Content-Type: multipart/mixed; boundary="-------------0406041739868" This is a multi-part message in MIME format. ---------------0406041739868 Content-Type: text/plain; name="04-179.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-179.keywords" spectral shift function, magnetic Schroedinger opeartor, high energy asymptotics ---------------0406041739868 Content-Type: application/postscript; name="hauen11.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hauen11.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: hauen11.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips hauen11 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.06.04:1838 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S 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b(odinger)34 b(op)s(erator)f(with)g(constan)m(t)h(magnetic)e(\014eld,)i (p)s(erturb)s(ed)118 3023 y(b)m(y)29 b(an)f(electric)f(p)s(oten)m(tial) f(whic)m(h)j(deca)m(ys)g(fast)f(enough)h(at)e(in\014nit)m(y)-8 b(.)41 b(The)29 b(note)f(could)g(b)s(e)g(regarded)118 3143 y(as)41 b(a)f(supplemen)m(t)h(to)f(the)g(articles)g([5])g(where)i (the)e(asymptotic)g(b)s(eha)m(viour)g(of)g(the)h(SSF)f(in)g(the)118 3263 y(strong)35 b(magnetic)e(\014eld)i(regime)f(w)m(as)h(considered,)i (and)e([6)o(])g(where)h(the)g(singularities)c(of)i(the)h(SSF)118 3384 y(at)d(the)h(Landau)g(lev)m(els)g(w)m(ere)h(in)m(v)m(estigated.) 118 3504 y(Let)28 b Fm(H)369 3519 y Fl(0)436 3504 y Fn(:=)g(\()p Fm(i)p Fk(r)12 b Fn(+)g Fm(A)p Fn(\))932 3468 y Fl(2)984 3504 y Fk(\000)g Fm(b)28 b Fn(b)s(e)g(the)g(unp)s(erturb)s(ed)h (three-dimensional)d(magnetic)g(Sc)m(hr\177)-49 b(odinger)28 b(op-)118 3625 y(erator,)c(essen)m(tially)f(self-adjoin)m(t)d(on)j Fm(C)1576 3588 y Fh(1)1569 3649 y Fl(0)1650 3625 y Fn(\()p Fj(R)1754 3588 y Fl(3)1800 3625 y Fn(\).)40 b(Here)23 b(the)g(magnetic)e(p)s(oten)m(tial)g Fm(A)28 b Fn(=)3297 3544 y Fe(\000)3342 3625 y Fk(\000)3429 3584 y Fi(bx)3499 3593 y Ff(2)p 3430 3602 105 4 v 3464 3659 a Fl(2)3544 3625 y Fm(;)3598 3584 y Fi(bx)3668 3593 y Ff(1)p 3598 3602 V 3632 3659 a Fl(2)3712 3625 y Fm(;)17 b Fn(0)3805 3544 y Fe(\001)118 3745 y Fn(generates)41 b(the)g(constan)m(t)g (magnetic)e(\014eld)h Fm(B)46 b Fn(=)40 b(curl)h Fm(A)f Fn(=)h(\(0)p Fm(;)17 b Fn(0)p Fm(;)g(b)p Fn(\),)41 b Fm(b)h(>)e Fn(0.)66 b(It)40 b(is)g(w)m(ell-kno)m(wn)118 3865 y(that)34 b Fm(\033)t Fn(\()p Fm(H)509 3880 y Fl(0)548 3865 y Fn(\))d(=)f Fm(\033)778 3880 y Fl(ac)849 3865 y Fn(\()p Fm(H)968 3880 y Fl(0)1008 3865 y Fn(\))g(=)h([0)p Fm(;)17 b Fk(1)p Fn(\))33 b(\(see)i([1]\),)g(where)g Fm(\033)t Fn(\()p Fm(H)2335 3880 y Fl(0)2375 3865 y Fn(\))f(denotes)h (the)g(sp)s(ectrum)f(of)g Fm(H)3592 3880 y Fl(0)3632 3865 y Fn(,)g(and)118 3986 y Fm(\033)173 4001 y Fl(ac)244 3986 y Fn(\()p Fm(H)363 4001 y Fl(0)403 3986 y Fn(\))g(its)h (absolutely)f(con)m(tin)m(uous)h(sp)s(ectrum.)51 b(Moreo)m(v)m(er,)37 b(the)e(so-called)e(Landau)i(lev)m(els)g(2)p Fm(bq)t Fn(,)118 4106 y Fm(q)d Fk(2)c Fj(Z)356 4121 y Fl(+)440 4106 y Fn(:=)f Fk(f)p Fn(0)p Fm(;)17 b Fn(1)p Fm(;)g(:)g(:)g(:)o Fk(g)p Fn(,)32 b(pla)m(y)h(the)g(role)e(of)h(thresholds)h(in)f Fm(\033)t Fn(\()p Fm(H)2468 4121 y Fl(0)2507 4106 y Fn(\).)118 4227 y(F)-8 b(or)31 b Fo(x)c Fn(=)h(\()p Fm(x)575 4242 y Fl(1)615 4227 y Fm(;)17 b(x)714 4242 y Fl(2)753 4227 y Fm(;)g(x)852 4242 y Fl(3)892 4227 y Fn(\))28 b Fk(2)g Fj(R)1118 4190 y Fl(3)1194 4227 y Fn(w)m(e)33 b(denote)f(b)m(y)g Fm(X)1865 4242 y Fh(?)1952 4227 y Fn(=)27 b(\()p Fm(x)2148 4242 y Fl(1)2188 4227 y Fm(;)17 b(x)2287 4242 y Fl(2)2326 4227 y Fn(\))32 b(the)f(v)-5 b(ariables)30 b(on)i(the)f(plane)g(p)s (erp)s(en-)118 4347 y(dicular)g(to)i(the)g(magnetic)e(\014eld.)43 b(W)-8 b(e)33 b(assume)g(that)g Fm(V)54 b Fn(satis\014es)292 4566 y Fm(V)49 b Fk(6\021)28 b Fn(0)p Fm(;)114 b(V)50 b Fk(2)28 b Fm(C)7 b Fn(\()p Fj(R)1075 4525 y Fl(3)1120 4566 y Fn(\))p Fm(;)114 b Fk(j)p Fm(V)21 b Fn(\()p Fo(x)p Fn(\))p Fk(j)28 b(\024)g Fm(C)1771 4581 y Fl(0)1810 4566 y Fk(h)p Fm(X)1930 4581 y Fh(?)1989 4566 y Fk(i)2028 4525 y Fh(\000)p Fi(m)2145 4537 y Fb(?)2201 4566 y Fk(h)p Fm(x)2295 4581 y Fl(3)2335 4566 y Fk(i)2374 4525 y Fh(\000)p Fi(m)2491 4534 y Ff(3)2530 4566 y Fm(;)114 b Fo(x)28 b Fn(=)f(\()p Fm(X)2980 4581 y Fh(?)3039 4566 y Fm(;)17 b(x)3138 4581 y Fl(3)3178 4566 y Fn(\))27 b Fk(2)h Fj(R)3403 4525 y Fl(3)3449 4566 y Fm(;)174 b Fn(\(1.1\))118 4786 y(with)32 b Fm(C)410 4801 y Fl(0)477 4786 y Fm(>)c Fn(0,)33 b Fm(m)775 4801 y Fh(?)862 4786 y Fm(>)27 b Fn(2,)33 b Fm(m)1159 4801 y Fl(3)1226 4786 y Fm(>)28 b Fn(1,)33 b(and)f Fk(h)p Fm(x)p Fk(i)c Fn(:=)g(\(1)22 b(+)g Fk(j)p Fm(x)p Fk(j)2238 4750 y Fl(2)2277 4786 y Fn(\))2315 4750 y Fl(1)p Fi(=)p Fl(2)2425 4786 y Fn(,)33 b Fm(x)28 b Fk(2)g Fj(R)2728 4750 y Fi(d)2775 4786 y Fn(,)k Fm(d)c Fk(\025)g Fn(1.)44 b(By)34 b(\(1.1\))e(and)h(the)118 4906 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b(F)-8 b(or)41 b(almost)g(ev)m(ery)j Fm(E)50 b(>)44 b Fn(0)e(the)g(SSF)g Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)3025 1918 y Fl(0)3063 1903 y Fn(\))43 b(is)e(related)h(to)g(the)118 2023 y(scattering)30 b(determinan)m(t)g(det)e Fm(S)6 b Fn(\()p Fm(E)g Fn(;)17 b Fm(H)r(;)g(H)1711 2038 y Fl(0)1750 2023 y Fn(\))30 b(for)g(the)h(pair)e Fm(H)r(;)17 b(H)2537 2038 y Fl(0)2606 2023 y Fn(b)m(y)32 b(the)f(Birman-Krein)c(form)m(ula)1311 2243 y(det)h Fm(S)6 b Fn(\()p Fm(E)g Fn(;)17 b Fm(H)r(;)g(H)1908 2258 y Fl(0)1947 2243 y Fn(\))27 b(=)h Fm(e)2161 2202 y Fh(\000)p Fl(2)p Fi(\031)r(i\030)s Fl(\()p Fi(E)t Fl(;)p Fi(H)q(;H)2592 2211 y Ff(0)2626 2202 y Fl(\))118 2463 y Fn(\(see)34 b([3])e(or)g([14,)h(Section)f(8.4]\).)118 2584 y(A)g(priori,)e(the)i(SSF)g Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)1302 2599 y Fl(0)1340 2584 y Fn(\))32 b(is)f(de\014ned)j(only)d(for)g(almost)f(ev)m(ery)k Fm(E)g Fk(2)28 b Fj(R)5 b Fn(.)49 b(In)32 b(Subsection)h(3.3)118 2704 y(b)s(elo)m(w)j(w)m(e)h(in)m(tro)s(duce)e(a)h(represen)m(tativ)m (e)h(of)f(the)g(equiv)-5 b(alence)36 b(class)g(determined)f(b)m(y)i Fm(\030)5 b Fn(\()p Fm(:)p Fn(;)17 b Fm(H)r(;)g(H)3747 2719 y Fl(0)3785 2704 y Fn(\),)118 2825 y(de\014ned)41 b(on)e Fj(R)f Fk(n)26 b Fn(2)p Fm(b)p Fj(Z)937 2840 y Fl(+)994 2825 y Fn(,)41 b(whic)m(h)e(is)g(b)s(ounded)h(on)f(eac)m(h)h (compact)f(subset)i(of)e(the)h(complemen)m(t)e(of)118 2945 y(the)g(Landau)f(lev)m(els,)i(and)e(is)g(con)m(tin)m(uous)h(on)f Fj(R)f Fk(n)25 b(f)p Fn(2)p Fm(b)p Fj(Z)2254 2960 y Fl(+)2336 2945 y Fk([)h Fm(\033)2483 2960 y Fi(p)2523 2945 y Fn(\()p Fm(H)8 b Fn(\))p Fk(g)36 b Fn(where)j Fm(\033)3116 2960 y Fi(p)3156 2945 y Fn(\()p Fm(H)8 b Fn(\))36 b(denotes)j(the)118 3065 y(set)33 b(of)g(the)g(eigen)m(v)-5 b(alues)32 b(of)g(the)i(op)s (erator)d Fm(H)8 b Fn(.)44 b(In)33 b(this)f(note)h(w)m(e)g(will)e(iden) m(tify)h(the)h(SSF)f(with)h(this)118 3186 y(particular)e(represen)m (tativ)m(e)j(of)f(its)f(equiv)-5 b(alence)32 b(class.)118 3306 y(The)37 b(main)d(goal)g(of)h(the)i(pap)s(er)f(is)f(the)h(study)h (of)f(the)g(asymptotics)f(of)h Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)3245 3321 y Fl(0)3283 3306 y Fn(\))36 b(as)g Fm(E)j Fk(!)33 b(1)p Fn(,)118 3426 y Fm(E)h Fk(2)28 b(O)397 3441 y Fi(r)435 3426 y Fn(,)33 b(where)887 3646 y Fk(O)966 3661 y Fi(r)1032 3646 y Fn(:=)28 b Fk(f)p Fm(E)33 b Fk(2)c Fn(\(0)p Fm(;)17 b Fk(1)p Fn(\))p Fk(j)p Fn(dist)26 b(\()p Fm(E)6 b(;)17 b Fn(2)p Fm(b)p Fj(Z)2213 3661 y Fl(+)2269 3646 y Fn(\))28 b Fm(>)f(r)s Fk(g)p Fm(;)114 b(r)30 b Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fm(:)569 b Fn(\(1.5\))118 3866 y(The)31 b(pap)s(er)e(is)g(organized)g (as)h(follo)m(ws.)41 b(In)30 b(Section)f(2)h(w)m(e)g(form)m(ulate)e (our)h(main)f(result.)43 b(In)29 b(Section)118 3987 y(3)g(w)m(e)h (obtain)e(some)h(preliminary)e(estimates,)j(while)e(the)i(pro)s(of)e (of)h(our)g(main)e(result)i(can)h(b)s(e)f(found)118 4107 y(in)j(Section)g(4.)118 4440 y Fc(2)161 b(F)-13 b(orm)l(ulation)54 b(of)f(the)g(Main)i(Result)118 4659 y Fo(Theorem)37 b(2.1.)42 b Fa(Assume)35 b(that)g Fm(V)57 b Fa(satis\014es)42 b Fn(\(1.1\))o Fa(.)j(Then)34 b(we)h(have)868 4927 y Fn(lim)731 4989 y Fi(E)t Fh(!1)p Fi(;E)t Fh(2O)1108 4997 y Fg(r)1157 4927 y Fm(E)1235 4886 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)1400 4927 y Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)1816 4942 y Fl(0)1854 4927 y Fn(\))28 b(=)2082 4860 y(1)p 2033 4905 148 4 v 2033 4996 a(4)p Fm(\031)2141 4967 y Fl(2)2207 4792 y Fe(Z)2262 5017 y Fd(R)2310 4999 y Ff(3)2365 4927 y Fm(V)22 b Fn(\()p Fo(x)p Fn(\))p Fm(d)p Fo(x)p Fm(;)116 b(r)30 b Fk(2)f Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fm(:)412 b Fn(\(2.1\))1960 5375 y(2)p eop %%Page: 3 3 3 2 bop 118 171 a Fa(R)-5 b(emarks)p Fn(:)69 b(i\))45 b(It)h(is)f(essen)m(tial)h(to)f(a)m(v)m(oid)g(the)i(Landau)e(lev)m(els) h(in)f(\(2.1\),)k(i.e.)82 b(to)46 b(supp)s(ose)h(that)118 291 y Fm(E)38 b Fk(2)33 b(O)406 306 y Fi(r)444 291 y Fn(,)j Fm(r)e Fk(2)f Fn(\(0)p Fm(;)17 b(b)p Fn(\),)35 b(as)h Fm(E)i Fk(!)31 b(1)p Fn(,)36 b(since)f(the)h(SSF)f(has)h (singularities)c(at)j(the)h(Landau)f(lev)m(els,)h(at)118 411 y(least)c(in)g(the)h(case)h(where)f Fm(V)54 b Fn(has)33 b(a)g(\014xed)g(sign)f(\(see)i([6]\).)118 532 y(ii\))d(F)-8 b(or)32 b Fm(E)h Fk(2)28 b Fj(R)44 b Fn(set)390 766 y Fm(\030)433 781 y Fl(cl)488 766 y Fn(\()p Fm(E)6 b Fn(\))27 b(:=)800 630 y Fe(Z)855 856 y Fi(T)906 837 y Fb(\003)942 856 y Fd(R)990 837 y Ff(3)1046 685 y Fe(\000)1091 766 y Fm(\022)s Fn(\()p Fm(E)i Fk(\000)22 b(j)p Fo(p)g Fn(+)g Fo(A)p Fn(\()p Fo(x)p Fn(\))p Fk(j)1835 725 y Fl(2)1874 766 y Fn(\))g Fk(\000)h Fm(\022)s Fn(\()p Fm(E)28 b Fk(\000)23 b(j)p Fo(p)f Fn(+)g Fo(A)p Fn(\()p Fo(x)p Fn(\))p Fk(j)2778 725 y Fl(2)2839 766 y Fk(\000)g Fm(V)g Fn(\()p Fo(x)p Fn(\)\))3190 685 y Fe(\001)3252 766 y Fm(d)p Fo(x)p Fm(d)p Fo(p)28 b Fn(=)1266 998 y(4)p Fm(\031)p 1266 1042 108 4 v 1295 1134 a Fn(3)1400 930 y Fe(Z)1455 1155 y Fd(R)1503 1136 y Ff(3)1558 955 y Fe(\020)1618 1065 y Fm(E)1696 1014 y Fl(3)p Fi(=)p Fl(2)1690 1087 y(+)1828 1065 y Fk(\000)23 b Fn(\()p Fm(E)28 b Fk(\000)22 b Fm(V)g Fn(\()p Fo(x)p Fn(\)\))2417 1014 y Fl(3)p Fi(=)p Fl(2)2417 1087 y(+)2527 955 y Fe(\021)2603 1065 y Fm(d)p Fo(x)118 1348 y Fn(where)i Fm(\022)s Fn(\()p Fm(t)p Fn(\))k(:=)708 1208 y Fe(\032)824 1287 y Fn(0)97 b(if)104 b Fm(t)28 b Fk(\024)g Fn(0)p Fm(;)824 1407 y Fn(1)97 b(if)104 b Fm(t)28 b(>)f Fn(0)p Fm(;)1450 1348 y Fn(is)22 b(the)h(Hea)m(viside)g(function.)40 b(Note)23 b(that)f Fm(\030)3010 1363 y Fl(cl)3065 1348 y Fn(\()p Fm(E)6 b Fn(\))23 b(is)f(indep)s(enden)m(t)118 1521 y(of)32 b(the)h(magnetic)f(\014eld)g Fm(b)c Fk(\025)g Fn(0.)43 b(Eviden)m(tly)-8 b(,)33 b(under)h(the)f(assumptions)f(of)g (Theorem)h(2.1)f(w)m(e)i(ha)m(v)m(e)1279 1750 y(lim)1248 1812 y Fi(E)t Fh(!1)1461 1750 y Fm(E)1539 1709 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)1704 1750 y Fm(\030)1747 1765 y Fl(cl)1802 1750 y Fn(\()p Fm(E)6 b Fn(\))27 b(=)h(2)p Fm(\031)2211 1614 y Fe(Z)2267 1840 y Fd(R)2315 1821 y Ff(3)2370 1750 y Fm(V)22 b Fn(\()p Fo(x)p Fn(\))p Fm(d)p Fo(x)p Fm(:)118 1994 y Fn(Hence,)34 b(in)e(the)h(case)923 1913 y Fe(R)970 2028 y Fd(R)1018 2009 y Ff(3)1074 1994 y Fm(V)21 b Fn(\()p Fo(x)p Fn(\))p Fm(d)p Fo(x)28 b Fk(6)p Fn(=)f(0,)33 b(relation)d (\(2.1\))j(is)f(equiv)-5 b(alen)m(t)32 b(to)360 2179 y Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)776 2194 y Fl(0)814 2179 y Fn(\))28 b(=)f(\(2)p Fm(\031)t Fn(\))1167 2138 y Fh(\000)p Fl(3)1261 2179 y Fm(\030)1304 2194 y Fl(cl)1359 2179 y Fn(\()p Fm(E)6 b Fn(\)\(1)22 b(+)g Fm(o)p Fn(\(1\)\))p Fm(;)113 b(E)34 b Fk(!)27 b(1)p Fm(;)114 b(E)34 b Fk(2)28 b(O)2823 2194 y Fi(r)2861 2179 y Fm(;)115 b(r)30 b Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fm(:)242 b Fn(\(2.2\))118 2365 y(Asymptotic)26 b(relations)f(of)h(the)h (t)m(yp)s(e)h(\(2.2\))e(are)g(kno)m(wn)i(in)e(the)h(case)g(of)f(deca)m (ying)h(magnetic)f(p)s(oten-)118 2486 y(tials)j Fm(A)j Fn(\(see)g(e.g.)43 b([13]\).)g(Ho)m(w)m(ev)m(er,)33 b(in)e(the)g(last)f (case)i(the)g(magnetic)d(part)i Fm(iA:)p Fk(r)20 b Fn(+)f Fm(i)e Fn(div)28 b Fm(A)20 b Fn(+)e Fk(j)p Fm(A)p Fk(j)3811 2449 y Fl(2)118 2606 y Fn(of)28 b(the)g(op)s(erator)f Fm(H)857 2621 y Fl(0)924 2606 y Fn(is)h(a)f(relativ)m(ely)g(compact)h (p)s(erturbation)f(of)g(the)h(Laplacian)e Fk(\000)p Fn(\001,)k(so)e (that)g(the)118 2726 y(resem)m(blance)k(with)f(the)h(case)h(of)e(a)g (constan)m(t)h(magnetic)f(\014eld,)g(and,)h(hence,)h(of)e(a)h(linear)e (magnetic)118 2847 y(p)s(oten)m(tial,)36 b(considered)h(in)e(the)h (presen)m(t)i(note,)f(is)f(only)g(formal.)51 b(In)37 b(particular,)e(the)i(metho)s(ds)f(of)118 2967 y([13])c(are)h(not)f (directly)g(applicable.)118 3087 y(iii\))25 b(As)k(far)f(as)g(the)g (authors)h(are)f(informed,)g(the)g(high)f(energy)i(asymptotics)f(of)g (the)g(SSF)h(for)e(three-)118 3208 y(dimensional)33 b(Sc)m(hr\177)-49 b(odinger)35 b(op)s(erators)g(with)g(constan)m(t)g(magnetic)f(\014elds) i(is)e(in)m(v)m(estigated)h(for)g(the)118 3328 y(\014rst)g(time)e(in)h (the)h(presen)m(t)h(note.)50 b(Ho)m(w)m(ev)m(er,)38 b(w)m(e)d(w)m(ould) g(lik)m(e)f(to)g(men)m(tion)g(a)g(result)g(con)m(tained)h(in)118 3449 y([9])d(where)i(an)e(axisymmetric)f Fm(V)49 b Fn(=)27 b Fm(V)22 b Fn(\()p Fk(j)p Fm(X)1707 3464 y Fh(?)1766 3449 y Fk(j)p Fm(;)17 b(x)1893 3464 y Fl(3)1932 3449 y Fn(\))32 b(is)g(considered.)44 b(It)32 b(is)g(w)m(ell-kno)m(wn)h (\(see)g(e.g.)43 b([1]\))118 3569 y(that)28 b(in)g(this)g(case)h(the)g (op)s(erators)f Fm(H)35 b Fn(and)29 b Fm(H)1797 3584 y Fl(0)1864 3569 y Fn(are)g(unitarily)d(equiv)-5 b(alen)m(t)28 b(to)g(the)g(orthogonal)f(sums)118 3615 y Fe(P)223 3718 y Fi(m)p Fh(2)p Fd(Z)398 3689 y Fk(\010)p Fm(H)564 3653 y Fl(\()p Fi(m)p Fl(\))718 3689 y Fn(and)907 3615 y Fe(P)1013 3718 y Fi(m)p Fh(2)p Fd(Z)1187 3689 y Fk(\010)p Fm(H)1353 3638 y Fl(\()p Fi(m)p Fl(\))1345 3714 y(0)1507 3689 y Fn(resp)s(ectiv)m(ely)-8 b(,)34 b(where)g(the)f(op)s(erators)285 3957 y Fm(H)374 3906 y Fl(\()p Fi(m)p Fl(\))366 3982 y(0)522 3957 y Fn(:=)28 b Fk(\000)741 3890 y Fn(1)p 740 3934 51 4 v 740 4026 a Fm(\045)836 3890 y(@)p 811 3934 108 4 v 811 4026 a(@)5 b(\045)928 3957 y(\045)1013 3890 y(@)p 988 3934 V 988 4026 a(@)g(\045)1128 3957 y Fk(\000)1265 3890 y Fm(@)1321 3854 y Fl(2)p 1237 3934 152 4 v 1237 4026 a Fm(@)g(x)1348 3991 y Fl(2)1348 4050 y(3)1421 3957 y Fn(+)1519 3817 y Fe(\022)1602 3890 y Fm(b\045)p 1602 3934 92 4 v 1624 4026 a Fn(2)1726 3957 y(+)1834 3890 y Fm(m)p 1834 3934 86 4 v 1852 4026 a(\045)1930 3817 y Fe(\023)2003 3839 y Fl(2)2059 3957 y Fm(;)114 b(H)2289 3916 y Fl(\()p Fi(m)p Fl(\))2438 3957 y Fn(:=)27 b Fm(H)2657 3906 y Fl(\()p Fi(m)p Fl(\))2649 3982 y(0)2801 3957 y Fn(+)22 b Fm(V)f Fn(\()p Fm(\045;)c(x)3164 3972 y Fl(3)3204 3957 y Fn(\))p Fm(;)114 b(m)28 b Fk(2)g Fj(Z)p Fm(;)118 4205 y Fn(are)35 b(self-adjoin)m(t)d(in)i Fm(L)968 4168 y Fl(2)1008 4205 y Fn(\()p Fj(R)1112 4220 y Fl(+)1200 4205 y Fk(\002)24 b Fj(R)5 b Fn(;)17 b Fm(\045d\045dx)1668 4220 y Fl(3)1714 4205 y Fn(\).)49 b(F)-8 b(or)33 b(an)i(arbitrary)e (\014xed)j Fm(m)31 b Fk(2)g Fj(Z)h Fn(the)j(authors)f(of)g([9])118 4339 y(studied)40 b(the)g(asymptotics)g(as)g Fm(E)46 b Fk(!)39 b(1)p Fn(,)i Fm(E)46 b Fk(2)40 b(O)2045 4354 y Fi(r)2123 4339 y Fn(of)f(the)h(SSF)g Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)2925 4302 y Fl(\()p Fi(m)p Fl(\))3045 4339 y Fm(;)g(H)3178 4288 y Fl(\()p Fi(m)p Fl(\))3170 4363 y(0)3299 4339 y Fn(\).)65 b(Note)40 b(that)118 4459 y(\(2.1\))31 b(cannot)h(b)s(e)g(deduced)h(from)d(the)i(results)g(of)f ([9])h(ev)m(en)h(in)e(the)h(case)g(of)f(axial)f(symmetry)i(of)f Fm(V)21 b Fn(.)118 4787 y Fc(3)161 b(Auxiliary)57 b(results)118 5006 y Fo(3.1.)44 b Fn(In)33 b(this)f(subsection)i(w)m(e)f(in)m(tro)s (duce)g(some)f(basic)h(notations)e(used)j(throughout)e(the)h(pap)s(er.) 118 5126 y(As)27 b(ab)s(o)m(v)m(e,)h(w)m(e)f(denote)g(b)m(y)g Fm(S)1188 5141 y Fh(1)1288 5126 y Fn(the)g(class)f(of)g(linear)e (compact)i(op)s(erators)g(acting)f(in)g(a)h(\014xed)h(Hilb)s(ert)1960 5375 y(3)p eop %%Page: 4 4 4 3 bop 118 171 a Fn(space.)45 b(Let)33 b Fm(T)42 b Fn(=)28 b Fm(T)866 134 y Fh(\003)933 171 y Fk(2)g Fm(S)1087 186 y Fh(1)1162 171 y Fn(.)44 b(Denote)33 b(b)m(y)g Fj(P)1762 186 y Fi(I)1804 171 y Fn(\()p Fm(T)14 b Fn(\))32 b(the)h(sp)s(ectral)g (pro)5 b(jection)32 b(of)h Fm(T)46 b Fn(asso)s(ciated)33 b(with)118 291 y(the)g(in)m(terv)-5 b(al)31 b Fm(I)36 b Fk(\032)28 b Fj(R)5 b Fn(.)49 b(F)-8 b(or)32 b Fm(s)c(>)f Fn(0)33 b(set)1371 500 y Fm(n)1429 515 y Fh(\006)1489 500 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))27 b(:=)g(rank)h Fj(P)2162 515 y Fl(\()p Fi(s;)p Fh(1)p Fl(\))2346 500 y Fn(\()p Fk(\006)p Fm(T)14 b Fn(\))p Fm(:)118 709 y Fn(F)-8 b(or)32 b(an)g(arbitrary)g(\(not)g(necessarily)h(self-adjoin)m (t\))e(op)s(erator)h Fm(T)42 b Fk(2)28 b Fm(S)2726 724 y Fh(1)2833 709 y Fn(put)1289 918 y Fm(n)1347 933 y Fh(\003)1387 918 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))27 b(:=)h Fm(n)1840 933 y Fl(+)1899 918 y Fn(\()p Fm(s)1983 877 y Fl(2)2022 918 y Fn(;)17 b Fm(T)2137 877 y Fh(\003)2176 918 y Fm(T)d Fn(\))p Fm(;)114 b(s)28 b(>)f Fn(0)p Fm(:)971 b Fn(\(3.1\))118 1127 y(If)33 b Fm(T)41 b Fn(=)28 b Fm(T)489 1091 y Fh(\003)528 1127 y Fn(,)k(then)i(eviden)m(tly)1141 1336 y Fm(n)1199 1351 y Fh(\003)1239 1336 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))27 b(=)g Fm(n)1664 1351 y Fl(+)1723 1336 y Fn(\()p Fm(s;)17 b(T)d Fn(\))22 b(+)g Fm(n)2138 1351 y Fh(\000)2197 1336 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))p Fm(;)114 b(s)27 b(>)h Fn(0)p Fm(:)822 b Fn(\(3.2\))118 1545 y(F)-8 b(urther,)51 b(w)m(e)d(denote)g(b)m(y)g Fm(S)1214 1560 y Fi(p)1254 1545 y Fn(,)j Fm(p)i Fk(2)47 b Fn([1)p Fm(;)17 b Fk(1)p Fn(\),)50 b(the)e(Sc)m(hatten-v)m(on)g(Neumann)g (class)f(of)g(compact)118 1689 y(op)s(erators)32 b(for)g(whic)m(h)h (the)f(norm)g Fk(k)p Fm(T)14 b Fk(k)1570 1704 y Fi(p)1641 1689 y Fn(:)43 b(=)1819 1609 y Fe(\000)1865 1689 y Fm(p)1931 1609 y Fe(R)1997 1635 y Fh(1)1978 1724 y Fl(0)2088 1689 y Fm(s)2134 1653 y Fi(p)p Fh(\000)p Fl(1)2264 1689 y Fm(n)2322 1704 y Fh(\003)2362 1689 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))j Fm(ds)2713 1609 y Fe(\001)2757 1631 y Fl(1)p Fi(=p)2899 1689 y Fn(is)32 b(\014nite.)43 b(In)33 b(particular,)118 1810 y(as)f(already)f(indicated,)g Fm(S)1089 1825 y Fl(1)1160 1810 y Fn(stands)h(for)f(the)h(trace)g (class,)g(and)f Fm(S)2524 1825 y Fl(2)2595 1810 y Fn(for)g(the)h(Hilb)s (ert-Sc)m(hmidt)d(class.)118 1930 y(If)k Fm(T)41 b Fk(2)28 b Fm(S)468 1945 y Fi(p)508 1930 y Fn(,)k Fm(p)c Fk(2)g Fn([1)p Fm(;)17 b Fk(1)p Fn(\),)32 b(then)h(the)g(follo)m(wing)d (elemen)m(tary)j(inequalit)m(y)1576 2139 y Fm(n)1634 2154 y Fh(\003)1673 2139 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))27 b Fk(\024)h Fm(s)2088 2098 y Fh(\000)p Fi(p)2183 2139 y Fk(k)p Fm(T)14 b Fk(k)2354 2098 y Fi(p)2354 2164 y(p)3650 2139 y Fn(\(3.3\))118 2348 y(holds)32 b(for)g(ev)m(ery)j Fm(s)27 b(>)h Fn(0.)43 b(If)33 b Fm(T)41 b Fn(=)28 b Fm(T)1447 2312 y Fh(\003)1514 2348 y Fk(2)g Fm(S)1668 2363 y Fi(p)1707 2348 y Fn(,)33 b Fm(p)28 b Fk(2)g Fn([1)p Fm(;)17 b Fk(1)p Fn(\),)31 b(then)j(\(3.2\))e(and)g(\(3.3\))g(imply) 1369 2557 y Fm(n)1427 2572 y Fh(\006)1486 2557 y Fn(\()p Fm(s)p Fn(;)17 b Fm(T)d Fn(\))27 b Fk(\024)h Fm(s)1901 2516 y Fh(\000)p Fi(p)1996 2557 y Fk(k)p Fm(T)14 b Fk(k)2167 2516 y Fi(p)2167 2582 y(p)2206 2557 y Fm(;)114 b(s)27 b(>)h Fn(0)p Fm(:)1050 b Fn(\(3.4\))118 2766 y(Finally)-8 b(,)29 b(w)m(e)i(de\014ne)h(the)f(self-adjoin)m(t)e(op)s(erators)i(Re)d Fm(T)41 b Fn(:=)2367 2727 y Fl(1)p 2367 2743 36 4 v 2367 2800 a(2)2412 2766 y Fn(\()p Fm(T)32 b Fn(+)18 b Fm(T)2704 2730 y Fh(\003)2743 2766 y Fn(\))31 b(and)g(Im)c Fm(T)41 b Fn(:=)3395 2727 y Fl(1)p 3383 2743 60 4 v 3383 2800 a(2)p Fi(i)3452 2766 y Fn(\()p Fm(T)32 b Fk(\000)19 b Fm(T)3746 2730 y Fh(\003)3785 2766 y Fn(\).)118 3007 y Fo(3.2.)43 b Fn(In)31 b(this)f(subsection)h(w)m(e)g(in)m(tro)s(duce)f (the)h(concepts)h(of)e(index)g(of)g(a)g(F)-8 b(redholm)29 b(pair)g(of)g(orthog-)118 3127 y(onal)35 b(pro)5 b(jections,)37 b(and)f(index)h(for)e(a)h(pair)f(of)g(selfadjoin)m(t)g(op)s(erators,)i (and)f(describ)s(e)h(some)e(basic)118 3248 y(prop)s(erties)c(related)g (to)f(these)i(concepts)h(whic)m(h)f(will)c(b)s(e)j(often)g(used)i(in)d (the)h(sequel.)44 b(More)31 b(details)118 3368 y(can)i(b)s(e)g(found)f (in)g([2].)118 3488 y(A)h(pair)e(of)h(orthogonal)f(pro)5 b(jections)33 b Fm(P)s(;)17 b(Q)32 b Fn(is)g(said)g(to)g(b)s(e)h(F)-8 b(redholm)31 b(if)1417 3697 y Fk(f)o(\000)p Fn(1)p Fm(;)17 b Fn(1)p Fk(g)22 b(\\)h Fm(\033)1901 3712 y Fl(ess)1992 3697 y Fn(\()p Fm(P)36 b Fk(\000)22 b Fm(Q)p Fn(\))28 b(=)g Fk(;)p Fm(:)118 3906 y Fn(In)33 b(particular,)e(if)g Fm(P)36 b Fk(\000)23 b Fm(Q)k Fk(2)i Fm(S)1264 3921 y Fh(1)1338 3906 y Fn(,)k(then)g(the)g(pair)f Fm(P)s(;)17 b(Q)32 b Fn(is)g(F)-8 b(redholm.)118 4027 y(Assume)34 b(that)e(the)h(pair)e(of)i(orthogonal)d(pro)5 b(jections)33 b Fm(P)s(;)17 b(Q)32 b Fn(is)g(F)-8 b(redholm.)42 b(Set)686 4236 y(index\()p Fm(P)s(;)17 b(Q)p Fn(\))28 b(:=)f(dim)f(Ker)i(\()p Fm(P)36 b Fk(\000)22 b Fm(Q)h Fk(\000)f Fm(I)8 b Fn(\))22 b Fk(\000)h Fn(dim)j(Ker)i(\()p Fm(P)35 b Fk(\000)23 b Fm(Q)g Fn(+)f Fm(I)8 b Fn(\))p Fm(:)118 4445 y Fn(Let)337 4419 y(~)301 4445 y Fm(M)j Fn(,)43 b Fm(M)10 b Fn(,)44 b(b)s(e)d(b)s(ounded)g(self-adjoin)m(t)e(op)s(erators.)69 b(If)40 b(the)i(sp)s(ectral)e(pro)5 b(jections)41 b Fj(P)3428 4460 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))3670 4445 y Fn(\()3744 4419 y(~)3708 4445 y Fm(M)10 b Fn(\))118 4565 y(and)33 b Fj(P)367 4581 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))608 4565 y Fn(\()p Fm(M)10 b Fn(\))33 b(form)f(a)g(F)-8 b(redholm)31 b(pair,)g(w)m(e)j(shall)d(use)j(the)f(short-hand)f(notation)1003 4785 y(ind\()1213 4760 y(~)1177 4785 y Fm(M)11 b(;)17 b(M)10 b Fn(\))28 b(:=)f(index)q(\()p Fj(P)1954 4801 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))2195 4785 y Fn(\()2269 4760 y(~)2233 4785 y Fm(M)11 b Fn(\))p Fm(;)17 b Fj(P)2479 4801 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))2719 4785 y Fn(\()p Fm(M)10 b Fn(\)\))p Fm(:)118 5006 y Fn(A)29 b(su\016cien)m(t)g (condition)e(that)i(the)g(pair)e Fj(P)1676 5021 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))1918 5006 y Fn(\()1992 4981 y(~)1956 5006 y Fm(M)10 b Fn(\))p Fm(;)17 b Fj(P)2201 5021 y Fl(\()p Fh(\0001)p Fi(;)p Fl(0\))2442 5006 y Fn(\()p Fm(M)10 b Fn(\))29 b(b)s(e)g(F)-8 b(redholm,)28 b(is)3369 4981 y(~)3333 5006 y Fm(M)38 b Fn(=)28 b Fm(M)d Fn(+)14 b Fm(A)118 5126 y Fn(where)34 b Fm(M)43 b Fn(is)32 b(a)h(b)s(ounded)g (self-adjoin)m(t)e(op)s(erator)h(suc)m(h)i(that)e(0)c Fk(62)g Fm(\033)2667 5141 y Fl(ess)2758 5126 y Fn(\()p Fm(M)10 b Fn(\),)34 b(and)e Fm(A)c Fn(=)g Fm(A)3466 5090 y Fh(\003)3533 5126 y Fk(2)g Fm(S)3687 5141 y Fh(1)3762 5126 y Fn(.)1960 5375 y(4)p eop %%Page: 5 5 5 4 bop 118 171 a Fo(Lemma)38 b(3.1.)43 b Fn(\(see)34 b([12,)g(Lemma)e(5.2]\))k Fa(L)-5 b(et)36 b Fm(M)k Fn(=)29 b Fm(M)2202 134 y Fh(\003)2243 171 y Fa(,)36 b Fn(0)29 b Fk(62)h Fm(\033)t Fn(\()p Fm(M)10 b Fn(\))p Fa(,)36 b Fn(0)29 b Fk(\024)h Fm(A)g Fn(=)f Fm(A)3254 134 y Fh(\003)3323 171 y Fk(2)h Fm(S)3479 186 y Fh(1)3554 171 y Fa(.)47 b(Then)118 291 y(for)35 b Fm(t)28 b Fk(2)g Fn(\(0)p Fm(;)17 b Fk(1)p Fn(\))34 b Fa(we)g(have)909 498 y Fn(ind)o(\()p Fm(M)f Fn(+)22 b Fm(tA;)17 b(M)10 b Fn(\))28 b(=)g Fk(\000)17 b Fn(lim)1843 561 y Fi(")p Fh(#)p Fl(0)1979 498 y Fm(n)2037 513 y Fh(\000)2096 498 y Fn(\(1)22 b Fk(\000)h Fm(")p Fn(;)17 b Fm(tA)2503 457 y Fl(1)p Fi(=)p Fl(2)2613 498 y Fm(M)2717 457 y Fh(\000)p Fl(1)2812 498 y Fm(A)2885 457 y Fl(1)p Fi(=)p Fl(2)2995 498 y Fn(\))p Fm(;)590 b Fn(\(3.5\))1115 769 y(ind)o(\()p Fm(M)33 b Fk(\000)23 b Fm(tA;)17 b(M)10 b Fn(\))28 b(=)g Fm(n)1999 784 y Fl(+)2058 769 y Fn(\(1;)17 b Fm(tA)2297 728 y Fl(1)p Fi(=)p Fl(2)2407 769 y Fm(M)2511 728 y Fh(\000)p Fl(1)2606 769 y Fm(A)2679 728 y Fl(1)p Fi(=)p Fl(2)2789 769 y Fn(\))p Fm(:)796 b Fn(\(3.6\))118 961 y Fo(Lemma)49 b(3.2.)e Fn(\(see)c([5,)i (Subsection)f(3.2,)g(Prop)s(ert)m(y)g(\(g\)]\))g Fa(L)-5 b(et)44 b Fm(M)55 b Fa(b)-5 b(e)43 b(a)h(b)-5 b(ounde)g(d)44 b(self-adjoint)118 1082 y(op)-5 b(er)g(ator)36 b(such)g(that)g Fn(0)30 b Fk(62)g Fm(\033)t Fn(\()p Fm(M)10 b Fn(\))p Fa(.)49 b(L)-5 b(et)36 b Fm(A)g Fa(and)g Fm(B)41 b Fa(b)-5 b(e)36 b(c)-5 b(omp)g(act)35 b(self-adjoint)g(op)-5 b(er)g(ators.)47 b(Then)36 b(for)118 1202 y Fm(s)28 b Fk(2)g Fn(\(0)p Fm(;)17 b Fk(1)p Fn(\))34 b Fa(such)g(that)i Fn([)p Fk(\000)p Fm(s;)17 b(s)p Fn(])22 b Fk(\\)h Fm(\033)t Fn(\()p Fm(M)10 b Fn(\))28 b(=)f Fk(;)35 b Fa(we)g(have)118 1409 y Fn(ind\()p Fm(M)14 b Fn(+)t Fm(s)t Fn(+)t Fm(B)5 b(;)17 b(M)d Fn(+)t Fm(s)p Fn(\))t Fk(\000)t Fm(n)1148 1424 y Fl(+)1207 1409 y Fn(\()p Fm(s)p Fn(;)j Fm(A)p Fn(\))27 b Fk(\024)h Fn(ind)o(\()p Fm(M)14 b Fn(+)t Fm(A)t Fn(+)t Fm(B)5 b(;)17 b(M)10 b Fn(\))28 b Fk(\024)h Fn(ind)o(\()p Fm(M)14 b Fk(\000)t Fm(s)t Fn(+)t Fm(B)5 b(;)17 b(M)d Fk(\000)t Fm(s)p Fn(\))t(+)t Fm(n)3525 1424 y Fh(\000)3585 1409 y Fn(\()p Fm(s)p Fn(;)j Fm(A)p Fn(\))p Fm(:)3650 1529 y Fn(\(3.7\))118 1721 y Fo(Lemma)35 b(3.3.)41 b Fn(\(see)32 b([11)o(,)g(Lemma)d(2.1],)i([5,)g (Subsection)h(3.3]\))g Fa(L)-5 b(et)34 b Fm(M)44 b Fa(b)-5 b(e)33 b(a)g(b)-5 b(ounde)g(d)33 b(self-adjoint)118 1842 y(op)-5 b(er)g(ator)42 b(such)f(that)i Fn(0)d Fk(62)h Fm(\033)t Fn(\()p Fm(M)10 b Fn(\))p Fa(.)67 b(L)-5 b(et)42 b Fm(T)1708 1857 y Fl(1)1788 1842 y Fn(=)f Fm(T)1976 1805 y Fh(\003)1962 1866 y Fl(1)2055 1842 y Fk(2)h Fm(S)2223 1857 y Fh(1)2339 1842 y Fa(and)f Fm(T)2592 1857 y Fl(2)2673 1842 y Fn(=)f Fm(T)2860 1805 y Fh(\003)2846 1866 y Fl(2)2940 1842 y Fk(2)h Fm(S)3107 1857 y Fl(1)3147 1842 y Fa(.)65 b(Then)41 b(for)h(e)-5 b(ach)118 1962 y Fm(s)164 1977 y Fl(1)231 1962 y Fm(>)28 b Fn(0)p Fa(,)34 b Fm(s)494 1977 y Fl(2)561 1962 y Fm(>)28 b Fn(0)34 b Fa(such)h(that)g Fn([)p Fk(\000)p Fm(s;)17 b(s)p Fn(])23 b Fk(\\)f Fm(\033)t Fn(\()p Fm(M)10 b Fn(\))29 b(=)e Fk(;)35 b Fa(with)g Fm(s)27 b Fn(=)h Fm(s)2438 1977 y Fl(1)2499 1962 y Fn(+)22 b Fm(s)2643 1977 y Fl(2)2718 1962 y Fa(we)34 b(have)767 2082 y Fe(Z)822 2307 y Fd(R)891 2217 y Fk(j)o Fn(ind\()p Fm(M)f Fn(+)22 b Fm(T)1374 2232 y Fl(1)1436 2217 y Fn(+)g Fm(t)17 b(T)1643 2232 y Fl(2)1682 2217 y Fm(;)g(M)10 b Fn(\))p Fk(j)33 b Fm(d\026)p Fn(\()p Fm(t)p Fn(\))27 b Fk(\024)i Fm(n)2341 2232 y Fh(\003)2380 2217 y Fn(\()p Fm(s)2464 2232 y Fl(1)2504 2217 y Fn(;)17 b Fm(T)2605 2232 y Fl(1)2644 2217 y Fn(\))22 b(+)2860 2150 y(1)p 2812 2194 145 4 v 2812 2286 a Fm(\031)t(s)2917 2301 y Fl(2)2966 2217 y Fk(k)p Fm(T)3073 2232 y Fl(2)3113 2217 y Fk(k)3163 2232 y Fl(1)3650 2217 y Fn(\(3.8\))118 2487 y Fa(wher)-5 b(e)34 b Fm(d\026)p Fn(\()p Fm(t)p Fn(\))27 b(:=)786 2448 y Fl(1)p 782 2464 43 4 v 782 2522 a Fi(\031)889 2448 y(dt)p 845 2464 151 4 v 845 2522 a Fl(1+)p Fi(t)960 2503 y Ff(2)1005 2487 y Fa(.)118 2679 y Fo(3.3.)54 b Fn(In)37 b(this)e(subsection)j(w)m(e)f(describ)s(e)f(a)g(represen)m (tation)h(of)f(the)g(SSF)g Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)3305 2694 y Fl(0)3343 2679 y Fn(\))36 b(whic)m(h)h(is)f(a)118 2800 y(sp)s(ecial)j(case)i(of)f(the)h(general)e (represen)m(tation)i(of)f(the)g(SSF)g(for)g(a)g(pair)f(of)h(lo)m(w)m (er-b)s(ounded)g(self-)118 2920 y(adjoin)m(t)28 b(op)s(erators,)h (obtained)g(b)m(y)g(F.)g(Gesztesy)-8 b(,)31 b(K.)e(Mak)-5 b(aro)m(v,)30 b(and)f(A.)g(Pushnitski)g(\(see)h([11],)f([8],)118 3040 y([12]\).)118 3161 y(F)-8 b(or)32 b Fm(z)g Fk(2)c Fj(C)20 b Fn(,)39 b(Im)16 b Fm(z)32 b(>)c Fn(0,)k(set)i Fm(T)14 b Fn(\()p Fm(z)t Fn(\))28 b(:=)f Fk(j)p Fm(V)22 b Fk(j)1660 3125 y Fl(1)p Fi(=)p Fl(2)1769 3161 y Fn(\()p Fm(H)1888 3176 y Fl(0)1950 3161 y Fk(\000)g Fm(z)t Fn(\))2136 3125 y Fh(\000)p Fl(1)2231 3161 y Fk(j)p Fm(V)f Fk(j)2365 3125 y Fl(1)p Fi(=)p Fl(2)2475 3161 y Fn(.)118 3353 y Fo(Lemma)44 b(3.4.)h Fn(\(see)39 b([5,)h(Lemma)d(3.1]\))j Fa(L)-5 b(et)50 b Fn(\(1.1\))40 b Fa(hold.)60 b(Then)40 b(for)g(every)g Fm(E)k Fk(2)38 b Fj(R)f Fk(n)26 b Fn(2)p Fm(b)p Fj(Z)3595 3368 y Fl(+)3652 3353 y Fa(,)41 b(the)118 3473 y(op)-5 b(er)g(ator-norm)34 b(limit)1416 3594 y Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))28 b(:=)f(n)c Fk(\000)f Fn(lim)1991 3656 y Fi(\016)r Fh(#)p Fl(0)2127 3594 y Fm(T)14 b Fn(\()p Fm(E)28 b Fn(+)22 b Fm(i\016)t Fn(\))1098 b(\(3.9\))118 3811 y Fa(exists,)35 b(and)f(by)44 b Fn(\(1.3\))34 b Fa(we)g(have)h Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))27 b Fk(2)h Fm(S)1750 3826 y Fh(1)1825 3811 y Fa(.)44 b(Mor)-5 b(e)g(over,)35 b Fn(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))27 b Fk(2)h Fm(S)2897 3826 y Fl(1)2937 3811 y Fa(.)118 4003 y Fn(Lemma)j(3.4)h(follo)m(ws)g(easily)g(from)f(Prop)s(ositions)g(3.2)h (and)h(3.3)f(b)s(elo)m(w)h(\(see)g(Corollary)e(3.2\).)118 4123 y(Denote)i(b)m(y)g Fm(J)42 b Fn(the)33 b(m)m(ultiplier)c(b)m(y)k (the)g(function)1227 4382 y(sign)27 b Fm(V)22 b Fn(\()p Fo(x)p Fn(\))27 b(=)1768 4242 y Fe(\032)1884 4321 y Fn(1)98 b(if)103 b Fm(V)21 b Fn(\()p Fo(x)p Fn(\))28 b Fk(\025)g Fn(0)p Fm(;)1884 4442 y Fk(\000)p Fn(1)98 b(if)103 b Fm(V)22 b Fn(\()p Fo(x)p Fn(\))28 b Fm(<)f Fn(0)p Fm(:)118 4642 y Fn(In)m(tro)s(duce)34 b(the)f(function)298 4846 y(~)288 4872 y Fm(\030)t Fn(\()p Fm(E)6 b Fn(;)17 b Fm(H)r(;)g(H)703 4887 y Fl(0)742 4872 y Fn(\))27 b(=)911 4737 y Fe(Z)966 4962 y Fd(R)1035 4872 y Fn(ind)o(\()p Fm(J)32 b Fn(+)22 b(Re)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))22 b(+)g Fm(t)28 b Fn(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))p Fm(;)17 b(J)9 b Fn(\))27 b Fm(d\026)p Fn(\()p Fm(t)p Fn(\))p Fm(;)114 b(E)34 b Fk(2)28 b Fj(R)33 b Fk(n)22 b Fn(2)p Fm(b)p Fj(Z)3348 4887 y Fl(+)3405 4872 y Fm(;)169 b Fn(\(3.10\))118 5126 y(whic)m(h)33 b(is)f(w)m(ell-de\014ned)h(b)m(y)h(Lemmas)d(3.3)h (and)h(3.4.)1960 5375 y(5)p eop %%Page: 6 6 6 5 bop 118 171 a Fo(Prop)s(osition)38 b(3.1.)43 b Fn(\(see)35 b([5,)g(Prop)s(osition)e(2.5]\))k Fa(The)f(function)2628 144 y Fn(~)2618 171 y Fm(\030)t Fn(\()p Fm(E)6 b Fn(;)17 b Fm(H)r(;)g(H)3033 186 y Fl(0)3072 171 y Fn(\))37 b Fa(is)f(c)-5 b(ontinuous)37 b(on)118 291 y Fj(R)c Fk(n)22 b(f)p Fn(2)p Fm(b)p Fj(Z)493 306 y Fl(+)572 291 y Fk([)g Fm(\033)715 306 y Fi(p)755 291 y Fn(\()p Fm(H)8 b Fn(\))p Fk(g)p Fa(,)34 b(and)h(is)f(b)-5 b(ounde)g(d)34 b(on)h(every)g(c)-5 b(omp)g(act)34 b(subset)h(of)f Fj(R)f Fk(n)22 b Fn(2)p Fm(b)p Fj(Z)3201 306 y Fl(+)3258 291 y Fa(.)118 468 y(R)-5 b(emark)p Fn(:)48 b(Note)36 b(that,)f(in)g(con)m(trast)h(to)e(the)i (case)g Fm(b)d Fn(=)f(0,)k(w)m(e)g(cannot)f(rule)g(out)g(the)h(p)s (ossibilit)m(y)d(of)118 588 y(existence)j(of)d(em)m(b)s(edded)i(eigen)m (v)-5 b(alues,)35 b(b)m(y)g(imp)s(osing)c(short-range)j(assumptions)g (of)f(the)i(t)m(yp)s(e)g(of)118 709 y(\(1.1\):)59 b(Theorem)41 b(5.1)f(of)f([1])i(sho)m(ws)h(that)e(there)h(are)g(axisymmetric)e(p)s (oten)m(tials)g Fm(V)62 b Fn(of)40 b(compact)118 829 y(supp)s(ort)30 b(suc)m(h)h(that)e(b)s(elo)m(w)h(eac)m(h)g(Landau)g (lev)m(el)f(2)p Fm(bq)t Fn(,)h Fm(q)h Fk(2)e Fj(Z)2393 844 y Fl(+)2449 829 y Fn(,)h(there)g(exists)h(an)e(in\014nite)g (sequence)118 950 y(of)40 b(eigen)m(v)-5 b(alues)41 b(of)f Fm(H)48 b Fn(whic)m(h)41 b(con)m(v)m(erges)i(to)d(2)p Fm(bq)t Fn(.)68 b(On)40 b(the)h(other)g(hand,)i(generically)-8 b(,)41 b(the)g(only)118 1070 y(p)s(ossible)28 b(accum)m(ulation)f(p)s (oin)m(ts)h(of)f(the)i(eigen)m(v)-5 b(alues)29 b(of)f(the)g(op)s (erators)h Fm(H)36 b Fn(are)28 b(the)h(Landau)f(lev)m(els)118 1190 y(\(see)40 b([1,)g(Theorem)f(4.7],)g([7,)h(Theorem)f(3.5.3)f (\(iii\)]\).)59 b(F)-8 b(urther)38 b(information)e(of)i(the)h(lo)s (cation)d(of)118 1311 y(these)e(eigen)m(v)-5 b(alues)32 b(can)h(b)s(e)g(found)g(in)f([5)o(].)118 1488 y Fo(Theorem)42 b(3.1.)i Fn(\(see)37 b([11],)h([8)o(],)g([12],)f([5,)h(Subsection)f (3.3]\))h Fa(L)-5 b(et)49 b Fn(\(1.1\))38 b Fa(hold.)55 b(Then)38 b(for)g(almost)118 1608 y(every)d Fm(E)f Fk(2)28 b Fj(R)46 b Fa(we)34 b(have)1413 1728 y Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)1829 1743 y Fl(0)1867 1728 y Fn(\))28 b(=)2046 1702 y(~)2036 1728 y Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)2452 1743 y Fl(0)2490 1728 y Fn(\))p Fm(:)1046 b Fn(\(3.11\))118 1905 y Fa(R)-5 b(emark)p Fn(:)63 b(As)44 b(explained)e(in)g(the)i(In)m(tro)s(duction,) h(w)m(e)f(iden)m(tify)e Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)2980 1920 y Fl(0)3018 1905 y Fn(\))43 b(with)3341 1879 y(~)3331 1905 y Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)3747 1920 y Fl(0)3785 1905 y Fn(\).)118 2026 y(The)35 b(iden)m(ti\014cation)d(on)i(the)g(set)h Fj(R)f Fk(n)22 b(f)p Fn(2)p Fm(b)p Fj(Z)1745 2041 y Fl(+)1825 2026 y Fk([)h Fm(\033)1969 2041 y Fi(p)2009 2026 y Fn(\()p Fm(H)8 b Fn(\))p Fk(g)33 b Fn(where)2551 2000 y(~)2540 2026 y Fm(\030)39 b Fn(is)33 b(con)m(tin)m(uous,)i(is)e(natural.)46 b(On)118 2146 y(the)27 b(other)g(hand,)h(the)f(v)-5 b(alues)27 b(prescrib)s(ed)g(to)f(the)h(SSF)g(at)f(the)h(eigen)m(v)-5 b(alues)27 b Fm(E)33 b Fk(2)28 b Fm(\033)3207 2161 y Fi(p)3248 2146 y Fn(\()p Fm(H)8 b Fn(\))26 b(ma)m(y)g(seem)118 2267 y(somewhat)k(arbitrary;)f(in)g(an)m(y)h(case,)h(as)f(Theorem)f (2.1)g(sho)m(ws,)j(these)f(v)-5 b(alues)29 b(are)g(consisten)m(t)i (with)118 2387 y(the)i(asymptotics)f(of)h Fm(\030)5 b Fn(\()p Fm(E)h Fn(;)17 b Fm(H)r(;)g(H)1354 2402 y Fl(0)1392 2387 y Fn(\))32 b(as)h Fm(E)h Fk(!)27 b(1)p Fn(,)32 b Fm(E)i Fk(2)28 b(O)2253 2402 y Fi(r)2291 2387 y Fn(,)33 b Fm(r)d Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\),)33 b Fa(and)f Fm(E)h Fk(62)c Fm(\033)3231 2402 y Fi(p)3271 2387 y Fn(\()p Fm(H)8 b Fn(\).)118 2628 y Fo(3.4.)44 b Fn(In)m(tro)s(duce)33 b(the)g(Landau)g(Hamiltonian)995 2866 y Fm(h)p Fn(\()p Fm(b)p Fn(\))28 b(:=)1327 2725 y Fe(\022)1400 2866 y Fm(i)1491 2799 y(@)p 1443 2843 152 4 v 1443 2934 a(@)5 b(x)1554 2949 y Fl(1)1627 2866 y Fk(\000)1737 2799 y Fm(bx)1833 2814 y Fl(2)p 1737 2843 137 4 v 1780 2934 a Fn(2)1883 2725 y Fe(\023)1956 2748 y Fl(2)2018 2866 y Fn(+)2116 2725 y Fe(\022)2189 2866 y Fm(i)2280 2799 y(@)p 2232 2843 152 4 v 2232 2934 a(@)g(x)2343 2949 y Fl(2)2416 2866 y Fn(+)2524 2799 y Fm(bx)2620 2814 y Fl(1)p 2524 2843 137 4 v 2568 2934 a Fn(2)2671 2725 y Fe(\023)2744 2748 y Fl(2)2806 2866 y Fk(\000)22 b Fm(b;)628 b Fn(\(3.12\))118 3107 y(i.e.)42 b(the)31 b(t)m(w)m(o-dimensional)d(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erator)e(with)h(constan)m(t)h(scalar)f(magnetic)f (\014eld)h Fm(b)e(>)g Fn(0,)118 3227 y(essen)m(tially)h(self-adjoin)m (t)f(on)i Fm(C)1291 3191 y Fh(1)1284 3252 y Fl(0)1365 3227 y Fn(\()p Fj(R)1469 3191 y Fl(2)1515 3227 y Fn(\).)42 b(It)30 b(is)f(w)m(ell-kno)m(wn)h(that)f Fm(\033)t Fn(\()p Fm(h)p Fn(\()p Fm(b)p Fn(\)\))f(=)g Fk([)3038 3191 y Fh(1)3038 3252 y Fi(q)r Fl(=0)3183 3227 y Fk(f)p Fn(2)p Fm(bq)t Fk(g)o Fn(,)j(and)e(eac)m(h)118 3348 y(eigen)m(v)-5 b(alue)32 b(2)p Fm(bq)t Fn(,)h Fm(q)e Fk(2)d Fj(Z)1018 3363 y Fl(+)1075 3348 y Fn(,)33 b(has)f(in\014nite)g(m)m(ultiplicit)m (y)d(\(see)34 b(e.g.)43 b([1]\).)118 3468 y(F)-8 b(or)29 b Fo(x)p Fm(;)17 b Fo(x)452 3432 y Fh(0)503 3468 y Fk(2)28 b Fj(R)663 3432 y Fl(2)737 3468 y Fn(denote)i(b)m(y)h Fk(P)1250 3483 y Fi(q)r(;b)1338 3468 y Fn(\()p Fo(x)p Fm(;)17 b Fo(x)1538 3432 y Fh(0)1561 3468 y Fn(\))29 b(the)h(in)m(tegral)e(k)m(ernel)i(of)f(the)g(orthogonal)f(pro)5 b(jection)29 b Fm(p)3695 3483 y Fi(q)3733 3468 y Fn(\()p Fm(b)p Fn(\))118 3588 y(on)m(to)k(the)g(subspace)h(Ker)28 b(\()p Fm(h)p Fn(\()p Fm(b)p Fn(\))23 b Fk(\000)f Fn(2)p Fm(bq)t Fn(\),)33 b Fm(q)e Fk(2)d Fj(Z)1903 3603 y Fl(+)1960 3588 y Fn(.)43 b(It)33 b(is)f(w)m(ell-kno)m(wn)h(that)333 3837 y Fk(P)402 3852 y Fi(q)r(;b)490 3837 y Fn(\()p Fo(x)p Fm(;)17 b Fo(x)690 3796 y Fh(0)714 3837 y Fn(\))27 b(=)926 3769 y Fm(b)p 893 3814 108 4 v 893 3905 a Fn(2)p Fm(\031)1010 3837 y(L)1076 3852 y Fi(q)1131 3696 y Fe(\022)1215 3769 y Fm(b)p Fk(j)p Fo(x)22 b Fk(\000)h Fo(x)1524 3733 y Fh(0)1547 3769 y Fk(j)1575 3733 y Fl(2)p 1215 3814 400 4 v 1390 3905 a Fn(2)1624 3696 y Fe(\023)1714 3837 y Fn(exp)1880 3696 y Fe(\022)1953 3837 y Fk(\000)2044 3769 y Fm(b)p 2040 3814 49 4 v 2040 3905 a Fn(4)2099 3837 y(\()p Fk(j)p Fo(x)f Fk(\000)h Fo(x)2405 3796 y Fh(0)2428 3837 y Fk(j)2456 3796 y Fl(2)2518 3837 y Fn(+)f(2)p Fm(i)p Fn(\()p Fm(x)2791 3852 y Fl(1)2830 3837 y Fm(x)2885 3796 y Fh(0)2885 3861 y Fl(2)2947 3837 y Fk(\000)h Fm(x)3102 3796 y Fh(0)3102 3861 y Fl(1)3142 3837 y Fm(x)3197 3852 y Fl(2)3237 3837 y Fn(\)\))3313 3696 y Fe(\023)3601 3837 y Fn(\(3.13\))118 4114 y(\(see)39 b([10]\))e(where)i Fm(L)899 4129 y Fi(q)938 4114 y Fn(\()p Fm(t)p Fn(\))d(:=)1225 4039 y Fe(P)1330 4066 y Fi(q)1330 4143 y(k)r Fl(=0)1479 4033 y Fe(\000)1527 4070 y Fi(q)1525 4149 y(k)1564 4033 y Fe(\001)1619 4067 y Fl(\()p Fh(\000)p Fi(t)p Fl(\))1753 4043 y Fg(k)p 1619 4091 174 4 v 1677 4149 a Fi(k)r Fl(!)1803 4114 y Fn(,)j Fm(t)d Fk(2)h Fj(R)5 b Fn(,)45 b Fm(q)c Fk(2)c Fj(Z)2437 4129 y Fl(+)2493 4114 y Fn(,)i(are)f(the)g(Laguerre)g (p)s(olynomials.)118 4235 y(Note)33 b(that)1657 4379 y Fk(P)1726 4394 y Fi(q)r(;b)1814 4379 y Fn(\()p Fo(x)p Fm(;)17 b Fo(x)p Fn(\))28 b(=)2227 4312 y Fm(b)p 2194 4357 108 4 v 2194 4448 a Fn(2)p Fm(\031)3601 4379 y Fn(\(3.14\))118 4576 y(for)c(eac)m(h)i Fm(q)31 b Fk(2)d Fj(Z)708 4591 y Fl(+)789 4576 y Fn(and)d Fo(x)j Fk(2)g Fj(R)1218 4540 y Fl(2)1263 4576 y Fn(.)41 b(In)m(tro)s(duce)26 b(the)f(orthogonal)e (pro)5 b(jections)25 b Fm(P)2962 4591 y Fi(q)3027 4576 y Fn(:)j Fm(L)3148 4540 y Fl(2)3188 4576 y Fn(\()p Fj(R)3292 4540 y Fl(3)3337 4576 y Fn(\))g Fk(!)f Fm(L)3596 4540 y Fl(2)3636 4576 y Fn(\()p Fj(R)3740 4540 y Fl(3)3785 4576 y Fn(\),)118 4696 y Fm(q)39 b Fk(2)d Fj(Z)371 4711 y Fl(+)428 4696 y Fn(,)i(b)m(y)g Fm(P)696 4711 y Fi(q)769 4696 y Fn(:=)e Fm(p)957 4711 y Fi(q)1020 4696 y Fk(\012)26 b Fm(I)45 b Fn(where)38 b Fm(I)45 b Fn(denotes)38 b(the)g(iden)m(tit)m (y)f(op)s(erator)f(in)h Fm(L)3062 4660 y Fl(2)3102 4696 y Fn(\()p Fj(R)3206 4711 y Fi(x)3246 4720 y Ff(3)3290 4696 y Fn(\).)57 b(F)-8 b(or)36 b Fm(z)41 b Fk(2)35 b Fj(C)118 4863 y Fn(with)e(Im)c Fm(z)34 b(>)29 b Fn(0,)34 b(de\014ne)h(the)f(op)s(erator)e Fm(R)q Fn(\()p Fm(z)t Fn(\))f(:=)1989 4753 y Fe(\020)2048 4863 y Fk(\000)2156 4824 y Fi(d)2192 4801 y Ff(2)p 2136 4841 111 4 v 2136 4900 a Fi(dx)2212 4877 y Ff(2)2212 4921 y(3)2278 4863 y Fk(\000)23 b Fm(z)2427 4753 y Fe(\021)2487 4775 y Fh(\000)p Fl(1)2615 4863 y Fn(b)s(ounded)34 b(in)f Fm(L)3195 4827 y Fl(2)3235 4863 y Fn(\()p Fj(R)5 b Fn(\),)40 b(as)33 b(w)m(ell)g(as)118 5006 y(the)g(op)s(erators)1038 5126 y Fm(T)1095 5141 y Fi(q)1133 5126 y Fn(\()p Fm(z)t Fn(\))c(:=)e Fk(j)p Fm(V)22 b Fk(j)1552 5085 y Fl(1)p Fi(=)p Fl(2)1661 5126 y Fm(P)1724 5141 y Fi(q)1762 5126 y Fn(\()p Fm(H)1881 5141 y Fl(0)1942 5126 y Fk(\000)h Fm(z)t Fn(\))2129 5085 y Fh(\000)p Fl(1)2224 5126 y Fk(j)p Fm(V)e Fk(j)2358 5085 y Fl(1)p Fi(=)p Fl(2)2468 5126 y Fm(;)114 b(q)32 b Fk(2)c Fj(Z)2847 5141 y Fl(+)2903 5126 y Fm(;)1960 5375 y Fn(6)p eop %%Page: 7 7 7 6 bop 118 171 a Fn(b)s(ounded)41 b(in)e Fm(L)711 134 y Fl(2)751 171 y Fn(\()p Fj(R)855 134 y Fl(3)900 171 y Fn(\).)66 b(The)41 b(op)s(erator)e Fm(R)q Fn(\()p Fm(z)t Fn(\))i(admits)d(the)j(in)m(tegral)d(k)m(ernel)j Fk(R)3124 186 y Fi(z)3164 171 y Fn(\()p Fm(x)3257 186 y Fl(3)3324 171 y Fk(\000)27 b Fm(x)3483 134 y Fh(0)3483 195 y Fl(3)3523 171 y Fn(\))40 b(where)118 291 y Fk(R)202 306 y Fi(z)242 291 y Fn(\()p Fm(x)p Fn(\))46 b(=)f Fm(ie)618 255 y Fi(i)642 207 y Fh(p)p 702 207 36 3 v 702 255 a Fi(z)r Fh(j)p Fi(x)p Fh(j)820 291 y Fm(=)p Fn(\(2)956 219 y Fk(p)p 1039 219 50 4 v 72 x Fm(z)5 b Fn(\),)45 b Fm(x)h Fk(2)g Fj(R)5 b Fn(,)52 b(the)43 b(branc)m(h)h(of)2189 219 y Fk(p)p 2272 219 V 72 x Fm(z)k Fn(b)s(eing)42 b(c)m(hosen)j(so)e(that)g(Im)3475 219 y Fk(p)p 3558 219 V 72 x Fm(z)50 b(>)c Fn(0.)118 439 y(Moreo)m(v)m(er,)34 b Fm(T)630 454 y Fi(q)669 439 y Fn(\()p Fm(z)t Fn(\))28 b(=)f Fk(j)p Fm(V)22 b Fk(j)1060 403 y Fl(1)p Fi(=)p Fl(2)1197 328 y Fe(\020)1256 439 y Fm(p)1305 454 y Fi(q)1343 439 y Fn(\()p Fm(b)p Fn(\))h Fk(\012)g Fm(R)q Fn(\()p Fm(z)j Fk(\000)d Fn(2)p Fm(bq)t Fn(\))2042 328 y Fe(\021)2129 439 y Fk(j)p Fm(V)e Fk(j)2263 403 y Fl(1)p Fi(=)p Fl(2)2373 439 y Fn(.)118 587 y(F)-8 b(or)32 b Fm(\025)27 b Fk(2)i Fj(R)t Fn(,)39 b Fm(\025)28 b Fk(6)p Fn(=)f(0,)33 b(de\014ne)g Fm(R)q Fn(\()p Fm(\025)p Fn(\))g(as)g(the)g(op)s(erator)e(with)i(in)m(tegral)e(k)m(ernel)i Fk(R)3051 602 y Fi(\025)3097 587 y Fn(\()p Fm(x)3190 602 y Fl(3)3252 587 y Fk(\000)22 b Fm(x)3406 550 y Fh(0)3406 611 y Fl(3)3446 587 y Fn(\))33 b(where)747 880 y Fk(R)831 895 y Fi(\025)877 880 y Fn(\()p Fm(x)p Fn(\))28 b(:=)f(lim)1182 943 y Fi(\016)r Fh(#)p Fl(0)1318 880 y Fk(R)1402 895 y Fi(\025)p Fl(+)p Fi(i\016)1561 880 y Fn(\()p Fm(x)p Fn(\))h(=)1823 710 y Fe(\()1955 764 y Fi(e)1988 741 y Fb(\000)2036 702 y(p)p 2085 702 85 3 v 2085 741 a(\000)p Fg(\025)p Fb(j)p Fg(x)p Fb(j)p 1955 780 292 4 v 2006 844 a Fl(2)2041 790 y Fh(p)p 2100 790 97 3 v 54 x(\000)p Fi(\025)2354 803 y Fn(if)104 b Fm(\025)27 b(<)h Fn(0)p Fm(;)1955 925 y Fi(ie)2012 901 y Fg(i)2034 859 y Fb(p)p 2084 859 37 3 v 42 x Fg(\025)o Fb(j)p Fg(x)p Fb(j)p 1955 941 243 4 v 2008 1008 a Fl(2)2043 951 y Fh(p)p 2103 951 42 3 v 2103 1008 a Fi(\025)2305 964 y Fn(if)103 b Fm(\025)28 b(>)f Fn(0)p Fm(;)2945 880 y(x)h Fk(2)h Fj(R)5 b Fm(:)385 b Fn(\(3.15\))118 1179 y(Eviden)m(tly)-8 b(,)37 b(if)e Fm(w)h Fk(2)e Fm(L)949 1142 y Fl(2)989 1179 y Fn(\()p Fj(R)5 b Fn(\))42 b(and)36 b Fm(\025)d Fk(6)p Fn(=)h(0,)i(then)h Fm(w)s(R)q Fn(\()p Fm(\025)p Fn(\))19 b(\026)-68 b Fm(w)35 b Fk(2)f Fm(S)2450 1194 y Fl(2)2489 1179 y Fn(.)54 b(F)-8 b(or)35 b Fm(E)40 b Fk(2)34 b Fj(R)5 b Fn(,)43 b Fm(E)c Fk(6)p Fn(=)34 b(2)p Fm(bq)t Fn(,)j Fm(q)g Fk(2)d Fj(Z)3767 1194 y Fl(+)3823 1179 y Fn(,)118 1299 y(set)1029 1419 y Fm(T)1086 1434 y Fi(q)1124 1419 y Fn(\()p Fm(E)6 b Fn(\))28 b(:=)f Fk(j)p Fm(V)22 b Fk(j)1571 1378 y Fl(1)p Fi(=)p Fl(2)1708 1309 y Fe(\020)1767 1419 y Fm(p)1816 1434 y Fi(q)1854 1419 y Fn(\()p Fm(b)p Fn(\))h Fk(\012)g Fm(R)q Fn(\()p Fm(E)28 b Fk(\000)22 b Fn(2)p Fm(bq)t Fn(\))2581 1309 y Fe(\021)2669 1419 y Fk(j)p Fm(V)f Fk(j)2803 1378 y Fl(1)p Fi(=)p Fl(2)2912 1419 y Fm(:)118 1630 y Fo(Prop)s(osition)36 b(3.2.)42 b Fn([6,)33 b(Prop)s(osition)e(4.1,)i (Corollary)e(4.1,)h(Corollary)f(4.2])k Fa(L)-5 b(et)36 b Fm(E)e Fk(2)28 b Fj(R)5 b Fa(,)42 b Fm(q)31 b Fk(2)e Fj(Z)3764 1645 y Fl(+)3821 1630 y Fa(,)118 1751 y Fm(E)34 b Fk(6)p Fn(=)27 b(2)p Fm(bq)t Fa(.)45 b(Assume)35 b(that)45 b Fn(\(1.1\))34 b Fa(holds.)118 1871 y(i\))h(We)g(have)f Fm(T)681 1886 y Fi(q)719 1871 y Fn(\()p Fm(E)6 b Fn(\))28 b Fk(2)g Fm(S)1055 1886 y Fl(2)1094 1871 y Fa(,)35 b(and)f Fn(lim)1484 1886 y Fi(\016)r Fh(#)p Fl(0)1609 1871 y Fk(k)p Fm(T)1716 1886 y Fi(q)1754 1871 y Fn(\()p Fm(E)28 b Fn(+)22 b Fm(i\016)t Fn(\))g Fk(\000)h Fm(T)2287 1886 y Fi(q)2325 1871 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)2529 1886 y Fl(2)2596 1871 y Fn(=)28 b(0)p Fa(.)118 1992 y(ii\))37 b(We)g(have)f Fn(Im)31 b Fm(T)865 2007 y Fi(q)903 1992 y Fn(\()p Fm(E)6 b Fn(\))32 b Fk(\025)g Fn(0)p Fa(,)37 b(and)g(if)f Fm(E)i(<)32 b Fn(2)p Fm(bq)t Fa(,)37 b(then)g Fn(Im)31 b Fm(T)2448 2007 y Fi(q)2486 1992 y Fn(\()p Fm(E)6 b Fn(\))32 b(=)f(0)p Fa(.)51 b(Mor)-5 b(e)g(over,)37 b Fn(Im)16 b Fm(T)3560 2007 y Fi(q)3598 1992 y Fn(\()p Fm(E)6 b Fn(\))32 b Fk(2)118 2112 y Fm(S)178 2127 y Fl(1)218 2112 y Fa(.)118 2296 y Fo(Prop)s(osition)50 b(3.3.)e Fn(\(see)f([6,)h(Prop)s(osition)c(4.2]\))i Fa(L)-5 b(et)47 b Fm(b)i(>)g Fn(0)p Fa(,)g Fm(E)56 b Fk(62)49 b Fn(2)p Fm(b)p Fj(Z)3019 2311 y Fl(+)3076 2296 y Fa(.)80 b(Assume)46 b(that)h Fm(V)118 2416 y Fa(satis\014es)39 b Fn(\(1.1\))o Fa(.)44 b(Then)30 b(the)i(op)-5 b(er)g(ator)30 b(series)h Fm(T)1880 2431 y Fl(+)1939 2416 y Fn(\()p Fm(E)20 b Fn(+)14 b Fm(i\016)t Fn(\))28 b(:=)2436 2341 y Fe(P)2541 2368 y Fh(1)2541 2445 y Fi(l)q Fl(=[)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(]+1)2924 2416 y Fm(T)2981 2431 y Fi(l)3007 2416 y Fn(\()p Fm(E)20 b Fn(+)14 b Fm(i\016)t Fn(\))p Fa(,)32 b Fm(\016)g(>)27 b Fn(0)p Fa(,)32 b(and)118 2547 y Fm(T)175 2562 y Fl(+)234 2547 y Fn(\()p Fm(E)6 b Fn(\))28 b(:=)547 2472 y Fe(P)652 2499 y Fh(1)652 2576 y Fi(l)q Fl(=[)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(]+1)1035 2547 y Fm(T)1092 2562 y Fi(l)1118 2547 y Fn(\()p Fm(E)6 b Fn(\))p Fa(,)34 b(wher)-5 b(e)34 b Fn([)p Fm(x)p Fn(])h Fa(denotes)f(the)g(inte)-5 b(ger)34 b(p)-5 b(art)35 b(of)f(the)h(r)-5 b(e)g(al)34 b(numb)-5 b(er)34 b Fm(x)p Fa(,)h(ar)-5 b(e)118 2668 y(c)g(onver)g(gent)34 b(in)h Fm(S)780 2683 y Fl(2)819 2668 y Fa(.)45 b(Mor)-5 b(e)g(over,)867 2935 y Fk(k)p Fm(T)974 2950 y Fl(+)1033 2935 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1237 2894 y Fl(2)1237 2960 y(2)1304 2935 y Fk(\024)1419 2868 y Fm(C)1489 2883 y Fl(0)1528 2868 y Fm(b)p 1419 2912 151 4 v 1440 3003 a Fn(8)p Fm(\031)1742 2811 y Fh(1)1705 2840 y Fe(X)1596 3056 y Fi(l)q Fl(=[)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(]+1)1958 2935 y Fn(\(2)p Fm(bl)25 b Fk(\000)e Fm(E)6 b Fn(\))2356 2894 y Fh(\000)p Fl(3)p Fi(=)p Fl(2)2537 2800 y Fe(Z)2592 3025 y Fd(R)2640 3006 y Ff(3)2696 2935 y Fk(j)p Fm(V)21 b Fn(\()p Fo(x)p Fn(\))p Fk(j)p Fm(d)p Fo(x)p Fm(:)499 b Fn(\(3.16\))118 3243 y Fa(Final)5 b(ly,)34 b Fn(lim)619 3258 y Fi(\016)r Fh(#)p Fl(0)744 3243 y Fk(k)p Fm(T)851 3258 y Fl(+)910 3243 y Fn(\()p Fm(E)28 b Fn(+)22 b Fm(i\016)t Fn(\))g Fk(\000)h Fm(T)1443 3258 y Fl(+)1502 3243 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1706 3258 y Fl(2)1773 3243 y Fn(=)27 b(0)p Fa(.)118 3426 y Fo(Corollary)38 b(3.1.)j Fa(L)-5 b(et)36 b Fm(r)30 b Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fa(.)45 b(Then)34 b(we)g(have)1065 3624 y Fk(k)p Fn(Re)17 b Fm(T)1304 3639 y Fl(+)1363 3624 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1567 3583 y Fl(2)1567 3648 y(2)1634 3624 y Fn(=)28 b Fm(O)s Fn(\(1\))p Fm(;)115 b(E)34 b Fk(!)27 b(1)p Fm(;)116 b(E)34 b Fk(2)28 b(O)2838 3639 y Fi(r)2876 3624 y Fm(:)698 b Fn(\(3.17\))118 3821 y Fa(Pr)-5 b(o)g(of.)49 b Fn(Estimate)31 b(\(3.17\))h(follo)m(ws)f(immediately)f(from)h(\(3.16\))h(since)h(w)m (e)h(ha)m(v)m(e)1160 3969 y Fh(1)1123 3999 y Fe(X)1014 4215 y Fi(l)q Fl(=[)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(]+1)1377 4094 y Fn(\(2)p Fm(bl)24 b Fk(\000)f Fm(E)6 b Fn(\))1774 4053 y Fh(\000)p Fl(3)p Fi(=)p Fl(2)1966 4094 y Fk(\024)2108 3969 y Fh(1)2072 3999 y Fe(X)2081 4208 y Fi(p)p Fl(=1)2215 4094 y Fn(\(2)p Fm(bp)p Fn(\))2430 4053 y Fh(\000)p Fl(3)p Fi(=)p Fl(2)2618 4094 y Fn(+)22 b Fm(r)2763 4053 y Fh(\000)p Fl(3)p Fi(=)p Fl(2)2927 4094 y Fm(:)p 3780 4393 4 66 v 3784 4330 59 4 v 3784 4393 V 3842 4393 4 66 v 118 4581 a Fn(F)-8 b(or)36 b(su\016cien)m(tly)h(large)e Fm(E)41 b Fk(2)35 b(O)1324 4596 y Fi(r)1398 4581 y Fn(with)i Fm(r)g Fk(2)d Fn(\(0)p Fm(;)17 b(b)p Fn(\),)38 b(set)f Fm(T)2294 4596 y Fh(\000)2353 4581 y Fn(\()p Fm(E)6 b Fn(\))35 b(:=)2679 4506 y Fe(P)2784 4530 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])2784 4609 y Fi(l)q Fl(=0)3000 4581 y Fm(T)3057 4596 y Fi(l)3083 4581 y Fn(\()p Fm(E)6 b Fn(\).)56 b(Prop)s(ositions)118 4702 y(3.2)32 b(and)h(3.3)f(imply)f (the)i(follo)m(wing)118 4885 y Fo(Corollary)46 b(3.2.)g Fa(F)-7 b(or)40 b Fm(E)47 b Fk(2)40 b(O)1338 4900 y Fi(r)1418 4885 y Fa(with)i Fm(r)h Fk(2)d Fn(\(0)p Fm(;)17 b(b)p Fn(\))42 b Fa(the)f(op)-5 b(er)g(ator-norm)41 b(limit)51 b Fn(\(3.9\))41 b Fa(exists,)i(and)118 5006 y Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))39 b(=)h Fm(T)555 5021 y Fh(\000)614 5006 y Fn(\()p Fm(E)6 b Fn(\))27 b(+)f Fm(T)954 5021 y Fl(+)1013 5006 y Fn(\()p Fm(E)6 b Fn(\))p Fa(.)64 b(Mor)-5 b(e)g(over,)43 b Fn(Re)c Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))39 b(=)h(Re)f Fm(T)2473 5021 y Fh(\000)2532 5006 y Fn(\()p Fm(E)6 b Fn(\))27 b(+)g Fm(T)2873 5021 y Fl(+)2932 5006 y Fn(\()p Fm(E)6 b Fn(\))p Fa(,)43 b(and)d Fn(Im)f Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))40 b(=)118 5126 y(Im)27 b Fm(T)319 5141 y Fh(\000)378 5126 y Fn(\()p Fm(E)6 b Fn(\))28 b Fk(2)g Fm(S)714 5141 y Fl(1)754 5126 y Fa(.)1960 5375 y Fn(7)p eop %%Page: 8 8 8 7 bop 118 171 a Fn(F)-8 b(or)32 b Fm(n)c Fn(=)f(0)p Fm(;)17 b Fn(1)32 b(and)h Fm(E)g Fk(2)c(O)1125 186 y Fi(r)1163 171 y Fn(,)j Fm(r)f Fk(2)d Fn(\(0)p Fm(;)17 b(b)p Fn(\),)32 b(set)i Fm(')1877 186 y Fi(n)1923 171 y Fn(\()p Fm(E)6 b Fn(\))28 b(:=)2236 96 y Fe(P)2341 122 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])2341 200 y Fi(q)r Fl(=0)2540 171 y Fn(\()p Fm(E)h Fk(\000)22 b Fn(2)p Fm(bq)t Fn(\))2953 134 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)3216 171 y Fn(.)118 374 y Fo(Lemma)37 b(3.5.)42 b Fa(L)-5 b(et)35 b Fm(r)c(>)c Fn(0)p Fa(.)45 b(Then)34 b(the)h(asymptotic)f(r)-5 b(elations)1612 594 y Fm(')1676 609 y Fl(0)1715 594 y Fn(\()p Fm(E)6 b Fn(\))28 b(=)f Fm(O)s Fn(\(ln)15 b Fm(E)6 b Fn(\))p Fm(;)1245 b Fn(\(3.18\))1432 858 y Fm(')1496 873 y Fl(1)1535 858 y Fn(\()p Fm(E)6 b Fn(\))28 b(=)f Fm(E)1898 817 y Fl(1)p Fi(=)p Fl(2)2018 791 y Fn(1)p 2018 836 49 4 v 2022 927 a Fm(b)2093 858 y Fn(\(1)22 b(+)g Fm(o)p Fn(\(1\)\))p Fm(;)1064 b Fn(\(3.19\))118 1058 y Fa(hold)34 b(as)h Fm(E)f Fk(!)27 b(1)p Fa(,)34 b Fm(E)g Fk(2)28 b(O)1128 1073 y Fi(r)1166 1058 y Fa(.)118 1262 y(Pr)-5 b(o)g(of.)49 b Fn(Eviden)m(tly)-8 b(,)33 b(for)f Fm(E)h(>)28 b Fn(0)k(large)g(enough)204 1581 y Fm(')268 1596 y Fi(n)315 1581 y Fn(\()p Fm(E)6 b Fn(\))27 b(=)h Fm(E)678 1540 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)957 1449 y([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])p Fh(\000)p Fl(1)1028 1486 y Fe(X)1038 1696 y Fi(q)r Fl(=0)1259 1441 y Fe(\022)1333 1581 y Fn(1)22 b Fk(\000)1513 1514 y Fn(2)p Fm(bq)p 1513 1558 138 4 v 1543 1649 a(E)1660 1441 y Fe(\023)1734 1463 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)2018 1581 y Fn(+)g(\()p Fm(E)29 b Fk(\000)22 b Fn(2)p Fm(b)p Fn([)p Fm(E)6 b(=)p Fn(2)p Fm(b)p Fn(]\))2753 1540 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)3016 1581 y Fm(;)114 b(n)28 b Fn(=)g(0)p Fm(;)17 b Fn(1)p Fm(:)85 b Fn(\(3.20\))118 1935 y(Since)48 b(the)g(functions)f (\(0)p Fm(;)17 b(E)6 b(=)p Fn(2)p Fm(b)p Fn(\))53 b Fk(3)h Fm(x)f Fk(7!)1826 1855 y Fe(\000)1872 1935 y Fn(1)22 b Fk(\000)2052 1896 y Fl(2)p Fi(bx)p 2052 1912 106 4 v 2077 1970 a(E)2167 1855 y Fe(\001)2213 1876 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)2476 1935 y Fn(,)51 b Fm(n)i Fn(=)g(0)p Fm(;)17 b Fn(1,)51 b(are)d(increasing,)i(and)118 2056 y Fm(E)34 b Fk(2)28 b(O)397 2071 y Fi(r)435 2056 y Fn(,)33 b(w)m(e)g(ha)m(v)m(e)745 2243 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])p Fh(\000)p Fl(1)816 2280 y Fe(X)826 2490 y Fi(q)r Fl(=0)1048 2235 y Fe(\022)1121 2375 y Fn(1)22 b Fk(\000)1302 2308 y Fn(2)p Fm(bq)p 1302 2352 138 4 v 1331 2443 a(E)1449 2235 y Fe(\023)1522 2257 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)1812 2375 y Fk(\024)1917 2239 y Fe(Z)2017 2266 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])1973 2465 y(0)2233 2235 y Fe(\022)2306 2375 y Fn(1)g Fk(\000)2487 2308 y Fn(2)p Fm(bx)p 2487 2352 146 4 v 2521 2443 a(E)2643 2235 y Fe(\023)2716 2257 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)3012 2375 y Fm(dx)28 b Fk(\024)327 2653 y Fe(Z)426 2679 y Fl(\()p Fi(E)t Fh(\000)p Fi(r)r Fl(\))p Fi(=)p Fl(\(2)p Fi(b)p Fl(\))382 2878 y(0)802 2648 y Fe(\022)875 2788 y Fn(1)22 b Fk(\000)1056 2721 y Fn(2)p Fm(bx)p 1056 2765 V 1090 2857 a(E)1211 2648 y Fe(\023)1285 2670 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)1581 2788 y Fm(dx)27 b Fn(=)1834 2721 y Fm(E)p 1828 2765 91 4 v 1828 2857 a Fn(2)p Fm(b)1945 2653 y Fe(Z)2044 2679 y Fl(1)p Fh(\000)p Fi(r)r(=E)2000 2878 y Fl(0)2263 2788 y Fn(\(1)22 b Fk(\000)h Fm(t)p Fn(\))2545 2747 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)2807 2788 y Fm(dt;)115 b(n)27 b Fn(=)h(0)p Fm(;)17 b Fn(1)p Fm(:)208 b Fn(\(3.21\))118 3014 y(F)-8 b(urther,)747 3066 y Fe(Z)846 3092 y Fl(1)p Fh(\000)p Fi(r)r(=E)802 3291 y Fl(0)1065 3201 y Fn(\(1)22 b Fk(\000)h Fm(t)p Fn(\))1347 3160 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)1609 3201 y Fm(dt)28 b Fn(=)1827 3061 y Fe(\032)1943 3139 y Fn(ln)o(\()p Fm(E)6 b(=r)s Fn(\))97 b(if)103 b Fm(n)28 b Fn(=)g(0)p Fm(;)1943 3261 y Fn(2)2009 3181 y Fe(\000)2054 3261 y Fn(1)22 b Fk(\000)g Fn(\()p Fm(r)s(=E)6 b Fn(\))2474 3225 y Fl(1)p Fi(=)p Fl(2)2584 3181 y Fe(\001)2744 3261 y Fn(if)103 b Fm(n)28 b Fn(=)f(1)p Fm(:)3601 3201 y Fn(\(3.22\))118 3435 y(Finally)-8 b(,)30 b(w)m(e)k(estimate)d(the)i(second)h(term)e(on) h(the)g(r.h.s.)44 b(of)39 b(\(3.20\))1064 3655 y(\()p Fm(E)28 b Fk(\000)23 b Fn(2)p Fm(b)p Fn([)p Fm(E)6 b(=)p Fn(2)p Fm(b)p Fn(]\))1701 3613 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)1991 3655 y Fk(\024)28 b Fm(r)2143 3613 y Fh(\000)p Fl(1+)p Fi(n=)p Fl(2)2406 3655 y Fm(;)114 b(n)28 b Fn(=)f(0)p Fm(;)17 b Fn(1)p Fm(:)696 b Fn(\(3.23\))118 3875 y(Putting)22 b(together)h(\(3.20\))f({)g(\(3.23\))o(,)j(w)m(e)f(obtain)d(\(3.18\))o (,)k(as)e(w)m(ell)e(as)i(lim)17 b(sup)2920 3898 y Fi(E)t Fh(!1)p Fi(;E)t Fh(2O)3297 3906 y Fg(r)3350 3875 y Fm(E)3428 3838 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)3593 3875 y Fm(')3657 3890 y Fl(1)3696 3875 y Fn(\()p Fm(E)6 b Fn(\))118 4005 y Fk(\024)234 3966 y Fl(1)p 234 3982 36 4 v 237 4040 a Fi(b)280 4005 y Fn(.)45 b(In)34 b(order)f(to)g(pro)m(v)m(e)h (\(3.19\),)f(it)f(remains)h(to)f(sho)m(w)j(that)e(lim)17 b(inf)2730 4020 y Fi(E)t Fh(!1)p Fi(;E)t Fh(2O)3107 4028 y Fg(r)3160 4005 y Fm(E)3238 3969 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)3403 4005 y Fm(')3467 4020 y Fl(1)3506 4005 y Fn(\()p Fm(E)6 b Fn(\))29 b Fk(\025)3805 3966 y Fl(1)p 3805 3982 V 3808 4040 a Fi(b)118 4126 y Fn(whic)m(h)k(follo)m(ws)e (immediately)f(from)300 4421 y Fm(')364 4436 y Fl(1)403 4421 y Fn(\()p Fm(E)6 b Fn(\))28 b Fk(\025)g Fm(E)768 4380 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)949 4285 y Fe(Z)1049 4312 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])1005 4511 y Fh(\000)p Fl(1)1265 4280 y Fe(\022)1339 4421 y Fn(1)21 b Fk(\000)1519 4354 y Fn(2)p Fm(bx)p 1519 4398 146 4 v 1553 4489 a(E)1675 4280 y Fe(\023)1748 4303 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)1930 4421 y Fm(dx)27 b Fk(\025)h Fm(E)2246 4380 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)2428 4285 y Fe(Z)2528 4312 y Fi(E)t(=)p Fl(2)p Fi(b)p Fh(\000)p Fl(1)2483 4511 y(0)2795 4280 y Fe(\022)2868 4421 y Fn(1)22 b Fk(\000)3049 4354 y Fn(2)p Fm(bx)p 3049 4398 V 3082 4489 a(E)3204 4280 y Fe(\023)3278 4303 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)3459 4421 y Fm(dx)28 b Fn(=)877 4733 y Fm(E)955 4697 y Fl(1)p Fi(=)p Fl(2)p 877 4777 189 4 v 926 4868 a Fn(2)p Fm(b)1091 4664 y Fe(Z)1191 4691 y Fl(1)p Fh(\000)p Fl(2)p Fi(b=E)1147 4890 y Fl(0)1458 4800 y Fn(\(1)22 b Fk(\000)h Fm(t)p Fn(\))1740 4752 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)1921 4800 y Fm(dt)28 b Fn(=)2148 4733 y Fm(E)2226 4697 y Fl(1)p Fi(=)p Fl(2)p 2148 4777 V 2222 4868 a Fm(b)2363 4630 y Fe( )2442 4800 y Fn(1)22 b Fk(\000)2612 4660 y Fe(\022)2696 4733 y Fn(2)p Fm(b)p 2696 4777 91 4 v 2702 4868 a(E)2796 4660 y Fe(\023)2869 4682 y Fl(1)p Fi(=)p Fl(2)2979 4630 y Fe(!)3075 4800 y Fm(:)p 3780 5053 4 66 v 3784 4990 59 4 v 3784 5053 V 3842 5053 4 66 v 1960 5375 a Fn(8)p eop %%Page: 9 9 9 8 bop 118 171 a Fo(Corollary)38 b(3.3.)j Fa(L)-5 b(et)36 b Fm(r)30 b Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fa(.)45 b(Then)34 b(the)h(asymptotic)f(estimate)1550 383 y Fk(k)p Fm(T)1657 398 y Fh(\000)1716 383 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1920 342 y Fl(2)1920 408 y(2)1986 383 y Fn(=)28 b Fm(O)s Fn(\(ln)15 b Fm(E)6 b Fn(\))1182 b(\(3.24\))118 596 y Fa(holds)34 b(as)h Fm(E)e Fk(!)28 b(1)p Fa(,)34 b Fm(E)g Fk(2)28 b(O)1168 611 y Fi(r)1206 596 y Fa(.)118 793 y(Pr)-5 b(o)g(of.)49 b Fn(W)-8 b(e)32 b(ha)m(v)m(e)850 1134 y Fk(k)p Fm(T)957 1149 y Fh(\000)1016 1134 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1220 1093 y Fl(2)1220 1159 y(2)1287 1134 y Fn(=)1391 930 y Fe(\015)1391 990 y(\015)1391 1049 y(\015)1391 1109 y(\015)1391 1169 y(\015)1391 1229 y(\015)1446 1134 y Fk(j)p Fm(V)21 b Fk(j)1580 1093 y Fl(1)p Fi(=)p Fl(2)1707 1002 y([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])1732 1039 y Fe(X)1742 1249 y Fi(q)r Fl(=0)1902 1134 y Fn(\()p Fm(p)1989 1149 y Fi(q)2049 1134 y Fk(\012)i Fm(R)q Fn(\()p Fm(E)28 b Fk(\000)22 b Fn(2)p Fm(bq)t Fn(\)\))p Fk(j)p Fm(V)g Fk(j)2809 1093 y Fl(1)p Fi(=)p Fl(2)2918 930 y Fe(\015)2918 990 y(\015)2918 1049 y(\015)2918 1109 y(\015)2918 1169 y(\015)2918 1229 y(\015)2974 956 y Fl(2)2974 1293 y(2)3041 1134 y Fk(\024)618 1614 y Fm(C)695 1573 y Fl(2)688 1639 y(0)778 1410 y Fe(\015)778 1470 y(\015)778 1529 y(\015)778 1589 y(\015)778 1649 y(\015)778 1709 y(\015)834 1482 y Fl([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])859 1519 y Fe(X)869 1729 y Fi(q)r Fl(=0)1029 1614 y Fn(\()p Fm(p)1116 1629 y Fi(q)1154 1614 y Fk(h)p Fm(X)1274 1629 y Fh(?)1333 1614 y Fk(i)1372 1573 y Fh(\000)p Fi(m)1489 1585 y Fb(?)1541 1573 y Fi(=)p Fl(2)1616 1614 y Fn(\))g Fk(\012)1775 1533 y Fe(\000)1821 1614 y Fk(h)p Fm(x)1915 1629 y Fl(3)1955 1614 y Fk(i)1994 1573 y Fh(\000)p Fi(m)2111 1582 y Ff(3)2145 1573 y Fi(=)p Fl(2)2220 1614 y Fm(R)q Fn(\()p Fm(E)28 b Fk(\000)23 b Fn(2)p Fm(bq)t Fn(\))p Fk(h)p Fm(x)2802 1629 y Fl(3)2841 1614 y Fk(i)2880 1573 y Fh(\000)p Fi(m)2997 1582 y Ff(3)3032 1573 y Fi(=)p Fl(2)3107 1533 y Fe(\001)3152 1410 y(\015)3152 1470 y(\015)3152 1529 y(\015)3152 1589 y(\015)3152 1649 y(\015)3152 1709 y(\015)3208 1436 y Fl(2)3208 1773 y(2)3275 1614 y Fn(=)670 2024 y Fm(C)747 1983 y Fl(2)740 2049 y(0)831 1892 y([)p Fi(E)t(=)p Fl(2)p Fi(b)p Fl(])857 1929 y Fe(X)867 2138 y Fi(q)r Fl(=0)1043 1939 y Fe(\015)1043 1999 y(\015)1099 2024 y Fm(p)1148 2039 y Fi(q)1186 2024 y Fk(h)p Fm(X)1306 2039 y Fh(?)1364 2024 y Fk(i)1403 1983 y Fh(\000)p Fi(m)1520 1995 y Fb(?)1572 1983 y Fi(=)p Fl(2)1647 1939 y Fe(\015)1647 1999 y(\015)1702 1959 y Fl(2)1702 2063 y(2)1759 1939 y Fe(\015)1759 1999 y(\015)1814 2024 y Fk(h)p Fm(x)1908 2039 y Fl(3)1947 2024 y Fk(i)1986 1983 y Fh(\000)p Fi(m)2103 1992 y Ff(3)2138 1983 y Fi(=)p Fl(2)2213 2024 y Fm(R)q Fn(\()p Fm(E)28 b Fk(\000)23 b Fn(2)p Fm(bq)t Fn(\))p Fk(h)p Fm(x)2795 2039 y Fl(3)2834 2024 y Fk(i)2873 1983 y Fh(\000)p Fi(m)2990 1992 y Ff(3)3025 1983 y Fi(=)p Fl(2)3100 1939 y Fe(\015)3100 1999 y(\015)3155 1959 y Fl(2)3155 2063 y(2)3222 2024 y Fn(=)949 2381 y Fm(C)1026 2340 y Fl(2)1019 2406 y(0)1109 2314 y Fm(b)p 1076 2359 108 4 v 1076 2450 a Fn(8)p Fm(\031)1237 2246 y Fe(Z)1293 2471 y Fd(R)1341 2453 y Ff(2)1380 2381 y Fk(h)p Fm(X)1500 2396 y Fh(?)1558 2381 y Fk(i)1597 2340 y Fh(\000)p Fi(m)1714 2352 y Fb(?)1771 2381 y Fm(dX)1903 2396 y Fh(?)1978 2241 y Fe(\022)2051 2246 y(Z)2107 2471 y Fd(R)2159 2381 y Fk(h)p Fm(x)2253 2396 y Fl(3)2292 2381 y Fk(i)2331 2340 y Fh(\000)p Fi(m)2448 2349 y Ff(3)2487 2381 y Fm(dx)2593 2396 y Fl(3)2633 2241 y Fe(\023)2706 2263 y Fl(2)2762 2381 y Fm(')2826 2396 y Fl(0)2865 2381 y Fn(\()p Fm(E)6 b Fn(\))582 b(\(3.25\))118 2610 y(\(see)48 b(\(3.13\))f(for)f(the)i(de\014nition)e(of)g(the)i(in)m(tegral)d(k)m (ernel)j(of)e Fm(p)2572 2625 y Fi(q)2610 2610 y Fn(,)51 b(\(3.14\))46 b(for)h(its)f(v)-5 b(alue)47 b(on)g(the)118 2730 y(diagonal,)32 b(and)i(\(3.15\))f(for)g(the)i(de\014nition)d(of)i (the)g(in)m(tegral)e(k)m(ernel)j(of)e Fm(R)q Fn(\()p Fm(E)c Fk(\000)24 b Fn(2)p Fm(bq)t Fn(\)\).)47 b(Bearing)33 b(in)118 2851 y(mind)e(\(3.18\),)h(w)m(e)i(\014nd)f(that)f(\(3.25\))g (implies)e(\(3.24\).)p 3780 2851 4 66 v 3784 2788 59 4 v 3784 2851 V 3842 2851 4 66 v 118 3048 a Fo(Prop)s(osition)36 b(3.4.)41 b Fa(L)-5 b(et)36 b Fm(r)30 b Fk(2)e Fn(\(0)p Fm(;)17 b(b)p Fn(\))p Fa(.)45 b(Then)34 b(we)g(have)1107 3261 y Fk(k)p Fn(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))p Fk(k)27 b Fn(=)h Fm(O)s Fn(\(1\))p Fm(;)115 b(E)34 b Fk(!)27 b(1)p Fm(;)116 b(E)34 b Fk(2)28 b(O)2796 3276 y Fi(r)2834 3261 y Fm(:)740 b Fn(\(3.26\))118 3474 y Fa(Pr)-5 b(o)g(of.)49 b Fn(Estimate)32 b(\(3.26\))h(follo)m(ws)f (immediately)e(from)i([5,)i(Lemma)e(4.2])h(according)f(to)h(whic)m(h)h (w)m(e)118 3594 y(ha)m(v)m(e)g Fk(k)p Fm(T)14 b Fn(\()p Fm(E)6 b Fn(\))p Fk(k)32 b(\024)h Fm(r)857 3558 y Fh(\000)p Fl(1)p Fi(=)p Fl(2)1031 3554 y Fi(C)1081 3563 y Ff(0)p 1031 3571 85 4 v 1056 3628 a Fl(2)1143 3514 y Fe(R)1190 3628 y Fd(R)1242 3594 y Fk(h)p Fm(x)1336 3609 y Fl(3)1376 3594 y Fk(i)1415 3558 y Fh(\000)p Fi(m)1532 3567 y Ff(3)1570 3594 y Fm(dx)1676 3609 y Fl(3)1716 3594 y Fn(.)p 3780 3594 4 66 v 3784 3531 59 4 v 3784 3594 V 3842 3594 4 66 v 118 3926 a Fc(4)161 b(Pro)t(of)54 b(of)g(the)f(Main)i(Result)118 4145 y Fn(Fix)31 b(an)h(arbitrary)f Fm(")d Fk(2)g Fn(\(0)p Fm(;)17 b Fn(1\).)42 b(Applying)32 b(\(3.5\),)g(\(3.6\),)g(and)g (\(3.7\))o(,)h(and)f(arguing)e(as)j(in)e(the)h(pro)s(of)118 4265 y(of)g([5,)h(Lemma)e(5.1],)h(w)m(e)i(easily)e(get)528 4455 y(1)p 523 4499 59 4 v 523 4591 a Fm(\031)591 4522 y Fn(T)-8 b(r)17 b(arctan)1012 4442 y Fe(\000)1058 4522 y Fn(\(Im)f Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))1492 4481 y Fl(1)p Fi(=)p Fl(2)1602 4522 y Fn(\()p Fm(J)31 b Fn(+)22 b Fm(")p Fn(\))1907 4481 y Fh(\000)p Fl(1)2001 4522 y Fn(\(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))2435 4481 y Fl(1)p Fi(=)p Fl(2)2544 4442 y Fe(\001)2612 4522 y Fk(\000)23 b Fm(n)2770 4537 y Fl(+)2829 4522 y Fn(\()p Fm(")p Fn(;)17 b(Re)f Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))27 b Fk(\024)1057 4691 y Fe(Z)1113 4917 y Fd(R)1181 4827 y Fn(ind\()p Fm(J)k Fn(+)22 b(Re)17 b Fm(T)d Fn(\()p Fm(E)6 b Fn(\))22 b(+)g Fm(t)17 b Fn(Im)f Fm(T)e Fn(\()p Fm(E)6 b Fn(\);)17 b Fm(J)9 b Fn(\))17 b Fm(d\026)p Fn(\()p Fm(t)p Fn(\))26 b Fk(\024)567 5040 y Fn(1)p 562 5085 V 562 5176 a Fm(\031)630 5108 y Fn(T)-8 b(r)17 b(arctan)1051 5027 y Fe(\000)1097 5108 y Fn(\(Im)f Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))1531 5066 y Fl(1)p Fi(=)p Fl(2)1641 5108 y Fn(\()p Fm(J)31 b Fk(\000)23 b Fm(")p Fn(\))1948 5066 y Fh(\000)p Fl(1)2042 5108 y Fn(\(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))2476 5066 y Fl(1)p Fi(=)p Fl(2)2585 5027 y Fe(\001)2653 5108 y Fn(+)22 b Fm(n)2809 5123 y Fh(\000)2868 5108 y Fn(\()p Fm(")p Fn(;)17 b(Re)f Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))p Fm(:)233 b Fn(\(4.1\))1960 5375 y(9)p eop %%Page: 10 10 10 9 bop 118 171 a Fn(Set)590 291 y Fm(G)667 306 y Fi(s)732 291 y Fn(=)27 b Fm(G)912 306 y Fi(s)949 291 y Fn(\()p Fm(E)6 b Fn(\))27 b(:=)h(\(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))1695 250 y Fl(1)p Fi(=)p Fl(2)1805 291 y Fn(\()p Fm(J)31 b Fn(+)22 b Fm(s)p Fn(\))2110 250 y Fh(\000)p Fl(1)2204 291 y Fn(\(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\)\))2638 250 y Fl(1)p Fi(=)p Fl(2)2748 291 y Fm(;)114 b(s)28 b Fk(2)g Fn(\()p Fk(\000)p Fn(1)p Fm(;)17 b Fn(1\))p Fm(:)118 455 y Fn(Eviden)m(tly)-8 b(,)33 b(for)f(eac)m(h)i Fm(s)27 b Fk(2)h Fn(\()p Fk(\000)p Fn(1)p Fm(;)17 b Fn(1\))32 b(w)m(e)i(ha)m(v)m(e)506 694 y Fk(j)p Fn(T)-8 b(r)17 b(arctan)f Fm(G)1015 709 y Fi(s)1052 694 y Fn(\()p Fm(E)6 b Fn(\))22 b Fk(\000)h Fn(T)-8 b(r)33 b Fm(G)1538 709 y Fi(s)1575 694 y Fn(\()p Fm(E)6 b Fn(\))p Fk(j)27 b(\024)1899 627 y Fn(1)p 1899 671 49 4 v 1899 763 a(3)1958 694 y Fk(k)p Fm(G)2085 709 y Fi(s)2122 694 y Fn(\()p Fm(E)6 b Fn(\))p Fk(k)2326 653 y Fl(3)2326 719 y(3)2393 694 y Fk(\024)2508 627 y Fn(1)p 2508 671 V 2508 763 a(3)2566 694 y Fk(k)p Fm(G)2693 709 y Fi(s)2730 694 y Fn(\()p Fm(E)g Fn(\))p Fk(k)2934 653 y Fl(2)2934 719 y(2)2990 694 y Fk(k)p Fm(G)3117 709 y Fi(s)3153 694 y Fn(\()p Fm(E)g Fn(\))p Fk(k)28 b(\024)276 910 y Fn(1)p 276 955 V 276 1046 a(3)335 978 y Fk(k)p Fn(\()p Fm(J)j Fn(+)22 b Fm(s)p Fn(\))690 936 y Fh(\000)p Fl(1)784 978 y Fk(k)834 936 y Fl(3)890 978 y Fk(k)p Fn(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))p Fk(k)1348 936 y Fl(2)1348 1002 y(2)1403 978 y Fk(k)p Fn(Im)16 b Fm(T)e Fn(\()p Fm(E)6 b Fn(\))p Fk(k)27 b(\024)2003 910 y Fn(1)p 2003 955 V 2003 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b(to)g(the)h(problem)f(ab)s(out)g(the)h(high)f(energy)h(asymptotics)g (of)f(the)h(magnetic)e(SSF.)118 4885 y(Both)23 b(authors)g(w)m(ere)g (partially)d(supp)s(orted)k(b)m(y)g(the)f(Chilean)e(Science)j(F)-8 b(oundation)21 b Fa(F)-7 b(onde)i(cyt)22 b Fn(under)118 5006 y(Gran)m(ts)k(1020737)f(and)h(7020737.)40 b(Moreo)m(v)m(er,)29 b(the)d(second)i(author)e(w)m(as)h(supp)s(orted)g(b)m(y)g(the)f Fa(CNRS-)118 5126 y(Conicyt)32 b Fn(Pro)5 b(ject)34 b(\\)p Fa(Sp)-5 b(e)g(ctr)g(al)34 b(and)g(Sc)-5 b(attering)35 b(The)-5 b(ory)34 b(for)h(Partial)g(Di\013er)-5 b(ential)34 b(Op)-5 b(er)g(ators)p Fn(".)1935 5375 y(10)p eop %%Page: 11 11 11 10 bop 118 171 a Fc(References)221 369 y Fn([1])49 b Fp(J.A)-12 b(vr)n(on,)54 b(I.Herbst,)e(B.Simon)p Fn(,)47 b Fa(Schr\177)-50 b(odinger)46 b(op)-5 b(er)g(ators)48 b(with)g(magnetic)f(\014elds.)g(I.)373 468 y(Gener)-5 b(al)34 b(inter)-5 b(actions)p Fn(,)32 b(Duk)m(e)i(Math.)f(J.)f Fo(45)h Fn(\(1978\),)f(847-883.)221 651 y([2])49 b Fp(J.A)-12 b(vr)n(on,)47 b(R.)g(Seiler,)e(B.)i(Simon)p Fn(,)42 b Fa(The)g(index)f(of)i(a)f(p)-5 b(air)42 b(of)h(pr)-5 b(oje)g(ctions)p Fn(,)40 b(J.)h(F)-8 b(unct.)373 751 y(Anal.)32 b Fo(120)g Fn(\(1994\),)g(220{237)221 955 y([3])49 b Fp(M.)507 930 y(\024)504 955 y(S.Birman,)35 b(M.)g(G.)h(Kre)1534 947 y(\025)1546 955 y(\020n)p Fn(,)31 b Fa(On)i(the)g(the)-5 b(ory)34 b(of)f(wave)g(op)-5 b(er)g(ators)33 b(and)g(sc)-5 b(attering)33 b(op-)373 1054 y(er)-5 b(ators)p Fn(,)31 b(Dokl.)g(Ak)-5 b(ad.)32 b(Nauk)g(SSSR)g Fo(144)f Fn(\(1962\),)g(475{478)f([in)h(Russian];)h(English)f(trans-)373 1154 y(lation)f(in)i(So)m(viet)h(Math.)g(Doklady)e Fo(3)i Fn(\(1962\).)221 1358 y([4])49 b Fp(M.)507 1333 y(\024)504 1358 y(S.Birman,)j(D.R.Y)-9 b(af)g(aev)p Fn(,)46 b Fa(The)h(sp)-5 b(e)g(ctr)g(al)48 b(shift)f(function.)h(The)f(p)-5 b(ap)g(ers)47 b(of)g(M.)h(G.)373 1458 y(Kr)-5 b(e)-10 b(\025)-40 b(\020n)39 b(and)h(their)g(further)g(development)p Fn(,)d(\(Russian\))h(Algebra)f (i)g(Analiz)g Fo(4)h Fn(\(1992\),)f(1{44;)373 1557 y(English)32 b(translation)e(in)i(St.)h(P)m(etersburg)h(Math.)f(J.)g Fo(4)f Fn(\(1993\),)g(833{870.)221 1740 y([5])49 b Fp(V.)39 b(Br)n(unea)n(u,)f(A.)h(Pushnitski,)g(G.)f(D.)h(Raik)n(o)n(v)p Fn(,)34 b Fa(Sp)-5 b(e)g(ctr)g(al)35 b(shift)h(function)g(in)g(str)-5 b(ong)373 1840 y(magnetic)34 b(\014elds)p Fn(,)e(Algebra)f(i)h(Analiz)f Fo(16)i Fn(\(2004\),)e(207)h(-)g(238.)221 2033 y([6])49 b Fp(C.)h(Fern)784 2025 y(\023)781 2033 y(andez,)f(G.)g(D.)h(Raik)n(o)n (v)p Fn(,)44 b Fa(On)g(the)i(singularities)e(of)h(the)g(magnetic)g(sp) -5 b(e)g(ctr)g(al)373 2133 y(shift)35 b(function)f(at)h(the)g(L)-5 b(andau)35 b(levels)p Fn(,)c Fo(5)i Fn(\(2004\),)e(381)h(-)g(403.)221 2327 y([7])49 b Fp(C.)d(G)614 2319 y(\023)614 2327 y(erard,)f(I.)51 b( )-36 b(L)o(aba)p Fn(,)41 b Fa(Multip)-5 b(article)42 b(Quantum)g(Sc)-5 b(attering)42 b(in)f(Constant)h(Magnetic)373 2426 y(Fields)p Fn(,)31 b(Mathematical)g(Surv)m(eys)k(and)d (Monographs,)i Fo(90)p Fn(,)e(AMS,)i(Pro)m(vidence,)g(RI,)e(2002.)221 2609 y([8])49 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