Content-Type: multipart/mixed; boundary="-------------0209301216151" This is a multi-part message in MIME format. ---------------0209301216151 Content-Type: text/plain; name="02-407.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-407.keywords" Discrete spectrum, spectral asymptotics, spectral gap, large coupling constant ---------------0209301216151 Content-Type: application/postscript; name="lara2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lara2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: lara2.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o lara2.ps lara2.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.09.30:1148 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false 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b Fx(=)37 b([0)p Fw(;)17 b Fx(1\))1734 4831 y Fl(d)1812 4867 y Fx(and)39 b Fw(B)2082 4882 y Fl(r)2157 4867 y Fx(=)e Fo(f)h Fw(x)g Fo(2)g Fh(R)2621 4831 y Fl(d)2705 4867 y Fx(:)f Fo(j)p Fw(x)p Fo(j)h Fw(<)f(r)s Fo(g)p Fw(;)17 b(r)39 b(>)f Fx(0.)456 4983 y(Notation)43 b Fw(H)966 4947 y Fl(s)1002 4983 y Fx(\(\012\))p Fw(;)62 b Fx(\012)50 b Fo(\032)f Fh(R)1549 4947 y Fl(d)1595 4983 y Fx(,)f(is)d(tak)m(en)h(for)e(the)h(Sob)s(olev)g(classes)h(of)e(order) 456 5099 y Fw(s)27 b Fo(2)h Fh(N)9 b Fx(;)37 b Fw(H)842 5063 y Fl(s)834 5124 y Fv(0)878 5099 y Fx(\(\012\))30 b(is)f(the)h(closure)f(of)g(the)h(class)g Fw(C)2212 5063 y Fm(1)2205 5124 y Fv(0)2286 5099 y Fx(\(\012\))g(in)f(the)g(metric)g (of)g Fw(H)3235 5063 y Fl(s)3271 5099 y Fx(\(\012\).)456 5216 y(F)-8 b(or)28 b(Laplace)h(op)s(erators)f(on)i(domains)d(\012)i Fo(\032)f Fh(R)2194 5179 y Fl(d)2269 5216 y Fx(w)m(e)i(use)g(the)g (notation)e(\001.)42 b(The)p eop %%Page: 3 3 3 2 bop 1129 251 a Fn(THE)34 b(AMOUNT)f(OF)g(DISCRETE)g(SPECTR)n(UM)669 b(3)456 450 y Fx(gradien)m(t)41 b(op)s(erator)g(is)g(denoted)i(b)m(y)f Fo(r)p Fx(.)71 b(Di\013eren)m(t)41 b(estimate)g(constan)m(ts)i(are)456 566 y(denoted)33 b(b)m(y)h Fw(C)r(;)48 b(c)33 b Fx(\(sometimes)e(with)i (indices\).)555 683 y Fy(2.)44 b Fx(Let)33 b(a)f(function)g Fw(p)c Fo(2)g Fw(L)1561 646 y Fm(1)1636 683 y Fx(\()p Fh(R)1740 646 y Fl(d)1786 683 y Fx(\))k(b)s(e)h(real)f(v)-5 b(alued)32 b(and)h(let)1221 868 y Fw(p)p Fx(\()p Fw(x)22 b Fx(+)g Fw(n)p Fx(\))28 b(=)g Fw(p)p Fx(\()p Fw(x)p Fx(\))p Fw(;)114 b(x)28 b Fo(2)g Fh(R)2275 827 y Fl(d)2322 868 y Fw(;)17 b(n)27 b Fo(2)h Fh(Z)2614 827 y Fl(d)2652 868 y Fw(:)456 1049 y Fx(Consider)33 b(the)g(selfadjoin)m(t)e(op)s (erator)1088 1231 y Fw(Au)c Fx(=)h Fo(\000)p Fx(\001)p Fw(u)22 b Fx(+)g Fw(pu;)114 b(u)27 b Fo(2)h Fw(D)s Fx(\()p Fw(A)p Fx(\))f(=)h Fw(H)2558 1190 y Fv(2)2597 1231 y Fx(\()p Fh(R)2701 1190 y Fl(d)2747 1231 y Fx(\))p Fw(;)456 1412 y Fx(on)52 b Fw(L)677 1376 y Fv(2)717 1412 y Fx(\()p Fh(R)821 1376 y Fl(d)867 1412 y Fx(\).)103 b(Ob)m(viously)53 b(A)f(is)g(an)h(elliptic)c Fh(Z)2321 1376 y Fl(d)2359 1412 y Fx(-p)s(erio)s(dic)i(op)s(erator.)102 b(The)456 1528 y(sp)s(ectrum)58 b Fw(\033)t Fx(\()p Fw(A)p Fx(\))g(can)h(ha)m(v)m (e)g(gaps)f(b)s(esides)h(the)g(semi-in\014nite)d(one.)120 b(Let)456 1644 y(\003)81 b(=)g(\()p Fw(\025)857 1659 y Fm(\000)916 1644 y Fw(;)17 b(\025)1017 1659 y Fv(+)1075 1644 y Fx(\))64 b(b)s(e)h(a)f(\014xed)h(gap)f(in)f(the)i(sp)s(ectrum)f Fw(\033)t Fx(\()p Fw(A)p Fx(\))g(\(if)f Fw(\025)3228 1659 y Fv(+)3369 1644 y Fx(=)456 1761 y(inf)6 b Fw(\033)t Fx(\()p Fw(A)p Fx(\))p Fw(;)52 b Fx(then)37 b Fw(\025)1144 1776 y Fm(\000)1236 1761 y Fx(=)c Fo(\0001)p Fx(\).)54 b(Belo)m(w)36 b(w)m(e)h(p)s(erturb)f(the)g(op)s(erator)g Fw(A)g Fx(so,)h(that)456 1877 y(the)c(sp)s(ectrum)g(of)f(the)h(p)s (erturb)s(ed)g(op)s(erator)f(is)g(discrete)h(in)f(\003.)555 1993 y(Let)h Fw(V)49 b Fx(:)28 b Fh(R)957 1957 y Fl(d)1031 1993 y Fo(!)f Fh(R)1224 2008 y Fv(+)1322 1993 y Fx(b)s(e)33 b(a)f(smo)s(oth)g(function)g(and)g(for)g(ev)m(ery)j Fw(\022)c Fo(2)d Fh(S)799 2174 y Fx(lim)780 2224 y Fl(r)r Fm(!1)971 2174 y Fw(r)1018 2133 y Fl(s)1055 2174 y Fw(V)21 b Fx(\()p Fw(\022)s(r)s Fx(\))28 b(=)f(\010\()p Fw(\022)s Fx(\))p Fw(;)115 b Fx(2)27 b Fw(<)h(s)g(<)f Fx(2)22 b(+)g(2)p Fw(=)p Fx(\()p Fw(d)f Fo(\000)i Fx(1\))p Fw(;)17 b Fx(\010\()p Fw(\022)s Fx(\))28 b Fw(>)f Fx(0)p Fw(:)183 b Fx(\(4\))456 2381 y(The)42 b(limit)37 b(in)j(\(4\))h(is)g(assumed)g(to)g(b)s(e)g (uniform)f(with)g(resp)s(ect)j(to)d Fw(\022)s Fx(.)69 b(Belo)m(w)456 2497 y(w)m(e)43 b(systematically)e(use)j(the)f(follo)m (wing)d(notations)i(for)g(the)h(op)s(erators)f(on)h(a)456 2613 y(domain.)58 b(Let)38 b(\012)g Fo(\032)g Fh(R)1326 2577 y Fl(d)1410 2613 y Fx(b)s(e)h(an)f(op)s(en)g(domain.)58 b(W)-8 b(e)39 b(consider)f(the)h(follo)m(wing)456 2729 y(op)s(erator)32 b(on)g Fw(L)1050 2693 y Fv(2)1090 2729 y Fx(\(\012\))1183 2918 y Fw(A)1256 2877 y Fv(\()p Fl(")p Fv(\))1348 2918 y Fx(\()p Fw(\013)q(;)17 b Fx(\012\))27 b(=)h Fo(\000)p Fx(\001)22 b(+)g Fw(p)h Fo(\000)f Fw(\013)q(V)5 b(;)114 b(\013)29 b(>)e Fx(0;)456 3100 y(the)38 b(index)h Fw(")f Fx(tak)m(es)i(the)e(v)-5 b(alues)39 b Fo(D)h Fx(or)f Fo(N)52 b Fx(and)39 b(c)m(haracterizes)g(the)g(b)s(oundary)456 3216 y(\(Diric)m(hlet)e(or)h(Neumann\))i(conditions)e(on)h Fw(@)5 b Fx(\012.)65 b(In)39 b(order)g(to)g(form)m(ulate)f(the)456 3332 y(main)29 b(result)h(w)m(e)i(need)g(to)f(in)m(tro)s(duce)f(the)i (densit)m(y)f(of)g(states)g(for)g(the)g(op)s(erator)456 3448 y Fw(A)p Fx(.)62 b(Let)40 b Fw(x)855 3463 y Fv(1)894 3448 y Fw(;)17 b(x)993 3463 y Fv(2)1072 3448 y Fo(2)39 b Fh(R)1243 3412 y Fl(d)1328 3448 y Fx(and)g Fw(l)i(>)d Fx(0.)63 b(Let)39 b(also)f(\012\()p Fw(l)r Fx(\))h(:=)g(\012\()p Fw(l)r Fx(;)17 b Fw(x)2826 3463 y Fv(1)2866 3448 y Fw(;)g(x)2965 3463 y Fv(2)3004 3448 y Fx(\))39 b(=)f Fw(l)r Fx(\()p Fh(Q)45 b Fx(+)456 3565 y Fw(x)511 3580 y Fv(1)550 3565 y Fx(\))23 b(+)f Fw(x)764 3580 y Fv(2)803 3565 y Fx(.)44 b(Then)34 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4 4 4 3 bop 456 251 a Fn(4)1184 b(O.)26 b(SAFR)n(ONO)n(V)456 450 y Fj(and)44 b Fx(\(5\))35 b Fj(ar)-5 b(e)34 b(ful\014l)5 b(le)-5 b(d.)44 b(Then)1653 626 y Fx(lim)1628 676 y Fl(\013)p Fm(!1)1831 626 y Fw(\013)1894 585 y Fm(\000)p Fl(d=s)2057 626 y Fx(rank)p Fw(E)2321 642 y Fl(A)p Fv(\()p Fl(\013)p Fv(\))2479 626 y Fx(\()p Fw(\025)2574 641 y Fv(1)2613 626 y Fw(;)17 b(\025)2714 641 y Fv(2)2753 626 y Fx(\))28 b(=)978 722 y Fk(Z)1024 911 y Fg(R)1072 892 y Ff(d)1112 839 y Fx(\()p Fw(\032)p Fx(\()p Fw(\025)1295 854 y Fv(2)1357 839 y Fx(+)22 b(\010\()p Fw(\022)s Fx(\))p Fo(j)p Fw(x)p Fo(j)1760 798 y Fm(\000)p Fl(s)1852 839 y Fx(\))g Fo(\000)g Fw(\032)p Fx(\()p Fw(\025)2156 854 y Fv(1)2218 839 y Fx(+)g(\010\()p Fw(\022)s Fx(\))p Fo(j)p Fw(x)p Fo(j)2621 798 y Fm(\000)p Fl(s)2713 839 y Fx(\)\))p Fw(dx:)3320 748 y Fx(\(6\))555 1053 y Fy(Remark)37 b(1.1.)44 b Fx(F)-8 b(or)32 b(a)g(\014xed)i(real)d(n)m(um)m(b)s(er)i Fw(\026)1062 1223 y(\032)p Fx(\()p Fw(\025)22 b Fx(+)g Fw(\026)p Fx(\))g Fo(\000)g 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b(A)1902 2696 y Fv(\()p Fm(N)10 b Fv(\))2029 2732 y Fx(\()p Fw(\013)q Fx(\)\))34 b(for)f(an)i(arbitrary)e Fw(\025)e Fo(2)g Fh(R)5 b Fx(.)54 b(The)456 2849 y(follo)m(wing)30 b(statemen)m(t)j(giv)m(es)g(an)f(estimate)g(of)g(the)h(di\013erence)h (of)e(the)h(distribu-)456 2965 y(tion)e(functions)i(of)f(the)h(sp)s (ectrum)g(of)f(the)h(op)s(erators)f Fw(A)2565 2929 y Fv(\()p Fm(D)r Fv(\))2681 2965 y Fx(\()p Fw(\013)q Fx(\))g(and)h Fw(A)3115 2929 y Fv(\()p Fm(N)10 b Fv(\))3241 2965 y Fx(\()p Fw(\013)q Fx(\).)456 3262 y Fy(Prop)s(osition)29 b(2.1.)38 b Fj(L)-5 b(et)30 b Fw(p)d Fo(2)h Fw(L)1666 3226 y Fm(1)1742 3262 y Fx(\()p Fh(R)1845 3226 y Fl(d)1892 3262 y Fx(\))p Fw(;)46 b Fx(2)28 b Fw(<)f(s)h(<)f Fx(2)11 b(+)g Fw(s=)p Fx(\()p Fw(d)g Fo(\000)g Fx(1\))29 b Fj(and)h Fw(\025)3148 3277 y Fv(1)3215 3262 y Fw(<)d(\025)3375 3277 y Fv(2)3415 3262 y Fj(.)456 3378 y(Assume)37 b(that)g Fw(V)59 b Fj(satis\014es)36 b(the)h(c)-5 b(onditions)36 b(of)h(The)-5 b(or)g(em)36 b(1.1.)50 b(Then)37 b(ther)-5 b(e)37 b(is)456 3494 y Fw(\021)504 3509 y Fv(0)571 3494 y Fw(>)27 b Fx(0)35 b Fj(such)g(that)g(for)g(al)5 b(l)34 b Fx(0)28 b Fw(<)f(\021)k(<)d(\021)1885 3509 y Fv(0)642 3665 y Fw(\026)p Fx(\()p Fw(\021)t Fx(\))f(:=)h(lim)17 b(sup)1043 3734 y Fl(\013)p Fm(!1)1302 3665 y Fw(\013)1365 3624 y Fm(\000)p Fl(d=s)1528 3665 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025)1749 3680 y Fv(2)1789 3665 y Fw(;)17 b(A)1906 3624 y Fv(\()p Fm(N)10 b Fv(\))2032 3665 y Fx(\()p Fw(\013)q Fx(\)\))22 b Fo(\000)h Fw(N)10 b Fx(\()p Fw(\025)2514 3680 y Fv(1)2553 3665 y Fw(;)17 b(A)2670 3624 y Fv(\()p Fm(D)r Fv(\))2786 3665 y Fx(\()p Fw(\013)q Fx(\)\)\))27 b Fw(<)g Fo(1)p Fw(;)456 3872 y Fj(wher)-5 b(e)1677 4000 y Fx(lim)1673 4057 y Fl(\021)r Fm(!)p Fv(0)1833 4000 y Fw(\026)p Fx(\()p Fw(\021)t Fx(\))27 b(=)h(0)p Fw(:)456 4187 y Fj(In)34 b(p)-5 b(articular)35 b(one)f(has)g(the)h(fol)5 b(lowing)34 b(estimate)746 4358 y Fx(lim)17 b(inf)788 4408 y Fl(\013)p Fm(!1)1032 4358 y Fw(\013)1095 4317 y Fm(\000)p Fl(d=s)1258 4358 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025)1479 4373 y Fv(2)1519 4358 y Fw(;)17 b(A)1636 4317 y Fv(\()p Fm(D)r Fv(\))1752 4358 y Fx(\()p Fw(\013)q Fx(\)\))k Fo(\000)i Fw(N)10 b Fx(\()p Fw(\025)2233 4373 y Fv(1)2273 4358 y Fw(;)17 b(A)2390 4317 y Fv(\()p Fm(N)10 b Fv(\))2516 4358 y Fx(\()p Fw(\013)q Fx(\)\)\))27 b Fo(\025)h(\000)p Fw(\026)p Fx(\()p Fw(\021)t Fx(\))p Fw(;)555 4558 y Fj(Pr)-5 b(o)g(of.)43 b Fx(W)-8 b(e)33 b(set)1181 4730 y(^)1153 4755 y Fw(N)10 b Fx(\()p Fw(\025;)17 b(\013)q Fx(\))27 b(=)h(\(2)p Fw(\031)t Fx(\))1796 4714 y Fm(\000)p Fl(d)1890 4755 y Fw(!)1951 4770 y Fl(d)2008 4638 y Fk(Z)2054 4827 y Fm(Y)2111 4755 y Fx(\()p Fw(\013)q(V)21 b Fx(\()p Fw(x)p Fx(\))i(+)f Fw(\025)p Fx(\))2637 4704 y Fl(d=)p Fv(2)2637 4774 y(+)456 4968 y Fx(\()p Fw(!)555 4983 y Fl(d)627 4968 y Fx(is)32 b(the)h(v)m(olume)f(of)g(the)h(unit)f (ball\),)1178 5171 y Fw(F)1241 5186 y Fv(1)1281 5171 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))27 b(=)g Fo(j)p Fw(\025)p Fo(j)1764 5130 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=)p Fv(2)2036 5054 y Fk(Z)2082 5242 y Fm(Y)2139 5171 y Fx(\(1)22 b(+)g Fo(j)p Fw(x)p Fo(j)p Fx(\))2495 5130 y Fm(\000)p Fv(1)2589 5171 y Fw(dx;)p eop %%Page: 5 5 5 4 bop 1129 251 a Fn(THE)34 b(AMOUNT)f(OF)g(DISCRETE)g(SPECTR)n(UM)669 b(5)993 484 y Fw(F)1056 499 y Fv(2)1096 484 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))27 b(=)1466 367 y Fk(Z)1512 556 y Fm(Y)1569 484 y Fx(\()p Fw(\013)q(V)21 b Fx(\()p Fw(x)p Fx(\))i(+)f Fw(\025)p Fx(\))2095 433 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=)p Fv(2)2095 503 y(+)2351 484 y Fx(\(1)f(+)h Fo(j)p Fw(x)p Fo(j)p Fx(\))2706 443 y Fm(\000)p Fv(1)2800 484 y Fw(dx)456 662 y Fx(and)950 790 y Fw(F)1013 805 y Fv(3)1052 790 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))27 b(=)1423 673 y Fk(Z)1469 862 y Fm(Y)1525 790 y Fx(\()p Fw(\013)q Fx(\(1)22 b(+)g Fo(j)p Fw(x)p Fo(j)p Fx(\))1982 749 y Fm(\000)p Fl(s)2073 790 y Fx(\))2111 749 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=)p Fv(2)2367 790 y Fx(\(1)g(+)g Fo(j)p Fw(x)p Fo(j)p Fx(\))2723 749 y Fm(\000)p Fv(1)2817 790 y Fw(dx:)456 974 y Fx(Let)30 b Fw(N)10 b Fx(\()p Fw(\025;)17 b(";)g(\013)q Fx(\))30 b(denote)h(the)g(n)m(um)m(b)s(er)g(of)f(negativ)m(e)h(eigen)m(v)-5 b(alues)30 b(of)g(the)h(op)s(era-)456 1090 y(tor)1446 1209 y Fo(\000)p Fx(\001)23 b Fo(\000)f Fw(\025)g Fo(\000)h Fw(\013)q(V)5 b(;)147 b(\025)27 b Fo(2)h Fh(R)456 1348 y Fx(on)34 b(the)i(domain)d Fo(Y)42 b Fx(with)35 b(the)g Fw(")p Fx(-b)s(oundary)g(conditions.)49 b(The)36 b(follo)m(wing)c (esti-)456 1464 y(mate)g(for)g(the)h(distribution)e(function)i(has)g(b) s(een)g(established)g(b)m(y)h(V.)f(Ivrii.)43 b(In)456 1580 y(the)33 b(presen)m(t)h(form)d(this)h(estimate)g(follo)m(ws)f (from)h(Prop)s(osition)f(3)h(of)g([5].)456 1872 y Fy(Theorem)40 b(2.1.)j Fj(L)-5 b(et)38 b Fw(d)32 b Fo(\025)h Fx(3)k Fj(and)g Fw(V)59 b Fj(satis\014es)45 b Fx(\(5\))37 b Fj(for)h Fw(s)32 b(>)h Fx(2)p Fj(.)52 b(Then)37 b(for)g(al)5 b(l)456 1988 y Fw(\013)28 b(>)g(\013)712 2003 y Fv(0)779 1988 y Fw(>)f Fx(0)p Fw(;)52 b(\025)27 b Fo(2)h Fh(R)46 b Fj(we)35 b(have)f(the)h(estimate)759 2153 y Fo(j)p Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(D)s Fw(;)g(\013)q Fx(\))k Fo(\000)1388 2128 y Fx(^)1360 2153 y Fw(N)10 b Fx(\()p Fw(\025;)17 b(\013)q Fx(\))p Fo(j)27 b(\024)h Fw(F)1911 2168 y Fv(1)1951 2153 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))k(+)h Fw(F)2373 2168 y Fv(2)2412 2153 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))22 b(+)g Fw(F)2835 2168 y Fv(3)2874 2153 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))p Fw(:)555 2329 y Fx(W)-8 b(e)33 b(also)f(need)h(the)g(follo)m (wing)d(result:)456 2621 y Fy(Lemma)37 b(2.1.)42 b Fj(L)-5 b(et)35 b Fw(d)27 b Fo(\025)h Fx(3)35 b Fj(and)f Fw(V)57 b Fj(satis\014es)42 b Fx(\(5\))35 b Fj(for)f Fw(s)28 b(>)f Fx(2)p Fj(.)45 b(Then)900 2788 y Fx(lim)17 b(sup)956 2858 y Fl(\013)p Fm(!1)1215 2788 y Fw(\013)1278 2747 y Fl(d=s)1386 2788 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(N)e Fw(;)i(\013)q Fx(\))k Fo(\000)i Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(D)s Fw(;)g(\013)q Fx(\)\))27 b Fo(\024)h Fw(C)2706 2803 y Fv(1)2745 2788 y Fx(\()p Fw(\021)t(;)17 b(\025)p Fx(\))p Fw(:)456 2992 y Fj(wher)-5 b(e)34 b Fw(C)801 3007 y Fv(1)840 2992 y Fx(\()p Fw(\021)t(;)17 b(\025)p Fx(\))27 b Fo(!)g Fx(0)35 b Fj(as)f Fw(\021)e Fo(!)27 b Fx(0)p Fw(:)555 3168 y Fj(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(2.1)p Fx(.)43 b(Let)33 b Fw(\014)38 b Fx(b)s(e)33 b(a)f Fw(C)2035 3131 y Fm(1)2110 3168 y Fx(-function)g(of)g(one)h(v)-5 b(ariable)30 b(and)1300 3413 y Fw(\014)6 b Fx(\()p Fw(t)p Fx(\))28 b(=)1603 3238 y Fk(8)1603 3313 y(<)1603 3463 y(:)1677 3346 y Fx(1)162 b(if)169 b Fw(t)28 b Fo(\024)g(\000)p Fx(1)p Fw(=)p Fx(2)p Fw(;)1677 3485 y Fx(0)162 b(if)169 b Fw(t)28 b Fo(\025)g Fx(0)p Fw(:)456 3658 y Fx(De\014ne)1171 3776 y Fw(\020)1214 3791 y Fv(1)1253 3776 y Fx(\()p Fw(x)p Fx(\))g(=)f Fw(\014)6 b Fx(\()p Fo(j)p Fw(x)p Fo(j)22 b(\000)g Fw(\021)t(\013)1961 3735 y Fv(1)p Fl(=s)2090 3776 y Fx(+)g(1\))p Fw(;)147 b(x)28 b Fo(2)g Fh(R)15 b Fw(;)456 3915 y Fx(and)33 b(\014nd)h Fw(\020)885 3930 y Fv(2)958 3915 y Fx(so)g(that)f Fw(\020)1342 3879 y Fv(2)1334 3940 y(1)1404 3915 y Fx(+)22 b Fw(\020)1553 3879 y Fv(2)1545 3940 y(2)1621 3915 y Fx(=)29 b(1.)47 b(Ob)m(viously)33 b Fw(\014)39 b Fx(can)34 b(b)s(e)g(c)m(hosen)h(so)f (that)f Fw(\020)3405 3930 y Fv(2)456 4032 y Fx(is)f(di\013eren)m (tiable.)42 b(Then)968 4083 y Fk(Z)1015 4272 y Fm(Y)1088 4200 y Fo(jr)p Fw(u)p Fo(j)1283 4159 y Fv(2)1338 4200 y Fw(dx)27 b Fx(=)1575 4083 y Fk(Z)1621 4272 y Fm(Y)1694 4200 y Fo(jr)p Fw(\020)1848 4215 y Fv(1)1887 4200 y Fw(u)p Fo(j)1971 4159 y Fv(2)2026 4200 y Fw(dx)22 b Fx(+)2252 4083 y Fk(Z)2299 4272 y Fm(Y)2372 4200 y Fo(jr)p Fw(\020)2526 4215 y Fv(2)2564 4200 y Fw(u)p Fo(j)2648 4159 y Fv(2)2704 4200 y Fw(dx)g Fo(\000)1813 4315 y Fk(Z)1859 4504 y Fm(Y)1916 4433 y Fx(\()p Fo(jr)p Fw(\020)2108 4448 y Fv(1)2146 4433 y Fo(j)2174 4391 y Fv(2)2236 4433 y Fx(+)g Fo(jr)p Fw(\020)2488 4448 y Fv(2)2526 4433 y Fo(j)2554 4391 y Fv(2)2594 4433 y Fx(\))p Fo(j)p Fw(u)p Fo(j)2744 4391 y Fv(2)2798 4433 y Fw(dx:)3320 4322 y Fx(\(7\))456 4631 y(Since)32 b(the)h(mapping)1072 4790 y Fw(L)1138 4749 y Fv(2)1178 4790 y Fx(\()p Fo(Y)8 b Fx(\))27 b Fo(3)h Fw(u)f Fo(7!)g Fx(\()p Fw(\020)1741 4805 y Fv(1)1780 4790 y Fw(u;)33 b(\020)1939 4805 y Fv(2)1978 4790 y Fw(u)p Fx(\))27 b Fo(2)i Fw(L)2260 4749 y Fv(2)2299 4790 y Fx(\()p Fo(Y)8 b Fx(\))22 b Fo(\002)h Fw(L)2638 4749 y Fv(2)2678 4790 y Fx(\()p Fo(Y)8 b Fx(\))456 4950 y(is)31 b(a)g(one)h(to)f(one)h (corresp)s(ondence)i(on)m(to)e(the)g(range)f(and)h(for)f(an)m(y)i (function)e Fw(W)785 5020 y Fk(Z)831 5209 y Fm(Y)904 5138 y Fw(W)14 b Fx(\()p Fw(x)p Fx(\))p Fo(j)p Fw(u)p Fo(j)1253 5096 y Fv(2)1308 5138 y Fw(dx)28 b Fx(=)1545 5020 y Fk(Z)1591 5209 y Fm(Y)1665 5138 y Fw(W)14 b Fx(\()p Fw(x)p Fx(\))p Fo(j)p Fw(\020)1973 5153 y Fv(1)2011 5138 y Fw(u)p Fo(j)2095 5096 y Fv(2)2150 5138 y Fw(dx)23 b Fx(+)2377 5020 y Fk(Z)2423 5209 y Fm(Y)2496 5138 y Fw(W)14 b Fx(\()p Fw(x)p Fx(\))p Fo(j)p Fw(\020)2804 5153 y Fv(2)2843 5138 y Fw(u)p Fo(j)2927 5096 y Fv(2)2982 5138 y Fw(dx;)205 b Fx(\(8\))p eop %%Page: 6 6 6 5 bop 456 251 a Fn(6)1184 b(O.)26 b(SAFR)n(ONO)n(V)456 450 y Fx(w)m(e)33 b(obtain)f(from)f(the)i(v)-5 b(ariational)29 b(principle)i(that)906 619 y Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(N)e Fw(;)i(\013)q Fx(\))27 b Fo(\024)h Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(c;)17 b Fo(D)s Fw(;)g(\013)q Fx(\))j(+)i Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(c;)17 b Fo(D)s Fw(;)g Fo(N)c Fw(;)k(\013)q Fx(\))326 b(\(9\))456 798 y(where)35 b Fw(c)c Fx(=)f(sup)18 b Fo(jr)p Fw(\020)1236 813 y Fv(1)1274 798 y Fo(j)1302 762 y Fv(2)1365 798 y Fx(+)23 b Fo(jr)p Fw(\020)1618 813 y Fv(2)1657 798 y Fo(j)1685 762 y Fv(2)1758 798 y Fx(and)35 b Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)h Fw(c;)17 b Fo(D)s Fw(;)g Fo(N)c Fw(;)k(\013)q Fx(\))34 b(is)f(the)i(n)m(um)m(b)s(er)g(of)456 914 y(the)d(negativ)m(e)g(eigen)m(v)-5 b(alues)31 b(of)h(the)g(op)s (erator)f Fo(\000)p Fx(\001)21 b Fo(\000)g Fw(\025)f Fo(\000)h Fw(c)g Fo(\000)g Fw(\013)q(V)53 b Fx(with)31 b(mixed)456 1030 y(b)s(oundary)25 b(conditions)f(on)i(the)f(b)s (oundary)h(of)e(the)i(domain)e(\012)2741 1045 y Fv(0)2780 1030 y Fx(\()p Fw(\013)q Fx(\))k(=)f Fo(f)p Fw(x)h Fo(2)g Fh(R)3343 994 y Fl(d)3417 1030 y Fx(:)456 1149 y Fw(\021)t(\013)571 1113 y Fv(1)p Fl(=s)703 1149 y Fo(\000)e Fx(2)35 b Fw(<)h Fo(j)p Fw(x)p Fo(j)g Fw(<)f(\021)t(\013)1375 1113 y Fv(1)p Fl(=s)1482 1149 y Fo(g)p Fx(.)58 b(It)37 b(is)g(in)m(teresting)g(to)g (giv)m(e)g(a)g(more)g(lucid)f(pro)s(of)456 1266 y(of)j(the)g(inequalit) m(y)h(\(9\).)64 b(F)-8 b(or)38 b(that)h(purp)s(ose)h(w)m(e)h(denote)f (b)m(y)g Fi(H)2883 1281 y Fv(1)2962 1266 y Fx(and)f Fi(H)3230 1281 y Fv(2)3309 1266 y Fx(the)456 1382 y(spaces)i(of)f(eigenfunctions) g(corresp)s(onding)f(to)h(negativ)m(e)g(eigen)m(v)-5 b(alues)40 b(of)g(the)456 1498 y(op)s(erators)31 b Fo(\000)p Fx(\001)20 b Fo(\000)g Fw(\025)g Fo(\000)g Fw(c)g Fo(\000)g Fw(\013)q(V)53 b Fx(on)31 b(the)h(domains)e Fo(Y)39 b Fx(and)32 b(\012)2717 1513 y Fv(0)2756 1498 y Fx(\()p Fw(\013)q Fx(\))f(resp)s(ectiv)m(ely)-8 b(.)456 1614 y(In)28 b(the)g(\014rst)g(case)h(w)m(e)g(imp)s(ose)e(the)h(Diric)m (hlet)d(b)s(oundary)k(condition)d(and)i(in)f(the)456 1731 y(second)35 b(case)h(the)e(conditions)g(on)g(the)h(b)s(oundary)g (are)f(mixed.)49 b(If)41 b(\(9\))34 b(is)g(false,)456 1847 y(then)40 b(there)g(is)f(a)g(function)g Fw(u)g Fo(2)h Fw(H)1813 1811 y Fv(1)1852 1847 y Fx(\()p Fo(Y)8 b Fx(\))39 b(whic)m(h)i(is)e(orthogonal)e(to)j Fw(\020)3098 1862 y Fv(1)3137 1847 y Fi(H)3209 1862 y Fv(1)3287 1847 y Fx(and)456 1963 y Fw(\020)499 1978 y Fv(2)538 1963 y Fi(H)610 1978 y Fv(2)681 1963 y Fx(and)33 b(satisfyies)g(the)g (inequalit)m(y)1182 2043 y Fk(Z)1228 2232 y Fm(Y)1301 2160 y Fo(jr)p Fw(u)p Fo(j)1496 2119 y Fv(2)1551 2160 y Fw(dx)28 b Fo(\024)1790 2043 y Fk(Z)1836 2232 y Fm(Y)1893 2160 y Fx(\()p Fw(\013)q(V)21 b Fx(\()p Fw(x)p Fx(\))h(+)g Fw(\025)p Fx(\))p Fo(j)p Fw(u)p Fo(j)2530 2119 y Fv(2)2585 2160 y Fw(dx;)456 2374 y Fx(whic)m(h)41 b(is)g(imp)s(ossible)e(b)s (ecause)k(of)48 b(\(7\))41 b(and)g(\(8\).)70 b(Indeed,)45 b(in)c(this)g(case)h(the)456 2490 y(function)32 b Fw(\020)881 2505 y Fl(j)917 2490 y Fw(u)g Fx(will)e(b)s(e)j(orthogonal)e(to)h Fi(H)2001 2505 y Fl(j)2037 2490 y Fx(,)h(whic)m(h)g(means)f(that)768 2579 y Fk(Z)814 2767 y Fm(Y)887 2696 y Fo(jr)p Fw(\020)1041 2711 y Fl(j)1077 2696 y Fw(u)p Fo(j)1161 2655 y Fv(2)1216 2696 y Fw(dx)c(>)1454 2579 y Fk(Z)1500 2767 y Fm(Y)1556 2696 y Fx(\()p Fw(\013)q(V)22 b Fx(\()p Fw(x)p Fx(\))g(+)g Fw(\025)g Fx(+)g Fw(c)p Fx(\))p Fo(j)p Fw(\020)2315 2711 y Fl(j)2351 2696 y Fw(u)p Fo(j)2435 2655 y Fv(2)2490 2696 y Fw(dx;)147 b(j)34 b Fx(=)27 b(1)p Fw(;)33 b Fx(2)p Fw(:)555 2904 y Fx(One)g(can)g(sho)m(w)h(that)923 3073 y Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(c;)17 b Fo(D)s Fw(;)g Fo(N)d Fw(;)j(\013)q Fx(\))26 b(=)i Fw(O)s Fx(\(v)m(ol)15 b(\012)2133 3088 y Fv(0)2173 3073 y Fx(\()p Fw(\013)q Fx(\)\))28 b(=)f Fw(O)s Fx(\()p Fw(\013)2660 3032 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=s)2912 3073 y Fx(\))p Fw(;)456 3242 y Fx(as)32 b Fw(\013)d Fo(!)e(1)p Fw(:)32 b Fx(Therefore)1068 3410 y(lim)17 b(sup)1125 3479 y Fl(\013)p Fm(!1)1384 3410 y Fw(\013)1447 3369 y Fm(\000)p Fl(d=s)1610 3410 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(N)e Fw(;)i(\013)q Fx(\))k Fo(\000)h Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(D)s Fw(;)g(\013)q Fx(\)\))27 b Fo(\024)1035 3651 y Fx(lim)17 b(sup)1091 3721 y Fl(\013)p Fm(!1)1350 3651 y Fw(\013)1413 3610 y Fm(\000)p Fl(d=s)1576 3651 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(c;)17 b Fo(D)s Fw(;)g(\013)q Fx(\))k Fo(\000)h Fw(N)10 b Fx(\()p Fw(\025;)17 b Fo(D)s Fw(;)g(\013)q Fx(\)\))p Fw(:)456 3855 y Fx(Since)33 b(for)h(an)m(y)g Fw(\025)f Fx(and)h(index)g Fw(j)40 b Fx(it)32 b(holds)i Fw(F)2090 3870 y Fl(j)2127 3855 y Fx(\()p Fw(\025;)17 b(\013)q Fx(\))28 b(=)i Fw(o)p Fx(\()p Fw(\013)2649 3819 y Fl(d=s)2757 3855 y Fx(\))p Fw(;)83 b(\013)31 b Fo(!)e(1)p Fw(;)k Fx(and)456 3972 y(starting)e(from)h(a)g(certain)g(0)27 b Fw(<)h(\021)j(<)d(\021)1867 3987 y Fv(0)911 4146 y Fx(lim)17 b(sup)967 4215 y Fl(\013)p Fm(!1)1227 4146 y Fw(\013)1290 4105 y Fm(\000)p Fl(d=s)1453 4146 y Fo(f)1530 4121 y Fx(^)1503 4146 y Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(c;)17 b(\013)q Fx(\))k Fo(\000)2142 4121 y Fx(^)2114 4146 y Fw(N)11 b Fx(\()p Fw(\025;)17 b(\013)q Fx(\))p Fo(g)26 b(\024)i Fw(C)2694 4161 y Fv(1)2734 4146 y Fx(\()p Fw(\021)t(;)17 b(\025)p Fx(\))p Fw(;)1391 4371 y(C)1461 4386 y Fv(1)1501 4371 y Fx(\()p Fw(\021)t(;)g(\025)p Fx(\))26 b Fo(!)i Fx(0)p Fw(;)114 b Fx(as)65 b Fw(\021)31 b Fo(!)d Fx(0)p Fw(;)456 4520 y Fx(the)33 b(statemen)m(t)g(of)f(Lemma)f (2.1)h(follo)m(ws.)64 b Fe(2)555 4688 y Fj(The)45 b(r)-5 b(est)46 b(of)f(the)g(pr)-5 b(o)g(of)45 b(of)g(Pr)-5 b(op)g(osition)45 b(2.1)e Fx(Since)i Fw(F)2674 4703 y Fl(j)2710 4688 y Fx(\()p Fw(\025)30 b Fo(\006)h(jj)p Fw(p)p Fo(jj)3104 4703 y Fm(1)3177 4688 y Fw(;)17 b(\013)q Fx(\))47 b(=)456 4811 y Fw(o)p Fx(\()p Fw(\013)604 4774 y Fl(d=s)711 4811 y Fx(\))p Fw(;)82 b(\013)29 b Fo(!)e(1)p Fw(;)32 b Fx(and)h(starting)e(from)h(a)g(certain)g(0)c Fw(<)f(\021)32 b(<)27 b(\021)2836 4826 y Fv(0)664 4991 y Fx(lim)17 b(sup)720 5060 y Fl(\013)p Fm(!1)979 4991 y Fw(\013)1042 4950 y Fm(\000)p Fl(d=s)1205 4991 y Fo(f)1283 4965 y Fx(^)1255 4991 y Fw(N)11 b Fx(\()p Fw(\025)1439 5006 y Fv(2)1500 4991 y Fx(+)22 b Fo(jj)p Fw(p)p Fo(jj)1759 5006 y Fm(1)1833 4991 y Fw(;)17 b(\013)q Fx(\))k Fo(\000)2127 4965 y Fx(^)2099 4991 y Fw(N)10 b Fx(\()p Fw(\025)2282 5006 y Fv(1)2344 4991 y Fo(\000)22 b(jj)p Fw(p)p Fo(jj)2604 5006 y Fm(1)2678 4991 y Fw(;)17 b(\013)q Fx(\))p Fo(g)26 b(\024)j Fw(C)7 b Fx(\()p Fw(\021)t Fx(\))p Fw(;)1458 5216 y(C)g Fx(\()p Fw(\021)t Fx(\))27 b Fo(!)g Fx(0)p Fw(;)114 b Fx(as)66 b Fw(\021)31 b Fo(!)c Fx(0)p Fw(;)p eop %%Page: 7 7 7 6 bop 1129 251 a Fn(THE)34 b(AMOUNT)f(OF)g(DISCRETE)g(SPECTR)n(UM)669 b(7)456 450 y Fx(w)m(e)50 b(come)f(to)g(the)h(statemen)m(t)g(of)f(Prop) s(osition)f(2.1)h(with)g Fw(\026)p Fx(\()p Fw(\021)t Fx(\))55 b(=)h Fw(C)7 b Fx(\()p Fw(\021)t Fx(\))34 b(+)456 566 y Fw(C)526 581 y Fv(1)565 566 y Fx(\()p Fw(\021)t(;)17 b(\025)756 581 y Fv(2)817 566 y Fx(+)22 b Fo(jj)p Fw(p)p Fo(jj)1076 581 y Fm(1)1149 566 y Fx(\))p Fw(:)65 b Fe(2)1322 818 y Fx(3.)49 b Fp(Pr)n(oof)38 b(of)g(Theorem)g(1.1)555 992 y Fy(1.)80 b Fx(In)45 b(this)f(section)h(w)m(e)h(pro)m(v)m(e)f(the) h(relation)c(\(6\).)80 b(Necessary)46 b(auxiliary)456 1109 y(statemen)m(ts)37 b(for)e(this)h(are)h(giv)m(en)f(in)f (Subsections)j(1)e(and)g(2.)54 b(In)37 b(Subsection)g(3)456 1225 y(the)d(pro)s(of)e(of)i(Theorem)f(1.1)h(is)f(completed.)46 b(Here)34 b(w)m(e)h(follo)m(w)d(the)i(w)m(a)m(y)h(of)e(the)456 1341 y(article)e([2])h(but)h(additionally)c(w)m(e)34 b(use)f(Prop)s(osition)e(2.1.)555 1457 y(F)-8 b(or)24 b Fw(\025)j Fo(2)h Fx(\003)c(and)h Fw(\013)j(>)g Fx(0)c(w)m(e)h (de\014ne)h(a)e(quan)m(tit)m(y)h Fw(N)2383 1472 y Fv(0)2422 1457 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))24 b(as)g(a)g(n)m(um)m(b)s(er)h(of)456 1574 y(eigen)m(v)-5 b(alues)27 b(of)g(the)i(op)s(erator)e Fw(A)p Fx(\()p Fw(t)p Fx(\))h(whic)m(h)g(cross)g(the)h(p)s(oin)m(t)d Fw(\025)i Fx(with)f Fw(t)h Fx(gro)m(wing)456 1690 y(from)j(zero)i(to)f Fw(\013)h Fx(\(excluding)f Fw(\013)q Fx(\).)44 b(It)32 b(is)g(clear)g(that)h(for)f Fw(\025)2606 1705 y Fv(1)2645 1690 y Fw(;)17 b(\025)2746 1705 y Fv(2)2813 1690 y Fo(2)28 b Fx(\003)p Fw(;)17 b(\025)3076 1705 y Fv(1)3142 1690 y Fw(<)28 b(\025)3303 1705 y Fv(2)903 1866 y Fx(rank)q Fw(E)1168 1881 y Fl(A)p Fv(\()p Fl(\013)p Fv(\))1325 1866 y Fx([)p Fw(\025)1409 1881 y Fv(1)1448 1866 y Fw(;)17 b(\025)1549 1881 y Fv(2)1588 1866 y Fx(\))28 b(=)g Fw(N)1836 1881 y Fv(0)1875 1866 y Fx(\()p Fw(\025)1970 1881 y Fv(2)2009 1866 y Fw(;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))22 b Fo(\000)h Fw(N)2503 1881 y Fv(0)2542 1866 y Fx(\()p Fw(\025)2637 1881 y Fv(1)2676 1866 y Fw(;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))p Fw(:)456 2042 y Fx(Therefore)27 b(it)f(is)g(enough)h(to)g (obtain)e(the)j(asymptotics)e(of)g(the)h(righ)m(t-hand)f(side,)456 2158 y(as)c Fw(\013)28 b Fo(!)g(1)p Fw(:)22 b Fx(The)h(follo)m(wing)d (statemen)m(t)i(pla)m(ys)h(an)f(imp)s(ortan)m(t)f(role)g(in)g(the)i (pro)s(of)456 2274 y(of)32 b(Theorem)h(1.1.)456 2575 y Fy(Theorem)26 b(3.1.)35 b Fj(L)-5 b(et)27 b Fw(\025)g Fo(2)h Fx(\003)e Fj(and)g Fw(")h(>)h Fx(0)p Fw(:)e Fj(Assume)g(that)h Fw(\025)s Fo(\000)s Fw(";)17 b(\025)s Fx(+)s Fw(")28 b Fo(2)g Fx(\003)p Fw(:)e Fj(Then)456 2692 y(ther)-5 b(e)39 b(exist)g(p)-5 b(ositive)38 b(numb)-5 b(ers)39 b Fw(r)1725 2707 y Fv(0)1764 2692 y Fw(;)17 b(\013)1870 2707 y Fv(0)1948 2692 y Fj(and)39 b Fw(c)g Fj(such)g(that)g(for)g Fw(r)g(>)c(r)3050 2707 y Fv(0)3090 2692 y Fw(;)17 b(\013)35 b(>)h(\013)3405 2707 y Fv(0)456 2808 y Fj(the)e(fol)5 b(lowing)34 b(ine)-5 b(qualities)34 b(ar)-5 b(e)35 b(ful\014l)5 b(le)-5 b(d:)900 2986 y Fw(N)978 3001 y Fv(0)1018 2986 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))26 b Fo(\024)j Fw(N)10 b Fx(\(\()p Fw(\025)22 b Fx(+)g Fw(")p Fx(\))g(+)g(0)p Fw(;)17 b(A)2250 2945 y Fv(\()p Fm(N)10 b Fv(\))2376 2986 y Fx(\()p Fw(\013)q(;)17 b(\013)2584 2945 y Fv(1)p Fl(=s)2690 2986 y Fw(B)2764 3001 y Fl(r)2802 2986 y Fx(\)\))22 b Fo(\000)1236 3153 y Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(";)17 b(A)1703 3112 y Fv(\()p Fm(N)10 b Fv(\))1829 3153 y Fx(\(0)p Fw(;)17 b(\013)2023 3112 y Fv(1)p Fl(=s)2129 3153 y Fw(B)2203 3168 y Fl(r)2241 3153 y Fx(\)\))23 b(+)f Fw(cr)2527 3112 y Fl(d)p Fm(\000)p Fv(1)2657 3153 y Fw(\013)2720 3112 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=s)2973 3153 y Fw(;)3271 3069 y Fx(\(10\))1027 3411 y Fw(N)1105 3426 y Fv(0)1145 3411 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))26 b Fo(\025)j Fw(N)10 b Fx(\()p Fw(\025)22 b Fo(\000)h Fw(";)17 b(A)2134 3370 y Fv(\()p Fm(D)r Fv(\))2249 3411 y Fx(\()p Fw(\013)q(;)g(\013)2457 3370 y Fv(1)p Fl(=s)2563 3411 y Fw(B)2637 3426 y Fl(r)2675 3411 y Fx(\)\))22 b Fo(\000)1117 3578 y Fw(N)10 b Fx(\()p Fw(\025)23 b Fo(\000)f Fw(";)17 b(A)1585 3537 y Fv(\()p Fm(D)r Fv(\))1700 3578 y Fx(\(0)p Fw(;)g(\013)1894 3537 y Fv(1)p Fl(=s)2001 3578 y Fw(B)2075 3593 y Fl(r)2113 3578 y Fx(\)\))22 b Fo(\000)g Fw(cr)2399 3537 y Fl(d)p Fm(\000)p Fv(1)2530 3578 y Fw(\013)2593 3537 y Fv(\()p Fl(d)p Fm(\000)p Fv(1\))p Fl(=s)2846 3578 y Fw(;)3271 3494 y Fx(\(11\))456 3755 y Fj(wher)-5 b(e)34 b Fw(c)h Fj(do)-5 b(es)34 b(not)h(dep)-5 b(end)34 b(on)g Fw(r)k Fj(and)c Fw(\013)q(:)555 3940 y Fx(W)-8 b(e)33 b(p)s(ostp)s(one)g(the)g (pro)s(of)f(of)g(Theorem)g(3.1)h(un)m(til)e(Subsection)i(4.)43 b(Actually)456 4056 y(all)32 b(the)i(ideas)h(of)e(the)i(pro)s(of)e(are) i(giv)m(en)f(in)f([2].)49 b(In)34 b Fo(x)p Fx(4)h(w)m(e)g(repro)s(duce) g(them)f(in)456 4172 y(the)f(exact)g(form.)555 4288 y(The)i(follo)m (wing)30 b(statemen)m(t)k(giv)m(es)g(an)g(asymptotics)f(of)g(the)h(sp)s (ectral)f(distri-)456 4405 y(bution)c(function)h(of)g(the)h(op)s (erator)f(on)g(the)h(increasing)f(with)g(the)h(gro)m(wth)g(of)f Fw(\013)456 4521 y Fx(spherical)i(la)m(y)m(er.)456 4822 y Fy(Prop)s(osition)k(3.1.)42 b Fj(L)-5 b(et)35 b Fx(0)29 b Fw(<)f(\016)1671 4837 y Fv(1)1739 4822 y Fw(<)h(\016)1887 4837 y Fv(2)1955 4822 y Fw(<)f Fo(1)p Fj(,)35 b Fx(\012)29 b(=)g Fo(f)p Fw(x)f Fo(2)h Fh(R)2722 4786 y Fl(d)2797 4822 y Fx(:)g Fw(\016)2896 4837 y Fv(1)2964 4822 y Fw(<)f Fo(j)p Fw(x)p Fo(j)h Fw(<)f(\016)3355 4837 y Fv(2)3395 4822 y Fo(g)456 4938 y Fj(and)34 b Fw(\025)27 b Fo(2)i Fh(R)5 b Fw(:)40 b Fj(Then)639 5127 y Fx(lim)614 5177 y Fl(\013)p Fm(!1)817 5127 y Fw(\013)880 5086 y Fm(\000)p Fl(d=s)1043 5127 y Fw(N)10 b Fx(\()p Fw(\025;)17 b(A)1343 5086 y Fv(\()p Fl(")p Fv(\))1434 5127 y Fx(\()p Fw(\013)q(;)g(\013)1642 5086 y Fv(1)p Fl(=s)1749 5127 y Fx(\012\)\))28 b(=)2026 5010 y Fk(Z)2072 5199 y Fl(\016)2103 5208 y Fd(1)2138 5199 y Fl(<)p Fm(j)p Fl(x)p Fm(j)p Fl(<\016)2359 5208 y Fd(2)2414 5127 y Fw(\032)p Fx(\()p Fw(\025)2559 5142 y Fv(2)2620 5127 y Fx(+)22 b(\010\()p Fw(\022)s Fx(\))p Fo(j)p Fw(x)p Fo(j)3023 5086 y Fm(\000)p Fl(s)3115 5127 y Fx(\))p Fw(dx:)p eop %%Page: 8 8 8 7 bop 456 251 a Fn(8)1184 b(O.)26 b(SAFR)n(ONO)n(V)555 450 y Fx(The)31 b Fj(pr)-5 b(o)g(of)30 b Fx(of)f(this)h(statemen)m(t)h (is)e(similar)e(to)j(the)g(pro)s(of)g(of)f(Prop)s(osition)g(3.3)456 566 y(in)g([4],)i(where)h(the)f(case)h(of)e(spherically)g(symmetric)g Fw(V)52 b Fx(is)30 b(considered.)44 b(In)31 b(fact)456 683 y(all)c(the)k(ideas)f(of)f(this)h(statemen)m(t)g(are)g(giv)m(en)g (in)g(Theorem)g(2.1)f(of)h([1].)42 b(See)31 b(also)456 799 y(the)i(\014rst)g(remark)f(after)g(Theorem)h(2.1)f([1].)555 915 y(Let)43 b(\012)j Fo(\032)g Fh(R)1044 879 y Fl(d)1134 915 y Fx(b)s(e)d(a)f(domain)f(with)i(a)g(piecewise)g(smo)s(oth)f(b)s (oundary)h(and)456 1031 y Fw(\025)513 1046 y Fl(j)577 1031 y Fo(2)28 b Fh(R)5 b Fw(;)17 b(j)39 b Fx(=)28 b(1)p Fw(;)17 b Fx(2)p Fw(:)31 b Fx(W)-8 b(e)33 b(put)456 1228 y Fw(n)p Fx(\()p Fw(\025)609 1243 y Fv(1)648 1228 y Fw(;)17 b(\025)749 1243 y Fv(2)788 1228 y Fw(;)g Fx(\012)p Fw(;)g(\013)q Fx(\))27 b(=)h Fw(N)10 b Fx(\()p Fw(\025)1361 1243 y Fv(2)1401 1228 y Fw(;)17 b(A)1518 1187 y Fv(\()p Fm(N)10 b Fv(\))1644 1228 y Fx(\()p Fw(\013)q(;)17 b(\013)1852 1187 y Fv(1)p Fl(=s)1958 1228 y Fx(\012\)\))t Fo(\000)t Fw(N)10 b Fx(\()p Fw(\025)2372 1243 y Fv(1)2414 1228 y Fw(;)17 b(A)2531 1187 y Fv(\()p Fm(D)r Fv(\))2646 1228 y Fx(\()p Fw(\013)q(;)g(\013)2854 1187 y Fv(1)p Fl(=s)2961 1228 y Fx(\012\)\))p Fw(;)41 b(\013)28 b(>)g Fx(0)p Fw(:)456 1416 y Fx(Ob)m(viously)k(if)p 1004 1338 71 4 v 32 w(\012)c(=)p 1205 1338 331 4 v 27 w(\012)1275 1431 y Fv(1)1337 1416 y Fo([)23 b Fx(\012)1496 1431 y Fv(2)1536 1416 y Fx(,)32 b(then)h(the)g(follo)m(wing)d(inequalities)h(are)i(ful\014lled)858 1602 y Fw(n)p Fx(\()p Fw(\025)1011 1617 y Fv(1)1050 1602 y Fw(;)17 b(\025)1151 1617 y Fv(2)1191 1602 y Fw(;)g Fx(\012)p Fw(;)g(\013)q Fx(\))27 b Fo(\025)h(\000)p Fw(n)p Fx(\()p Fw(\025)1812 1617 y Fv(2)1852 1602 y Fw(;)17 b(\025)1953 1617 y Fv(1)1992 1602 y Fw(;)g Fx(\012)2106 1617 y Fv(1)2145 1602 y Fw(;)g(\013)q Fx(\))22 b Fo(\000)h Fw(n)p Fx(\()p Fw(\025)2565 1617 y Fv(2)2604 1602 y Fw(;)17 b(\025)2705 1617 y Fv(1)2744 1602 y Fw(;)g Fx(\012)2858 1617 y Fv(2)2898 1602 y Fw(;)g(\013)q Fx(\))228 b(\(12\))884 1864 y Fw(n)p Fx(\()p Fw(\025)1037 1879 y Fv(1)1076 1864 y Fw(;)17 b(\025)1177 1879 y Fv(2)1217 1864 y Fw(;)g Fx(\012)p Fw(;)g(\013)q Fx(\))27 b Fo(\024)h Fw(n)p Fx(\()p Fw(\025)1761 1879 y Fv(1)1800 1864 y Fw(;)17 b(\025)1901 1879 y Fv(2)1940 1864 y Fw(;)g Fx(\012)2054 1879 y Fv(1)2094 1864 y Fw(;)g(\013)q Fx(\))k(+)h Fw(n)p Fx(\()p Fw(\025)2511 1879 y Fv(1)2551 1864 y Fw(;)17 b(\025)2652 1879 y Fv(2)2691 1864 y Fw(;)g Fx(\012)2805 1879 y Fv(2)2845 1864 y Fw(;)g(\013)q Fx(\))p Fw(:)254 b Fx(\(13\))456 2172 y Fy(Prop)s(osition)35 b(3.2.)42 b Fj(L)-5 b(et)35 b Fw(r)c(>)c Fx(0)35 b Fj(and)f Fw(\025)1953 2187 y Fv(1)2020 2172 y Fw(<)28 b(\025)2181 2187 y Fv(2)2220 2172 y Fw(:)35 b Fj(Then)1797 2360 y Fx(lim)1771 2410 y Fl(\013)p Fm(!1)1974 2360 y Fw(\013)2037 2319 y Fm(\000)p Fl(d=s)2200 2360 y Fw(n)p Fx(\()p Fw(\025)2353 2375 y Fv(1)2393 2360 y Fw(;)17 b(\025)2494 2375 y Fv(2)2533 2360 y Fw(;)g(B)2651 2375 y Fl(r)2688 2360 y Fw(;)g(\013)q Fx(\))27 b(=)936 2456 y Fk(Z)982 2644 y Fm(j)p Fl(x)p Fm(j)p Fl()h Fx(0)p Fw(:)e Fj(Assume)h(that)g Fw(\025)3186 581 y Fl(j)3225 566 y Fx(+)s Fw(")h Fo(2)456 683 y Fx(\003)p Fw(;)17 b(\025)625 698 y Fl(j)683 683 y Fo(\000)22 b Fw(")28 b Fo(2)g Fx(\003)p Fw(;)17 b(j)33 b Fx(=)27 b(1)p Fw(;)17 b Fx(2)p Fw(:)34 b Fj(Then)g(ther)-5 b(e)35 b(exists)g Fw(r)30 b(>)e Fx(0)34 b Fj(such)h(that)994 868 y Fx(lim)17 b(sup)1293 892 y Fl(\013)p Fm(!1)1484 868 y Fw(\013)1547 827 y Fm(\000)p Fl(d=s)1710 868 y Fx(\()p Fw(N)1826 883 y Fv(0)1865 868 y Fx(\()p Fw(\025)1960 883 y Fv(2)2000 868 y Fw(;)g(A)p Fx(\()p Fw(\013)q Fx(\)\))k Fo(\000)i Fw(N)2493 883 y Fv(0)2532 868 y Fx(\()p Fw(\025)2627 883 y Fv(1)2667 868 y Fw(;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\)\))26 b Fo(\024)769 945 y Fk(Z)816 1133 y Fm(j)p Fl(x)p Fm(j)p Fl()27 b(b;)50 b Fx([)p Fw(a)2119 1626 y Fm(0)2143 1667 y Fw(;)17 b(b)2228 1626 y Fm(0)2251 1667 y Fx(])28 b Fo(\032)g Fx(\003)p Fw(:)765 b Fx(\(25\))456 1848 y(Let)36 b(the)h(op)s(erators)f Fw(A)1314 1812 y Fv(\()p Fl(")p Fv(\))1406 1848 y Fx(\()p Fw(\013)q(;)17 b Fx(\012\))36 b(b)s(e)h(the)g(same)f(as)h(in)f(the)h(section)f(1.)55 b(W)-8 b(e)37 b(put)456 1964 y(\005)529 1979 y Fl(r)594 1964 y Fx(=)28 b Fw(E)770 1987 y Fl(A)823 1967 y Fd(\()p Fc(D)r Fd(\))925 1987 y Fv(\(0)p Fl(;B)1060 1995 y Ff(r)1095 1987 y Fv(\))1126 1964 y Fx(\()p Fw(a)1215 1928 y Fm(0)1239 1964 y Fw(;)17 b(b)1324 1928 y Fm(0)1347 1964 y Fx(\))p Fw(;)g(r)30 b(>)e Fx(0)p Fw(:)i Fx(Also)f(let)h(the)g(function)g Fw( )h Fo(2)d Fw(C)2879 1928 y Fm(1)2954 1964 y Fx(\()p Fh(R)3058 1928 y Fl(d)3104 1964 y Fx(\))i(satisfy)456 2081 y(the)j(follo)m(wing)c(conditions)1000 2331 y(0)f Fo(\024)g Fw( )t Fx(\()p Fw(x)p Fx(\))g Fo(\024)g Fx(1)p Fw(;)114 b( )t Fx(\()p Fw(x)p Fx(\))28 b(=)2032 2157 y Fk(8)2032 2231 y(<)2032 2381 y(:)2148 2271 y Fx(0)64 b(for)h Fo(j)p Fw(x)p Fo(j)27 b Fw(<)h Fx(1)p Fw(=)p Fx(2)p Fw(;)2148 2387 y Fx(1)64 b(for)h Fo(j)p Fw(x)p Fo(j)27 b Fw(>)h Fx(3)p Fw(=)p Fx(4)p Fw(:)456 2586 y Fx(W)-8 b(e)34 b(denote)h(b)m(y)f(\011)1153 2601 y Fl(r)1225 2586 y Fx(the)h(m)m(ultiplication)29 b(op)s(erator)k(b)m(y)i(the)f (function)g Fw( )3169 2601 y Fl(r)3207 2586 y Fx(\()p Fw(x)p Fx(\))d(=)456 2703 y Fw( )t Fx(\()p Fw(x=r)s Fx(\))p Fw(;)17 b(r)29 b(>)f Fx(0)p Fw(;)k Fx(on)h Fw(L)1281 2666 y Fv(2)1321 2703 y Fx(\()p Fh(R)1425 2666 y Fl(d)1471 2703 y Fx(\))f(and)h(w)m(e)h(put)1063 2850 y(~)1031 2875 y Fw(A)1104 2890 y Fl(r)1170 2875 y Fx(=)28 b Fw(A)1347 2834 y Fv(\()p Fm(D)r Fv(\))1462 2875 y Fx(\(0)p Fw(;)17 b(B)1667 2890 y Fl(r)1705 2875 y Fx(\))22 b(+)g(\()p Fw(b)1942 2834 y Fm(0)1988 2875 y Fo(\000)g Fw(a)2138 2834 y Fm(0)2162 2875 y Fx(\)\011)2276 2890 y Fl(r)2314 2875 y Fx(\005)2387 2890 y Fl(r)2425 2875 y Fx(\011)2501 2890 y Fl(r)2539 2875 y Fw(;)49 b(r)30 b(>)e Fx(0)p Fw(:)402 b Fx(\(26\))456 3061 y(In)33 b([1)o(])g(the)g(follo)m(wing)d(statemen)m (t)j(concerning)2265 3036 y(~)2234 3061 y Fw(A)2307 3076 y Fl(r)2377 3061 y Fx(is)f(pro)m(v)m(ed.)456 3356 y Fy(Prop)s(osition) 37 b(4.1.)43 b Fj(L)-5 b(et)37 b(the)g(c)-5 b(onditions)43 b Fx(\(24\))p Fj(,)37 b Fx(\(25\))e Fj(b)-5 b(e)37 b(ful\014l)5 b(le)-5 b(d)36 b(and)g(let)3364 3331 y Fx(~)3333 3356 y Fw(A)3406 3371 y Fl(r)456 3472 y Fj(b)-5 b(e)34 b(de\014ne)-5 b(d)34 b(as)g(in)42 b Fx(\(26\))p Fj(.)j(Then)34 b(ther)-5 b(e)34 b(exists)52 b Fx(~)-66 b Fw(r)2211 3487 y Fv(0)2278 3472 y Fw(>)28 b Fx(0)34 b Fj(such)h(that)1441 3645 y Fx([)p Fw(a;)17 b(b)p Fx(])28 b Fo(\032)g Fw(\032)p Fx(\()1884 3619 y(~)1852 3645 y Fw(A)1925 3660 y Fl(r)1964 3645 y Fx(\))p Fw(;)86 b Fo(8)p Fw(r)31 b(>)45 b Fx(~)-67 b Fw(r)2392 3660 y Fv(0)2432 3645 y Fw(:)555 3823 y Fx(W)-8 b(e)33 b(put)1598 3965 y(~)1567 3990 y Fw(A)1640 4005 y Fl(r)1678 3990 y Fx(\()p Fw(\013)q Fx(\))27 b(=)1979 3965 y(~)1948 3990 y Fw(A)2021 4005 y Fl(r)2081 3990 y Fo(\000)c Fw(\013)q(V)5 b(:)938 b Fx(\(27\))456 4160 y(The)27 b(next)g(statemen)m(t)g(allo)m(ws)e(to)h(appro)m(ximate)f(the) i(quan)m(tit)m(y)g Fw(N)2888 4175 y Fv(0)2927 4160 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))25 b(b)m(y)456 4276 y(similar)31 b(quan)m(tities)j(for)f(op)s(erators)h (on)g(a)g(ball)f Fw(B)2299 4291 y Fl(r)2337 4276 y Fw(;)17 b(r)32 b(>)f Fx(0)p Fw(:)j Fx(W)-8 b(e)34 b(refer)h(to)f([2])g(for)456 4392 y(its)e(pro)s(of.)456 4687 y Fy(Prop)s(osition)52 b(4.2.)d Fj(L)-5 b(et)49 b Fw(\025)j Fo(2)h Fx(\()p Fw(a;)17 b(b)p Fx(\))49 b Fj(and)e(the)i(c)-5 b(onditions)55 b Fx(\(24\))o Fj(,)d Fx(\(25\))c Fj(ar)-5 b(e)456 4811 y(ful\014l)5 b(le)-5 b(d.)43 b(Assume)34 b(that)h(the)f(op)-5 b(er)g(ator)1972 4786 y Fx(~)1941 4811 y Fw(A)2014 4826 y Fl(r)2086 4811 y Fj(in)41 b Fx(\(27\))34 b Fj(is)f(de\014ne)-5 b(d)33 b(in)41 b Fx(\(26\))p Fj(.)j(Then)904 4984 y Fw(N)982 4999 y Fv(0)1021 4984 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))27 b Fo(\025)h Fx(lim)17 b(sup)1604 5053 y Fl(r)r Fm(!1)1841 4984 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025;)2137 4958 y Fx(~)2106 4984 y Fw(A)2179 4999 y Fl(r)2217 4984 y Fx(\()p Fw(\013)2318 4943 y Fm(0)2341 4984 y Fx(\)\))22 b Fo(\000)h Fw(N)10 b Fx(\()p Fw(\025;)2797 4958 y Fx(~)2766 4984 y Fw(A)2839 4999 y Fl(r)2877 4984 y Fx(\)\))p Fw(;)2509 5177 y Fx(0)28 b Fw(<)f(\013)2752 5135 y Fm(0)2803 5177 y Fw(<)h(\013)q Fx(;)3271 5079 y(\(28\))p eop %%Page: 11 11 11 10 bop 1129 251 a Fn(THE)34 b(AMOUNT)f(OF)g(DISCRETE)g(SPECTR)n(UM) 631 b(11)918 458 y Fw(N)996 473 y Fv(0)1036 458 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))26 b Fo(\024)j Fx(lim)17 b(inf)1604 508 y Fl(r)r Fm(!1)1827 458 y Fx(\()p Fw(N)10 b Fx(\()p Fw(\025;)2123 433 y Fx(~)2092 458 y Fw(A)2165 473 y Fl(r)2203 458 y Fx(\()p Fw(\013)2304 417 y Fm(0)2327 458 y Fx(\)\))22 b Fo(\000)g Fw(N)10 b Fx(\()p Fw(\025;)2783 433 y Fx(~)2751 458 y Fw(A)2824 473 y Fl(r)2862 458 y Fx(\)\))p Fw(;)2495 632 y Fx(0)27 b Fw(<)h(\013)g(<)g(\013)2932 591 y Fm(0)2955 632 y Fw(:)3271 544 y Fx(\(29\))555 828 y(W)-8 b(e)36 b(in)m(tro)s(duce)f(the)h (selfadjoin)m(t)e(op)s(erators)h Fw(A)2312 780 y Fl(")2345 789 y Fd(1)2379 780 y Fl(;")2432 789 y Fd(2)2312 852 y Fl(R;r)2502 828 y Fx(=)d Fo(\000)p Fx(\001)25 b(+)f Fw(p)35 b Fx(on)g Fw(L)3181 791 y Fv(2)3221 828 y Fx(\()p Fw(B)3333 843 y Fl(r)3395 828 y Fo(n)456 948 y Fw(B)530 963 y Fl(R)587 948 y Fx(\))p Fw(;)54 b(r)39 b(>)c(R)i(>)f Fx(0,)i(where)h(the)f(indexes)g Fw(")2092 963 y Fv(1)2169 948 y Fx(and)f Fw(")2409 963 y Fv(2)2448 948 y Fx(,)i(whic)m(h)f(tak)m (e)g(the)f(v)-5 b(alues)456 1064 y Fo(D)36 b Fx(or)e Fo(N)15 b Fx(,)34 b(c)m(haracterize)h(b)s(oundary)f(conditions)g (\(Diric)m(hlet)d(or)j(Neumann\))g(on)456 1189 y Fw(@)5 b(B)586 1204 y Fl(R)690 1189 y Fx(and)45 b Fw(@)5 b(B)1022 1204 y Fl(r)1106 1189 y Fx(corresp)s(ondingly)-8 b(.)81 b(W)-8 b(e)45 b(put)h(also)2485 1163 y(~)2459 1189 y Fw(A)2532 1141 y Fl(")2565 1150 y Fd(1)2600 1141 y Fl(;")2653 1150 y Fd(2)2532 1213 y Fl(R;r)2740 1189 y Fx(=)j Fw(A)2938 1141 y Fl(")2971 1150 y Fd(1)3005 1141 y Fl(;")3058 1150 y Fd(2)2938 1213 y Fl(R;r)3127 1189 y Fx(+)30 b(\()p Fw(b)3312 1152 y Fm(0)3367 1189 y Fo(\000)456 1309 y Fw(a)507 1273 y Fm(0)530 1309 y Fx(\)\011)644 1324 y Fl(r)682 1309 y Fx(\005)755 1324 y Fl(r)793 1309 y Fx(\011)869 1324 y Fl(r)907 1309 y Fw(;)46 b(r)30 b(>)d(R)i(>)f Fx(0)p Fw(:)h Fx(The)h(next)g(statemen)m(t)g(is)e(con)m(tained)i(in)e(Lemma)g (3.2)456 1425 y(and)k(Lemma)f(4.2)i(of)f([2].)456 1727 y Fy(Prop)s(osition)g(4.3.)40 b Fj(L)-5 b(et)50 b Fx(~)-66 b Fw(r)1482 1742 y Fv(0)1554 1727 y Fj(b)-5 b(e)32 b(as)g(in)g(Pr)-5 b(op)g(osition)32 b(4.1)g(and)g Fw(\025)27 b Fo(2)h Fx(\()p Fw(a;)17 b(b)p Fx(\))p Fw(:)33 b Fj(Then)456 1843 y(for)h Fw(r)d(>)44 b Fx(~)-66 b Fw(r)833 1858 y Fv(0)873 1843 y Fw(;)17 b Fx(1)27 b Fo(\024)h Fw(R)h Fo(\024)f Fw(r)s(=)p Fx(2)34 b Fj(the)h(fol)5 b(lowing)33 b(ine)-5 b(qualities)34 b(hold)781 2029 y Fw(N)10 b Fx(\()p Fw(\025;)17 b(A)1081 1988 y Fv(\()p Fm(D)r Fv(\))1197 2029 y Fx(\(0)p Fw(;)g(B)1402 2044 y Fl(R)1459 2029 y Fx(\)\))22 b(+)g Fw(N)10 b Fx(\()p Fw(\025;)1908 2004 y Fx(~)1882 2029 y Fw(A)1955 1982 y Fm(D)r Fl(;)p Fm(D)1955 2054 y Fl(R;r)2092 2029 y Fx(\))28 b Fo(\025)g Fw(N)10 b Fx(\()p Fw(\025;)2515 2004 y Fx(~)2490 2029 y Fw(A)2563 2044 y Fl(r)2601 2029 y Fx(\))22 b Fo(\000)h Fw(C)2831 2044 y Fv(1)2870 2029 y Fw(R)2945 1988 y Fl(d)p Fm(\000)p Fv(1)3075 2029 y Fw(;)2637 2180 y(C)2707 2195 y Fv(1)2774 2180 y Fx(=)k Fw(C)2947 2195 y Fv(1)2986 2180 y Fx(\()p Fw(\025)p Fx(\))3271 2219 y(\(30\))771 2480 y Fw(N)10 b Fx(\()p Fw(\025;)17 b(A)1071 2439 y Fv(\()p Fm(N)10 b Fv(\))1197 2480 y Fx(\(0)p Fw(;)17 b(B)1402 2495 y Fl(R)1460 2480 y Fx(\)\))22 b(+)g Fw(N)10 b Fx(\()p Fw(\025;)1908 2455 y Fx(~)1883 2480 y Fw(A)1956 2433 y Fm(N)g Fl(;)p Fm(D)1956 2504 y Fl(R;r)2104 2480 y Fx(\))27 b Fo(\024)i Fw(N)10 b Fx(\()p Fw(\025;)2527 2455 y Fx(~)2502 2480 y Fw(A)2575 2495 y Fl(r)2613 2480 y Fx(\))22 b(+)g Fw(C)2841 2495 y Fv(2)2880 2480 y Fw(R)2955 2439 y Fl(d)p Fm(\000)p Fv(1)3086 2480 y Fw(;)2620 2631 y(C)2690 2646 y Fv(2)2757 2631 y Fx(=)27 b Fw(C)2930 2646 y Fv(2)2970 2631 y Fx(\()p Fw(\025)p Fx(\))p Fw(;)3271 2670 y Fx(\(31\))456 2847 y Fj(wher)-5 b(e)34 b Fw(C)801 2862 y Fv(1)875 2847 y Fj(and)g Fw(C)1134 2862 y Fv(2)1208 2847 y Fj(do)h(not)g(dep)-5 b(end)33 b(on)i Fw(r)i Fj(and)e Fw(R)q(:)555 3032 y Fy(2.)43 b Fx(No)m(w)31 b(let)f(us)i(complete)e (the)h(pro)s(of)e(of)i(Theorem)f(3.1.)43 b(First)30 b(of)g(all)e(w)m(e) k(es-)456 3148 y(tablish)26 b(the)h(relation)f(\(10\))o(.)42 b(Denote)27 b(b)m(y)h Fw(v)2034 3163 y Fv(+)2121 3148 y Fx(the)f(quan)m(tit)m(y)h(sup)2815 3172 y Fm(j)p Fl(x)p Fm(j)p Fl(>)p Fv(1)3005 3148 y Fo(j)p Fw(x)p Fo(j)3116 3112 y Fm(\000)p Fl(s)3207 3148 y Fw(V)22 b Fx(\()p Fw(x)p Fx(\).)456 3283 y(Let)37 b Fw(\013)698 3247 y Fm(0)755 3283 y Fw(>)e(\013)i Fx(and)g Fw(r)1203 3298 y Fv(0)1277 3283 y Fx(=)e(\()p Fw(v)1473 3298 y Fv(+)1532 3283 y Fw(=")p Fx(\))1665 3247 y Fv(1)p Fl(=s)1772 3283 y Fw(:)h Fx(W)-8 b(e)38 b(put)f Fw(R)e Fx(=)g Fw(m\013)2559 3247 y Fv(1)p Fl(=s)2666 3283 y Fw(;)17 b(m)35 b(>)g(r)2985 3298 y Fv(0)3024 3283 y Fw(:)i Fx(W)-8 b(e)37 b(note)456 3399 y(that)30 b Fw(\013)q(V)21 b Fx(\()p Fw(x)p Fx(\))28 b Fw(<)f(")j Fx(for)g Fo(j)p Fw(x)p Fo(j)d Fw(>)h(R)q(:)i Fx(Therefore)h(from)e(the)i(v)-5 b(ariational)27 b(principle)h(it)456 3515 y(follo)m(ws)j(that)1134 3675 y Fw(N)10 b Fx(\()p Fw(\025;)1387 3650 y Fx(~)1361 3675 y Fw(A)1434 3690 y Fl(r)1472 3675 y Fx(\()p Fw(\013)1573 3634 y Fm(0)1596 3675 y Fx(\)\))27 b Fo(\024)i Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(";)17 b(A)p Fx(\()p Fw(\013)2372 3634 y Fm(0)2395 3675 y Fw(;)g(B)2513 3690 y Fl(R)2570 3675 y Fx(\)\))22 b(+)1753 3844 y Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(";)2171 3819 y Fx(~)2146 3844 y Fw(A)2219 3797 y Fm(N)10 b Fl(;)p Fm(D)2219 3869 y Fl(R;r)2366 3844 y Fx(\))p Fw(;)82 b(R)29 b(<)e(r)3271 3759 y Fx(\(32\))456 4027 y(Since)32 b Fw(\025)22 b Fx(+)g Fw(")28 b Fo(2)g Fx(\()p Fw(a;)17 b(b)p Fx(\),)33 b(w)m(e)g(obtain)f(from)f(\(31\))h(that)1860 4212 y Fw(N)10 b Fx(\()p Fw(\025;)2112 4186 y Fx(~)2087 4212 y Fw(A)2160 4227 y Fl(r)2198 4212 y Fx(\))27 b(=)h Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(";)2785 4186 y Fx(~)2760 4212 y Fw(A)2833 4227 y Fl(r)2871 4212 y Fx(\))27 b Fo(\025)858 4381 y Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(";)17 b(A)1325 4339 y Fv(\()p Fm(N)10 b Fv(\))1451 4381 y Fx(\(0)p Fw(;)17 b(B)1656 4396 y Fl(R)1713 4381 y Fx(\)\))22 b(+)g Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(";)2328 4355 y Fx(~)2303 4381 y Fw(A)2376 4333 y Fm(N)10 b Fl(;)p Fm(D)2376 4405 y Fl(R;r)2523 4381 y Fx(\))22 b Fo(\000)h Fw(C)2753 4396 y Fv(2)2792 4381 y Fw(R)2867 4339 y Fl(d)p Fm(\000)p Fv(1)2998 4381 y Fw(;)1484 4532 y(C)1554 4547 y Fv(2)1621 4532 y Fx(=)k Fw(C)1794 4547 y Fv(2)1833 4532 y Fx(\()p Fw(\025)22 b Fx(+)g Fw(")p Fx(\))p Fw(;)17 b(r)30 b(>)i Fx(~)-53 b Fw(r)2398 4547 y Fv(0)2437 4532 y Fw(;)17 b Fx(1)27 b Fo(\024)h Fw(R)h Fo(\024)f Fw(r)s(=)p Fx(2)p Fw(:)3271 4370 y Fx(\(33\))456 4714 y(Estimating)34 b(the)j(righ)m(t-hand)f(side)h(of)43 b(\(29\))36 b(with)h(the)g(help)g(of)43 b(\(32\))o(,)38 b(\(33\))e(w)m(e)456 4830 y(get)1278 5009 y Fw(N)1356 5024 y Fv(0)1395 5009 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(\013)q Fx(\)\))27 b Fo(\024)h Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(";)17 b(A)2383 4968 y Fv(\()p Fm(N)10 b Fv(\))2509 5009 y Fx(\()p Fw(\013)2610 4968 y Fm(0)2633 5009 y Fw(;)17 b(B)2751 5024 y Fl(R)2809 5009 y Fx(\)\))22 b Fo(\000)894 5177 y Fw(N)10 b Fx(\()p Fw(\025)22 b Fx(+)g Fw(";)17 b(A)1360 5135 y Fv(\()p Fm(N)10 b Fv(\))1486 5177 y Fx(\(0)p Fw(;)17 b(B)1691 5192 y Fl(R)1749 5177 y Fx(\)\))22 b(+)g Fw(C)2015 5192 y Fv(2)2054 5177 y Fw(R)2129 5135 y Fl(d)p Fm(\000)p Fv(1)2260 5177 y Fw(;)49 b(R)29 b Fx(=)e Fw(m\013)2690 5135 y Fv(1)p Fl(=s)2825 5177 y Fo(\025)h Fx(1)p Fw(:)3271 5092 y Fx(\(34\))p eop %%Page: 12 12 12 11 bop 456 251 a Fn(12)1146 b(O.)26 b(SAFR)n(ONO)n(V)456 450 y Fx(T)-8 b(aking)32 b Fw(\013)844 414 y Fm(0)894 450 y Fo(!)c Fw(\013)33 b Fx(w)m(e)h(come)e(to)g(the)h(upp)s(er)g (estimate)936 609 y Fw(N)1014 624 y Fv(0)1054 609 y Fx(\()p Fw(\025;)17 b(A)p Fx(\()p Fw(E)6 b(\013)q Fx(\)\))27 b Fo(\024)h Fw(N)10 b Fx(\(\()p Fw(\025)22 b Fx(+)g Fw(")p Fx(\))g(+)g(0)p Fw(;)17 b(A)2364 568 y Fv(\()p Fm(N)10 b Fv(\))2490 609 y Fx(\()p Fw(\013)q(;)17 b(B)2709 624 y Fl(R)2766 609 y Fx(\)\))22 b Fo(\000)1033 777 y Fw(N)10 b Fx(\()p Fw(\025)23 b Fx(+)f Fw(";)17 b(A)1500 736 y Fv(\()p Fm(N)10 b Fv(\))1626 777 y Fx(\(0)p Fw(;)17 b(B)1831 792 y Fl(R)1888 777 y Fx(\)\))22 b(+)g Fw(C)2154 792 y Fv(2)2193 777 y Fw(R)2268 736 y Fl(d)p Fm(\000)p Fv(1)2399 777 y Fw(;)49 b(R)29 b Fx(=)e Fw(m\013)2829 736 y Fv(1)p Fl(=s)2937 777 y Fw(;)3271 692 y Fx(\(35\))456 939 y(whic)m(h)35 b(coincides)f(with)h(\(10\))f(for)g Fw(r)g Fx(=)d Fw(m)k Fx(and)g Fw(c)c Fx(=)g Fw(C)2482 954 y Fv(2)2522 939 y Fw(:)j Fx(Moreo)m(v)m(er)j(\(35\))d(holds)456 1056 y(for)e(all)f Fw(m)e(>)f(r)1003 1071 y Fv(0)1043 1056 y Fw(;)17 b(\013)29 b(>)f(\013)1344 1071 y Fv(0)1412 1056 y Fx(=)g Fw(r)1563 1014 y Fm(\000)p Fl(s)1560 1077 y Fv(0)1655 1056 y Fw(:)33 b Fx(The)h(relation)d(\(11\))h(can)i(b)s(e)f (obtained)g(in)f(the)456 1172 y(same)g(w)m(a)m(y)i(from)d(\(28\))h(and) h(\(30\))o(.)44 b Fe(2)1660 1370 y Fp(References)497 1527 y Fu([1])d(Alama,)35 b(S.,)g(Deift,)h(P)-7 b(.A.,)36 b(Hemp)r(el,)f(R.:)49 b(Eigen)n(v)-5 b(alue)33 b(branc)n(hes)f(of)h (the)h(Sc)n(hr\177)-42 b(odinger)626 1627 y(op)r(erator)20 b Fs(H)12 b Fr(\000)6 b Fs(\025W)33 b Fu(in)21 b(a)g(gap)g(of)g Fs(\033)s Fu(\()p Fs(H)7 b Fu(\).)22 b Fb(Commun.)i(Math.)h(Phys.)e Fa(121)p Fu(,)f(291-321)c(\(1989\))497 1727 y([2])41 b(Hemp)r(el,)31 b(R.:)40 b(Eigen)n(v)-5 b(alues)27 b(in)j(gaps)e(and)h (decoupling)g(b)n(y)f(Neumann)i(b)r(oundary)e(con-)626 1826 y(ditions.)g Fb(J.)i(Math.)h(A)n(nal.)f(Appl.)f Fa(169)p Fu(,)e(no.)g(1,)g(229-259)e(\(1992\))497 1926 y([3])41 b(Ivrii,)20 b(V.:Estimations)f(p)r(our)f(le)h(nom)n(bre)f(de)h (v)-5 b(aleurs)18 b(propres)f(negativ)n(es)h(de)h(l'op)r(erateur)626 2025 y(de)k(Sc)n(hr\177)-42 b(odinger)20 b(a)n(v)n(ec)h(p)r(oten)n (tiels)h(singuliers.)g Fb(C.R.A)l(c)l(ad.Sci.Paris)28 b(Ser.I)d(Math.)f Fa(302)p Fu(,)626 2125 y(467-470)h(\(1986\))497 2225 y([4])41 b(Safrono)n(v,)d(O.:)55 b(The)36 b(discrete)h(sp)r (ectrum)g(of)f(the)i(p)r(erturb)r(ed)f(p)r(erio)r(dic)f(Sc)n(hr\177)-42 b(odinger)626 2324 y(op)r(erator)22 b(in)h(the)h(large)e(coupling)g (constan)n(t)h(limit.)h Fb(Commun.)i(Math.)h(Phys.)e Fa(218)p Fu(,)e(217-)626 2424 y(232)j(\(2001\))497 2524 y([5])41 b(Sob)r(olev,)22 b(A.:)33 b(Quasiclassical)18 b(asymptotics)i(b)r(eha)n(vior)f(of)h(the)h(scattering)e(cross)g (section)626 2623 y(for)j(asymptotically)e(homogeneous)g(p)r(oten)n (tials.)i Fb(J.)i(Soviet)h(Math.)f Fa(40)p Fu(,)e(652-672)d(\(1988\)) 497 2723 y([6])41 b(Sob)r(olev,)26 b(A.:)36 b(W)-7 b(eyl)25 b(asymptotics)g(for)g(the)g(discrete)g(sp)r(ectrum)g(of)h(the)f(p)r (erturb)r(ed)h(Hill)626 2822 y(op)r(erator.)g Fb(A)l(dv.)k(Sov.)h (Math.)e Fa(7)p Fu(,)e(159-178,)e(\(1991\))555 3009 y Ft(Oleg)47 b(Safr)n(ono)n(v,)i(Dep)-6 b(ar)g(tment)48 b(of)e(Ma)-6 b(thema)g(tics,)51 b(KTH,)46 b(10044)g(Stock-)456 3109 y(holm,)30 b(Sweden;)h(safr)n(ono)n(v@ma)-6 b(th.kth.se)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0209301216151--