Content-Type: multipart/mixed; boundary="-------------9901250749900" This is a multi-part message in MIME format. ---------------9901250749900 Content-Type: text/plain; name="99-29.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-29.keywords" Dirac operators, singular perturbations, Coulomb potentials, limiting absorption principle, Mourre theory, spectral theory, scattering theory, matrix valued pseudo-differential operators ---------------9901250749900 Content-Type: application/postscript; name="dirarch.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dirarch.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: dirac10.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips dirac10.dvi -o dirarch.ps %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.01.25:1057 %%BeginProcSet: tex.pro 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Fy(Hyp)s(othesis)43 b(1.2.)j Fl(The)c(op)l(er)l(ator)f Ft(V)61 b Fw(:)42 b Fs(H)1468 4434 y Fr(1)p Fp(=)p Fr(2)1614 4464 y Fs(!)g(H)1810 4434 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)2006 4464 y Fl(c)l(an)e(b)l(e)h(written)f(as)g(the)g(sum)0 4564 y Ft(V)i Fw(=)23 b Ft(V)226 4576 y Fp(S)277 4564 y Fw(+)s Ft(V)393 4576 y Fp(L)465 4564 y Fl(of)g(a)g(short-r)l(ange)g (op)l(er)l(ator)h Ft(V)1418 4576 y Fp(S)1489 4564 y Fl(and)f(a)g (long-r)l(ange)f(smal)t(l)i(at)e(in\014nity)h(op)l(er)l(ator)0 4663 y Ft(V)48 4675 y Fp(L)98 4663 y Fl(.)42 b(The)32 b(op)l(er)l(ator)g Ft(H)g Fw(=)25 b Ft(H)924 4675 y Fr(0)980 4663 y Fw(+)19 b Ft(V)50 b Fl(is)31 b(such)g(that)g Ft(H)26 b Fs(\000)19 b Ft(z)28 b Fw(:)d Fs(H)1978 4633 y Fr(1)p Fp(=)p Fr(2)2108 4663 y Fs(!)g(H)2287 4633 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)2474 4663 y Fl(is)31 b(bije)l(ctive)i(for)0 4763 y(some)g Ft(z)t Fl(;)f(we)h(use)f(the)g(same)g(notation)h Ft(H)39 b Fl(for)33 b(the)f(self-adjoint)i(op)l(er)l(ator)f(in)f Fs(H)h Fl(asso)l(ciate)l(d)0 4862 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Ft(V)1628 1450 y Fr(\()p Fp(b)p Fr(\))1609 1518 y Fp(k)1736 1493 y Fs(k)1778 1508 y Fr(\(1)p Fp(=)p Fr(2\))1934 1493 y Ft(<)g Fs(1)p Ft(:)0 1680 y Fw(The)28 b(follo)n(wing)e(is)i(a)f (quite)h(explicit)g(su\016cien)n(t)f(condition)h(for)f(\(1-4\))g(to)g (hold:)506 1823 y Fm(")555 1852 y(Z)601 2041 y Fo(j)p Fp(x)p Fo(\000)p Fp(x)749 2024 y Fj(0)770 2041 y Fo(j\024)p Fr(1)893 1965 y Ft(dx)983 1931 y Fo(0)1030 1965 y Fs(k)22 b Ft(V)1161 1922 y Fr(\()p Fp(b)p Fr(\))1142 1990 y Fp(k)1247 1965 y Fw(\()p Ft(x)1326 1931 y Fo(0)1350 1965 y Fw(\))h Fs(k)1447 1931 y Fp(n)1447 1988 y(B)s Fr(\()p Fh(E)p Fr(\))1605 1823 y Fm(#)1653 1840 y Fr(1)p Fp(=n)1788 1965 y Fs(\024)g Ft(C)6 b Fs(j)p Ft(x)p Fs(j)2034 1931 y Fo(\000)p Fp(s)p Fo(\000)p Fr(1)p Fp(=)p Fr(2+)p Fp(k)q Fo(\000j)p Fp(b)p Fo(j)0 2227 y Fw(for)37 b(large)e Ft(x)j Fw(\(stronger)d(singularities)h(are)g(allo)n(w)n(ed)g(lo)r(cally\).)65 b(F)-7 b(or)36 b(the)i(de\014nition)f(of)g(the)0 2326 y(H\177)-42 b(older-Zygm)n(und)34 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Fw(is)j(compact)f(\(b)r(ecause)h Ft(w)r Fw(\()p Ft(P)12 b Fw(\))p Ft(')p Fw(\()p Ft(Q)p Fw(\))32 b(is)e(a)g(compact)f(op)r(erator)f(in)j Fs(H)q Fw(\))f(and)0 2385 y Ft(')p Fw(\()p Ft(Q)p Fw(\))24 b(:)f Fs(H)325 2355 y Fp(!)396 2385 y Fs(!)g(H)573 2355 y Fp(!)648 2385 y Fw(is)28 b(b)r(ounded.)37 b(Hence)28 b(for)f(0)c Ft(<)f(\022)k(<)c Fw(1)624 2605 y Ft(')p Fw(\()p Ft(Q)p Fw(\))i(:)f([)p Fs(H)q Ft(;)14 b Fs(H)1080 2571 y Fp(!)1128 2605 y Fw(])1151 2617 y Fp(\022)1211 2605 y Fw(=)23 b Fs(H)1370 2571 y Fp(!)1414 2546 y Fi(\022)1475 2605 y Fs(!)g Fw([)p Fs(H)1675 2571 y Fp(w)1728 2605 y Ft(;)14 b Fs(H)1836 2571 y Fp(!)1884 2605 y Fw(])1907 2617 y Fp(\022)1968 2605 y Fw(=)23 b Fs(H)2127 2571 y Fp(w)2177 2546 y Fk(1)p Fj(\000)p Fi(\022)2283 2571 y Fp(!)2327 2546 y Fi(\022)0 2808 y Fw(is)k(compact.)37 b(The)28 b(result)f(follo)n(ws)f(b)n(y)i(taking)f Ft(\022)e Fw(=)e(1)p Ft(=)p Fw(2.)100 2909 y(The)34 b(particular)g(case)f(of)i(the)g(usual)f 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Fw(is)c(a)e(lo)r(cal)h(op) r(erator)f(\(i.e)h(supp)14 b Ft(S)5 b(f)39 b Fs(\032)31 b Fw(supp)14 b Ft(f)41 b Fw(for)31 b(eac)n(h)h Ft(f)39 b Fs(2)32 b(H)2513 211 y Fp(!)2555 219 y Fk(1)2591 241 y Fw(\),)i(then)e(the)0 340 y(recipro)r(cal)26 b(assertion)g(is)i(also) e(true,)i(cf.)37 b(the)28 b(pro)r(of)f(of)g(Theorem)g(7.6.10)f(in)i ([ABG].)0 497 y Fl(Pr)l(o)l(of)g(of)f(the)g(the)l(or)l(em.)43 b Fw(Since)25 b Fs(S)6 b Fw(,)26 b(hence)e Ft(C)1411 467 y Fo(1)1405 517 y Fp(c)1481 497 y Fw(\()p Fq(R)1568 467 y Fp(n)1619 497 y Fw(;)14 b Fy(E)p Fw(\),)25 b(is)f(dense)g(in)2220 476 y(\025)2196 497 y Fs(H)2267 460 y Fp(!)2309 468 y Fk(1)2266 525 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)p Fp(;)p Fo(1)2508 497 y Fw(,)h(it)g(su\016ces)f(to)0 606 y(sho)n(w)j(that)h (there)f(is)g Ft(C)j(>)22 b Fw(0)28 b(suc)n(h)f(that)0 801 y(\(2-21\))792 b Fs(k)p Ft(S)5 b(f)k Fs(k)1200 822 y Fo(H)1257 790 y Fi(!)1294 802 y Fk(2)1257 845 y(1)p Fi(=)p Fk(2)p Fi(;)p Fk(1)1421 801 y Fs(\024)22 b Ft(C)6 b Fs(k)p Ft(f)j Fs(k)1707 822 y Fo(H)1764 790 y Fi(!)1801 802 y Fk(1)1764 845 y Fj(\000)p Fk(1)p Fi(=)p Fk(2)p Fi(;)p Fj(1)0 1021 y Fw(for)27 b(all)h Ft(f)k Fs(2)24 b Ft(C)460 991 y Fo(1)454 1042 y Fp(c)530 1021 y Fw(\()p Fq(R)617 991 y Fp(n)668 1021 y Fw(;)14 b Fy(E)p Fw(\).)37 b(W)-7 b(e)29 b(can)e(further)h(simplify)g(later)f(argumen)n(ts)g(b)n (y)g(observing)g(that)0 1121 y(it)f(is)g(enough)f(to)h(pro)n(v)n(e)e (this)j(under)e(the)i(assumption)e Ft(f)9 b Fw(\()p Ft(x)p Fw(\))24 b(=)e(0)k(if)g Fs(j)p Ft(x)p Fs(j)e Ft(<)f Fw(1.)36 b(Indeed,)26 b(c)n(ho)r(ose)0 1220 y Ft( )g Fs(2)e Ft(C)224 1190 y Fo(1)218 1241 y Fp(c)294 1220 y Fw(\()p Fq(R)380 1190 y Fp(n)432 1220 y Fw(\))k(with)g Ft( )s Fw(\()p Ft(x)p Fw(\))c(=)e(1)28 b(if)g Fs(j)p Ft(x)p Fs(j)23 b(\024)g Fw(1;)k(then)513 1415 y Fs(k)p Ft(S)5 b(f)k Fs(k)703 1437 y Fo(H)760 1405 y Fi(!)797 1417 y Fk(2)760 1459 y(1)p Fi(=)p Fk(2)p Fi(;)p Fk(1)924 1415 y Fs(\024)23 b(k)p Ft(S)5 b Fw([1)17 b Fs(\000)h Ft( )s Fw(\()p Ft(Q)p Fw(\)])p Ft(f)9 b Fs(k)1577 1437 y 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Fs(\032)g Ft(B)1392 4918 y Fm(\020)1441 5010 y Fs(H)1512 4967 y Fr(1)p Fp(=!)1511 5039 y Fr(1)p Fp(=)p Fr(2)p Fp(;)p Fr(1)1669 5010 y Ft(;)14 b Fs(H)1777 4976 y Fp(!)1776 5033 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)p Fp(;)p Fo(1)2018 4918 y Fm(\021)2081 5010 y Ft(:)p eop %%Page: 22 22 22 21 bop 0 42 a Fu(22)785 b(V.)30 b(GEOR)n(GESCU)h(AND)g(M.)e(M)1822 25 y(\025)1813 42 y(ANTOIU)0 241 y Fw(Under)f(suitable)f(h)n(yp)r (otheses)g(the)h(resolv)n(en)n(t)e(\()p Ft(H)f Fs(\000)18 b Ft(z)t Fw(\))1760 211 y Fo(\000)p Fr(1)1849 241 y Fw(,)27 b(whic)n(h)h(do)r(es)f(not)g(satisfy)h(an)f(uni-)0 340 y(form)d(estimate)h(in)f Ft(B)t Fw(\()p Fs(G)769 310 y Fo(\003)808 340 y Ft(;)14 b Fs(G)5 b Fw(\))25 b(\(or)f(in)h Ft(B)t Fw(\()p Fs(H)q Fw(\)\))g(when)g Ft(z)j Fw(approac)n(hes)22 b(the)j(sp)r(ectrum)f(of)h Ft(H)7 b Fw(,)25 b(will)0 465 y(ha)n(v)n(e)h(b)r(oundary)h(v)-5 b(alues)27 b(in)h Ft(B)990 373 y Fm(\020)1040 465 y Fs(H)1111 422 y Fr(1)p Fp(=!)1110 493 y Fr(1)p Fp(=)p Fr(2)p Fp(;)p Fr(1)1267 465 y Ft(;)14 b Fs(H)1375 435 y Fp(!)1374 492 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)p Fp(;)p Fo(1)1616 373 y Fm(\021)1666 465 y Fw(.)100 585 y(The)27 b(next)g(theorem)f(is)h(a)g(consequence)f (of)h(the)g(existence)g(of)g(standard)e(conjugate)i(op)r(er-)0 684 y(ators,)32 b(see)g(Theorem)g(2.10)f(ab)r(o)n(v)n(e,)i(and)f(of)g (Prop)r(ositions)f(7.5.6)g(and)h(7.5.7)f(and)i(Theorem)0 784 y(7.5.8)f(from)h([ABG].)g(The)g(details)g(of)g(the)g(pro)r(of)g (are)f(similar)g(to)h(those)g(of)g(Theorem)f(7.6.8)0 884 y(\([ABG]\))i(and)e(will)h(not)f(b)r(e)h(rep)r(eated)f(here.)52 b(Note,)34 b(ho)n(w)n(ev)n(er,)d(that)i(ev)n(en)f(if)h Ft(h)g Fw(is)f(a)h(scalar)0 983 y(function,)24 b(the)e(next)g(result)g (is)g(m)n(uc)n(h)g(stronger)e(than)i(that)g(of)g(Theorem)f(7.6.8.)34 b(F)-7 b(or)21 b(example,)0 1083 y(the)h(h)n(yp)r(o)r(ellipticit)n(y)g (t)n(yp)r(e)g(conditions)f(imp)r(osed)h(on)f Ft(h)h Fw(at)f(page)g(343) f(in)i([ABG])g(are)f(no)g(longer)0 1183 y(needed.)0 1320 y Fy(Theorem)39 b(2.17.)46 b Fl(Assume)35 b(that)h Ft(h)g Fl(is)h(of)g(class)g Ft(C)1705 1290 y Fr(2)1742 1320 y Fl(.)58 b(L)l(et)36 b Ft(!)i Fs(\025)c Fw(1)i Fl(b)l(e)g(a)h(weight)g (function)0 1419 y(satisfying)f(\(2-22\).)52 b(Denote)34 b Fs(G)40 b Fl(the)34 b(form)h(domain)h(of)f Ft(H)1874 1431 y Fr(0)1942 1419 y Fw(=)c Ft(h)p Fw(\()p Ft(P)12 b Fw(\))34 b Fl(and)h(let)f Ft(H)k Fw(:)31 b Fs(G)36 b(!)31 b(G)2950 1389 y Fo(\003)0 1519 y Fl(b)l(e)f(a)g(symmetric)g(op)l (er)l(ator)g(such)g(that)f Fw(\()p Ft(H)c Fw(+)18 b Ft(i)p Fw(\))p Fs(G)28 b Fw(=)23 b Fs(G)1751 1489 y Fo(\003)1819 1519 y Fl(and)30 b Fw(\()p Ft(H)25 b Fw(+)17 b Ft(i)p Fw(\))2249 1489 y Fo(\000)p Fr(1)2356 1519 y Fs(\000)h Fw(\()p Ft(H)2540 1531 y Fr(0)2595 1519 y Fw(+)g Ft(i)p Fw(\))2739 1489 y Fo(\000)p Fr(1)2857 1519 y Fl(is)30 b(a)0 1619 y(c)l(omp)l(act)i(op)l(er)l(ator)g(in)f Fs(H)q Fl(.)42 b(Assume)30 b(that)h Ft(H)26 b Fs(\000)19 b Ft(H)1621 1631 y Fr(0)1684 1619 y Fw(=)24 b Ft(V)1821 1631 y Fp(S)1889 1619 y Fw(+)19 b Ft(V)2021 1631 y Fp(L)2071 1619 y 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Ft(x)20 b Fs(\000)g Ft(a)p Fs(j)27 b Ft(<)f(")32 b Fl(one)g(has)g Ft(U)9 b Fw(\()p Ft(x)p Fw(\))28 b(=)e Ft(u)1989 942 y Fp(a)2029 930 y Fw(\()p Fs(j)p Ft(x)21 b Fs(\000)e Ft(a)p Fs(j)p Fw(\))p Fs(j)p Ft(x)i Fs(\000)e Ft(a)p Fs(j)2577 900 y Fo(\000)p Fr(1)2686 930 y Fw(+)h Ft(v)2811 942 y Fp(a)2851 930 y Fw(\()p Ft(x)p Fw(\))p Fl(,)0 1029 y(wher)l(e)33 b Ft(v)277 1041 y Fp(a)350 1029 y Fl(is)g(a)g(b)l(ounde)l (d)g(matrix)f(value)l(d)h(function)g(and)g Ft(u)1902 1041 y Fp(a)1969 1029 y Fw(:)c(\(0)p Ft(;)14 b(")p Fw(\))27 b Fs(!)h Fq(R)39 b Fl(is)33 b(a)g(c)l(ontinuous)0 1129 y(function)23 b(such)g(that)g Ft(u)713 1141 y Fp(a)753 1129 y Fw(\(0)827 1099 y Fr(+)882 1129 y Fw(\))g(=)g Ft(l)r(im)1154 1141 y Fp(r)r Fo(&)p Fr(0)1289 1129 y Ft(u)1337 1141 y Fp(a)1376 1129 y Fw(\()p Ft(r)r Fw(\))i Fl(exists.)36 b(Mor)l(e)l(over,)26 b(assume)d(that)g Ft(U)32 b Fl(is)23 b(lo)l(c)l(al)t(ly)0 1242 y(r)l(e)l(gular)30 b(on)g Fq(R)449 1212 y Fr(3)510 1242 y Fs(n)18 b Fw(\000)p Fl(.)39 b(Then)30 b(any)g(self-adjoint)i(extension)1879 1221 y Fm(e)1859 1242 y Ft(H)37 b Fl(of)30 b Ft(H)2131 1254 y Fp(min)2289 1242 y Fl(is)g(lo)l(c)l(al)t(ly)h(c)l(omp)l(act.)0 1387 y(Pr)l(o)l(of.)44 b Fw(By)21 b(Lemma)g(4.5)g(it)h(will)f(b)r(e)h (enough)f(to)g(sho)n(w)g(that)h Ft(U)30 b Fw(is)21 b(quasi-regular)e (on)i Ft(B)t Fw(\()p Ft(a)p Fw(;)14 b Ft(")p Fw(\))23 b(=)0 1487 y Fs(f)p Ft(x)g Fs(2)g Fq(R)244 1457 y Fr(3)311 1487 y Fs(j)g(j)p Ft(x)s Fs(\000)s Ft(a)p Fs(j)g Ft(<)g(")p Fs(g)p Fw(.)33 b(By)20 b(a)g(translation,)g(w)n(e)f(reduce)h(ourselv)n (es)e(to)i(the)g(case)f Ft(a)k Fw(=)g(0.)34 b(Since)0 1587 y(quasi-regularit)n(y)29 b(on)i Ft(B)t Fw(\()p Ft(a)p Fw(;)14 b Ft(")p Fw(\))32 b(dep)r(ends)g(only)g(on)f Ft(U)9 b Fs(j)1720 1602 y Fp(B)s Fr(\()p Fp(a)p Fr(;)p Fp(")p Fr(\))1947 1587 y Fw(\(cf.)50 b(the)32 b(de\014nition)g(of)g Fs(D)2797 1599 y Fp(max)2933 1587 y Fw(\),)0 1686 y(w)n(e)d(only)g (need)h(to)f(sho)n(w)g(that)g(the)h(p)r(oten)n(tial)f(considered)g(in)h 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b(4.9.)46 b Fl(L)l(et)37 b Ft(U)45 b Fw(:)37 b Fq(R)1089 3752 y Fr(3)1169 3783 y Fs(!)f Ft(B)t Fw(\()p Fy(E)p Fw(\))i Fl(b)l(e)f(a)h(lo)l(c)l(al) t(ly)h(squar)l(e)e(inte)l(gr)l(able)h(symmetric)0 3882 y(op)l(er)l(ator)23 b(value)l(d)f(function.)36 b(Denote)21 b Ft(H)1267 3894 y Fp(min)1416 3882 y Fl(the)h(closur)l(e)g(of)g(the)g (symmetric)g(op)l(er)l(ator)h Ft(H)2820 3894 y Fr(0)2858 3882 y Fw(+)q Ft(U)0 3982 y Fl(with)34 b(domain)h Ft(C)550 3952 y Fo(1)544 4002 y Fp(c)621 3982 y Fw(\()p Fq(R)707 3952 y Fr(3)751 3982 y Fw(;)14 b Fy(E)p Fw(\))p Fl(.)50 b(Assume)33 b(that)h Ft(U)39 b Fw(=)30 b Ft(U)1694 3994 y Fr(1)1752 3982 y Fw(+)21 b Ft(\014)t(U)1946 3994 y Fr(2)2005 3982 y Fw(+)g Ft(U)2148 3994 y Fr(3)2218 3982 y Fl(wher)l(e)35 b Ft(U)2514 3994 y Fr(1)2551 3982 y Ft(;)14 b(U)2645 3994 y Fr(2)2715 3982 y Fl(ar)l(e)34 b(r)l(e)l(al)0 4081 y(value)l(d)c(functions)f(and)h Ft(U)830 4093 y Fr(3)896 4081 y Fl(is)g(a)f(b)l(ounde)l(d)h(matrix)f(value)l(d)h (function.)39 b(Then)29 b Ft(H)2535 4093 y Fp(min)2692 4081 y Fl(has)h(self-)0 4181 y(adjoint)h(extensions.)0 4326 y(Pr)l(o)l(of.)44 b Fw(Since)35 b(an)n(y)e(represen)n(tation)g(of) h(the)g(relations)f(\(1-1\))h(is)g(unitarily)g(equiv)-5 b(alen)n(t)34 b(to)g(a)0 4426 y(direct)c(sum)h(of)f(Ma)5 b(jorana)29 b(represen)n(tations)f(\(cf.)46 b(page)30 b(36)f(in)i([T]\),)g(w)n(e)f(ma)n(y)g(assume)g(that)0 4526 y Fy(E)23 b Fw(=)f Fq(C)227 4495 y Fr(4)297 4526 y Fw(\(equipp)r(ed)28 b(with)f(the)g(usual)f(conjugation)g(op)r (eration\))g(and)g(that)h(the)g(matrices)f Ft(\013)2948 4538 y Fp(k)0 4629 y Fw(are)g(real)g(if)i(1)22 b Fs(\024)h Ft(k)j Fs(\024)d Fw(3)j(while)i Ft(\013)1022 4641 y Fr(0)1082 4629 y Fw(=)23 b Ft(\014)31 b Fw(is)c(purely)g(imaginary:)p 2009 4583 54 4 v 35 w Ft(\013)2062 4641 y Fp(k)2126 4629 y Fw(=)c Ft(\013)2267 4641 y Fp(k)2308 4629 y Ft(;)p 2345 4561 52 4 v 14 w(\014)k Fw(=)c Fs(\000)p Ft(\014)t Fw(.)37 b(If)27 b(w)n(e)g(set)0 4729 y Ft(\015)h Fw(=)22 b Ft(i\013)240 4741 y Fr(0)277 4729 y Ft(\013)330 4741 y Fr(1)368 4729 y Ft(\013)421 4741 y Fr(2)458 4729 y Ft(\013)511 4741 y Fr(3)574 4729 y Fw(then)p 761 4683 48 4 v 26 w Ft(\015)27 b Fw(=)c Ft(\015)5 b(;)14 b(\015)1052 4698 y Fo(\003)1112 4729 y Fw(=)23 b Fs(\000)p Ft(\015)5 b(;)14 b(\015)1398 4698 y Fr(2)1457 4729 y Fw(=)23 b Fs(\000)p Fw(1.)35 b(Moreo)n(v)n(er,)23 b(w)n(e)i(ha)n(v)n(e)f Ft(\015)5 b(\013)2506 4741 y Fp(k)2570 4729 y Fw(=)23 b Fs(\000)p Ft(\013)2776 4741 y Fp(k)2816 4729 y Ft(\015)30 b Fw(for)0 4832 y(0)24 b Fs(\024)h Ft(k)i Fs(\024)e Fw(3.)40 b(No)n(w)28 b(w)n(e)g(de\014ne)h Ft(J)k Fw(:)25 b Fs(H)g(!)g(H)30 b Fw(b)n(y)e(\()p Ft(J)8 b(f)h Fw(\)\()p Ft(x)p Fw(\))26 b(=)f Ft(\015)p 1964 4760 162 4 v 5 w(f)9 b Fw(\()p Ft(x)p Fw(\).)40 b(Cleary)28 b Ft(J)36 b Fw(is)29 b(an)n(tilinear,)0 4932 y Ft(J)54 4901 y Fr(2)127 4932 y Fw(=)35 b Fs(\000)p Fw(1)g(and)g Fs(h)p Ft(J)8 b(f)t(;)14 b(g)s Fs(i)36 b Fw(=)f Fs(\000h)p Ft(J)8 b(g)s(;)14 b(f)9 b Fs(i)35 b Fw(for)f(all)h Ft(f)t(;)14 b(g)s Fw(.)59 b(Then)36 b(w)n(e)f(ha)n(v)n (e)f Ft(\013)24 b Fs(\001)f Ft(P)12 b(J)c(g)38 b Fw(=)e Ft(J)8 b(\013)24 b Fs(\001)f Ft(P)12 b(g)0 5031 y Fw(and)30 b(\()p Ft(U)253 5043 y Fr(1)310 5031 y Fw(+)20 b Ft(\014)t(U)503 5043 y Fr(2)540 5031 y Fw(\))p Ft(J)8 b(g)30 b Fw(=)d Ft(J)8 b Fw(\()p Ft(U)931 5043 y Fr(1)988 5031 y Fw(+)20 b Ft(\014)t(U)1181 5043 y Fr(2)1218 5031 y Fw(\))p Ft(g)33 b Fw(if)d Ft(g)g Fs(2)e Ft(C)1619 5001 y Fo(1)1613 5052 y Fp(c)1689 5031 y Fw(.)44 b(Denote)31 b Ft(S)j Fw(the)d(symmetric)f (op)r(erator)p eop %%Page: 30 30 30 29 bop 0 42 a Fu(30)785 b(V.)30 b(GEOR)n(GESCU)h(AND)g(M.)e(M)1822 25 y(\025)1813 42 y(ANTOIU)0 241 y Ft(\013)22 b Fs(\001)f Ft(P)33 b Fw(+)20 b Ft(U)347 253 y Fr(1)406 241 y Fw(+)g Ft(\014)t(U)599 253 y Fr(2)668 241 y Fw(with)32 b(domain)g Ft(C)1226 211 y Fo(1)1220 261 y Fp(c)1296 241 y Fw(\()p Fq(R)1382 211 y Fr(3)1426 241 y Fw(;)14 b Fy(E)p Fw(\).)49 b(If)32 b Ft(f)38 b Fs(2)30 b Ft(D)r Fw(\()p Ft(S)2040 211 y Fo(\003)2079 241 y Fw(\))i(and)f Ft(g)h Fs(2)f Ft(C)2531 211 y Fo(1)2525 261 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Fp(c)2554 1254 y Fw(\()p Fq(R)2641 1224 y Fr(3)2693 1254 y Fs(n)9 b Ft(M)g Fw(;)14 b Fy(E)p Fw(\).)0 1354 y(Then,)36 b Ft(\014)i Fw(can)c(b)r(e)h(replaced)e(b)n(y)h(an)n(y)f(matrix)g Ft(\016)38 b Fw(whic)n(h)c(has,)h(in)f(the)h(preceding)e(Ma)5 b(jorana)0 1454 y(represen)n(tation,)37 b(the)g(prop)r(ert)n(y)e Ft(\016)s(\015)42 b Fw(=)37 b Ft(\015)p 1352 1386 41 4 v 5 w(\016)i Fw(\(for)d(example)g Ft(\016)41 b Fw(=)c Ft(\015)2153 1466 y Fr(5)2190 1454 y Fw(,)i(cf.)63 b(page)35 b(37)h(in)g([T]\).)0 1553 y(Finally)-7 b(,)26 b(it)g(su\016ces)f(that)h (the)g(op)r(erator)e(b)r(ound)i(of)g Ft(U)1714 1565 y Fr(3)1776 1553 y Fw(with)g(resp)r(ect)g(to)f Ft(H)2414 1565 y Fr(0)2466 1553 y Fw(+)14 b Ft(U)2602 1565 y Fr(1)2653 1553 y Fw(+)h Ft(\014)t(U)2841 1565 y Fr(2)2903 1553 y Fw(b)r(e)0 1653 y(strictly)27 b(less)g(than)h(1.)0 1805 y Fy(Ac)m(kno)m(wledgemen)m(ts.)k Fw(P)n(art)18 b(of)h(this)g(w)n(ork)f(w)n(as)g(done)h(while)g(the)g(\014rst)g(author) f(w)n(as)g(visiting)0 1905 y(the)26 b(Departmen)n(t)f(of)g(Ph)n(ysics)g (of)g(the)h(Univ)n(ersit)n(y)e(of)h(Genev)-5 b(a.)36 b(Both)26 b(authors)e(are)g(indebted)0 2005 y(to)35 b(the)g(Swiss)g (National)f(Science)h(F)-7 b(oundation)35 b(for)f(\014nancial)h(supp)r (ort)g(and)g(V.G.)g(thanks)0 2104 y(W)-7 b(erner)27 b(Amrein)h(for)f (his)g(kind)h(hospitalit)n(y)-7 b(.)1252 2309 y Fx(References)0 2442 y Fu([ABG])39 b(W.)28 b(Amrein,)f(A.)g(Boutet)i(de)f(Mon)n(v)n(el) h(and)f(V.)f(Georgescu,)j Fe(C)1977 2451 y Fd(0)2012 2442 y Fc(-Gr)l(oups,)i(Commutator)f(Metho)l(ds)237 2525 y(and)c(Sp)l(e)l(ctr)l(al)g(The)l(ory)f(of)g(N-Bo)l(dy)g(Hamiltonians)p Fu(,)e(Birkh\177)-35 b(auser,)23 b(1996.)0 2609 y([Ar])116 b(M.)22 b(Arai,)g Fc(On)j(Essential)g(Self-A)l(djointness,)g (Distinguishe)l(d)h(Selfadjoint)g(Extensions)f(and)h(Essen-)237 2692 y(tial)21 b(Sp)l(e)l(ctrum)g(of)g(Dir)l(ac)g(Op)l(er)l(ators)h (with)e(Matrix)h(V)-5 b(alue)l(d)21 b(Potentials)p Fu(,)f(Publ.)d (RIMS,)h(Ky)n(oto)h(Univ.)237 2775 y Fb(19)k Fu(\(1983\),)i(33{57.)0 2859 y([A)-6 b(Y])97 b(M.)20 b(Arai)g(and)h(O.)f(Y)-6 b(amada,)21 b Fc(Essential)i(Self-A)l(djointness)g(and)h(Invarianc)l(e) g(of)f(the)g(Essential)g(Sp)l(e)l(c-)237 2943 y(trum)j(for)g(Dir)l(ac)g (Op)l(er)l(ators)p Fu(,)e(Publ.)f(RIMS,)g(Ky)n(oto)i(Univ.)e Fb(18)g Fu(\(1982\),)i(973{985.)0 3027 y([BMP])34 b(A.)23 b(Boutet)i(de)e(Mon)n(v)n(el,)h(D.)e(Manda)i(and)g(R.)f(Purice,)f Fc(Limiting)j(A)n(bsorption)h(for)f(the)h(Dir)l(ac)f(Op)l(er-)237 3110 y(ator)p Fu(,)f(Annales)g(Inst.)g(Henry)f(P)n(oincar)n(\023)-33 b(e,)24 b(Ph)n(ysique)g(Th)n(\023)-33 b(eorique)25 b Fb(58)e Fu(\(1993\),)i(413{431.)0 3194 y([BGS])53 b(A.)17 b(Boutet)j(de)e(Mon)n(v)n(el,)h(V.)e(Georgescu)j(and)e(J.)g(Sah)n (bani,)h Fc(Boundary)j(V)-5 b(alues)21 b(of)f(R)l(e)l(gular)i(R)l (esolvent)237 3277 y(F)-5 b(amilies)p Fu(,)24 b(Helv.)f(Ph)n(ys.)g (Acta)i Fb(71)e Fu(\(1998\),)i(518{553.)0 3361 y([GM])77 b(V.)23 b(Georgescu)i(and)f(M.)e(M\025)-35 b(an)n(toiu,)24 b Fc(Asymptotic)h(V)-5 b(elo)l(city)25 b(and)h(Conjugate)g(Op)l(er)l (ators)g(for)g(Matrix)237 3444 y(V)-5 b(alue)l(d)27 b(Hamiltonians,)f Fu(in)e(preparation.)0 3528 y([GN])89 b(C.)25 b(G)n(\023)-33 b(erard)26 b(and)g(F.)f(Nier,)h Fc(The)h(Mourr)l(e)h(The)l(ory)h(for)e (A)n(nalytic)l(al)t(ly)h(Fib)l(er)l(e)l(d)h(Op)l(er)l(ators)p Fu(,)e(J.)e(F)-6 b(unc.)237 3611 y(Analys.)23 b Fb(152)g Fu(\(1998\),)i(202{219.)0 3695 y([H])144 b(L.)27 b(H\177)-35 b(ormander,)26 b Fc(The)j(A)n(nalysis)g(of)g(Line)l(ar)g(Partial)h (Di\013er)l(ential)e(Op)l(er)l(ators,)j(V)-5 b(ol.)28 b(II)p Fu(,)g(Springer,)237 3778 y(Berlin,)22 b(1983.)0 3862 y([K])142 b(M.)35 b(Klaus,)k Fc(Dir)l(ac)e(Op)l(er)l(ators)i(with) e(Sever)l(al)h(Coulomb)h(Singularities)p Fu(,)g(Helv.)c(Ph)n(ys.)h (Acta)k Fb(53)237 3945 y Fu(\(1980\),)25 b(463{482.)0 4029 y([Kt])115 b(T.)23 b(Kato,)h Fc(Perturb)l(ation)i(The)l(ory)h(for) f(Line)l(ar)f(Op)l(er)l(ators)p Fu(,)g(Springer,)e(Berlin,)f(1966.)0 4113 y([LR])101 b(J.)35 b(Landgren)h(and)g(P)-6 b(.)35 b(Rejt\177)-35 b(o,)38 b Fc(A)n(n)f(Aplic)l(ation)g(of)g(the)f(Maximum) i(Principle)e(to)h(the)f(Study)g(of)237 4196 y(Essential)26 b(Self-A)l(djointness)g(of)g(Dir)l(ac)g(Op)l(er)l(ators,)g(I)p Fu(,)e(J.)f(Math.)h(Ph)n(ys.)f Fb(20)g Fu(\(1979\),)i(2204{2211.)0 4280 y([LRK])46 b(J.)19 b(Landgren)h(P)-6 b(.)19 b(Rejt\177)-35 b(o)19 b(and)h(M.)e(Klaus,)h Fc(A)n(n)i(Aplic)l(ation)i(of)e(the)g (Maximum)i(Principle)e(to)h(the)f(Study)237 4363 y(of)g(Essential)h (Self-A)l(djointness)f(of)g(Dir)l(ac)g(Op)l(er)l(ators,)i(II)p Fu(,)d(J.)e(Math.)g(Ph)n(ys.)i Fb(21)f Fu(\(1980\),)i(1210{1217.)0 4447 y([M])132 b(E.)20 b(Mourre,)g Fc(A)n(bsenc)l(e)i(of)h(Singular)g (Continuous)h(Sp)l(e)l(ctrum)f(for)g(Certain)f(Self-A)l(djoint)h(Op)l (er)l(ators)p Fu(,)237 4530 y(Comm.)e(Math.)j(Ph)n(ys.)f Fb(78)g Fu(\(1981\),)i(391{408.)0 4614 y([N1])109 b(G.)19 b(Nenciu,)g Fc(Self-A)l(djointness)i(and)h(Invarianc)l(e)g(of)g(the)e (Essential)i(Sp)l(e)l(ctrum)g(for)f(Dir)l(ac)g(Opr)l(e)l(ators)237 4697 y(De\014ne)l(d)26 b(as)g(Quadr)l(atic)g(F)-5 b(orms)p Fu(,)25 b(Comm.)c(Math.)i(Ph)n(ys.)h Fb(48)f Fu(\(1976\),)i(235{247.)0 4781 y([N2])109 b(G.)35 b(Nenciu,)j Fc(Distinguishe)l(d)e(Self-A)l (djoint)g(Extensions)h(for)f(Dir)l(ac)g(Op)l(er)l(ators)i(with)e (Potentials)237 4864 y(Dominate)l(d)27 b(by)e(Multic)l(enter)g(Coulomb) i(Potentials)p Fu(,)c(Helv.)h(Ph)n(ys.)f(Acta)h Fb(50)f Fu(\(1977\),)i(1{3.)0 4948 y([S])158 b(J.)23 b(Sah)n(bani,)g Fc(Pr)l(op)l(agation)28 b(The)l(or)l(ems)f(for)e(Some)g(Classes)h(of)f (Pseudo-Di\013er)l(ential)h(Op)l(er)l(ators)p Fu(,)e(J.)237 5031 y(Math.)g(Analys.)f(Appl.)g Fb(211)f Fu(\(1997\),)k(481{497.)p eop %%Page: 31 31 31 30 bop 146 42 a Fu(ON)30 b(THE)g(SPECTRAL)h(THEOR)-6 b(Y)30 b(OF)h(SINGULAR)g(DIRA)n(C)f(TYPE)h(HAMIL)-6 b(TONIANS)105 b(31)0 241 y([St])131 b(R.)32 b(S.)h(Stric)n(harts,)i Fc(Multipliers)g(on)f(F)-5 b(r)l(actional)36 b(Sob)l(olev)f(Sp)l(ac)l (es)p Fu(,)h(J.)c(Math.)h(Mec)n(h.)j Fb(16)e Fu(\(1967\),)237 324 y(1031{1060.)0 407 y([T])146 b(B.)16 b(Thaller,)g Fc(The)j(Dir)l(ac)g(Equation,)i(T)-5 b(exts)18 b(and)h(Mono)l(gr)l (aphs)j(in)c(Physics)p Fu(,)g(Springer-V)-6 b(erlag,)17 b(Berlin)237 490 y(Heidelb)r(erg,)23 b(1992.)0 573 y([V])144 b(V.)21 b(V)-6 b(ogelsang,)22 b Fc(Self-A)l(djoint)i(Extensions)g(of)g (Dir)l(ac)f(Op)l(er)l(ators)i(for)f(Non)g(Spheric)l(al)t(ly)g (Symmetric)237 656 y(Potentials)i(in)g(Coulomb)h(Sc)l(attering)p Fu(,)22 b(In)n(t.)j(Eq.)e(and)h(Op.)f(Theor.)h Fb(10)f Fu(\(1987\),)i(841{858.)0 739 y([X])144 b(J.)22 b(Xia,)g Fc(On)i(the)g(Contribution)h(to)f(the)h(Coulomb)g(Singularity)g(of)f(A) n(rbitr)l(ary)h(Char)l(ge)g(to)f(the)h(Dir)l(ac)237 822 y(Hamiltonian)p Fu(,)f(T)-6 b(rans.)24 b(Amer.)d(Math.So)r(c.,)j(to)g (app)r(ear.)100 1005 y Fv(V.)29 b(Geor)o(gescu,)g(D)696 1000 y(\023)696 1005 y(ep)l(ar)l(tement)i(de)e(Ma)l(th)1429 1000 y(\023)1429 1005 y(ema)l(tiques,)h(Universit)2161 1000 y(\023)2161 1005 y(e)g(de)f(Cer)o(gy-Pontoise,)h(2)0 1088 y(a)-6 b(venue)27 b(Adolphe)g(Cha)o(uvin,)f(95302)e(Cer)o (gy-Pontoise)j(Cedex,)g(France)100 1171 y Fc(E-mail)f(addr)l(ess)5 b Fu(:)37 b Fa(vlad@oara.pst.u-cergy.fr)100 1353 y Fv(M.)25 b(M)287 1348 y(\025)285 1353 y(antoiu,)h(D)623 1348 y(\023)623 1353 y(ep)l(ar)l(tement)g(de)g(Physique)f(Th)1593 1348 y(\023)1593 1353 y(eorique,)i(Universit)2245 1348 y(\023)2245 1353 y(e)f(de)g(Gen)2556 1348 y(\022)2556 1353 y(eve,)g(24)e(Quai)0 1436 y(Ernest)j(Ansermet,)g(CH-1211)d(Gen)1120 1431 y(\022)1120 1436 y(eve)j(4,)f(Suisse)100 1519 y Fc(E-mail)g(addr)l(ess)5 b Fu(:)37 b Fa(marius.mantoiu@physics.u)q(nig)q(e.ch)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9901250749900--