(cos)q(\))h(op)s(erators.)60 b(Eliasson)37 b([7])h(pro)m(v)m(ed)h(purely)g(ac)328 2852 y(sp)s(ectrum)33 b(for)f(analytic)f Fn(f)43 b Fo(in)32 b(the)h(con)m(tin)m(uous)h(case.) 474 2972 y(While)28 b(w)m(e)i(do)e(not)h(claim)d(that)i(our)h(constan)m (t)g Fn(\025)2275 2987 y Fl(0)2343 2972 y Fo(is)f(optimal,)f(there)i (is,)h(b)s(esides)f(a)328 3093 y(simple)35 b(pro)s(of,)j(an)f(imp)s (ortan)m(t)e(di\013erence)i(b)s(et)m(w)m(een)j(our)c(\(nonp)s (erturbativ)m(e\))i(result)328 3213 y(and)31 b(the)g(\(p)s(erturbativ)m (e\))f(theorems)h(ab)s(o)m(v)m(e)g(in)f(that)h Fn(\025)2385 3228 y Fl(0)2454 3213 y Fo(do)s(es)g(not)g(dep)s(end)g(on)g Fn(!)j Fo(and)328 3333 y(also)e(dep)s(ends)i(on)e Fn(f)43 b Fo(in)32 b(a)h(constructiv)m(e)g(w)m(a)m(y)-8 b(.)474 3454 y(The)38 b(\014rst)e(nonp)s(erturbativ)m(e)h(results)g(on)f(the)h (ac)f(sp)s(ectrum)h(w)m(ere)g(obtained,)g(for)328 3574 y(the)i(case)g(of)f(the)h(almost)d(Mathieu)j(op)s(erator,)g(b)m(y)g (Last)g([15)o(],)h(who)f(pro)m(v)m(ed)h(the)e(ex-)328 3694 y(istence)e(of)e(a)h(large)f(absolutely)g(con)m(tin)m(uous)i(comp) s(onen)m(t)f(for)f(an)m(y)i Fn(\025)c(>)f Fo(1)p Fn(=)p Fo(2)k(for)f(an)m(y)328 3815 y Fn(!)39 b Fo(\(and)d(all)f Fn(\022)s(;)h Fo(as)g(sho)m(wn)i(b)m(y)f(Gestesy-Simon)e([10])h(or)g (as)g(follo)m(ws)f(from)g([17]\).)54 b(The)328 3935 y(result)37 b(of)f(Last)h(extends)i(to)e(a)f(general)h(class)g(of)f(op)s(erators)h (\(1.1\))g(with)f Fn(f)46 b Fj(2)35 b Fn(C)3372 3899 y Fl(3)p Fm(=)p Fl(2+)p Fm(\017)328 4056 y Fo(and)e(ev)m(en)h([16)o(].) 474 4176 y(In)28 b([11)o(])g(Gordon)e(et)h(al.)41 b(pro)m(v)m(ed)29 b(the)e(strong)g(v)m(ersion)h(of)f(Aubry)h(dualit)m(y)e(theorem:)328 4296 y(that)32 b(pure)i(p)s(oin)m(t)d(sp)s(ectrum)i(for)g(the)g(dual)e (mo)s(del)g(implies)g(purely)i(ac)f(sp)s(ectrum)h(for)328 4417 y(a.e.)56 b Fn(\022)s Fo(.)h(While)36 b(the)h(theorem)g(in)f([11)o (])h(w)m(as)h(stated)g(for)e(almost)f(Mathieu)i(op)s(erators)328 4537 y(only)-8 b(,)37 b(it)e(holds)h(in)g(higher)f(generalit)m(y)-8 b(,)37 b(whic)m(h)g(will)d(b)s(e)i(discussed)i(b)s(elo)m(w.)55 b(The)37 b(only)328 4658 y(previous)c(nonp)s(erturbativ)m(e)g(result)f (on)h(pure)g(ac)f(sp)s(ectrum)h(w)m(as)h(established)e(in)g([14])328 4778 y(for)i(the)i(almost)d(Mathieu)i(op)s(erator)f(for)h(an)m(y)g Fn(\025)d(>)f Fo(1)p Fn(=)p Fo(2)k(and)g(Diophan)m(tine)e Fn(!)t Fo(,)i(using)328 4898 y(the)e(dualit)m(y)f(theorem,)g(as)h(a)f (corollary)f(of)h(lo)s(calization)c(for)k Fn(\025)c(<)g Fo(1)p Fn(=)p Fo(2)p Fn(:)1922 5214 y Fo(1)p eop %%Page: 2 3 2 2 bop 474 631 a Fo(In)38 b(this)f(pap)s(er)g(w)m(e)h(follo)m(w)e(the) h(same)g(strategy)h(in)f(that)g(w)m(e)h(deduce)h(the)f(purely)328 751 y(ac)30 b(sp)s(ectrum)h(b)m(y)f(dualit)m(y)g(from)e(the)j(pure)f(p) s(oin)m(t)g(sp)s(ectrum)g(for)g(the)g(dual)f(mo)s(del.)41 b(In)328 872 y(case)31 b(of)f(general)g Fn(f)41 b Fo(the)31 b(dual)e(mo)s(del)g(is)h(going)f(to)h(b)s(e)h(long-range)d(whic)m(h)j (creates)h(cer-)328 992 y(tain)k(obstacles)i(in)e(applying)g (previously)i(dev)m(elop)s(ed)g(nonp)s(erturbativ)m(e)g(argumen)m(ts) 328 1112 y([14,)28 b(3)o(])g(based)h(on)f(Ly)m(apuno)m(v)h(exp)s(onen)m (ts)h(and)e(transfer-matrices.)41 b(While)27 b(p)s(erturba-)328 1233 y(tiv)m(e)c(analysis)g(t)m(ypically)f(extends)k(without)d(substan) m(tial)g(problems)f(to)i(the)f(long-range)328 1353 y(op)s(erators,)28 b(it)e(is)g(not)h(the)h(case)g(with)e(nonp)s(erturbativ)m(e)i(pro)s (ofs.)41 b(The)28 b(main)d(tec)m(hnical)328 1474 y(ac)m(hiev)m(emen)m (t)33 b(of)f(this)g(pap)s(er)g(is)g(precisely)g(in)f(dev)m(eloping)h(a) g(nonp)s(erturbativ)m(e)h(argu-)328 1594 y(men)m(t)43 b(for)g(lo)s(calization)c(in)j(long-range)g(Hamiltonians.)72 b(W)-8 b(e)43 b(note)g(that)g(a)g(related)328 1714 y(tec)m(hnique)34 b(w)m(as)f(used)h(in)e([4])g(for)g(the)h(pro)s(of)f(of)g(lo)s (calization)d(in)j(the)h(band)g(mo)s(del.)474 1835 y(W)-8 b(e)33 b(will)e(no)m(w)i(recall)e(the)i(Aubry)h(dualit)m(y)d(setup.)45 b(Let)33 b Fj(H)h Fo(=)28 b Fn(L)2854 1799 y Fl(2)2893 1835 y Fo(\([0)p Fn(;)17 b Fo(1\))22 b Fj(\002)g Fk(Z)n Fo(\))p Fn(:)33 b Fo(Let)328 1955 y Fn(A)401 1970 y Fm(!)r(;\025)545 1955 y Fo(b)s(e)g(the)g(direct)f(in)m(tegral)f(in)h Fn(\022)j Fo(of)1808 1930 y(~)1782 1955 y Fn(H)1863 1970 y Fm(!)r(;\025;\022)2057 1955 y Fo(:)536 2197 y(\()p Fn(A)647 2212 y Fm(!)r(;\025)758 2197 y Fn(\036)p Fo(\)\()p Fn(\022)s(;)17 b(n)p Fo(\))28 b(=)f Fn(\036)p Fo(\()p Fn(\022)s(;)17 b(n)22 b Fo(+)g(1\))g(+)g Fn(\036)p Fo(\()p Fn(\022)s(;)17 b(n)22 b Fj(\000)h Fo(1\))e(+)2372 2130 y(1)p 2368 2174 57 4 v 2368 2265 a Fn(\025)2435 2197 y(f)11 b Fo(\()p Fn(!)t(n)22 b Fo(+)g Fn(\022)s Fo(\)\010\()p Fn(\022)s(;)17 b(n)p Fo(\))208 b(\(1.3\))474 2417 y(The)34 b(dualit)m(y)d(transform)h Fn(U)43 b Fo(is)32 b(a)h(unitary)f(op)s(erator)g(on)g Fj(H)i Fo(acting)d(b)m(y)1347 2611 y(\()p Fn(U)10 b(\036)p Fo(\)\()p Fn(\014)c(;)17 b(m)p Fo(\))27 b(=)1966 2585 y(^)1954 2611 y Fn(\036)p Fo(\()p Fn(m;)17 b(\014)27 b Fo(+)22 b Fn(!)t(m)p Fo(\))818 b(\(1.4\))328 2806 y(where)622 2780 y(^)610 2806 y Fn(\036)32 b Fo(is)g(the)h(F)-8 b(ourier)31 b(transform.)474 2926 y(It)j(is)f(straigh)m(tforw)m(ard)f(to)h(c)m(hec)m(k)j(that)d Fn(U)2038 2890 y Fh(\000)p Fl(1)2133 2926 y Fn(\025A)2263 2941 y Fm(!)r(;\025)2374 2926 y Fn(U)40 b Fo(=)29 b Fn(B)2659 2941 y Fm(!)r(;\025)2770 2926 y Fn(;)k Fo(where)i Fn(B)3187 2941 y Fm(!)r(;\025)3331 2926 y Fo(is)e(the)328 3047 y(direct)f(in)m(tegral)f(in)h Fn(\022)k Fo(of)c Fn(H)1346 3062 y Fm(!)r(;\025;\022)1511 3047 y Fn(;)h Fo(where)h Fn(H)1934 3062 y Fm(!)r(;\025;\022)2132 3047 y Fo(is)e(giv)m(en)h(b)m (y)797 3320 y(\()p Fn(H)916 3335 y Fm(!)r(;\025;\022)1081 3320 y Fo(\010\)\()p Fn(n)p Fo(\))c(=)1529 3212 y Fh(1)1505 3237 y Fi(X)1455 3422 y Fm(k)r Fl(=)p Fh(\0001)1691 3320 y Fn(a)1742 3335 y Fm(k)1784 3320 y Fo(\010\()p Fn(n)23 b Fj(\000)g Fn(k)s Fo(\))f(+)g(2)p Fn(\025cos)p Fo(\()p Fn(!)t(n)f Fo(+)h Fn(\022)s Fo(\)\010\()p Fn(n)p Fo(\))269 b(\(1.5\))474 3579 y(W)-8 b(e)33 b(will)d(sa)m(y)k(that)e Fn(H)40 b Fo(is)33 b(dual)e(to)1787 3554 y(~)1762 3579 y Fn(H)7 b(:)474 3700 y Fo(Let)39 b Fn(H)736 3715 y Fm(\022)813 3700 y Fo(b)s(e)g(a)f(measurable)f(\(in)h Fn(\022)s Fo(\))g(family)e (of)i(self-adjoin)m(t)f(op)s(erators)h(on)g Fn(`)3384 3664 y Fl(2)3424 3700 y Fo(\()p Fk(Z)m Fo(\))328 3820 y(\(w)m(eak)k(measurabilit)m(y)-8 b(,)40 b(e.g.)68 b(Borel)40 b(measurabilit)m(y)f(of)h(matrix)f(elemen)m(ts)i(\()p Fn(H)3321 3835 y Fm(\022)3359 3820 y Fn(\036;)17 b( )t Fo(\))328 3941 y(is)35 b(su\016cien)m(t)h(here.\))53 b(Supp)s(ose)37 b Fn(H)1604 3956 y Fm(\022)1678 3941 y Fo(has)f(pure)g(p)s(oin)m(t)e(sp)s(ectrum)i(for)f(a.e.)52 b Fn(\022)s Fo(.)g(W)-8 b(e)36 b(will)328 4061 y(sa)m(y)31 b(that)e Fn(H)783 4076 y Fm(\022)852 4061 y Fo(admits)f(a)i(measurable) f(en)m(umeration)g(of)g(eigenelemen)m(ts)h(if)f(there)h(exist)328 4181 y(Borel)i(measurable)g(functions)h Fn(e)1563 4196 y Fm(`)1597 4181 y Fo(\()p Fn(\022)s Fo(\))28 b(:)g([0)p Fn(;)17 b Fo(1\))28 b Fj(!)g Fk(R)44 b Fo(and)33 b Fn(u)2518 4196 y Fm(`)2551 4181 y Fo(\()p Fn(n;)17 b(\022)s Fo(\))28 b(:)g([0)p Fn(;)17 b Fo(1\))28 b Fj(!)g Fn(`)3264 4145 y Fl(2)3303 4181 y Fo(\()p Fk(Z)n Fo(\))33 b(so)328 4302 y(that)28 b Fn(u)591 4317 y Fm(`)623 4302 y Fo(\()p Fn(n;)17 b(\022)s Fo(\))29 b(is)e(an)h(eigenfunction)g(of)g Fn(H)1885 4317 y Fm(\022)1951 4302 y Fo(with)g(an)g(eigen)m(v)-5 b(alue)28 b Fn(e)2807 4317 y Fm(`)2840 4302 y Fo(\()p Fn(\022)s Fo(\))g(and)h(functions)328 4422 y Fn(u)384 4437 y Fm(`)416 4422 y Fo(\()p Fn(n;)17 b(\022)s Fo(\))33 b(form)e(an)i(orthonormal)d(basis)i(for)h Fn(`)2029 4386 y Fl(2)2068 4422 y Fo(\()p Fk(Z)m Fo(\))g(for)f(a.e.)43 b Fn(\022)s(:)474 4542 y Fo(The)f(follo)m(wing)c(is)j(a)g(part)f(of)h (a)f(general)h(theorem)g(in)f([12)o(])h(\(see)h(also)e(a)h(related)328 4663 y(result)32 b(in)g([8]\):)328 4844 y Fe(THEOREM)k(2)49 b Fd(Every)44 b(me)-5 b(asur)g(able)42 b(family)i(of)f(selfadjoint)g (op)-5 b(er)g(ators)43 b(with)g(a.e.)328 4965 y(pur)-5 b(e)35 b(p)-5 b(oint)34 b(sp)-5 b(e)g(ctrum)35 b(admits)f(a)h(me)-5 b(asur)g(able)34 b(enumer)-5 b(ation)34 b(of)h(eigenelements.)1922 5214 y Fo(2)p eop %%Page: 3 4 3 3 bop 474 631 a Fo(Let)40 b(\()p Fn(T)14 b(\036)p Fo(\)\()p Fn(n)p Fo(\))38 b(=)h Fn(\036)p Fo(\()p Fn(n)27 b Fo(+)f(1\))39 b(b)s(e)g(a)g(shift)g(op)s(erator)g(on)g Fn(`)2594 595 y Fl(2)2633 631 y Fo(\()p Fk(Z)m Fo(\))p Fn(:)h Fo(Let)f(\(\002)p Fn(;)17 b(S)6 b Fo(\))38 b(b)s(e)i(an)328 751 y(ergo)s(dic)d(dynamical) f(system.)60 b(A)38 b(measurable)f(family)e Fn(H)2533 766 y Fm(\022)2572 751 y Fn(;)53 b(\022)40 b Fj(2)d Fo(\002)p Fn(;)g Fo(of)g(self-adjoin)m(t)328 872 y(op)s(erators)31 b(on)g Fn(`)933 836 y Fl(2)972 872 y Fo(\()p Fk(Z)m Fo(\))g(will)e(b)s (e)i(called)f(ergo)s(dic)g(if)g Fn(T)2230 836 y Fh(\000)p Fl(1)2324 872 y Fn(H)2405 887 y Fm(\022)2444 872 y Fn(T)42 b Fo(=)27 b Fn(H)2727 887 y Fm(S)t(\022)2813 872 y Fn(:)k Fo(W)-8 b(e)31 b(will)e(sa)m(y)j(that)328 992 y(an)g(ergo)s(dic)g (family)e(of)h(selfadjoin)m(t)g(op)s(erators)i(with)f(a.e.)43 b(pure)33 b(p)s(oin)m(t)f(sp)s(ectrum)g(has)328 1112 y(a)h(co)m(v)-5 b(arian)m(t)33 b(sp)s(ectral)g(represen)m(tation)h(if)e (there)i(exist)g(Borel)e(measurable)h(functions)328 1233 y Fn(e)373 1248 y Fm(`)406 1233 y Fo(\()p Fn(\022)s Fo(\))28 b(:)g([0)p Fn(;)17 b Fo(1\))27 b Fj(!)g Fk(R)38 b Fo(and)28 b Fn(v)1305 1248 y Fm(`)1338 1233 y Fo(\()p Fn(n;)17 b(\022)s Fo(\))27 b(:)h([0)p Fn(;)17 b Fo(1\))27 b Fj(!)g Fn(`)2048 1197 y Fl(2)2088 1233 y Fo(\()p Fk(Z)m Fo(\))g(so)h(that)f Fn(v)2625 1248 y Fm(`)2658 1233 y Fo(\()p Fn(n;)17 b(\022)s Fo(\))28 b(is)f(an)g(eigenfunc-)328 1353 y(tion)32 b(of)h Fn(H)722 1368 y Fm(\022)794 1353 y Fo(with)g(an)g(eigen)m(v)-5 b(alue)33 b Fn(e)1665 1368 y Fm(`)1698 1353 y Fo(\()p Fn(\022)s Fo(\))g(and)g(functions)h Fj(f)p Fn(v)2564 1368 y Fm(`)2597 1353 y Fo(\()p Fn(n)22 b Fj(\000)h Fn(j;)17 b(S)2965 1317 y Fm(j)3002 1353 y Fn(\022)s Fo(\))p Fj(g)3138 1371 y Fm(`;j)t Fl(:)p Fm(v)3274 1383 y Fc(`)3303 1371 y Fl(\()p Fm(S)3377 1352 y Fc(j)3409 1371 y Fm(\022)r Fl(\))p Fh(6)p Fl(=0)328 1474 y Fo(form)31 b(an)i(orthonormal)d(basis)i (for)h Fn(`)1682 1437 y Fl(2)1721 1474 y Fo(\()p Fk(Z)m Fo(\))p Fn(;)g Fo(in)f(particular,)1333 1575 y Fi(X)1372 1750 y Fm(n)p 1470 1574 605 4 v 1470 1658 a Fn(v)1517 1673 y Fm(m)1584 1658 y Fo(\()p Fn(n)22 b Fj(\000)h Fn(j;)17 b(S)1952 1629 y Fm(j)1988 1658 y Fn(\022)s Fo(\))p Fn(v)2121 1673 y Fm(`)2154 1658 y Fo(\()p Fn(n;)g(\022)s Fo(\))28 b(=)g(0)328 1900 y(for)k(all)e Fn(m;)17 b(`)28 b Fj(2)g Fk(Z)i Fo(and)j Fn(j)g Fj(6)p Fo(=)28 b(0)p Fn(:)474 2020 y Fo(Co)m(v)-5 b(arian)m(t)33 b(sp)s(ectral)g(represen)m(tations)h (ha)m(v)m(e)g(b)s(een)g(obtained)e(constructiv)m(ely)i(for)328 2140 y(random)24 b(Sc)m(hr\177)-49 b(odinger)26 b(op)s(erators)f(with)f (Mark)m(o)m(v)j(p)s(oten)m(tials)d(and)h(quasip)s(erio)s(dic)f(cos-)328 2261 y(t)m(yp)s(e)35 b(p)s(oten)m(tials)d(at)i(large)e(coupling)h(in,)g (corresp)s(ondingly)-8 b(,)33 b([18,)h(20)o(].)48 b(In)34 b([11)o(])g(it)f(w)m(as)328 2381 y(sho)m(wn)49 b(that)f(families)d(of)j (op)s(erators)g(with)g(simple)f(sp)s(ectrum)h(alw)m(a)m(ys)h(ha)m(v)m (e)g(suc)m(h)328 2501 y(represen)m(tations.)474 2622 y(As)33 b(an)g(immediate)d(corollary)h(of)h(Theorem)h(2)f(w)m(e)h (obtain)328 2810 y Fe(THEOREM)j(3)49 b Fd(Every)35 b(er)-5 b(go)g(dic)34 b(family)h(of)f(selfadjoint)g(op)-5 b(er)g(ators)34 b(with)h(a.e.)45 b(pur)-5 b(e)328 2930 y(p)g(oint)35 b(sp)-5 b(e)g(ctrum)34 b(has)h(a)f(c)-5 b(ovariant)34 b(sp)-5 b(e)g(ctr)g(al)35 b(r)-5 b(epr)g(esentation.)474 3118 y Fe(Pro)s(of:)43 b Fo(Set)33 b Fn(v)1035 3133 y Fm(`)1068 3118 y Fo(\()p Fn(\022)s Fo(\))28 b(=)f Fn(u)1379 3133 y Fm(`)1412 3118 y Fo(\()p Fn(\022)s Fo(\))32 b(if)e Fn(u)1712 3133 y Fm(`)1745 3118 y Fo(\()p Fn(\022)s Fo(\))i(has)g(it's) g(leftmost)e(maxim)m(um)g(at)i(zero.)43 b(Set)328 3238 y Fn(v)375 3253 y Fm(`)408 3238 y Fo(\()p Fn(\022)s Fo(\))32 b(=)g(0)j(otherwise.)51 b(Since)36 b Fn(v)1540 3253 y Fm(`)1573 3238 y Fo(\()p Fn(n)24 b Fj(\000)g Fn(j;)17 b(S)1944 3202 y Fm(j)1981 3238 y Fn(\022)s Fo(\))35 b(is)g(an)g (eigenfunction)f(of)h Fn(H)3137 3253 y Fm(\022)3210 3238 y Fo(with)g(it's)328 3359 y(leftmost)c(maxim)m(um)f(at)j Fn(j;)g Fo(the)g(theorem)f(follo)m(ws.)p 2335 3367 42 59 v 474 3558 a(Finally)e(w)m(e)k(will)c(need)k(the)f(follo)m(wing)d (dualit)m(y)h(theorem)i(pro)m(v)m(ed)h(in)d([11]:)328 3746 y Fe(THEOREM)36 b(4)49 b Fd(Supp)-5 b(ose)46 b(for)g(some)f Fn(!)t(;)17 b(\025)46 b Fd(op)-5 b(er)g(ator)46 b Fn(H)2555 3761 y Fm(!)r(;\025;\022)2767 3746 y Fd(given)f(by)i(\(1.5\))e(has)328 3867 y(pur)-5 b(e)40 b(p)-5 b(oint)40 b(sp)-5 b(e)g(ctrum)40 b(for)g(a.e.)60 b Fn(\022)s(;)40 b Fd(and)g(the)g(family)g Fn(H)2454 3882 y Fm(!)r(;\025;\022)2660 3867 y Fd(has)f(a)h(c)-5 b(ovariant)40 b(sp)-5 b(e)g(c-)328 3987 y(tr)g(al)44 b(r)-5 b(epr)g(esentation.)72 b(Then)43 b(dual)h(op)-5 b(er)g(ators)2159 3962 y Fo(~)2134 3987 y Fn(H)2215 4002 y Fm(!)r(;\025;\022)2425 3987 y Fd(given)43 b(by)h(\(1.1\))f(have)g (pur)-5 b(ely)328 4107 y(absolutely)35 b(c)-5 b(ontinuous)34 b(sp)-5 b(e)g(ctrum)35 b(for)g(a.e.)44 b Fn(\022)s(:)474 4295 y Fo(Giv)m(en)33 b(Theorems)g(3)f(and)h(4,)f(Theorem)h(1)g(follo)m (ws)e(immediately)e(from)328 4483 y Fe(THEOREM)36 b(5)49 b Fd(F)-7 b(or)33 b(any)g(r)-5 b(e)g(al-analytic)33 b Fn(f)45 b Fd(ther)-5 b(e)34 b(exists)f Fn(\025)2633 4498 y Fl(0)2672 4483 y Fo(\()p Fn(f)11 b Fo(\))34 b Fd(such)f(that)h(for)g (any)328 4604 y(Diophantine)47 b Fn(!)52 b Fd(ther)-5 b(e)49 b(is)g(a)f(set)h Fo(\002)k Fj(\032)h Fo([0)p Fn(;)17 b Fo(2)p Fn(\031)t Fo(\))48 b Fd(of)h(zer)-5 b(o)48 b(L)-5 b(eb)g(esgue)49 b(me)-5 b(asur)g(e)48 b(such)328 4724 y(that)41 b(for)f Fn(\022)53 b(=)-61 b Fj(2)39 b Fo(\002)p Fn(;)54 b(\025)38 b(>)g(\025)1308 4739 y Fl(0)1348 4724 y Fn(;)i Fd(op)-5 b(er)g(ator)40 b Fn(H)1885 4739 y Fm(!)r(;\025;\022) 2091 4724 y Fd(given)g(by)h(\(1.5\))e(has)h(only)h(pur)-5 b(e)40 b(p)-5 b(oint)328 4844 y(sp)g(e)g(ctrum)43 b(with)f(exp)-5 b(onential)5 b(ly)42 b(de)-5 b(c)g(aying)42 b(eigenfunctions.)67 b(Mor)-5 b(e)g(over,)44 b(the)f(family)328 4965 y Fn(H)409 4980 y Fm(!)r(;\025;\022)609 4965 y Fd(is)35 b(str)-5 b(ongly)35 b(dynamic)-5 b(al)5 b(ly)34 b(lo)-5 b(c)g(alize)g(d.)1922 5214 y Fo(3)p eop %%Page: 4 5 4 4 bop 328 631 a Fe(Remark.)43 b Fo(The)33 b(set)h(of)e(resonan)m(t)h (phases)h(\002)f(ma)m(y)f(b)s(e)h(giv)m(en)g(explicitly)d(b)m(y)328 845 y(\002)d(=)h Fj(f)p Fn(\022)j Fo(:)55 b(the)33 b(relation)26 b Fj(j)17 b Fo(sin)f(2)p Fn(\031)t Fo(\()p Fn(\022)s Fo(+\()p Fn(j)6 b(=)p Fo(2\))p Fn(!)t Fo(\))p Fj(j)25 b Fn(<)j Fo(exp)q(\()p Fj(\000)p Fn(k)2583 769 y Fb(1)p 2523 781 150 4 v 2523 822 a(2)p Fc(r)r Fb(\()p Fc(!)r Fb(\))2687 845 y Fo(\))g(holds)k(for)g(in\014nitely)f(man)m(y)d Fn(k)3881 804 y Fh(0)3904 845 y Fo(s)q Fj(g)p Fn(:)3365 966 y Fo(\(1.6\))474 1086 y(Com)m(bining)e(the)h(tec)m(hnique)h(of)f (this)f(pap)s(er)h(with)g(the)g(metho)s(d)g(of)f([3],)i(as)f(adapted) 328 1207 y(in)40 b([4],)i(one)f(can)g(obtain)e(a)i(measure-theoretic)f (v)m(ersion)h(\(that)g(is)f(a)g(result)g(for)g(a.e.)328 1327 y Fn(!)t(;)17 b(\022)s(;)29 b Fo(without)h(precise)h(arithmetic)e (conditions\))g(of)h(Theorem)g(5)g(for)g(op)s(erators)g(\(1.5\))328 1447 y(with)k(cos)g(replaced)g(b)m(y)h(an)f(analytic)f(function.)48 b(This,)34 b(ho)m(w)m(ev)m(er,)j(will)32 b(not)i(lead)f(to)h(a)328 1568 y(theorem)h(on)g(pure)h(ac)f(sp)s(ectrum)h(for)e(the)i(dual)f(mo)s (del)e(\(whic)m(h)j(w)m(ould)f(b)s(e)g(op)s(erator)328 1688 y(\(1.1\))43 b(with)f(a)h(long-range)f(Laplacian\))f(since)i (Theorem)h(4)f(is)f(not)h(kno)m(wn)i(in)d(suc)m(h)328 1808 y(setting.)g(A)31 b(conclusion)e(one)i(can)f(mak)m(e)h(for)f(the)g (dual)g(mo)s(del)e(based)k(on)e(lo)s(calization)328 1929 y(in)j(this)g(situation)f(is)h(con)m(tin)m(uit)m(y)h(of)f(the)h(sp)s (ectrum)g(for)f(all)e(phases.)48 b(W)-8 b(e)34 b(also)f(w)m(ould)328 2049 y(lik)m(e)26 b(to)h(men)m(tion)g(that)g(curren)m(t)h(tec)m (hniques)h(do)f(not)f(extend)h(to)f(op)s(erators)g(\(1.5\))g(with)328 2170 y(m)m(ulti-frequency)32 b(p)s(oten)m(tials.)474 2290 y(It)38 b(therefore)f(remains)g(to)g(pro)m(v)m(e)h(Theorem)g(5)e (whic)m(h)i(w)m(e)h(do)e(in)f(Section)h(2.)57 b(W)-8 b(e)328 2410 y(omit)29 b(the)j(pro)s(of)f(of)g(dynamical)e(lo)s (calization)e(statemen)m(t)32 b(as)g(it)e(is)h(a)g(straigh)m(tforw)m (ard)328 2531 y(adaptation)24 b(of)i(the)g(argumen)m(t)g(of)f([9],)i (and)f(is)f(not)h(needed)i(for)d(the)h(pro)s(of)f(of)g(Theorem)328 2651 y(1.)63 b(Note,)42 b(that)d(uniformit)m(y)e(of)i(the)g(upp)s(er)h (estimates)f(in)g(energy)h(is)f(immediate)e(in)328 2771 y(the)48 b(curren)m(t)i(pro)s(of,)h(therefore)d(our)g(Diophan)m(tine)f (condition)f(\(1.2\))i(on)g Fn(!)j Fo(is)c(less)328 2892 y(restrictiv)m(e)33 b(than)f(that)h(of)f([9].)328 3222 y Fp(2)161 b(Nonp)t(erturbativ)l(e)35 b(lo)t(calization)j(for)e (long-range)570 3404 y(Hamiltonians)328 3623 y Fo(A)g(formal)d (solution)i(\011)1195 3638 y Fm(E)1254 3623 y Fo(\()p Fn(x)p Fo(\))h(of)g(the)g(equation)g Fn(H)2190 3638 y Fm(!)r(;\025;\022)2355 3623 y Fo(\011)2431 3638 y Fm(E)2524 3623 y Fo(=)d Fn(E)6 b Fo(\011)2787 3638 y Fm(E)2882 3623 y Fo(is)36 b(called)e Fe(a)42 b(gen-)328 3744 y(eralized)34 b(eigenfunction)c Fo(if)f Fj(j)p Fo(\011)1615 3759 y Fm(E)1675 3744 y Fo(\()p Fn(x)p Fo(\))p Fj(j)e(\024)h Fn(C)7 b Fo(\(1)18 b(+)g Fj(j)p Fn(x)p Fj(j)p Fo(\))31 b(for)f(some)g Fn(C)35 b Fo(=)28 b Fn(C)7 b Fo(\(\011)3211 3759 y Fm(E)3270 3744 y Fo(\))28 b Fn(<)f Fj(1)p Fo(.)328 3864 y(It)i(is)f(w)m(ell)g(kno)m(wn)i(that)f(to)f(pro)m(v)m(e)i (Theorem)f(2)g(it)e(su\016ces)k(to)e(pro)m(v)m(e)h(that)e(generalized) 328 3984 y(eigenfunctions)36 b(deca)m(y)i(exp)s(onen)m(tially)e([2,)h (19)o(].)56 b(Fix)36 b(an)h(in)m(terv)-5 b(al)35 b Fn(I)43 b Fj(2)35 b Fk(Z)m Fn(:)i Fo(Let)g(\000)g(b)s(e)328 4105 y(the)c(coupling)e(op)s(erator)h(b)s(et)m(w)m(een)j Fn(I)40 b Fo(and)33 b Fk(Z)m Fj(n)p Fn(I)j Fo(:)1117 4366 y(\000\()p Fn(i;)17 b(j)6 b Fo(\))27 b(=)1508 4220 y Fi(\()1727 4305 y Fn(a)1778 4320 y Fm(i)p Fh(\000)p Fm(j)1894 4305 y Fn(;)44 b(\037)2026 4320 y Fm(I)2066 4305 y Fo(\()p Fn(i)p Fo(\))22 b(+)g Fn(\037)2356 4320 y Fm(I)2396 4305 y Fo(\()p Fn(j)6 b Fo(\))28 b(=)f(1)p Fn(;)1924 4425 y Fo(0)p Fn(;)100 b Fo(otherwise)p Fn(:)474 4631 y Fo(Then)33 b(w)m(e)f(can)g(write)f(\011)1372 4646 y Fm(E)1459 4631 y Fo(=)d Fj(\000)p Fo(\()p Fn(H)f Fj(\000)21 b Fn(E)k Fj(\000)c Fo(\000\))2178 4595 y Fh(\000)p Fl(1)2272 4631 y Fo(\000\011)2409 4646 y Fm(E)2500 4631 y Fo(from)30 b(whic)m(h)i(for)f(an)m(y)h Fn(x)c Fj(2)g Fn(I)328 4751 y Fo(w)m(e)34 b(obtain)1074 4872 y(\011)1150 4887 y Fm(E)1210 4872 y Fo(\()p Fn(x)p Fo(\))28 b(=)f Fj(\000)1637 4789 y Fi(X)1566 4977 y Fm(y)r Fh(2)p Fm(I)5 b(;k)10 b(=)-43 b Fh(2)p Fm(I)1844 4872 y Fn(G)1921 4887 y Fm(I)1961 4872 y Fo(\()p Fn(E)6 b Fo(\)\()p Fn(x;)17 b(y)t Fo(\))p Fn(a)2393 4887 y Fm(y)r Fh(\000)p Fm(k)2527 4872 y Fo(\011)2603 4887 y Fm(E)2662 4872 y Fo(\()p Fn(k)s Fo(\))p Fn(;)546 b Fo(\(2.1\))1922 5214 y(4)p eop %%Page: 5 6 5 5 bop 328 631 a Fo(where)42 b Fn(G)695 646 y Fm(I)776 631 y Fo(is)f(the)h(Green's)g(function)e(of)h Fn(H)49 b Fo(restricted)41 b(to)g(the)h(in)m(terv)-5 b(al)39 b Fn(I)51 b Fo(:)41 b Fn(G)3408 646 y Fm(I)3490 631 y Fo(=)328 751 y(\()p Fn(R)440 766 y Fm(I)480 751 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)918 766 y Fm(I)958 751 y Fo(\))996 715 y Fh(\000)p Fl(1)1091 751 y Fn(:)474 962 y Fo(Fix)36 b Fn(E)k Fj(2)34 b Fk(R)11 b Fn(;)51 b(m)34 b Fj(2)g Fk(R)11 b Fn(:)37 b Fo(A)f(p)s(oin)m(t)f Fn(y)j Fj(2)c Fk(Z)g Fo(will)g(b)s(e)i(called)f(\()p Fn(m;)17 b(k)s Fo(\)-regular)35 b(if)g(there)328 1082 y(exists)e(an)g(in)m(terv)-5 b(al)31 b Fn(I)36 b Fo(=)27 b([)p Fn(x)1349 1097 y Fl(1)1389 1082 y Fn(;)17 b(x)1488 1097 y Fl(2)1528 1082 y Fo(])p Fn(;)44 b(x)1681 1097 y Fl(2)1749 1082 y Fo(=)27 b Fn(x)1907 1097 y Fl(1)1969 1082 y Fo(+)22 b Fn(k)j Fj(\000)e Fo(1)p Fn(;)32 b Fo(con)m(taining)g Fn(y)t(;)f Fo(suc)m(h)j(that)1404 1219 y Fi(X)1309 1403 y Fm(x)p Fh(2)p Fm(I)5 b(;i)p Fl(=1)p Fm(;)p Fl(2)1637 1302 y Fj(j)p Fn(G)1742 1317 y Fm(I)1781 1302 y Fo(\()p Fn(y)t(;)17 b(x)p Fo(\))p Fn(a)2059 1317 y Fm(y)r Fh(\000)p Fm(x)2191 1327 y Fc(i)2221 1302 y Fj(j)27 b Fn(<)h(e)2425 1261 y Fh(\000)p Fm(mk)328 1607 y Fo(Otherwise,)33 b Fn(y)j Fo(will)30 b(b)s(e)j(called)e(\()p Fn(m;)17 b(k)s Fo(\)-singular.)474 1727 y(F)-8 b(rom)38 b(\(2.1\))g(w)m(e)i(see)f (that)g(if)f(\011)1679 1742 y Fm(E)1777 1727 y Fo(is)g(a)h(generalized) f(eigenfunction,)h(then)h(ev)m(ery)328 1847 y(p)s(oin)m(t)35 b Fn(y)h Fj(2)e Fk(Z)f Fo(with)i(\011)1173 1862 y Fm(E)1233 1847 y Fo(\()p Fn(y)t Fo(\))d Fj(6)p Fo(=)h(0)i(is)g(\()p Fn(m;)17 b(k)s Fo(\)-singular)34 b(for)h Fn(k)k Fo(su\016cien)m(tly)e (large.)52 b(Sim-)328 1968 y(ilarly)-8 b(,)33 b(if)h Fn(k)k Fo(is)c(\()p Fn(m;)17 b(k)s Fo(\)-regular)34 b(then)h Fj(j)p Fo(\011)1836 1983 y Fm(E)1896 1968 y Fo(\()p Fn(k)s Fo(\))p Fj(j)f Fo(is)h(b)s(ounded)g(b)m(y)h Fn(C)7 b(e)2849 1932 y Fh(\000)p Fm(mk)3044 1968 y Fo(so)36 b(it)e(su\016ces)328 2088 y(to)j(pro)m(v)m(e)j(that)d(for)h(an)f(appropriate)g Fn(m)h Fo(and)g(su\016cien)m(tly)h(large)e Fn(k)s(;)h Fo(\()p Fn(m;)17 b(k)s Fo(\)-singular)328 2209 y(p)s(oin)m(ts)32 b(cannot)h(b)s(e)g(at)f(a)g(distance)h Fn(k)j Fo(from,)31 b(sa)m(y)-8 b(,)34 b(zero.)474 2329 y(Let)f(\002)f(b)s(e)h(de\014ned)h (b)m(y)g(\(1.6\).)43 b(W)-8 b(e)33 b(will)d(pro)m(v)m(e)328 2557 y Fe(LEMMA)37 b(6)49 b Fd(Ther)-5 b(e)31 b(exists)h Fn(\025)1506 2572 y Fl(0)1545 2557 y Fo(\()p Fn(f)11 b Fo(\))31 b Fd(such)h(that)g(for)f Fn(\025)d(>)f(\025)2523 2572 y Fl(0)2562 2557 y Fn(;)32 b Fd(for)g(smal)5 b(l)30 b(enough)h Fn(m;)h(!)328 2678 y Fd(satisfying)k(\(1.2\),)g(and)g(any)h Fn(\022)45 b(=)-60 b Fj(2)31 b Fo(\002)p Fn(;)37 b Fd(we)f(have)g(that) h(for)f(every)h Fn(y)d Fj(2)d Fk(Z)m Fn(;)48 b(\013)32 b(<)e Fo(2)p Fn(;)37 b Fd(ther)-5 b(e)328 2798 y(exists)41 b Fn(k)s Fo(\()p Fn(\022)s(;)17 b(!)t(;)g(y)t(;)g(m;)g(\025;)g(\013)q Fo(\))p Fn(;)38 b Fd(such)j(that)h(for)f(al)5 b(l)40 b Fn(k)j(>)c(k)s Fo(\()p Fn(\022)s(;)17 b(!)t(;)g(y)t(;)g(m;)g(\025;)g (\013)q Fo(\))p Fn(;)38 b Fd(if)j Fn(x)h Fd(and)f Fn(y)328 2918 y Fd(ar)-5 b(e)31 b(b)-5 b(oth)32 b Fo(\()p Fn(m;)17 b(k)s Fo(\))p Fj(\000)p Fd(singular)32 b(for)f(some)g Fn(E)38 b Fd(in)31 b(the)h(sp)-5 b(e)g(ctrum)32 b(and)f Fj(j)p Fn(x)15 b Fj(\000)g Fn(y)t Fj(j)27 b Fn(>)3187 2879 y Fm(k)r Fl(+1)p 3187 2895 129 4 v 3233 2953 a(2)3325 2918 y Fn(;)32 b Fd(then)328 3039 y Fj(j)p Fn(x)22 b Fj(\000)h Fn(y)t Fj(j)j Fn(>)i(k)797 3003 y Fm(\013)846 3039 y Fn(:)328 3267 y Fe(Remarks:)447 3471 y Fo(1.)49 b(Relaxing)26 b(the)i(nonresonan)m(t)h(condition)e(on)h(the)g(excluded) h(set)f(\002)g(\(with)f(\002)h(still)572 3591 y(ha)m(ving)46 b(measure)g(0)p Fn(;)h Fo(namely)e(replacing)g(the)i(sub)s(exp)s(onen)m (tial)f(function)g(b)m(y)572 3711 y(p)s(olynomial)25 b(deca)m(y)k(in)f(\(1.6\)\))g(one)g(can)h(obtain)e(exp)s(onen)m(tial)h (in)f Fn(k)32 b Fo(repulsion)27 b(of)572 3832 y(singular)k(clusters)i (in)f(Lemma)f(6.)447 4035 y(2.)49 b(An)22 b(explicit)e(estimate)h(on)h Fn(\025)1616 4050 y Fl(0)1677 4035 y Fo(is)f(giv)m(en)h(in)f(a)g(com)m (bination)f(of)h(\(2.13\),\(2.14\),\(2.18\),\(2.19\).)474 4239 y(Theorem)35 b(5)e(no)m(w)i(follo)m(ws)e(in)g(the)h(standard)h(w)m (a)m(y)g(from)e(Lemma)f(6)i(and)g(the)h(dis-)328 4359 y(cussion)e(ab)s(o)m(v)m(e.)p 1029 4367 42 59 v 474 4579 a(W)-8 b(e)38 b(pro)s(ceed)h(to)e(the)h(pro)s(of)f(of)g(Lemma)f(6.)59 b(De\014ne)38 b Fn(I)44 b Fj(\032)37 b Fk(Z)e Fo(b)m(y)k Fn(I)44 b Fo(=)36 b([0)p Fn(;)17 b(N)36 b Fj(\000)26 b Fo(1])p Fn(;)328 4699 y(N)41 b Fj(2)31 b Fk(N)15 b Fn(:)34 b Fo(Fix)g Fn(E)i Fj(2)31 b Fk(R)11 b Fn(:)35 b Fo(W)-8 b(e)34 b(will)e(omit)g(the)j Fn(E)40 b Fo(dep)s(endence)d(of) d(v)-5 b(arious)33 b(quan)m(tities)i(in)328 4820 y(what)23 b(follo)m(ws,)h(and)e(all)f(constan)m(ts,)26 b(unless)e(sp)s (eci\014cally)e(claimed)f(to)h(b)s(e)h(dep)s(enden)m(t)i(on)328 4940 y Fn(E)6 b(;)24 b Fo(will)e(b)s(e)i(uniform)e(for)i(all)e Fn(E)30 b Fo(in)23 b(the)h(sp)s(ectrum.)42 b(W)-8 b(e)24 b(will)e(also)h(often)h(assume)g Fn(f)39 b Fj(6\021)28 b Fo(0)p Fn(;)1922 5214 y Fo(5)p eop %%Page: 6 7 6 6 bop 328 631 a Fo(as)39 b(otherwise)h(Theorems)f(1,5)g(b)s(ecome)g (trivial.)60 b(Let)39 b Fn(P)2462 646 y Fm(N)2529 631 y Fo(\()p Fn(\022)s Fo(\))g(=)f Fn(detR)3011 646 y Fm(I)3052 631 y Fo(\()p Fn(H)c Fj(\000)27 b Fn(E)6 b Fo(\))p Fn(R)3499 646 y Fm(I)3539 631 y Fo(.)328 751 y Fn(P)391 766 y Fm(N)458 751 y Fo(\()p Fn(\022)s Fo(\))36 b(is)f(an)h(ev)m(en)i(function)d(of)h Fn(\022)27 b Fo(+)1767 712 y Fm(N)7 b Fh(\000)p Fl(1)p 1767 728 154 4 v 1826 786 a(2)1930 751 y Fn(!)39 b Fo(and)d(can)g(b)s (e)h(written)e(as)i(a)e(p)s(olynomial)328 872 y(of)d(degree)i Fn(N)43 b Fo(in)31 b(cos)18 b(2)p Fn(\031)t Fo(\()p Fn(\022)25 b Fo(+)1448 833 y Fm(N)7 b Fh(\000)p Fl(1)p 1448 848 V 1507 906 a(2)1612 872 y Fn(!)t Fo(\))27 b(:)32 b Fn(P)1864 887 y Fm(N)1931 872 y Fo(\()p Fn(\022)s Fo(\))c(=)2187 805 y Fi(P)2274 832 y Fm(N)2274 896 y(j)t Fl(=0)2418 872 y Fn(c)2460 887 y Fm(j)2513 872 y Fo(cos)2643 836 y Fm(j)2696 872 y Fo(2)p Fn(\031)t Fo(\()p Fn(\022)d Fo(+)3020 833 y Fm(N)7 b Fh(\000)p Fl(1)p 3020 848 V 3079 906 a(2)3184 872 y Fn(!)t Fo(\))p Fn(:)328 1067 y Fe(LEMMA)37 b(7)908 996 y Fi(R)964 1022 y Fl(1)947 1092 y(0)1044 1028 y(1)p 1030 1044 64 4 v 1030 1101 a Fm(N)1119 1067 y Fo(ln)16 b Fj(j)p Fn(P)1308 1082 y Fm(N)1375 1067 y Fo(\()p Fn(\022)s Fo(\))p Fj(j)p Fn(d\022)30 b Fj(\025)e Fo(ln)16 b Fn(\025:)328 1249 y Fe(Pro)s(of:)44 b Fo(It)34 b(is)e(pro)m(v)m(ed)j(b)m(y)f(a)f(standard)g(Herman's)h (subharmonicit)m(y)e(argumen)m(t)h([13].)p 328 1378 42 59 v 328 1576 a(Our)f(k)m(ey)j(tec)m(hnical)d(statemen)m(t)h(is)f(the)h (follo)m(wing)328 1758 y Fe(LEMMA)k(8)49 b Fd(Ther)-5 b(e)29 b(exists)g Fn(\025)1501 1773 y Fl(0)1541 1758 y Fo(\()p Fn(f)11 b Fo(\))29 b Fd(such)g(that)h(for)g Fn(\025)d(>)h(\025)2511 1773 y Fl(0)2550 1758 y Fn(;)i Fd(and)f(smal)5 b(l)28 b(enough)h Fn(m;)17 b(\017;)328 1878 y Fd(if)33 b Fj(j)p Fn(P)512 1893 y Fm(N)578 1878 y Fo(\()p Fn(\022)21 b Fo(+)d Fn(x!)t Fo(\))p Fj(j)27 b Fn(>)g Fo(\()p Fn(\025)18 b Fj(\000)g Fn(\017)p Fo(\))1377 1842 y Fm(N)1445 1878 y Fn(;)33 b Fd(then)f(al)5 b(l)33 b(p)-5 b(oints)32 b Fn(y)f Fj(2)d Fn(x)18 b Fo(+)g Fn(I)41 b Fd(with)32 b Fn(dist)p Fo(\()p Fn(y)t(;)17 b(@)5 b Fo(\()p Fn(x)18 b Fo(+)g Fn(I)8 b Fo(\)\))27 b Fj(\025)338 1960 y Fm(N)p 338 1976 64 4 v 352 2033 a Fl(4)433 1999 y Fj(\000)c Fo(1)34 b Fd(ar)-5 b(e)35 b Fo(\()p Fn(m;)17 b(N)10 b Fo(\))p Fj(\000)p Fd(r)-5 b(e)g(gular)36 b(for)e (su\016ciently)h(lar)-5 b(ge)35 b Fn(N)5 b(:)p Fd(.)474 2181 y Fo(F)-8 b(or)35 b(this)g(Lemma)g Fn(!)j Fo(is)e(assumed)g(to)f (b)s(e)h(Diophan)m(tine.)51 b(W)-8 b(e)36 b(will)d(p)s(ostp)s(one)j (the)328 2302 y(pro)s(of)e(of)h(Lemma)f(8)h(to)f(the)i(end)g(of)e(the)i (section,)g(and)f(\014nish)h(the)f(pro)s(of)f(of)h(Lemma)328 2422 y(6.)70 b(The)42 b(next)h(Lemma,)f(pro)m(v)m(ed)h(in)e([14],)i (states,)i(essen)m(tially)-8 b(,)43 b(a)f(simple)e(fact)h(that)328 2542 y(logarithm)25 b(of)j(an)h(ev)m(en)h(trigonometric)c(p)s (olynomial)f(of)j(degree)i Fn(k)h Fo(cannot)e(b)s(e)g(smaller)328 2663 y(than)j(its)f(mean)h(v)-5 b(alue,)32 b(along)e(a)i(uniformly)e (distributed)h(sequence)k(of)c Fn(k)24 b Fo(+)d(1)32 b(p)s(oin)m(ts.)328 2783 y(W)-8 b(e)33 b(will)d(form)m(ulate)h(here)i (a)g(particular)e(form)g(con)m(v)m(enien)m(t)j(to)e(us.)474 2904 y(Set)896 3061 y Fn(\022)941 3076 y Fm(j)1006 3061 y Fo(=)1109 2915 y Fi(\()1510 3000 y Fn(\022)26 b Fo(+)c Fn(j)6 b(!)t(;)43 b(j)34 b Fo(=)27 b(0)p Fn(;)17 b Fo(1)p Fn(;)g(:::;)g Fo([)2385 2961 y Fm(k)r Fl(+1)p 2385 2977 129 4 v 2431 3035 a(2)2523 3000 y Fo(])22 b Fj(\000)h Fo(1)1329 3122 y Fn(\022)i Fo(+)d(\()p Fn(j)28 b Fj(\000)22 b Fo([)1739 3082 y Fm(k)r Fl(+1)p 1739 3098 V 1786 3156 a(2)1878 3122 y Fo(])g(+)g Fn(d)p Fo(\))p Fn(!)t(;)44 b(j)33 b Fo(=)28 b([)2464 3082 y Fm(k)r Fl(+1)p 2464 3098 V 2511 3156 a(2)2603 3122 y Fo(])p Fn(;)17 b(:)g(:)g(:)f(;)h(k) 2970 3061 y(;)474 3298 y Fo(where)34 b Fj(j)p Fn(d)p Fj(j)27 b Fn(>)g Fo([)1030 3259 y Fm(k)r Fl(+1)p 1030 3275 V 1077 3332 a(2)1169 3298 y Fo(])p Fn(;)33 b Fo(so)g(all)d Fn(\022)1556 3313 y Fm(j)1593 3298 y Fn(;)j(j)h Fo(=)27 b(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(k)s(;)32 b Fo(are)h(di\013eren)m(t.)328 3507 y Fe(LEMMA)k(9)49 b Fd(L)-5 b(et)38 b Fn(P)1142 3522 y Fm(k)1185 3507 y Fo(\()p Fn(\022)s Fo(\))33 b(=)1451 3440 y Fi(P)1539 3467 y Fm(k)1539 3531 y(j)t Fl(=0)1682 3507 y Fn(b)1723 3522 y Fm(j)1777 3507 y Fo(cos)1907 3471 y Fm(j)1960 3507 y Fo(2)p Fn(\031)t(\022)41 b Fd(b)-5 b(e)38 b(an)f(even)g(trigonometric)g(p)-5 b(olyno-)328 3627 y(mial.)41 b(Supp)-5 b(ose)24 b Fn(!)k Fd(satis\014es)c(\(1.2\))g (and)h Fn(\022)42 b(=)-61 b Fj(2)29 b Fo(\002)p Fn(:)24 b Fd(Supp)-5 b(ose)25 b Fo(ln)15 b Fj(j)p Fn(P)2651 3642 y Fm(k)2694 3627 y Fo(\()p Fn(\022)2777 3642 y Fm(j)2814 3627 y Fo(\))p Fj(j)27 b Fn(<)g Fo(sup)3157 3651 y Fm(\022)3213 3627 y Fo(ln)16 b Fj(j)p Fn(P)3402 3642 y Fm(k)3444 3627 y Fo(\()p Fn(\022)s Fo(\))p Fj(j\000)328 3747 y Fn(\017k)s(;)34 b(j)f Fo(=)28 b(0)p Fn(;)17 b(:)g(:)g(:)e(;)i(k)s(:)32 b Fd(Then)f(for)h(every)g Fn(\017)c(>)f Fo(0)p Fn(;)17 b(\013)28 b(<)g Fo(2)p Fn(;)j Fd(for)h Fn(k)f(>)c(k)s Fo(\()p Fn(\022)s(;)17 b(r)s Fo(\()p Fn(!)t Fo(\))p Fn(;)g(c)p Fo(\()p Fn(!)t Fo(\))p Fn(;)g(\017;)g(\013)q Fo(\))29 b Fd(we)328 3868 y(have)34 b(that)h Fj(j)p Fn(d)p Fj(j)27 b Fn(>)h(k)1043 3832 y Fm(\013)1092 3868 y Fn(:)474 4068 y Fo(T)-8 b(o)37 b(\014nish)g(the)h(pro)s(of)e(of)g(Lemma)f(6)i(w)m(e)h (tak)m(e)g Fn(\025)d(>)g(\025)2496 4083 y Fl(0)2572 4068 y Fo(giv)m(en)i(b)m(y)h(Lemma)d(8)i(and)328 4188 y(observ)m(e)24 b(that)e(if)e Fn(y)25 b Fo(is)d(\()p Fn(m;)17 b(k)s Fo(\))p Fj(\000)p Fo(singular,)23 b(then,)i(b)m(y)e(Lemma)e(8,)i Fj(j)p Fn(P)2723 4203 y Fm(k)2766 4188 y Fo(\()p Fn(\022)s Fo(+)q Fn(x!)t Fo(\))p Fj(j)k Fn(<)g Fo(\()p Fn(\025)q Fj(\000)q Fn(\017)p Fo(\))3496 4152 y Fm(k)3539 4188 y Fn(;)328 4309 y Fo(for)k(all)e Fn(x)f Fj(2)g Fo([)p Fn(y)23 b Fj(\000)d Fo([)1019 4270 y Fl(3)p 1019 4286 36 4 v 1019 4343 a(4)1065 4309 y Fo(])p Fn(k)s(;)d(y)23 b Fj(\000)d Fo([)1395 4270 y Fl(3)p 1395 4286 V 1395 4343 a(4)1440 4309 y Fo(])p Fn(k)j Fo(+)d([)1674 4270 y Fm(k)r Fl(+1)p 1674 4286 129 4 v 1720 4343 a(2)1812 4309 y Fo(]])p Fn(:)32 b Fo(Therefore,)h(if)d(b)s(oth)h Fn(x)h Fo(and)f Fn(y)k Fo(are)c(singular)328 4429 y(w)m(e)46 b(can,)h(taking)d(in)m(to)g(accoun)m(t)h(Lemma)e(7,)48 b(apply)c(Lemma)f(9)h(\(replacing)f Fn(\022)48 b Fo(with)328 4550 y Fn(\022)25 b Fo(+)d(\()544 4510 y Fm(N)7 b Fh(\000)p Fl(1)p 544 4526 154 4 v 603 4584 a(2)730 4550 y Fo(+)22 b Fn(y)t Fo(\))p Fn(!)t Fo(\))30 b(with)j Fn(d)27 b Fo(=)g Fn(x)c Fj(\000)g Fn(y)t(;)31 b Fo(to)i(obtain)e(that)i Fj(j)p Fn(x)22 b Fj(\000)g Fn(y)t Fj(j)27 b Fn(>)g(k)2846 4513 y Fm(\013)2896 4550 y Fn(:)p 3005 4558 42 59 v 474 4756 a Fo(It)41 b(no)m(w)g(remains)e(to)h(pro)m(v)m(e)i(Lemma)d(8.)66 b(Without)40 b(loss,)i(set)f Fn(x)g Fo(=)g(0)p Fn(:)f Fo(Set)h Fn(x)3409 4771 y Fl(1)3490 4756 y Fo(=)328 4876 y Fj(\000)p Fo(1)p Fn(;)k(x)581 4891 y Fl(2)648 4876 y Fo(=)28 b Fn(N)5 b(:)27 b Fo(It)g(will)e(b)s(e)i(enough)h(to)f(sho)m (w)h(that)2282 4793 y Fi(X)2182 4977 y Fm(z)s Fh(2)p Fm(I)5 b(;)11 b(i)p Fl(=1)p Fm(;)p Fl(2)2518 4876 y Fj(j)p Fn(G)2623 4891 y Fm(I)2662 4876 y Fo(\()p Fn(y)t(;)17 b(z)t Fo(\))p Fj(jj)p Fn(a)2990 4891 y Fm(z)s Fh(\000)p Fm(x)3121 4901 y Fc(i)3150 4876 y Fj(j)27 b Fn(<)h(e)3354 4835 y Fh(\000)p Fm(mN)3539 4876 y Fn(:)1922 5214 y Fo(6)p eop %%Page: 7 8 7 7 bop 328 631 a Fo(By)23 b(Cramer's)g(rule)g Fn(G)1142 646 y Fm(I)1181 631 y Fo(\()p Fn(y)t(;)17 b(z)t Fo(\))28 b(=)1574 583 y Fm(\026)1616 591 y Fc(y)r(;z)p 1543 608 193 4 v 1543 665 a Fm(P)1588 676 y Fc(N)1646 665 y Fl(\()p Fm(\022)r Fl(\))1768 631 y Fo(where)c Fn(\026)2099 646 y Fm(y)r(;z)2218 631 y Fo(is)e(the)i(corresp)s(onding)e(minor.)39 b(The)328 751 y(follo)m(wing)33 b(Lemma)h(is)h(crucial)f(for)h(our)g (nonp)s(erturbativ)m(e)h(argumen)m(t,)g(as)g(it)e(reduces)328 872 y(the)f(study)h(of)f(Green's)h(function)e(to)h(the)g(study)h(of)e (determinan)m(ts)h(of)g(restrictions)f(of)328 992 y Fn(H)27 b Fj(\000)20 b Fn(E)6 b(:)31 b Fo(The)h(statemen)m(t)g(\(and)f(the)h (pro)s(of)7 b(\))30 b(holds)h(for)f(Sc)m(hro)s(dinger-t)m(yp)s(e)i(op)s (erators)328 1112 y(on)g(an)m(y)i(graph.)328 1319 y Fe(LEMMA)j(10)968 1582 y Fn(\026)1027 1597 y Fm(y)r(;z)1151 1582 y Fo(=)1255 1499 y Fi(X)1295 1674 y Fm(\015)1391 1582 y Fn(\013)1453 1597 y Fm(\015)1514 1582 y Fo(det)17 b Fn(R)1740 1598 y Fm(I)5 b Fh(n)p Fm(\015)1856 1582 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)2294 1598 y Fm(I)f Fh(n)p Fm(\015)2444 1465 y Fh(j)p Fm(\015)t Fh(j)2431 1499 y Fi(Y)2427 1681 y Fm(i)p Fl(=1)2558 1582 y Fj(j)p Fn(a)2637 1597 y Fm(\015)2673 1607 y Fc(i)p Fb(+1)2776 1597 y Fh(\000)p Fm(\015)2867 1607 y Fc(i)2898 1582 y Fj(j)439 b Fo(\(2.2\))328 1858 y Fd(wher)-5 b(e)42 b(the)h(sum)f(is)g (taken)h(over)f(al)5 b(l)42 b(or)-5 b(der)g(e)g(d)42 b(subsets)h Fn(\015)k Fo(=)42 b(\()p Fn(\015)2738 1873 y Fl(1)2777 1858 y Fn(;)17 b(:)g(:)g(:)e(;)i(\015)3046 1873 y Fm(n)3093 1858 y Fo(\))42 b Fd(of)g Fn(I)51 b Fd(with)328 1978 y Fn(\015)379 1993 y Fl(1)446 1978 y Fo(=)27 b Fn(y)38 b Fd(and)c Fn(\015)875 1993 y Fm(n)950 1978 y Fo(=)27 b Fn(z)t(;)45 b Fj(j)p Fn(\015)5 b Fj(j)27 b Fo(=)h Fn(n)22 b Fj(\000)h Fo(1)p Fn(;)35 b Fd(and)f Fn(\013)1959 1993 y Fm(\015)2031 1978 y Fj(2)28 b(f\000)p Fo(1)p Fn(;)17 b Fo(1)p Fj(g)p Fn(:)328 2184 y Fe(Remark:)39 b Fo(The)26 b(sign)f Fn(\013)1235 2199 y Fm(\015)1305 2184 y Fo(actually)f(only)g(dep)s(ends)j(on)e(the)h(parit)m(y)f(of)g Fj(j)p Fn(\015)5 b Fj(j)24 b Fo(and)i(of)e Fn(x)7 b Fj(\000)g Fn(y)t(:)328 2305 y Fo(W)-8 b(e)33 b(will)d(not)j(need)g(this)f(here)i (so)f(omit)d(the)j(pro)s(of.)474 2425 y Fe(Pro)s(of.)69 b Fo(Expand)42 b Fn(\026)1275 2440 y Fm(y)r(;z)1412 2425 y Fo(through)f(the)h Fn(y)t Fj(\000)p Fo(column:)59 b Fn(\026)2550 2440 y Fm(y)r(;z)2688 2425 y Fo(=)2828 2342 y Fi(X)2806 2527 y Fm(\015)2842 2536 y Fb(2)2877 2527 y Fh(6)p Fl(=)p Fm(y)2986 2425 y Fn(\016)3029 2440 y Fm(\015)3065 2449 y Fb(2)3104 2425 y Fn(a)3155 2440 y Fm(\015)3191 2449 y Fb(2)3226 2440 y Fh(\000)p Fm(y)3322 2425 y Fn(h)3378 2440 y Fm(z)s(;y)r(;\015)3527 2449 y Fb(2)328 2631 y Fo(where)44 b Fn(\016)663 2646 y Fm(\015)699 2655 y Fb(2)784 2631 y Fj(2)i(f\000)p Fo(1)p Fn(;)17 b Fo(1)p Fj(g)p Fn(;)42 b Fo(and)i Fn(h)1541 2646 y Fm(z)s(;y)r(;\015) 1690 2655 y Fb(2)1771 2631 y Fo(is)e(the)i(determinan)m(t)e(of)h(the)h (original)39 b(matrix)328 2751 y(with)i Fn(z)46 b Fo(and)41 b Fn(y)j Fo(ra)m(ws)e(and)f Fn(y)k Fo(and)c Fn(\015)1714 2766 y Fl(2)1794 2751 y Fo(columns)g(remo)m(v)m(ed.)70 b(Con)m(tin)m(ue)42 b(this)f(pro)s(cess)328 2872 y(with)31 b(eac)m(h)h(term)f(un)m(til)f Fn(\015)1279 2887 y Fm(n)1353 2872 y Fo(=)e Fn(z)t(:)k Fo(The)g(con)m(tribution)f(of)g(eac)m(h)h (term)f(is)g(equal,)g(up)h(to)f(a)328 3074 y(sign,)h(to)g(det)17 b Fn(R)901 3090 y Fm(I)5 b Fh(n)p Fm(\015)1017 3074 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)1455 3090 y Fm(I)f Fh(n)p Fm(\015)1605 2957 y Fh(j)p Fm(\015)t Fh(j)1592 2991 y Fi(Y)1588 3174 y Fm(i)p Fl(=1)1719 3074 y Fj(j)p Fn(a)1798 3089 y Fm(\015)1834 3099 y Fc(i)p Fb(+1)1937 3089 y Fh(\000)p Fm(\015)2028 3099 y Fc(i)2059 3074 y Fj(j)p Fn(:)p 2163 3082 42 59 v 474 3348 a Fo(Our)43 b(next)g(Lemma)e(pro)m(vides)i(an)f(upp)s(er)h(b)s(ound)g(necessary)i (to)d(estimate)f Fn(\026)3442 3363 y Fm(y)r(;z)3539 3348 y Fn(:)328 3468 y Fo(Note)33 b(that)f(already)g(an)h(ob)m(vious)f(b)s (ound)h(of)f(the)h(form)884 3709 y Fj(j)17 b Fo(det)g Fn(R)1155 3725 y Fm(I)5 b Fh(n)p Fl(\003)1279 3709 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)1717 3725 y Fm(I)f Fh(n)p Fl(\003)1842 3709 y Fj(j)27 b(\024)h Fo(\(2)p Fj(j)p Fn(\025)p Fj(j)21 b Fo(+)h Fj(j)p Fn(E)6 b Fj(j)22 b Fo(+)g Fj(k)p Fn(f)11 b Fj(k)2734 3724 y Fh(1)2808 3709 y Fo(\))2846 3668 y Fh(j)p Fm(I)5 b Fh(n)p Fl(\003)p Fh(j)3365 3709 y Fo(\(2.3\))474 3875 y(w)m(ould)41 b(pro)m(vide)g(a)f(result)h(for)f(su\016cien)m(tly)h(large)f Fn(\026;)g Fo(ho)m(w)m(ev)m(er)j(a)e(more)f(delicate)328 3995 y(estimate)32 b(is)g(needed)i(for)e(small)e Fn(\026:)j Fo(Assume)g Fn(!)j Fo(is)c(Diophan)m(tine.)328 4202 y Fe(LEMMA)37 b(11)49 b Fd(F)-7 b(or)43 b(any)h Fo(\003)h Fj(\032)g Fn(I)8 b(;)44 b Fd(uniformly)g(for)f Fn(E)50 b Fd(in)44 b(the)g(sp)-5 b(e)g(ctrum)44 b(of)g Fn(H)r(;)g Fd(for)328 4322 y(su\016ciently)35 b(lar)-5 b(ge)34 b Fn(N)5 b(;)1132 4532 y Fj(j)17 b Fo(det)f Fn(R)1402 4547 y Fm(I)5 b Fh(n)p Fl(\003)1526 4532 y Fo(\()p Fn(H)30 b Fj(\000)23 b Fn(E)6 b Fo(\))p Fn(R)1965 4547 y Fm(I)f Fh(n)p Fl(\003)2089 4532 y Fj(j)27 b(\024)h Fn(\025)2306 4490 y Fm(N)2374 4532 y Fn(e)2419 4490 y Fm(C)2469 4502 y Fc(f)2509 4490 y Fm(\025)2550 4467 y Fa(\000)p Fb(1)p Fc(=)p Fb(2)2695 4490 y Fm(N)3365 4532 y Fo(\(2.4\))328 4732 y Fd(and)34 b(if)h Fj(j)p Fo(\003)p Fj(j)27 b Fn(>)g(\025)923 4696 y Fh(\000)p Fl(1)p Fm(=)p Fl(10)1123 4732 y Fn(N)46 b Fd(then)1201 4933 y Fj(j)17 b Fo(det)g Fn(R)1472 4948 y Fm(I)5 b Fh(n)p Fl(\003)1596 4933 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)2034 4948 y Fm(I)f Fh(n)p Fl(\003)2159 4933 y Fj(j)27 b(\024)h Fn(\025)2376 4891 y Fm(N)7 b Fh(\000)p Fl(1)p Fm(=)p Fl(5)p Fh(j)p Fl(\003)p Fh(j)3365 4933 y Fo(\(2.5\))1922 5214 y(7)p eop %%Page: 8 9 8 8 bop 474 631 a Fe(Pro)s(of.)474 751 y Fo(By)33 b(Hadamard's)g(b)s (ound,)413 1081 y Fj(j)17 b Fo(det)f Fn(R)683 1096 y Fm(I)5 b Fh(n)p Fl(\003)807 1081 y Fo(\()p Fn(H)30 b Fj(\000)23 b Fn(E)6 b Fo(\))p Fn(R)1246 1096 y Fm(I)f Fh(n)p Fl(\003)1370 1081 y Fj(j)27 b(\024)1577 998 y Fi(Y)1531 1186 y Fm(j)t Fh(2)p Fm(I)5 b Fh(n)p Fl(\003)1730 1081 y Fo(\(\(2)p Fn(\025)17 b Fo(cos)f(2)p Fn(\031)t Fo(\()p Fn(\022)25 b Fo(+)d Fn(j)6 b(!)t Fo(\))22 b Fj(\000)g Fn(E)6 b Fo(\))2775 1039 y Fl(2)2837 1081 y Fo(+)22 b Fj(k)p Fn(f)11 b Fj(k)3094 1039 y Fl(2)3094 1105 y(2)3133 1081 y Fo(\))3171 1039 y Fl(1)p Fm(=)p Fl(2)3365 1081 y Fo(\(2.6\))474 1375 y(Hence)386 1724 y(log)16 b Fj(j)h Fo(det)g Fn(R)799 1740 y Fm(I)5 b Fh(n)p Fl(\003)923 1724 y Fo(\()p Fn(H)30 b Fj(\000)22 b Fn(E)6 b Fo(\))p Fn(R)1361 1740 y Fm(I)f Fh(n)p Fl(\003)1486 1724 y Fj(j)27 b(\024)1656 1657 y Fo(1)p 1656 1701 49 4 v 1656 1793 a(2)1731 1641 y Fi(X)1734 1825 y Fm(j)t Fh(2)p Fm(I)1868 1724 y Fo(log)17 b Fj(j)p Fo(\(2)p Fn(\025)g Fo(cos)f(2)p Fn(\031)t Fo(\()p Fn(\022)25 b Fo(+)d Fn(j)6 b(!)t Fo(\))21 b Fj(\000)i Fn(E)6 b Fo(\))3046 1683 y Fl(2)3107 1724 y Fo(+)22 b Fj(k)p Fn(f)11 b Fj(k)3364 1683 y Fl(2)3364 1749 y(2)3403 1724 y Fj(j\000)3365 1929 y Fo(\(2.7\))1012 2094 y Fj(\000)1099 2027 y Fo(1)p 1099 2071 V 1099 2162 a(2)1179 2011 y Fi(X)1174 2195 y Fm(j)t Fh(2)p Fl(\003)1319 2094 y Fo(log)17 b Fj(j)p Fo(\(2)p Fn(\025)g Fo(cos)f(2)p Fn(\031)t Fo(\()p Fn(\022)25 b Fo(+)d Fn(j)6 b(!)t Fo(\))21 b Fj(\000)i Fn(E)6 b Fo(\))2497 2053 y Fl(2)2559 2094 y Fo(+)22 b Fj(k)p Fn(f)11 b Fj(k)2816 2053 y Fl(2)2816 2119 y(2)2854 2094 y Fj(j)483 b Fo(\(2.8\))474 2348 y(Assuming)25 b Fj(k)p Fn(f)11 b Fj(k)1074 2363 y Fl(2)1140 2348 y Fj(\025)28 b Fo(1)d(\(otherwise)h(w)m(e)g(will)c(use)k(b)s(ound)g (\(2.6\))e(with)h Fj(k)p Fn(f)11 b Fj(k)3150 2363 y Fl(2)3214 2348 y Fo(replaced)328 2469 y(b)m(y)27 b(1\))f(w)m(e)i(can)e(estimate)g Fj(j)17 b Fo(det)f Fn(R)1535 2484 y Fm(I)5 b Fh(n)p Fl(\003)1660 2469 y Fo(\()p Fn(H)17 b Fj(\000)10 b Fn(E)c Fo(\))p Fn(R)2073 2484 y Fm(I)f Fh(n)p Fl(\003)2197 2469 y Fj(j)26 b Fo(b)m(y)i(dropping)d(the)i(negativ)m(e)g(terms)328 2589 y(in)e(\(2.8\).)41 b(Let)25 b Fn(u)p Fo(\()p Fn(\022)s Fo(\))j(=)f(log)17 b Fj(j)p Fo(\(2)p Fn(\025cos)p Fo(2)p Fn(\031)t(\022)11 b Fj(\000)d Fn(E)e Fo(\))1997 2553 y Fl(2)2044 2589 y Fo(+)i Fj(k)p Fn(f)j Fj(k)2287 2553 y Fl(2)2287 2614 y(2)2326 2589 y Fj(j)p Fn(:)25 b Fo(The)i(righ)m (t-hand)d(side)i(of)f(\(2.7\))328 2709 y(is)k(a)h(sum)f(of)h(v)-5 b(alues)29 b(of)h Fn(u)f Fo(along)f(the)i(tra)5 b(jectory)31 b(of)e(rotation)f(b)m(y)j Fn(!)t(:)e Fo(W)-8 b(e)30 b(use)h(Denjo)m(y-) 328 2830 y(Koksma)46 b(t)m(yp)s(e)h(b)s(ound)g(\(for)e(a)h(particular)f (statemen)m(t)i(used)g(here)g(see,)k(e.g.)85 b([14],)328 2950 y(Lemma)31 b(12\))h(to)g(estimate)g(the)h(di\013erence)g(b)s(et)m (w)m(een)i(\(2.7\))d(and)h(in)m(tegral)e(of)h Fn(u:)869 3108 y Fi(X)871 3292 y Fm(j)t Fh(2)p Fm(I)1006 3191 y Fn(u)p Fo(\()p Fn(\022)24 b Fo(+)e Fn(j)6 b(!)t Fo(\))27 b Fj(\024)h Fn(N)1653 3074 y Fi(Z)1753 3191 y Fn(u)p Fo(\()p Fn(\022)s Fo(\))p Fn(d\022)d Fo(+)d Fn(C)7 b(N)2317 3142 y Fl(1)p Fh(\000)2462 3114 y Fb(1)p 2417 3126 120 4 v 2417 3168 a Fc(r)r Fb(\()p Fc(!)r Fb(\))2567 3191 y Fo(log)17 b Fn(N)10 b Fo(V)p Fn(ar)s(u)340 b Fo(\(2.9\))328 3445 y(for)33 b Fn(!)k Fo(satisfying)c(\(1.2\))g(and)h(an)m(y)h(con)m (tin)m(uous)f(function)g Fn(u;)f Fo(where)i Fn(C)41 b Fo(dep)s(ends)35 b(only)328 3565 y(on)d Fn(!)t(:)474 3615 y Fi(R)546 3686 y Fn(u)p Fo(\()p Fn(\022)s Fo(\))p Fn(d\022)j Fo(can)e(b)s(e)g(computed)g(explicitly)d(b)m(y)k(writing)999 3809 y Fi(Z)1098 3926 y Fn(u)p Fo(\()p Fn(\022)s Fo(\))p Fn(d\022)d Fo(=)c(2)17 b(log)f(2)p Fn(\025)22 b Fo(+)g(2)2008 3809 y Fi(Z)2107 3926 y Fo(log)17 b Fj(j)g Fo(cos)f(2)p Fn(\031)t(\022)25 b Fj(\000)e Fn(z)t Fj(j)p Fn(d\022)328 4159 y Fo(where)34 b Fn(z)e Fo(=)801 4107 y Fm(E)t Fl(+)p Fm(i)p Fh(k)p Fm(f)7 b Fh(k)1047 4083 y Fb(2)1047 4129 y(2)p 801 4135 281 4 v 903 4193 a Fl(2)p Fm(\025)1091 4159 y Fn(:)474 4279 y Fo(Hence,)34 b(using)f(that)1102 4427 y Fi(Z)1201 4545 y Fo(log)17 b Fj(j)p Fn(e)1417 4503 y Fl(2)p Fm(\031)r(i\022)1580 4545 y Fj(\000)23 b Fn(z)t Fj(j)28 b Fo(=)1889 4399 y Fi(\()2180 4484 y Fo(0)p Fn(;)100 b Fj(j)p Fn(z)t Fj(j)27 b Fn(<)h Fo(1)p Fn(;)2108 4604 y Fo(log)16 b Fj(j)p Fn(z)t Fj(j)p Fn(;)45 b Fj(j)p Fn(z)t Fj(j)27 b(\025)i Fo(1)p Fn(;)328 4810 y Fo(w)m(e)34 b(obtain)1112 4813 y Fi(Z)1211 4931 y Fn(u)p Fo(\()p Fn(\022)s Fo(\))p Fn(d\022)c Fo(=)e(2)17 b(log)f Fn(\025)22 b Fo(+)g(2)17 b(log)f Fj(j)p Fn(z)27 b Fo(+)2412 4844 y Fj(p)p 2495 4844 260 4 v 87 x Fn(z)2544 4902 y Fl(2)2606 4931 y Fj(\000)c Fo(1)o Fj(j)1922 5214 y Fo(8)p eop %%Page: 9 10 9 9 bop 328 631 a Fo(where)43 b(the)g(branc)m(h)g(of)f(the)h(square)g (ro)s(ot)f(is)f(c)m(hosen)j(so)f(that)f Fj(j)p Fn(z)33 b Fo(+)2954 549 y Fj(p)p 3037 549 260 4 v 82 x Fn(z)3086 602 y Fl(2)3148 631 y Fj(\000)23 b Fo(1)o Fj(j)44 b(\025)h Fo(1)p Fn(:)328 751 y Fo(Finally)-8 b(,)35 b(for)h Fn(E)42 b Fo(in)36 b(the)h(sp)s(ectrum,)h(w)m(e)g(ha)m(v)m(e)g Fn(E)i Fj(\024)35 b Fo(2)p Fn(\025)25 b Fo(+)f Fj(k)p Fn(f)11 b Fj(k)2690 766 y Fh(1)2764 751 y Fn(;)37 b Fo(and)g (therefore,)h(b)m(y)328 872 y(the)g(prop)s(erties)g(of)f Fn(z)30 b Fo(+)1257 790 y Fj(p)p 1340 790 V 82 x Fn(z)1389 843 y Fl(2)1452 872 y Fj(\000)22 b Fo(1)p Fn(;)38 b Fo(obtain)1973 801 y Fi(R)2045 872 y Fn(u)p Fo(\()p Fn(\022)s Fo(\))p Fn(d\022)h Fj(\024)e Fo(2)17 b(log)f Fn(\025)25 b Fo(+)h Fn(C)7 b(\025)3000 836 y Fh(\000)p Fl(1)p Fm(=)p Fl(2)3202 872 y Fo(where)39 b Fn(C)328 992 y Fo(dep)s(ends)f(only)f(on)f Fn(f)5 b(;)37 b Fo(through)g Fj(k)p Fn(f)11 b Fj(k)1716 1007 y Fl(2)1792 992 y Fo(and)36 b Fj(k)p Fn(f)11 b Fj(k)2144 1007 y Fh(1)2219 992 y Fn(:)36 b Fo(Com)m(bining)f(it)h(with)h (\(2.7\),\(2.9\),)328 1112 y(w)m(e)d(obtain)891 1353 y(log)17 b Fj(j)g Fo(det)f Fn(R)1304 1369 y Fm(I)5 b Fh(n)p Fl(\003)1429 1353 y Fo(\()p Fn(H)29 b Fj(\000)23 b Fn(E)6 b Fo(\))p Fn(R)1867 1369 y Fm(I)f Fh(n)p Fl(\003)1991 1353 y Fj(j)28 b(\024)g Fn(N)f Fo(log)17 b Fn(\025)22 b Fo(+)g Fn(C)2647 1368 y Fm(f)2692 1353 y Fn(N)10 b(\025)2837 1312 y Fh(\000)p Fl(1)p Fm(=)p Fl(2)3317 1353 y Fo(\(2.10\))328 1527 y(for)32 b(su\016cien)m(tly)h(large)f Fn(N)5 b(;)32 b Fo(whic)m(h)i(is)e(\(2.4\).)474 1647 y(T)-8 b(o)33 b(obtain)e(\(2.5\))h(w)m(e)i(need)g(a)e(lo)m(w)m(er)h(b)s(ound)f(on) 1084 1782 y Fi(X)1080 1966 y Fm(j)t Fh(2)p Fl(\003)1224 1865 y Fo(log)17 b Fj(j)p Fo(\(2)p Fn(\025)g Fo(cos)f(2)p Fn(\031)t Fo(\()p Fn(\022)25 b Fo(+)d Fn(j)6 b(!)t Fo(\))21 b Fj(\000)i Fn(E)6 b Fo(\))2402 1824 y Fl(2)2464 1865 y Fo(+)22 b Fj(k)p Fn(f)11 b Fj(k)2721 1824 y Fl(2)2721 1890 y(2)2759 1865 y Fj(j)p Fn(;)503 b Fo(\(2.11\))328 2180 y(where)34 b(\003)27 b Fj(2)h Fo([0)p Fn(;)17 b(N)32 b Fj(\000)23 b Fo(1])32 b(is)h(an)f(arbitrary)g(set)h(of)f(size)h Fj(j)p Fo(\003)p Fj(j)27 b Fn(>)g(\025)2645 2143 y Fh(\000)p Fl(1)p Fm(=)p Fl(10)2845 2180 y Fn(N)5 b(:)474 2300 y Fo(F)-8 b(or)32 b(an)m(y)h Fn(\022)j Fo(and)d Fn(\016)e Fj(2)d Fo(\(0)p Fn(;)17 b Fo(1\))32 b(one)h(has)328 2688 y(#)p Fj(f)p Fn(j)h Fo(=)27 b(0)p Fn(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fj(\000)p Fo(1)28 b(:)g Fj(k)1261 2621 y Fn(j)6 b(!)p 1261 2665 111 4 v 1291 2756 a Fo(2)1381 2688 y(+)p Fn(\022)s Fj(k)27 b Fn(<)h(\016)t Fj(g)f Fo(=)1914 2580 y Fm(N)7 b Fh(\000)p Fl(1)1930 2605 y Fi(X)1929 2787 y Fm(j)t Fl(=0)2084 2688 y Fn(\037)2145 2704 y Fl([)p Fh(\000)p Fm(\016)o(;\016)r Fl(])2326 2688 y Fo(\()p Fn(\022)s Fo(+)2498 2621 y Fn(j)f(!)p 2498 2665 V 2529 2756 a Fo(2)2618 2688 y(\))28 b Fj(\024)g Fo(2)p Fn(\016)t(N)10 b Fo(+)p Fn(C)d(N)3214 2623 y Fl(1)p Fh(\000)3369 2596 y Fb(1)p 3315 2608 140 4 v 3315 2653 a Fc(r)r Fb(\()3379 2633 y Fc(!)p 3379 2642 41 4 v 3384 2677 a Fb(2)3430 2653 y(\))3485 2688 y Fo(log)16 b Fn(N)3317 2892 y Fo(\(2.12\))474 3012 y(b)m(y)34 b(\(2.9\))e(applied)f(to)h(the)h(\(smo)s(othed\))g Fn(\037:)474 3132 y Fo(Set)1292 3346 y Fn(\022)1337 3361 y Fl(0)1404 3346 y Fo(=)1508 3147 y Fi(8)1508 3222 y(>)1508 3247 y(<)1508 3396 y(>)1508 3421 y(:)1734 3225 y Fo(arccos)2032 3186 y Fm(E)p 2021 3202 77 4 v 2021 3260 a Fl(2)p Fm(\025)2108 3225 y Fn(;)17 b Fj(j)p Fn(E)6 b Fj(j)26 b(\024)j Fo(2)p Fn(\025;)1883 3346 y Fo(0)p Fn(;)100 b(E)34 b(>)27 b Fo(2)p Fn(\025;)1840 3466 y(\031)t(;)99 b(E)34 b(<)28 b Fj(\000)p Fo(2)p Fn(\025;)328 3660 y Fo(With)k Fn(\016)g Fo(=)773 3613 y Fh(j)p Fl(\003)p Fh(j)p 768 3637 99 4 v 768 3694 a Fl(4)p Fm(N)909 3660 y Fo(w)m(e)h(can)g(estimate)430 4019 y(\(2)p Fn(:)p Fo(11\))27 b Fn(>)1010 3936 y Fi(X)810 4139 y Fm(j)t Fh(2)p Fl(\003:)p Fh(k)1004 4108 y Fc(j)s(!)p 1003 4124 70 4 v 1023 4165 a Fb(2)1082 4139 y Fh(\000)p Fm(\022)1170 4148 y Fb(0)1205 4139 y Fh(k)p Fm(>\016)1384 4019 y Fj(\025)h Fo(#)p Fj(f)p Fn(j)34 b Fj(2)28 b Fo(\003)f(:)h Fj(k)1998 3952 y Fn(j)6 b(!)p 1998 3996 111 4 v 2029 4088 a Fo(2)2140 4019 y Fj(\000)23 b Fn(\022)2285 4034 y Fl(0)2325 4019 y Fj(k)k Fn(>)h(\016)t Fj(g)p Fo(\(log)16 b(4)p Fn(\025\016)2936 3978 y Fl(2)2997 4019 y Fo(+)22 b Fj(k)p Fn(f)11 b Fj(k)3254 3978 y Fl(2)3254 4044 y(2)3293 4019 y Fj(j)p Fo(\))27 b Fj(\025)608 4422 y(\025)i Fo(\()p Fj(j)p Fo(\003)p Fj(j)21 b(\000)1006 4354 y(j)p Fo(\003)p Fj(j)p 1006 4398 124 4 v 1043 4490 a Fo(2)1161 4422 y Fj(\000)i Fn(C)7 b(N)1426 4357 y Fl(1)p Fh(\000)1581 4330 y Fb(1)p 1527 4342 140 4 v 1527 4386 a Fc(r)r Fb(\()1591 4367 y Fc(!)p 1591 4376 41 4 v 1596 4411 a Fb(2)1642 4386 y(\))1696 4422 y Fo(log)17 b Fn(N)10 b Fo(\))17 b(log\()2156 4354 y(1)p 2156 4398 49 4 v 2156 4490 a(4)2215 4422 y Fn(\025)2272 4380 y Fl(4)p Fm(=)p Fl(5)2382 4422 y Fo(\))27 b Fj(\025)2587 4354 y Fo(3)p 2562 4398 98 4 v 2562 4490 a(10)2670 4422 y Fj(j)p Fo(\003)p Fj(j)17 b Fo(log)e Fn(\025;)281 b Fo(\(2.13\))328 4626 y(for)36 b Fn(\025)h Fo(large)f(\(the)h(largeness)h(condition)d(is)i(only)f (needed)j(for)d(the)i(last)e(inequalit)m(y)g(\).)328 4746 y(Finally)-8 b(,)30 b(for)i Fn(\025)g Fo(suc)m(h)i(that)1437 4965 y Fn(C)1507 4980 y Fm(f)1552 4965 y Fn(\025)1609 4924 y Fh(\000)1674 4896 y Fb(1)p 1673 4908 31 4 v 1673 4950 a(2)1746 4965 y Fn(<)27 b Fo(1)p Fn(=)p Fo(10)p Fn(\025)2102 4924 y Fh(\000)2181 4896 y Fb(1)p 2166 4908 61 4 v 2166 4950 a(10)2258 4965 y Fo(log)16 b Fn(\025)860 b Fo(\(2.14\))1922 5214 y(9)p eop %%Page: 10 11 10 10 bop 328 631 a Fo(estimate)32 b(\(2.5\))g(follo)m(ws)f (immediately)e(from)j(\(2.8\),\(2.10\))f(and)i(\(2.13\))p 3056 639 42 59 v 474 849 a Fe(Remark.)64 b Fo(By)40 b(c)m(ho)s(osing)f Fj(j)p Fo(\003)p Fj(j)f Fn(>)h(\025)1854 813 y Fh(\000)p Fm(\014)1956 849 y Fn(N)50 b Fo(w)m(e)41 b(can)f(obtain)e(a)h(b)s(ound) h(in)e(\(2.5\))h(of)328 969 y(the)d(form)e Fn(\025)789 933 y Fm(N)7 b Fh(\000)p Fl(\(1)p Fh(\000)p Fl(3)p Fm(\014)s Fl(\))p Fh(j)p Fl(\003)p Fh(j)1258 969 y Fo(for)34 b(an)m(y)i Fn(\014)i(>)33 b Fo(0)i(and)g(su\016cien)m(tly)h(large)e Fn(\025:)i Fo(Ha)m(ving)f(smaller)328 1090 y Fn(\014)49 b Fo(w)m(ould)44 b(impro)m(v)m(e,)i(for)d(small)f Fn(\026;)h Fo(the)h(b)s(ounds)h(in)e(what)h(follo)m(ws)e(\(\(2.18\),\(2.19\)\),) 328 1210 y(ho)m(w)m(ev)m(er)33 b(require)f(larger)e Fn(\025)h Fo(for)g(estimates)f(\(2.13\),\(2.14\).)42 b(Larger)31 b Fn(\014)37 b Fo(ma)m(y)31 b(pro)m(vide)g(a)328 1330 y(b)s(etter)h(o)m(v)m(erall)e(b)s(ound)i(in)f(case)i(of)e(large)f Fn(\026:)h Fo(This)h(pro)s(of)f(can)h(therefore)g(b)s(e,)g(for)f(eac)m (h)328 1451 y(individual)f Fn(f)11 b Fo(,)32 b(optimized)f(in)h Fn(\014)38 b Fo(to)32 b(yield)g(a)g(lo)m(w)m(er)h(constan)m(t)g Fn(\025)2702 1466 y Fl(0)2742 1451 y Fn(:)474 1571 y Fo(W)-8 b(e)33 b(can)g(no)m(w)g(complete)f(the)h(pro)s(of)f(of)g(Lemma) f(8.)474 1692 y(Assuming)h Fj(j)p Fn(P)1013 1707 y Fm(N)1080 1692 y Fo(\()p Fn(\022)s Fo(\))p Fj(j)c Fn(>)f Fo(\()p Fn(\025)22 b Fj(\000)h Fn(\017)p Fo(\))1657 1655 y Fm(N)1724 1692 y Fn(;)33 b Fo(w)m(e)h(estimate,)d(using)i(\(2.2\):)427 1995 y Fi(X)328 2178 y Fm(z)s Fh(2)p Fm(I)5 b(;)11 b(i)p Fl(=1)p Fm(;)p Fl(2)663 2078 y Fj(j)p Fn(G)768 2093 y Fm(I)808 2078 y Fo(\()p Fn(y)t(;)17 b(z)t Fo(\))p Fj(jj)p Fn(a)1136 2093 y Fm(z)s Fh(\000)p Fm(x)1267 2103 y Fc(i)1296 2078 y Fj(j)27 b(\024)h Fo(\()p Fn(\025)p Fj(\000)p Fn(\017)p Fo(\))1705 2037 y Fh(\000)p Fm(N)1845 1970 y(N)7 b Fh(\000)p Fl(1)1861 1995 y Fi(X)1855 2176 y Fm(n)p Fl(=1)2179 1995 y Fi(X)2015 2183 y Fm(i)p Fl(=1)p Fm(;)p Fl(2;)k Fm(\015)t Fl(:)h Fh(j)p Fm(\015)t Fh(j)p Fl(=)p Fm(n)2480 2078 y Fj(j)17 b Fo(det)g Fn(R)2751 2093 y Fm(I)5 b Fh(n)p Fm(\015)2866 2078 y Fo(\()p Fn(H)j Fj(\000)p Fn(E)e Fo(\))p Fn(R)3260 2093 y Fm(I)f Fh(n)p Fm(\015)3376 2078 y Fj(jj)p Fn(a)3483 2093 y Fm(x)3523 2103 y Fc(i)3549 2093 y Fh(\000)p Fm(\015)3640 2107 y Fa(j)p Fc(\015)s Fa(j)p Fb(+1)3799 2078 y Fj(j)3879 1970 y Fm(n)3847 1995 y Fi(Y)3843 2177 y Fm(i)p Fl(=1)3974 2078 y Fj(j)p Fn(a)4053 2093 y Fm(\015)4089 2103 y Fc(i)p Fb(+1)4193 2093 y Fh(\000)p Fm(\015)4284 2103 y Fc(i)4314 2078 y Fj(j)668 2512 y Fo(=)27 b(\()p Fn(\025)22 b Fj(\000)h Fn(\017)p Fo(\))1065 2471 y Fh(\000)p Fm(N)1187 2512 y Fo(\()1225 2404 y Fm(\025)1266 2372 y Fa(\000)1340 2352 y Fb(1)p 1324 2361 61 4 v 1324 2397 a(10)1395 2372 y Fc(N)1279 2429 y Fi(X)1273 2610 y Fm(n)p Fl(=1)1497 2512 y Fo(+)1697 2429 y Fi(X)1590 2650 y Fm(n>\025)1729 2619 y Fa(\000)1802 2599 y Fb(1)p 1787 2608 V 1787 2643 a(10)1862 2650 y Fm(N)1952 2512 y Fo(\))28 b(=)f(\()p Fn(\025)22 b Fj(\000)h Fn(\017)p Fo(\))2415 2471 y Fh(\000)p Fm(N)2538 2512 y Fo(\()2576 2445 y Fi(P)2663 2532 y Fl(1)2719 2512 y Fo(+)2812 2445 y Fi(P)2899 2532 y Fl(2)2939 2512 y Fo(\))340 b(\(2.15\))474 2798 y(Using)32 b(\(2.4\),)h(w)m(e)g(b)s (ound)1453 2732 y Fi(P)1541 2819 y Fl(1)1613 2798 y Fo(b)m(y)328 3156 y Fi(P)416 3243 y Fl(1)483 3223 y Fj(\024)28 b Fn(\025)645 3186 y Fm(N)712 3223 y Fn(e)757 3186 y Fm(c)788 3198 y Fc(f)828 3186 y Fm(\025)869 3155 y Fa(\000)927 3135 y Fb(1)p 927 3144 31 4 v 927 3179 a(2)971 3186 y Fm(N)1055 3115 y(\025)1096 3083 y Fa(\000)1169 3063 y Fb(1)p 1154 3072 61 4 v 1154 3107 a(10)1229 3115 y Fm(N)1114 3140 y Fi(X)1107 3321 y Fm(n)p Fl(=1)1309 3223 y Fj(k)p Fn(f)11 b Fj(k)1468 3181 y Fm(n)1468 3247 y Fl(1)1586 3140 y Fi(X)1531 3324 y Fm(i)p Fl(=1)p Fm(;)p Fl(2;)g Fm(b)1778 3223 y Fn(e)1823 3181 y Fh(\000)p Fm(\026b)1955 3223 y Fo(#)p Fj(f)p Fn(\015)33 b Fo(:)44 b Fj(j)p Fn(\015)5 b Fj(j)27 b Fo(=)h Fn(n;)33 b(\015)2653 3238 y Fl(1)2720 3223 y Fo(=)27 b Fn(y)t(;)33 b Fj(j)p Fn(\015)3014 3238 y Fm(n)p Fl(+1)3150 3223 y Fj(\000)p Fn(x)3282 3238 y Fm(i)3311 3223 y Fj(j)p Fo(+)3453 3115 y Fm(n)3415 3140 y Fi(X)3418 3322 y Fm(i)p Fl(=1)3551 3223 y Fj(j)p Fn(\015)3630 3238 y Fm(i)p Fl(+1)3748 3223 y Fj(\000)p Fn(\015)3876 3238 y Fm(i)3904 3223 y Fj(j)28 b Fo(=)f Fn(b)p Fj(g)1035 3657 y(\024)h Fn(\025)1197 3616 y Fm(N)1264 3657 y Fn(e)1309 3616 y Fm(c)1340 3628 y Fc(f)1380 3616 y Fm(\025)1421 3585 y Fa(\000)1479 3565 y Fb(1)p 1479 3574 31 4 v 1479 3609 a(2)1524 3616 y Fm(N)1744 3574 y Fi(X)1608 3763 y Fm(b)p Fh(\025)p Fm(dist)p Fl(\()p Fm(y)r(;@)t(I)5 b Fl(\))2016 3657 y Fn(e)2061 3616 y Fh(\000)p Fm(\026b)2210 3549 y(\025)2251 3518 y Fa(\000)2324 3498 y Fb(1)p 2309 3507 61 4 v 2309 3542 a(10)2384 3549 y Fm(N)2268 3574 y Fi(X)2262 3756 y Fm(n)p Fl(=1)2464 3657 y Fj(k)p Fn(f)11 b Fj(k)2623 3616 y Fm(n)2623 3682 y Fl(1)2669 3511 y Fi( )2743 3590 y Fn(b)2735 3726 y(n)2793 3511 y Fi(!)328 3923 y Fo(Assume)30 b Fj(k)p Fn(f)11 b Fj(k)845 3938 y Fl(1)911 3923 y Fj(\025)29 b Fo(1)g(\(otherwise)g(b)s(ound)h(it)e(b)m (y)i(1)p Fn(:)p Fo(\))f(F)-8 b(or)28 b Fn(y)k Fo(as)e(in)e(the)i (statemen)m(t)g(of)e(the)328 4043 y(Lemma)j(w)m(e)j(ha)m(v)m(e)g Fn(dist)p Fo(\()p Fn(y)t(;)17 b(@)5 b(I)j Fo(\))27 b Fn(>)g(N)r(=)p Fo(4)c Fj(\000)f Fo(1)p Fn(;)33 b Fo(therefore,)g(for)f (large)f Fn(N)5 b(;)740 4255 y Fi(P)827 4343 y Fl(1)894 4322 y Fj(\024)29 b Fn(\025)1057 4286 y Fm(N)1124 4322 y Fn(e)1169 4286 y Fm(c)1200 4298 y Fc(f)1239 4286 y Fm(\025)1280 4254 y Fa(\000)1338 4234 y Fb(1)p 1338 4243 31 4 v 1338 4279 a(2)1383 4286 y Fm(N)1450 4322 y Fn(\025)1507 4285 y Fh(\000)1587 4258 y Fb(1)p 1572 4270 61 4 v 1572 4311 a(10)1647 4322 y Fn(N)10 b Fj(k)p Fn(f)h Fj(k)1894 4286 y Fm(\025)1935 4254 y Fa(\000)2008 4234 y Fb(1)p 1993 4243 V 1993 4279 a(10)2068 4286 y Fm(N)1894 4346 y Fl(1)2288 4239 y Fi(X)2152 4427 y Fm(b)p Fh(\025)p Fm(dist)p Fl(\()p Fm(y)r(;@)t(I)5 b Fl(\))2560 4322 y Fn(e)2605 4281 y Fh(\000)p Fm(\026b)2737 4176 y Fi( )2925 4254 y Fn(b)2803 4390 y(\025)2860 4353 y Fh(\000)2940 4326 y Fb(1)p 2925 4338 V 2925 4379 a(10)3000 4390 y Fn(N)3088 4176 y Fi(!)328 4625 y Fo(By)33 b(Stierling)d(form)m(ula,)h (for)h(large)f Fn(N)5 b(;)1146 4752 y Fi( )1334 4831 y Fn(b)1212 4967 y(\025)1269 4930 y Fh(\000)1349 4903 y Fb(1)p 1333 4915 V 1333 4956 a(10)1408 4967 y Fn(N)1497 4752 y Fi(!)1590 4898 y Fj(\024)1696 4752 y Fi( )1884 4831 y Fn(b)1761 4967 y Fo(4)p Fn(\025)1867 4930 y Fh(\000)1947 4903 y Fb(1)p 1932 4915 V 1932 4956 a(10)2007 4967 y Fn(b)2048 4752 y Fi(!)2142 4898 y Fj(\024)28 b Fn(C)2317 4913 y Fm(\025)2362 4898 y Fn(e)2407 4857 y Fm(\036)p Fl(\(4)p Fm(\025)2552 4825 y Fa(\000)2626 4806 y Fb(1)p 2611 4815 V 2611 4850 a(10)2686 4857 y Fl(\))p Fm(b)1898 5214 y Fo(10)p eop %%Page: 11 12 11 11 bop 474 631 a Fo(where)34 b Fn(\036)p Fo(\()p Fn(x)p Fo(\))28 b(=)f Fj(\000)p Fn(x)17 b Fo(log)g Fn(x)23 b Fj(\000)g Fo(\(1)f Fj(\000)g Fn(x)p Fo(\))17 b(log)g Fn(x:)474 751 y Fo(Hence,)1005 926 y Fi(P)1093 1013 y Fl(1)1160 992 y Fj(\024)28 b Fn(\025)1322 956 y Fm(N)1389 992 y Fn(C)7 b(e)1511 955 y Fl(\()p Fm(c)1569 967 y Fc(f)1609 955 y Fm(\025)1650 924 y Fa(\000)1708 904 y Fb(1)p 1708 913 31 4 v 1708 948 a(2)1753 955 y Fl(+)p Fm(\025)1849 924 y Fa(\000)1922 904 y Fb(1)p 1907 913 61 4 v 1907 948 a(10)1994 955 y Fl(log)13 b Fh(k)p Fm(f)7 b Fh(k)2208 964 y Fb(1)2243 955 y Fl(+)2308 928 y Fb(1)p 2308 940 31 4 v 2308 981 a(4)2348 955 y Fm(\036)p Fl(\(4)p Fm(\025)2493 924 y Fa(\000)2567 904 y Fb(1)p 2552 913 61 4 v 2552 948 a(10)2627 955 y Fl(\))p Fh(\000)2719 924 y Fc(\026)p 2719 940 38 4 v 2722 981 a Fb(4)2767 955 y Fl(\))p Fm(N)2861 992 y Fn(;)429 b Fo(\(2.16\))474 1166 y(where)27 b Fn(C)33 b Fo(is)26 b(a)f(constan)m(t)i(dep)s(ending)f(on)g Fn(f)5 b(;)17 b(\026;)g(\025:)25 b Fo(Similarly)-8 b(,)23 b(using)j(\(2.5\),)g (w)m(e)h(ha)m(v)m(e)328 1478 y Fi(P)416 1565 y Fl(2)483 1544 y Fj(\024)h Fn(\025)645 1508 y Fm(N)712 1544 y Fn(e)757 1508 y Fm(c)788 1520 y Fc(f)828 1508 y Fm(\025)869 1476 y Fa(\000)927 1457 y Fb(1)p 927 1466 31 4 v 927 1501 a(2)971 1508 y Fm(N)1055 1461 y Fi(X)1094 1636 y Fm(n)1192 1544 y Fj(k)p Fn(f)11 b Fj(k)1351 1503 y Fm(n)1351 1569 y Fl(1)1397 1544 y Fn(\025)1454 1503 y Fh(\000)1519 1476 y Fb(1)p 1519 1488 V 1519 1529 a(5)1559 1503 y Fm(n)1759 1461 y Fi(X)1623 1649 y Fm(b)p Fh(\025)p Fm(dist)p Fl(\()p Fm(y)r(;@)t(I)5 b Fl(\))2031 1544 y Fn(e)2076 1503 y Fh(\000)p Fm(\026b)2208 1398 y Fi( )2282 1477 y Fn(b)2274 1613 y(n)2332 1398 y Fi(!)2426 1544 y Fj(\024)28 b Fn(\025)2588 1503 y Fm(N)2655 1544 y Fn(e)2700 1503 y Fm(c)2731 1515 y Fc(f)2770 1503 y Fm(\025)2811 1471 y Fa(\000)2870 1452 y Fb(1)p 2870 1461 V 2870 1496 a(2)2914 1503 y Fm(N)3134 1461 y Fi(X)2998 1649 y Fm(b)p Fh(\025)p Fm(dist)p Fl(\()p Fm(y)r(;@)t(I)5 b Fl(\))3407 1544 y Fn(e)3452 1503 y Fh(\000)p Fm(\026b)3584 1544 y Fo(\(1+)p Fj(k)p Fn(f)11 b Fj(k)3906 1559 y Fl(1)3944 1544 y Fn(\025)4001 1503 y Fh(\000)4066 1476 y Fb(1)p 4066 1488 V 4066 1529 a(5)4110 1544 y Fo(\))4148 1503 y Fm(b)3317 1757 y Fo(\(2.17\))474 1878 y(Substituting)39 b(\(2.16\),\(2.17\))g(in)m(to)g(\(2.15\))g(w)m (e)i(obtain)e(the)h(desired)g(exp)s(onen)m(tial)328 1998 y(deca)m(y)34 b(for)e(su\016cien)m(tly)h(small)d Fn(\017)j Fo(and)g(large)f Fn(N)43 b Fo(pro)m(vided)634 2363 y(log)867 2296 y Fn(\025)p 786 2340 219 4 v 786 2431 a(\025)23 b Fj(\000)f Fn(\017)1037 2363 y Fo(+)g Fn(c)1177 2378 y Fm(f)1222 2363 y Fn(\025)1279 2322 y Fh(\000)1344 2295 y Fb(1)p 1344 2307 31 4 v 1344 2348 a(2)1410 2363 y Fo(+)g Fn(\025)1565 2322 y Fh(\000)1645 2295 y Fb(1)p 1630 2307 61 4 v 1630 2348 a(10)1722 2363 y Fo(log)16 b Fj(k)p Fn(f)11 b Fj(k)2023 2378 y Fl(1)2084 2363 y Fo(+)2192 2296 y(1)p 2192 2340 49 4 v 2192 2431 a(4)2251 2363 y Fn(\036)p Fo(\(4)p Fn(\025)2453 2322 y Fh(\000)2532 2295 y Fb(1)p 2517 2307 61 4 v 2517 2348 a(10)2592 2363 y Fo(\))22 b Fj(\000)2762 2296 y Fn(\026)p 2762 2340 59 4 v 2767 2431 a Fo(4)2858 2363 y Fn(<)28 b Fo(0)306 b(\(2.18\))328 2617 y(and)909 2762 y(log)1142 2694 y Fn(\025)p 1062 2738 219 4 v 1062 2830 a(\025)22 b Fj(\000)h Fn(\017)1312 2762 y Fo(+)f Fn(c)1452 2777 y Fm(f)1497 2762 y Fn(\025)1554 2720 y Fh(\000)1619 2693 y Fb(1)p 1619 2705 31 4 v 1619 2747 a(2)1686 2762 y Fo(+)1794 2694 y(1)p 1794 2738 49 4 v 1794 2830 a(4)1869 2762 y(log\(1)g(+)g Fj(k)p Fn(f)11 b Fj(k)2361 2777 y Fl(1)2400 2762 y Fn(\025)2457 2720 y Fh(\000)2522 2693 y Fb(1)p 2522 2705 31 4 v 2522 2747 a(5)2566 2762 y Fo(\))22 b Fj(\000)2736 2694 y Fn(\026)p 2736 2738 59 4 v 2741 2830 a Fo(4)2832 2762 y Fn(<)28 b Fo(0)332 b(\(2.19\))p 378 2958 42 59 v 328 3383 a Fp(References)377 3602 y Fo([1])48 b(J.)36 b(Bellisard,)f(R.)h(Lima,)e(and)i(D.)g(T)-8 b(estard,)37 b(A)f(metal-insulator)c(transition)i(for)528 3722 y(the)f(almost)e(Mathieu)i(mo)s(del.)e(Comm)m(un.)h(Math.)h(Ph)m (ys.)h Fe(88)f Fo(\(1983\),)e(207{234)377 3926 y([2])48 b(Y.)d(Berezanskii,)k(Expansions)c(in)f(eigenfunctions)h(of)f (selfadjoin)m(t)g(op)s(erators.)528 4046 y(T)-8 b(ransl.)33 b(Math.)g(Monogr.,)f(V)-8 b(ol.)32 b(17.)g(Pro)m(vidence,)i(RI:)f(Am.)f (Math.)h(So)s(c.)f(1968.)377 4249 y([3])48 b(J.)70 b(Bourgain,)77 b(M.)70 b(Goldstein,)77 b(On)69 b(nonp)s(erturbativ)m(e)h(lo)s (calization)65 b(with)528 4370 y(quasip)s(erio)s(dic)31 b(p)s(oten)m(tial.)g(Annals)i(of)f(Math.,)h Fe(152)g Fo(\(2000\),)e(835-879)377 4573 y([4])48 b(J.)29 b(Bourgain,)g(S.)g (Jitomirsk)-5 b(a)m(y)m(a,)28 b(Anderson)i(lo)s(calization)25 b(for)j(the)h(band)g(mo)s(del,)528 4694 y(GAF)-11 b(A)54 b(\(1996-2000\),)i(67-79,)h(Lecture)e(Notes)f(in)f(Math.)h Fe(1745)p Fo(,)59 b(Springer,)528 4814 y(Berlin,)32 b(2000)1898 5214 y(11)p eop %%Page: 12 13 12 12 bop 377 631 a Fo([5])48 b(V.)40 b(Ch)m(ulaevsky)i(and)d(F.)g (Dely)m(on,)i(Purely)f(absolutely)e(con)m(tin)m(uous)i(sp)s(ectrum)528 751 y(for)32 b(almost)f(Mathieu)i(op)s(erators.)f(J.)h(Stat.)f(Ph)m (ys.)j Fe(55)e Fo(\(1989\),)e(1279{1284)377 955 y([6])48 b(E.)37 b(Dinaburg)e(and)h(Y)-8 b(a.)36 b(Sinai,)f(The)i (one-dimensional)d(Sc)m(hr\177)-49 b(odinger)36 b(equation)528 1075 y(with)d(a)f(quasi-p)s(erio)s(dic)e(p)s(oten)m(tial.)h(F)-8 b(unct.)33 b(Anal.)f(Appl.)g Fe(9)h Fo(\(1975\),)e(279{289)377 1279 y([7])48 b(L.)30 b(H.)g(Eliasson,)g(Flo)s(quet)f(solutions)g(for)g (the)i(one-dimensional)c(quasi-p)s(erio)s(dic)528 1399 y(Sc)m(hr\177)-49 b(odinger)33 b(equation.)g(Comm)m(un.)f(Math.)h(Ph)m (ys.)h Fe(146)f Fo(\(1992\),)e(447{482)377 1602 y([8])48 b(M.)29 b(F)-8 b(oreman,)28 b(A)g(descriptiv)m(e)h(view)g(of)f(ergo)s (dic)f(theory)-8 b(.)29 b(Descrptiv)m(e)g(set)g(theory)528 1723 y(and)f(dynamical)d(systems)k(\(Marseille-Lumin)m(y)c(1996\),)i (87-171,)g(London)g(Math.)528 1843 y(So)s(c.)33 b(Lect.)g(Notes)h(Ser.) f(277,)f(Cam)m(bridge)f(Univ.)i(Press,)h(Cam)m(bridge,)e(2000)377 2046 y([9])48 b(F.)34 b(Germinet,)g(S.)g(Jitomirsk)-5 b(a)m(y)m(a,)33 b(Strong)h(dynamical)e(lo)s(calization)e(for)k(the)g (al-)528 2167 y(most)e(Mathieu)h(mo)s(del,)e(Rev.)i(Math.)g(Ph)m(ys.,)i (to)d(app)s(ear.)328 2370 y([10])48 b(F.)43 b(Gesztesy)i(and)e(B.)g (Simon,)i(The)e(xi)g(function,)i(Acta)e(Math.)h Fe(176)f Fo(\(1996\),)528 2491 y(49{71)328 2694 y([11])48 b(A.)25 b(Gordon,)g(S.)f(Jitomirsk)-5 b(a)m(y)m(a,)25 b(Y.)f(Last,)i(and)e(B.)g (Simon,)h(Dualit)m(y)d(and)i(singular)528 2814 y(con)m(tin)m(uous)33 b(sp)s(ectrum)f(in)f(the)h(almost)e(Mathieu)i(equation.)f(Acta)h (Math.,)h Fe(178)528 2935 y Fo(\(1997\),)f(169{183)328 3138 y([12])48 b(A.)j(Gordon,)j(A.S.)d(Kec)m(hris,)k(Measurable)c(en)m (umeration)f(of)g(eigenelemen)m(ts.)528 3259 y(Appl.)33 b(Anal.)e Fe(71)i Fo(\(1999\),)f(41-61,)328 3462 y([13])48 b(M.)38 b(Herman,)f(Une)h(metho)s(de)e(p)s(our)h(minorer)e(les)i(exp)s (osan)m(ts)h(de)f(Ly)m(apuno)m(v)i(et)528 3582 y(quelques)f(exemples)e (mon)m(tran)m(t)g(le)f(caractere)h(lo)s(cal)e(d'un)i(theoreme)g (d'Arnold)528 3703 y(et)29 b(de)f(moser)f(sur)i(le)e(tore)h(en)g (dimension)f(2.Comm)m(un.)g(Math.)h(Helv.)g Fe(58)g Fo(\(1983\),)528 3823 y(453{502)328 4027 y([14])48 b(S.)37 b(Jitomirsk)-5 b(a)m(y)m(a,)36 b(Metal-Insulator)f(transition)g(for)h(the)g(almost)f (Mathieu)h(op-)528 4147 y(erator.)d(Annals)f(of)g(Math.)h Fe(150)g Fo(\(1999\),)e(1159-1175)328 4350 y([15])48 b(Y.)36 b(Last,)g(A)f(relation)e(b)s(et)m(w)m(een)k(absolutely)e(con)m (tin)m(uous)g(sp)s(ectrum)h(of)f(ergo)s(dic)528 4471 y(Jacobi)46 b(matrices)f(and)h(the)h(sp)s(ectra)f(of)g(p)s(erio)s(dic)e (appro)m(ximan)m(ts.)i(Comm)m(un.)528 4591 y(Math.)33 b(Ph)m(ys.)i Fe(151)d Fo(\(1993\),)g(183{192)328 4795 y([16])48 b(Y.)33 b(Last,)g(1993,)f(unpublished.)1898 5214 y(12)p eop %%Page: 13 14 13 13 bop 328 631 a Fo([17])48 b(Y.)38 b(Last)f(and)h(B.)f(Simon,)g (Eigenfuctions,)i(transfermatrices,)f(and)f(absolutely)528 751 y(con)m(tin)m(uous)e(sp)s(ectrum)f(of)g(one-dimensional)d(Sc)m (hr\177)-49 b(odinger)34 b(op)s(erators,)g(In)m(v)m(en)m(t.)528 872 y(Math.)f Fe(135)g Fo(\(1999\),)f(329{367)328 1075 y([18])48 b(S.A.Molc)m(hano)m(v,)33 b(The)f(structure)g(of)f (eigenfunctions)g(of)f(one-dimensional)f(dis-)528 1196 y(ordered)34 b(structures.)g(Math.)f(USSR)g(Izv.)h Fe(75)p Fo(,)e(179-196)f(\(1980\).)328 1399 y([19])48 b(B.)33 b(Simon,)e(Sc)m(hr\177)-49 b(odinger)33 b(semigroups,)f(Bull.)f(AMS)j Fe(7)e Fo(\(1982\),)g(447{526)328 1602 y([20])48 b(Y)-8 b(a.)92 b(Sinai,)105 b(Anderson)93 b(lo)s(calization)88 b(for)j(one-dimensional)e(di\013erence)528 1723 y(Sc)m(hr\177)-49 b(odinger)45 b(op)s(erator)f(with)g(quasi-p)s(erio)s(dic)e(p)s(oten)m (tial.)h(J.)i(Stat.)f(Ph)m(ys.)j Fe(46)528 1843 y Fo(\(1987\),)32 b(861{909)1898 5214 y(13)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0104181729718--