Content-Type: multipart/mixed; boundary="-------------0008251206635" This is a multi-part message in MIME format. ---------------0008251206635 Content-Type: text/plain; name="00-323.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-323.keywords" Spectral gap, spectral asymptotics, eigenvalues ---------------0008251206635 Content-Type: application/postscript; name="aaj.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="aaj.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: aaj.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o aaj.ps aaj.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.08.25:1853 %%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 N/vsize 11 72 mul N/hsize 8.5 72 mul 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Fj(2)671 1471 y Fq(\()p Fg(R)775 1435 y Fh(d)836 1471 y Fl(n)14 b Fo(G)p Fq(\))28 b(with)g(Diric)m(hlet)f(b)s(oundary)i(conditions)f(\(see)h([10]\),)g (where)h Fo(G)f Fq(is)456 1588 y(op)s(en)h(and)g(coincides)f(with)h (the)g(supp)s(ort)h(of)e Fo(V)51 b Fq(up)31 b(to)e(a)h(set)g(of)g (measure)g(zero)g(.)456 1704 y(Secondly)i(for)e(asymptotics)h(of)g (order)h Fo(\013)1977 1668 y Fh(p)2016 1704 y Fo(;)48 b(p)28 b(>)f Fq(1)k(the)h(relations)e(\(1\))o(,)i(\(2\))f(w)m(ere)456 1820 y(pro)m(v)m(ed)j(in)d([16].)555 1936 y(Note)g(that)f(the)h(form)m (ulae)e(\(1\))o(,)i(\(2\))f(allo)m(w)f(one)i(to)f(apply)g(the)h (results)g(of)f([15)o(])456 2052 y(where)g(the)f(di\013erence)h Fo(N)10 b Fq(\()p Fo(\025;)17 b(\013)q Fq(\))28 b(=)f Fo(N)1868 2067 y Fj(+)1927 2052 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))e Fl(\000)g Fo(N)2475 2067 y Fk(\000)2533 2052 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))28 b(w)m(as)i(studied.)456 2169 y(In)i(particular,)e(in)i (dimension)e Fo(d)e Fl(\025)g Fq(3)k(for)f(the)i(case)g(of)e(the)i(Sc)m (hr\177)-49 b(odinger)32 b(op)s(er-)456 2285 y(ator)k(\(3\))h(p)s (erturb)s(ed)h(b)m(y)g(a)f(non-signde\014nite)g(p)s(oten)m(tial)f (\(4\))g(the)i(asymptotic)456 2401 y(form)m(ula)863 2578 y Fo(N)10 b Fq(\()p Fo(\025;)17 b(\013)q Fq(\))27 b Fl(\030)h Fq(\(2)p Fo(\031)t Fq(\))1507 2537 y Fk(\000)p Fh(d)1602 2578 y Fo(!)1663 2593 y Fh(d)1703 2578 y Fo(\013)1766 2537 y Fh(d=)p Fj(2)1894 2461 y Ff(Z)1993 2578 y Fo(V)2072 2528 y Fh(d=)p Fj(2)2050 2598 y(+)2183 2578 y Fo(dx;)82 b Fq(2)p Fo(V)2504 2593 y Fj(+)2590 2578 y Fq(=)28 b Fl(j)p Fo(V)21 b Fl(j)h Fq(+)g Fo(V)5 b(;)456 2775 y Fq(w)m(as)31 b(obtained.)43 b(In)31 b(dimensions)f Fo(d)d Fq(=)h(1)p Fo(;)17 b Fq(2)30 b(similar)d(asymptotics)k(are)g(v)-5 b(alid)28 b(un-)456 2891 y(der)e(some)g(additional)d(conditions)i(on)g Fo(V)d Fq(.)41 b(Th)m(us)28 b(\(1\))o(,)f(\(2\))f(giv)m(e)g(an)g (asymptotics)456 3008 y(for)32 b Fo(N)683 3023 y Fj(+)742 3008 y Fq(\()p Fo(\025;)17 b(\013)q Fq(\))31 b(and)i(asymptotic)f (estimates)g(for)g Fo(N)2361 3023 y Fj(0)2401 3008 y Fq(\()p Fo(\025;)17 b(\013)q Fq(\))31 b(and)i Fo(N)2940 3023 y Fk(\000)2999 3008 y Fq(\()p Fo(\025;)17 b(\013)q Fq(\).)555 3124 y(It)26 b(is)g(a)g(pleasure)g(for)f(the)i(author)f(to)f (express)k(his)c(deep)i(gratitude)e(to)h(Profes-)456 3240 y(sor)33 b(M.)g(Sh.)45 b(Birman)31 b(for)h(suggesting)h(the)g (problem.)44 b(The)33 b(author)g(is)g(grateful)456 3356 y(to)k(M.)i(Sh.)61 b(Birman)36 b(and)i(A.)h(A.)f(Laptev)h(for)f(their)f (atten)m(tion)h(to)g(the)g(w)m(ork.)456 3473 y(The)e(author)g(w)m(an)m (ts)h(to)e(thank)i(Professor)f(M.)g(Klein)f(for)g(reading)g(the)h(man)m (u-)456 3589 y(script.)1121 3818 y(1.)55 b Fp(St)-7 b(a)g(tement)38 b(of)g(the)f(main)h(resul)-7 b(t)555 3993 y Fx(1.)58 b Fq(Let)37 b Fn(H)g Fq(b)s(e)h(a)f(separable)g(Hilb)s(ert)f(space.)58 b(Belo)m(w)38 b(\()p Fl(\001)p Fo(;)17 b Fl(\001)p Fq(\))p Fo(;)g Fl(k)23 b(\001)i(k)p Fq(,)39 b Fo(I)45 b Fq(are)37 b(the)456 4109 y(scalar)27 b(pro)s(duct,)h(the)g(norm)f(and)h(the)g (iden)m(tit)m(y)f(op)s(erator)g(in)g Fn(H)p Fq(.)41 b(W)-8 b(e)28 b(denote)g(b)m(y)456 4225 y Fn(R)c Fq(the)g(space)g(of)f(con)m (tin)m(uous)i(linear)d(op)s(erators)h(and)h(b)m(y)g Fn(S)2628 4240 y Fk(1)2726 4225 y Fq(the)g(space)g(of)f(com-)456 4341 y(pact)f(op)s(erators)g(acting)g(from)f Fn(H)h Fq(to)g Fn(H)p Fq(.)40 b(F)-8 b(or)21 b(a)h(densely)i(de\014ned)f(linear)e(op)s (erator)456 4458 y Fo(M)43 b Fq(w)m(e)33 b(denote)g(its)e(domain,)g (range,)h(k)m(ernel,)h(adjoin)m(t)f(op)s(erator,)f(resolv)m(en)m(t)j (set)456 4574 y(and)44 b(sp)s(ectrum)g(b)m(y)h(the)f(sym)m(b)s(ols)g Fl(D)s Fq(\()p Fo(M)10 b Fq(\))p Fo(;)17 b Fq(Ran)o Fo(M)5 b(;)17 b Fq(Ker)g Fo(M)5 b(;)17 b(M)2846 4538 y Fk(\003)2886 4574 y Fo(;)g(\032)p Fq(\()p Fo(M)10 b Fq(\))p Fo(;)17 b(\033)t Fq(\()p Fo(M)10 b Fq(\))456 4690 y(resp)s(ectiv)m(ely)-8 b(.)61 b(If)39 b Fo(M)48 b Fq(=)38 b Fo(M)1498 4654 y Fk(\003)1538 4690 y Fo(;)g Fq(w)m(e)i(put)e(2)p Fo(M)2080 4705 y Fk(\006)2177 4690 y Fq(=)f Fl(j)p Fo(M)10 b Fl(j)27 b(\006)f Fo(M)5 b(:)39 b Fq(F)-8 b(or)37 b Fo(T)52 b Fl(2)38 b Fn(S)3221 4705 y Fk(1)3333 4690 y Fq(w)m(e)456 4806 y(denote)d(b)m(y)h Fo(s)956 4821 y Fh(k)999 4806 y Fq(\()p Fo(T)14 b Fq(\))p Fo(;)51 b(k)35 b Fl(2)e Fg(N)9 b Fq(,)41 b(the)36 b(singular)d(n)m(um)m(b)s(ers)j(of)f Fo(T)14 b Fq(,)35 b(i.e.,)h(the)f(consecu-)456 4925 y(tiv)m(e)i(eigen)m (v)-5 b(alues)37 b(of)g(the)g(op)s(erator)g(\()p Fo(T)1954 4889 y Fk(\003)1993 4925 y Fo(T)14 b Fq(\))2102 4889 y Fj(1)p Fh(=)p Fj(2)2211 4925 y Fq(.)57 b(W)-8 b(e)38 b(in)m(tro)s(duce)f(the)h(coun)m(ting)456 5042 y(function)1091 5210 y Fo(n)p Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b(=)h(card)p Fl(f)k Fo(k)f Fq(:)60 b Fo(s)1998 5225 y Fh(k)2041 5210 y Fq(\()p Fo(T)14 b Fq(\))27 b Fo(>)h(s)k Fl(g)p Fo(;)82 b(s)27 b(>)h Fq(0)p Fo(;)511 b Fq(\(5\))p eop %%Page: 4 4 4 3 bop 456 236 a Fi(4)1180 b(O.)34 b(SAFR)n(ONO)n(V)456 425 y Fq(of)e(the)h Fo(s)p Fq(-n)m(um)m(b)s(ers.)44 b(If)32 b Fo(T)41 b Fq(=)28 b Fo(T)1615 389 y Fk(\003)1682 425 y Fl(2)g Fn(S)1859 440 y Fk(1)1933 425 y Fq(\()p Fn(H)p Fq(\),)k(w)m(e)i(denote)1542 583 y Fo(n)1600 598 y Fk(\006)1660 583 y Fq(\()p Fl(\001)p Fo(;)17 b(T)d Fq(\))26 b(=)i Fo(n)p Fq(\()p Fl(\001)p Fo(;)17 b(T)2234 598 y Fk(\006)2293 583 y Fq(\))p Fo(:)962 b Fq(\(6\))456 746 y(F)-8 b(or)34 b(0)e Fo(<)g(p)g(<)g Fl(1)j Fq(w)m(e)h(consider)f(the)h(class)f (\(ideal\))f(\006)2458 761 y Fh(p)2530 746 y Fl(\032)f Fn(S)2723 761 y Fk(1)2832 746 y Fq(de\014ned)k(b)m(y)f(the)456 862 y(condition)1130 979 y(\006)1200 994 y Fh(p)1268 979 y Fq(=)27 b Fl(f)p Fo(T)42 b Fl(2)28 b Fn(S)1697 994 y Fk(1)1799 979 y Fq(:)93 b(sup)1931 1055 y Fh(s>)p Fj(0)2082 979 y Fo(s)2128 938 y Fh(p)2168 979 y Fo(n)p Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b Fo(<)g Fl(1g)p Fo(:)456 1166 y Fq(W)-8 b(e)33 b(in)m(tro)s(duce)f(the)h(follo)m(wing)d (functionals)h(on)i(\006)2339 1181 y Fh(p)2379 1166 y Fq(:)1240 1244 y Ff(8)1240 1318 y(<)1240 1468 y(:)1355 1358 y Fq(\001)1436 1373 y Fh(p)1476 1358 y Fq(\()p Fo(T)14 b Fq(\))27 b(=)h(lim)17 b(sup)2053 1382 y Fh(s)p Fk(!)p Fj(0)2212 1358 y Fo(s)2258 1322 y Fh(p)2297 1358 y Fo(n)p Fq(\()p Fo(s;)g(T)d Fq(\))p Fo(;)1388 1474 y(\016)1431 1489 y Fh(p)1471 1474 y Fq(\()p Fo(T)g Fq(\))27 b(=)h(lim)17 b(inf)2019 1489 y Fh(s)p Fk(!)p Fj(0)2178 1474 y Fo(s)2224 1438 y Fh(p)2264 1474 y Fo(n)p Fq(\()p Fo(s;)g(T)d Fq(\))p Fo(:)3320 1418 y Fq(\(7\))555 1667 y Fx(2.)51 b Fq(Let)35 b Fo(a)p Fq([)p Fl(\001)p Fo(;)17 b Fl(\001)p Fq(])34 b(b)s(e)h(a)g(semib)s(ounded)g(from)e(b)s(elo)m(w)i(closed)g (sesquilinear)g(form)456 1783 y(in)e Fn(H)p Fo(:)g Fq(W)-8 b(e)34 b(assume)g(that)g(its)f(domain)f Fo(d)p Fq([)p Fo(a)p Fq(])i(is)f(a)g(dense)i(set)g(in)e Fn(H)p Fq(.)46 b(The)35 b(form)d Fo(a)456 1899 y Fq(induces)39 b(in)e Fn(H)h Fq(the)h(self-adjoin)m(t)e(op)s(erator)g Fo(A)p Fq(.)61 b(Fix)37 b(a)h(v)-5 b(alue)38 b(of)g Fo(\015)k Fl(2)c Fg(R)5 b Fq(,)46 b(suc)m(h)456 2016 y(that)32 b Fo(a)718 2031 y Fh(\015)790 2016 y Fq(:=)c Fo(a)22 b Fq(+)g Fo(\015)33 b Fl(\025)28 b Fq(1)p Fo(;)33 b Fq(i.e.)1019 2177 y Fo(a)1070 2192 y Fh(\015)1114 2177 y Fq([)p Fo(x;)17 b(x)p Fq(])29 b(=)e Fo(a)p Fq([)p Fo(x;)17 b(x)p Fq(])23 b(+)f Fo(\015)5 b Fl(k)p Fo(x)p Fl(k)2045 2136 y Fj(2)2113 2177 y Fl(\025)28 b(k)p Fo(x)p Fl(k)2373 2136 y Fj(2)2412 2177 y Fo(;)82 b(x)28 b Fl(2)g Fo(d)p Fq([)p Fo(a)p Fq(])p Fo(:)456 2334 y Fq(Denote)d(b)m(y)i Fo(H)993 2349 y Fh(\015)1037 2334 y Fq([)p Fo(a)p Fq(])f(the)g(\(complete\))f(Hilb)s(ert)f(space)i Fo(d)p Fq([)p Fo(a)p Fq(])g(with)f(the)h(metric)f(form)1157 2503 y Fo(a)1208 2518 y Fh(\015)1253 2503 y Fq([)p Fo(x;)17 b(x)p Fq(])28 b(=)g Fl(k)p Fq(\()p Fo(A)22 b Fq(+)g Fo(\015)5 b(I)j Fq(\))2019 2462 y Fj(1)p Fh(=)p Fj(2)2129 2503 y Fo(x)p Fl(k)2234 2462 y Fj(2)2274 2503 y Fo(;)81 b(x)28 b Fl(2)g Fo(d)p Fq([)p Fo(a)p Fq(])p Fo(:)555 2666 y Fq(Let)41 b Fo(V)62 b Fq(b)s(e)40 b(a)h(self-adjoin)m(t)d(op)s(erator)i (in)g Fn(H)p Fq(,)i(satisfying)d Fl(D)s Fq(\()p Fl(j)p Fo(V)21 b Fl(j)2954 2630 y Fj(1)p Fh(=)p Fj(2)3064 2666 y Fq(\))41 b Fl(\033)g Fo(d)p Fq([)p Fo(a)p Fq(].)456 2785 y(Put)33 b Fo(W)739 2800 y Fk(\006)825 2785 y Fq(=)28 b(\()977 2746 y Fj(1)p 977 2762 36 4 v 977 2820 a(2)1022 2785 y Fq(\()p Fl(j)p Fo(V)21 b Fl(j)h(\006)h Fo(V)e Fq(\)\))1470 2749 y Fj(1)p Fh(=)p Fj(2)1580 2785 y Fo(:)33 b Fq(Ob)m(viously)-8 b(,)32 b Fl(D)s Fq(\()p Fo(W)2327 2800 y Fk(\006)2386 2785 y Fq(\))c Fl(\033)g Fo(d)p Fq([)p Fo(a)p Fq(])k(and)1488 2960 y Fo(W)1580 2975 y Fk(\006)1639 2960 y Fq(\()p Fo(A)23 b Fq(+)f Fo(\015)5 b(I)j Fq(\))2016 2919 y Fk(\000)p Fj(1)p Fh(=)p Fj(2)2208 2960 y Fl(2)28 b Fn(R)p Fo(:)456 3118 y Fq(W)-8 b(e)44 b(imp)s(ose)e(on)h Fo(W)1212 3133 y Fk(\006)1315 3118 y Fq(the)h(more)f(restrictiv)m(e)h (conditions:)64 b(there)44 b(exists)h Fo(p)h Fl(2)456 3234 y Fq(\(0)p Fo(;)17 b Fl(1)p Fq(\),)31 b(suc)m(h)j(that)1457 3400 y Fo(W)1549 3415 y Fk(\006)1608 3400 y Fq(\()p Fo(A)22 b Fq(+)g Fo(\015)5 b(I)j Fq(\))1984 3359 y Fk(\000)p Fj(1)p Fh(=)p Fj(2)2177 3400 y Fl(2)28 b Fq(\006)2341 3415 y Fj(2)p Fh(p)2416 3400 y Fo(:)877 b Fq(\(8\))456 3557 y(Let)32 b(us)i(in)m(tro)s(duce)e(the)h(form)1154 3715 y Fo(v)t Fq([)p Fo(x;)17 b(y)t Fq(])27 b(=)g(\()p Fo(W)1670 3730 y Fj(+)1729 3715 y Fo(x;)17 b(W)1920 3730 y Fj(+)1980 3715 y Fo(y)t Fq(\))k Fl(\000)i Fq(\()p Fo(W)2321 3730 y Fk(\000)2380 3715 y Fo(x;)17 b(W)2571 3730 y Fk(\000)2630 3715 y Fo(y)t Fq(\))p Fo(:)573 b Fq(\(9\))456 3878 y(The)36 b(condition)f(\(8\))g(implies)e(that)j Fo(v)j Fq(is)c(compact)g(on)h Fo(d)p Fq([)p Fo(a)p Fq(])p Fo(:)g Fq(This)g(means)g(that)456 3995 y(the)g(form)f Fo(v)40 b Fq(is)c(con)m(tin)m(uous)h(on)f Fo(H)1762 4010 y Fh(\015)1807 3995 y Fq([)p Fo(a)p Fq(])g(and)h(the)f (corresp)s(onding)g(op)s(erator)g Fo(Q)p Fq(,)456 4111 y(determined)d(b)m(y)h(the)g(equation)f Fo(a)1721 4126 y Fh(\015)1765 4111 y Fq([)p Fo(Qx;)17 b(y)t Fq(])29 b(=)g Fo(v)t Fq([)p Fo(x;)17 b(y)t Fq(],)33 b Fo(x;)17 b(y)32 b Fl(2)d Fo(d)p Fq([)p Fo(a)p Fq(],)34 b(is)f(compact)456 4227 y(on)f Fo(H)672 4242 y Fh(\015)716 4227 y Fq([)p Fo(a)p Fq(].)44 b(Hence,)34 b(the)f(form)f Fo(v)k Fq(satis\014es)d(the) g(estimate)909 4388 y Fo(v)t Fq([)p Fo(x;)17 b(x)p Fq(])28 b Fl(\024)g Fo("a)1398 4403 y Fh(\015)1443 4388 y Fq([)p Fo(x;)17 b(x)p Fq(])23 b(+)f Fo(C)7 b Fq(\()p Fo(")p Fq(\))p Fl(k)p Fo(x)p Fl(k)2126 4347 y Fj(2)2165 4388 y Fo(;)81 b(x)28 b Fl(2)h Fo(d)p Fq([)p Fo(a)p Fq(])p Fo(;)49 b Fl(8)p Fo(")27 b(>)h Fq(0)p Fo(:)456 4546 y Fq(Therefore,)58 b(w)m(e)c(can)e(in)m(tro)s(duce)h(a)f(family)e(of)i (semib)s(ounded)g(from)f(b)s(elo)m(w,)456 4662 y(closed)32 b(forms)g Fo(a)p Fq(\()p Fo(\013)q Fq(\))g(on)h Fo(d)p Fq([)p Fo(a)p Fq(]:)1457 4820 y Fo(a)p Fq(\()p Fo(\013)q Fq(\))27 b(=)h Fo(a)22 b Fl(\000)h Fo(\013)q(v)t(;)81 b(\013)29 b(>)e Fq(0)p Fo(:)828 b Fq(\(10\))456 4983 y(Denote)39 b(b)m(y)h Fo(A)p Fq(\()p Fo(\013)q Fq(\))f(the)g (self-adjoin)m(t)f(op)s(erator)g(in)h Fn(H)p Fo(;)g Fq(whic)m(h)g (corresp)s(onds)i(to)456 5099 y(the)36 b(form)e(\(10\))o(.)53 b(Since)35 b(the)h(di\013erence)g(of)f(the)h(resolv)m(en)m(ts)h(of)e Fo(A)h Fq(and)g Fo(A)p Fq(\()p Fo(\013)q Fq(\))f(is)456 5216 y(compact,)43 b(the)f(sp)s(ectrum)g(of)g Fo(A)p Fq(\()p Fo(\013)q Fq(\))f(in)g(the)h(gaps)g(of)f Fo(\033)t Fq(\()p Fo(A)p Fq(\))h(is)f(discrete.)71 b(Let)p eop %%Page: 5 5 5 4 bop 905 236 a Fi(THE)34 b(DISCRETE)f(SPECTR)n(UM)h(IN)e(THE)i (SPECTRAL)f(GAPS)411 b(5)456 425 y Fq(the)39 b(in)m(terv)-5 b(al)38 b(\003)h(=)g(\()p Fo(\025)1306 440 y Fk(\000)1365 425 y Fo(;)17 b(\025)1466 440 y Fj(+)1524 425 y Fq(\))39 b(b)s(e)h(a)f(gap)g(in)f(the)i(sp)s(ectrum)g Fo(\033)t Fq(\()p Fo(A)p Fq(\))p Fo(:)f Fq(W)-8 b(e)40 b(\014x)f(an)456 541 y(\\observ)-5 b(ation)31 b(p)s(oin)m(t")h Fo(\025;)1661 669 y(\025)1718 684 y Fk(\000)1804 669 y Fo(<)c(\025)g(<)f(\025)2153 684 y Fj(+)2212 669 y Fo(:)456 816 y Fq(Assume)37 b(that)g Fo(\025)f Fq(is)h(an)f(eigen)m(v)-5 b(alue)36 b(of)h(m)m(ultiplicit)m (y)c Fo(k)40 b Fq(of)c(the)h(op)s(erator)f Fo(A)p Fq(\()p Fo(t)p Fq(\))456 933 y(for)29 b(some)h Fo(t)e(>)f Fq(0)p Fo(:)j Fq(There)h(is)e(a)h(neigh)m(b)s(orho)s(o)s(d)e(of)i Fo(t)g Fq(where)h(w)m(e)g(can)f(c)m(ho)s(ose)h(real-)456 1049 y(analytic)i(functions)h Fo(\025)1308 1064 y Fh(j)1345 1049 y Fq(\()p Fo(s)p Fq(\))p Fo(;)51 b(j)37 b Fq(=)31 b(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(k)s(;)34 b Fq(whose)i(v)-5 b(alues)35 b(are)f(eigen)m(v)-5 b(alues)35 b(of)456 1165 y Fo(A)p Fq(\()p Fo(s)p Fq(\))d(and)1436 1292 y Fo(\025)1493 1307 y Fh(j)1530 1292 y Fq(\()p Fo(t)p Fq(\))c(=)f Fo(\025;)82 b(j)34 b Fq(=)27 b(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(k)s(:)456 1442 y Fq(W)-8 b(e)33 b(coun)m(t)g(the)g(eigen)m(v)-5 b(alues)32 b Fl(f)p Fo(\025)1671 1457 y Fh(j)1708 1442 y Fq(\()p Fo(s)p Fq(\))p Fl(g)1880 1406 y Fh(k)1880 1467 y(j)t Fj(=1)2038 1442 y Fq(with)h(their)f(m)m(ultiplicities.)555 1558 y(Let)j(us)h(c)m(ho)s(ose)f(a)g(suitable)f(neigh)m(b)s(orho)s(o)s (d)g(of)g(p)s(oin)m(t)g Fo(t:)h Fq(Since)g(none)h(of)e(the)456 1675 y(functions)24 b Fo(\025)925 1690 y Fh(j)985 1675 y Fq(is)g(constan)m(t,)j(the)d(zeros)h(of)f(the)h(deriv)-5 b(ativ)m(es)24 b Fo(d\025)2732 1690 y Fh(j)2768 1675 y Fo(=ds)f Fq(are)i(isolated.)456 1791 y(Hence)34 b(w)m(e)f(can)g(c)m (ho)s(ose)g(suc)m(h)h(a)f(neigh)m(b)s(orho)s(o)s(d)e(of)h Fo(t)h Fq(that)1124 1959 y Fo(d\025)1232 1974 y Fh(j)1268 1959 y Fq(\()p Fo(s)p Fq(\))p Fo(=ds)27 b Fl(6)p Fq(=)g(0)65 b(for)32 b Fo(s)c Fl(6)p Fq(=)f Fo(t;)82 b(j)34 b Fq(=)28 b(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(k)s(:)555 2127 y Fq(Let)33 b Fo(k)781 2142 y Fj(+)839 2127 y Fq(\()p Fo(t)p Fq(\))p Fo(;)49 b(k)1077 2142 y Fk(\000)1136 2127 y Fq(\()p Fo(t)p Fq(\))p Fo(;)g(k)1374 2142 y Fj(0)1413 2127 y Fq(\()p Fo(t)p Fq(\))33 b(b)s(e)f(the)h(n)m(um)m(b)s(er)f(of)g(the)h(functions) f Fo(\025)2967 2142 y Fh(j)3036 2127 y Fq(whic)m(h)g(are)456 2243 y(monotone)26 b(decreasing,)k(monotone)c(increasing)h(and)h (non-monotone)f(\(resp)s(ec-)456 2359 y(tiv)m(ely\))40 b(in)g(the)i(c)m(hosen)g(neigh)m(b)s(orho)s(o)s(d.)67 b(If)41 b Fo(\025)h Fl(2)g Fo(\032)p Fq(\()p Fo(A)p Fq(\()p Fo(t)p Fq(\)\),)h(then)f(w)m(e)g(assume)456 2475 y(that)32 b Fo(k)718 2490 y Fj(+)777 2475 y Fq(\()p Fo(t)p Fq(\))c(=)60 b Fo(k)1103 2490 y Fk(\000)1161 2475 y Fq(\()p Fo(t)p Fq(\))28 b(=)60 b Fo(k)1487 2490 y Fj(0)1526 2475 y Fq(\()p Fo(t)p Fq(\))28 b(=)g(0)p Fo(:)k Fq(In)m(tro)s(duce)i(the)f(follo)m (wing)d(v)-5 b(alues)1011 2561 y Ff(8)1011 2636 y(>)1011 2661 y(>)1011 2686 y(>)1011 2711 y(<)1011 2860 y(>)1011 2885 y(>)1011 2910 y(>)1011 2935 y(:)1435 2642 y Fo(N)1513 2657 y Fk(\006)1572 2642 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)2129 2575 y Ff(P)2065 2715 y Fj(0)p Fh()g Fq(0)p Fo(;)3271 4920 y Fq(\(12\))456 5088 y(where)1048 5216 y Fo(X)8 b Fq(\()p Fo(\025)p Fq(\))p Fo(S)e(g)30 b Fq(=)d Fo(W)14 b Fq(\()p Fo(A)22 b Fl(\000)h Fo(\025I)8 b Fq(\))2001 5174 y Fk(\000)p Fj(1)2095 5216 y Fo(W)14 b(g)t(;)113 b(g)31 b Fl(2)d(D)s Fq(\()p Fo(W)14 b Fq(\))p Fo(:)p eop %%Page: 6 6 6 5 bop 456 236 a Fi(6)1180 b(O.)34 b(SAFR)n(ONO)n(V)555 425 y Fq(It)26 b(is)f(w)m(ell)f(kno)m(wn)j(that)e(the)h(description)f (of)g(the)h(discrete)g(sp)s(ectrum)g(of)f Fo(A)p Fq(\()p Fo(\013)q Fq(\))456 541 y(can)34 b(b)s(e)h(reduced)h(to)e(the)h(study)h (of)e(the)h(sp)s(ectrum)g(of)f(the)h(compact)f(op)s(erator)456 658 y Fo(X)8 b Fq(\()p Fo(\025)p Fq(\))34 b(\(the)i(Birman-Sc)m(h)m (winger)f(principle\).)50 b(W)-8 b(e)36 b(presen)m(t)h(here)f(a)f(v)m (ersion)h(of)456 774 y(this)c(reduction)g(suitable)g(for)g(p)s (erturbations)g(of)g(forms.)594 952 y Fx(Prop)s(osition)j(1.1.)49 b Fm(L)-5 b(et)32 b Fo(\025)27 b Fq(=)p 1775 873 57 4 v 28 w Fo(\025)g Fl(2)h Fo(\032)p Fq(\()p Fo(A)p Fq(\))p Fo(:)k Fm(The)e(fol)5 b(lowing)30 b(two)h(statements)456 1068 y(ar)-5 b(e)34 b(e)-5 b(quivalent:)555 1184 y(1\))26 b(the)h(p)-5 b(oint)26 b Fo(\025)g Fm(is)g(an)g(eigenvalue)g(of)g (multiplicity)h Fo(k)i Fm(for)d(the)h(op)-5 b(er)g(ator)26 b Fo(A)p Fq(\()p Fo(\013)q Fq(\))p Fm(;)555 1300 y(2\))35 b(the)h(p)-5 b(oint)35 b Fo(\013)1153 1264 y Fk(\000)p Fj(1)1283 1300 y Fm(is)g(an)g(eigenvalue)g(of)g(ge)-5 b(ometric)35 b(multiplicity)g Fo(k)k Fm(for)c(the)456 1417 y(op)-5 b(er)g(ator)34 b Fo(X)8 b Fq(\()p Fo(\025)p Fq(\))p Fo(:)555 1595 y Fq(Prop)s(osition)36 b(1.1)h(is)g(a)h(direct)f (consequence)k(of)c(Lemma)f(1)i(from)e([13].)58 b(F)-8 b(or)456 1711 y(additiv)m(e)41 b(op)s(erator)g(p)s(erturbations)h(a)f (simple)g(pro)s(of)g(can)h(b)s(e)g(found)g(in)g([12)o(].)456 1827 y(Note,)24 b(that)e(in)f(the)i(case)f Fo(V)50 b Fl(\025)28 b Fq(0)22 b(or)f Fo(V)49 b Fl(\024)29 b Fq(0)21 b(this)h(statemen)m(t)h(w)m(as)g(later)e(pro)m(v)m(ed)i(in)456 1943 y([2)o(])h(and)g(this)f(pro)s(of)g(can)h(b)s(e)g(also)f(applied)f (for)h(non-signde\014nite)h(p)s(erturbations.)555 2059 y(No)m(w)33 b(w)m(e)h(are)f(ready)g(to)f(form)m(ulate)f(the)i(main)e (result)h(of)h(the)g(pap)s(er.)594 2237 y Fx(Theorem)k(1.1.)49 b Fm(L)-5 b(et)32 b(the)h(form)f Fo(v)k Fm(in)j Fq(\(10\))32 b Fm(b)-5 b(e)32 b(de\014ne)-5 b(d)31 b(by)41 b Fq(\(9\))32 b Fm(and)g(let)g(the)456 2356 y(c)-5 b(ondition)35 b Fq(\(8\))28 b Fm(b)-5 b(e)28 b(ful\014l)5 b(le)-5 b(d)28 b(for)h(some)f Fo(p)f(>)h Fq(0)p Fm(.)42 b(Assume)29 b(that)g(for)f Fo(\025)g Fq(=)p 3066 2277 V 27 w Fo(\025)g Fl(2)g Fo(\032)p Fq(\()p Fo(A)p Fq(\))1443 2519 y(lim)1418 2569 y Fh(\013)p Fk(!1)1621 2519 y Fo(\013)1684 2478 y Fk(\000)p Fh(p)1778 2519 y Fo(N)10 b Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)i Fo(J)2413 2534 y Fj(+)2472 2519 y Fo(;)772 b Fq(\(13\))1418 2761 y(lim)1392 2811 y Fh(\013)p Fk(!1)1595 2761 y Fo(\013)1658 2720 y Fk(\000)p Fh(p)1753 2761 y Fo(N)10 b Fq(\()p Fl(\000)p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))27 b(=)g Fo(J)2465 2776 y Fk(\000)3271 2761 y Fq(\(14\))456 2943 y Fm(and)1351 3106 y Fq(lim)1350 3163 y Fh(s)p Fk(!)p Fj(0)1505 3106 y Fo(s)1551 3065 y Fh(p)1590 3106 y Fo(n)p Fq(\()p Fo(s;)17 b(X)8 b Fq(\()p Fo(\025)p Fq(\)\))27 b(=:)h Fo(J)2248 3121 y Fk(\000)2329 3106 y Fq(+)22 b Fo(J)2481 3121 y Fj(+)2540 3106 y Fo(:)704 b Fq(\(15\))456 3296 y Fm(Then)1410 3418 y Fq(lim)1385 3468 y Fh(\013)p Fk(!1)1588 3418 y Fo(\013)1651 3377 y Fk(\000)p Fh(p)1745 3418 y Fo(N)1823 3433 y Fj(+)1882 3418 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)i Fo(J)2429 3433 y Fj(+)2488 3418 y Fo(;)1372 3599 y Fq(lim)1346 3648 y Fh(\013)p Fk(!1)1549 3599 y Fo(\013)1612 3557 y Fk(\000)p Fh(p)1706 3599 y Fo(N)1784 3614 y Fj(+)1844 3599 y Fq(\()p Fl(\000)p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)i Fo(J)2468 3614 y Fk(\000)2527 3599 y Fo(;)1119 3779 y Fq(lim)1093 3829 y Fh(\013)p Fk(!1)1296 3779 y Fo(\013)1359 3738 y Fk(\000)p Fh(p)1453 3779 y Fq(\()p Fo(N)1569 3794 y Fj(0)1609 3779 y Fq(\()p Fo(V)21 b Fq(;)c Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)2168 3794 y Fk(\000)2228 3779 y Fq(\()p Fo(V)f Fq(;)c Fo(\025;)g(\013)q Fq(\)\))27 b(=)g(0)456 3941 y Fm(and)1028 4063 y Fq(lim)1002 4113 y Fh(\013)p Fk(!1)1205 4063 y Fo(\013)1268 4022 y Fk(\000)p Fh(p)1362 4063 y Fq(\()p Fo(N)1478 4078 y Fj(0)1518 4063 y Fq(\()p Fl(\000)p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)2155 4078 y Fk(\000)2214 4063 y Fq(\()p Fl(\000)p Fo(V)g Fq(;)17 b Fo(\025;)g(\013)q Fq(\)\))27 b(=)g(0)p Fo(:)468 4297 y Fq(2.)55 b Fp(Additional)39 b(inf)n(orma)-7 b(tion)40 b(concerning)d(comp)-7 b(a)n(ct)38 b(opera)-7 b(tors)555 4471 y Fx(1.)44 b Fq(Let)32 b Fo(T)41 b Fl(2)28 b Fn(S)1135 4486 y Fk(1)1241 4471 y Fq(and)k(let)g Fl(f)p Fo(\025)1678 4486 y Fh(k)1720 4471 y Fq(\()p Fo(T)14 b Fq(\))p Fl(g)1917 4435 y Fh(\027)1917 4496 y Fj(1)1991 4471 y Fq(b)s(e)33 b(the)f(sequence)j(of)c(the)h(eigen)m(v)-5 b(alues)456 4588 y(of)30 b(T,)i(en)m(umerated)g(so)g(that)f(their)f (absolute)h(v)-5 b(alues)31 b(are)h(non-increasing,)e(and)456 4704 y(coun)m(ted)j(with)f(their)g Fm(algebr)-5 b(aic)31 b Fq(m)m(ultiplicities.)39 b(Here)33 b Fo(\027)38 b Fq(is)32 b(a)g(p)s(ositiv)m(e)f(in)m(teger)456 4820 y(or)j Fo(\027)k Fq(=)32 b Fl(1)p Fq(.)50 b(If)35 b(the)h(sequence)h Fl(f)p Fo(\025)1732 4835 y Fh(k)1775 4820 y Fq(\()p Fo(T)14 b Fq(\))p Fl(g)1972 4784 y Fh(\027)1972 4845 y Fj(1)2049 4820 y Fq(is)35 b(\014nite,)g(w)m(e)h(con)m(tin)m(ue)g(it)e(b)m(y)h (zeros.)456 4936 y(After)41 b(this)f(w)m(e)i(can)f(assume)g(that)g Fo(\027)48 b Fq(=)42 b Fl(1)p Fq(.)67 b(Note)41 b(that)g Fo(T)55 b Fq(ma)m(y)40 b(b)s(e)h(a)g(non-)456 5052 y(selfadjoin)m(t)31 b(op)s(erator.)43 b(W)-8 b(e)33 b(in)m(tro)s(duce)f(the)h(distribution) e(function)1038 5216 y Fo(n)1096 5231 y Fj(0)1136 5216 y Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b(=)g(card)q Fl(f)32 b Fo(k)f Fq(:)60 b Fl(j)p Fo(\025)2024 5231 y Fh(k)2066 5216 y Fq(\()p Fo(T)14 b Fq(\))p Fl(j)27 b Fo(>)h(s)k Fl(g)p Fo(;)82 b(s)27 b(>)h Fq(0)p Fo(;)p eop %%Page: 7 7 7 6 bop 905 236 a Fi(THE)34 b(DISCRETE)f(SPECTR)n(UM)h(IN)e(THE)i (SPECTRAL)f(GAPS)411 b(7)456 425 y Fq(of)32 b(the)h(eigen)m(v)-5 b(alues)32 b(of)g Fo(T)8 b(:)555 541 y Fq(There)33 b(is)f(a)g(relation) e(\(disco)m(v)m(ered)j(b)m(y)g(H.)f(W)-8 b(eyl\))32 b(b)s(et)m(w)m(een) i(the)e(eigen)m(v)-5 b(alues)456 658 y(and)32 b(the)h(singular)e(n)m (um)m(b)s(ers)j(of)e Fo(T)594 833 y Fx(Prop)s(osition)j(2.1.)49 b Fm(L)-5 b(et)35 b Fo(T)42 b Fl(2)28 b Fn(S)1866 848 y Fk(1)1940 833 y Fm(.)45 b(Then)1244 956 y Fh(l)1202 981 y Ff(Y)1191 1166 y Fh(k)r Fj(=1)1336 1064 y Fl(j)p Fo(\025)1421 1079 y Fh(k)1463 1064 y Fq(\()p Fo(T)14 b Fq(\))p Fl(j)27 b(\024)1824 956 y Fh(l)1781 981 y Ff(Y)1770 1166 y Fh(k)r Fj(=1)1915 1064 y Fo(s)1961 1079 y Fh(k)2004 1064 y Fq(\()p Fo(T)14 b Fq(\))p Fo(;)86 b(l)30 b Fq(=)e(1)p Fo(;)17 b Fq(2)p Fo(;)g(:)g(:)g(:)543 b Fq(\(16\))555 1316 y(The)37 b(pro)s(of)e(of)g(this)g(prop)s(osition)f(can)i(b)s(e)f (found)h(in)f([3].)52 b(Let)36 b Fo(n)g Fq(b)s(e)g(de\014ned)456 1432 y(in)31 b(\(5\),)i(then)g(this)f(prop)s(osition)f(could)h(b)s(e)h (rewritten)f(in)g(the)h(follo)m(wing)d(form.)594 1607 y Fx(Prop)s(osition)35 b(2.2.)49 b Fm(L)-5 b(et)35 b Fo(T)42 b Fl(2)28 b Fn(S)1866 1622 y Fk(1)1975 1607 y Fm(and)34 b Fo(\021)e(>)27 b Fq(0)p Fo(:)35 b Fm(Then)1373 1713 y Fk(1)1348 1730 y Ff(Z)1352 1964 y Fh(\021)1448 1847 y Fo(n)1506 1862 y Fj(0)1545 1847 y Fq(\()p Fo(s;)17 b(T)d Fq(\))1792 1779 y Fo(ds)p 1792 1824 97 4 v 1817 1915 a(s)1926 1847 y Fl(\024)2056 1713 y Fk(1)2031 1730 y Ff(Z)2035 1964 y Fh(\021)2131 1847 y Fo(n)p Fq(\()p Fo(s;)j(T)d Fq(\))2436 1779 y Fo(ds)p 2436 1824 V 2460 1915 a(s)2542 1847 y(:)702 b Fq(\(17\))555 2127 y Fm(Pr)-5 b(o)g(of.)43 b Fq(Inequalit)m(y)34 b(\(16\))e(can)h(b)s(e)g(rewritten)f (in)g(the)h(form)1367 2256 y Fh(l)1318 2281 y Ff(X)1314 2466 y Fh(k)r Fj(=1)1459 2364 y Fq(log)17 b Fl(j)p Fo(\025)1687 2379 y Fh(k)1729 2364 y Fq(\()p Fo(T)d Fq(\))p Fl(j)27 b(\024)2089 2256 y Fh(l)2040 2281 y Ff(X)2036 2466 y Fh(k)r Fj(=1)2181 2364 y Fq(log)17 b Fo(s)2370 2379 y Fh(k)2413 2364 y Fq(\()p Fo(T)d Fq(\))p Fo(:)456 2593 y Fq(Hence,)34 b(for)e(ev)m(ery)i Fo(\021)d(>)d Fq(0)k(w)m(e)i(ha)m(v)m (e)1000 2717 y Fh(l)951 2742 y Ff(X)947 2926 y Fh(k)r Fj(=1)1075 2825 y Fq(\(log)17 b Fl(j)p Fo(\025)1341 2840 y Fh(k)1383 2825 y Fq(\()p Fo(T)d Fq(\))p Fl(j)22 b(\000)g Fq(log)17 b Fo(\021)t Fq(\))27 b Fl(\024)2098 2717 y Fh(l)2049 2742 y Ff(X)2044 2926 y Fh(k)r Fj(=1)2173 2825 y Fq(\(log)17 b Fo(s)2400 2840 y Fh(k)2442 2825 y Fq(\()p Fo(T)d Fq(\))22 b Fl(\000)h Fq(log)16 b Fo(\021)t Fq(\))p Fo(:)301 b Fq(\(18\))456 3059 y(Let)34 b(us)g(tak)m(e)h(the)f(n)m(um)m (b)s(er)g Fo(l)e Fq(=)d Fo(n)1720 3074 y Fj(0)1760 3059 y Fq(\()p Fo(\021)t(;)17 b(T)d Fq(\))p Fo(:)33 b Fq(Then)i(w)m(e)g(can) f(substitute)g(the)g(sum)456 3175 y(on)e(the)h(left-hand)f(side)g(of)39 b(\(18\))32 b(b)m(y)1458 3259 y Ff(X)1360 3447 y Fk(j)p Fh(\025)1421 3459 y Fe(k)1459 3447 y Fj(\()p Fh(T)10 b Fj(\))p Fk(j)p Fh(>\021)1677 3342 y Fq(\(log)16 b Fl(j)p Fo(\025)1942 3357 y Fh(k)1985 3342 y Fq(\()p Fo(T)e Fq(\))p Fl(j)21 b(\000)i Fq(log)16 b Fo(\021)t Fq(\))p Fo(:)456 3591 y Fq(Ob)m(viously)-8 b(,)955 3715 y Fh(l)906 3740 y Ff(X)901 3924 y Fh(k)r Fj(=1)1030 3823 y Fq(\(log)17 b Fo(s)1257 3838 y Fh(k)1299 3823 y Fq(\()p Fo(T)d Fq(\))22 b Fl(\000)h Fq(log)16 b Fo(\021)t Fq(\))27 b Fl(\024)2007 3740 y Ff(X)1933 3928 y Fh(s)1966 3940 y Fe(k)2003 3928 y Fj(\()p Fh(T)10 b Fj(\))p Fh(>\021)2202 3823 y Fq(\(log)16 b Fo(s)2428 3838 y Fh(k)2471 3823 y Fq(\()p Fo(T)e Fq(\))22 b Fl(\000)g Fq(log)17 b Fo(\021)t Fq(\))p Fo(:)456 4072 y Fq(Th)m(us)34 b(w)m(e)f(ha)m(v)m(e)881 4147 y Ff(X)782 4335 y Fk(j)p Fh(\025)843 4347 y Fe(k)881 4335 y Fj(\()p Fh(T)10 b Fj(\))p Fk(j)p Fh(>\021)1099 4230 y Fq(\(log)17 b Fl(j)p Fo(\025)1365 4245 y Fh(k)1407 4230 y Fq(\()p Fo(T)d Fq(\))p Fl(j)21 b(\000)i Fq(log)17 b Fo(\021)t Fq(\))27 b Fl(\024)2143 4147 y Ff(X)2068 4335 y Fh(s)2101 4347 y Fe(k)2139 4335 y Fj(\()p Fh(T)10 b Fj(\))p Fh(>\021)2337 4230 y Fq(\(log)17 b Fo(s)2564 4245 y Fh(k)2607 4230 y Fq(\()p Fo(T)d Fq(\))21 b Fl(\000)i Fq(log)17 b Fo(\021)t Fq(\))p Fo(:)3271 4346 y Fq(\(19\))456 4510 y(It)38 b(remains)g(to)h (note)f(that)h(the)g(left-)e(and)i(righ)m(t-hand)f(sides)h(in)f(\(17\)) g(are)h(the)456 4626 y(same)32 b(as)h(in)f(\(19\))g(resp)s(ectiv)m(ely) -8 b(.)44 b(The)34 b(pro)s(of)d(is)h(completed.)43 b Fd(2)555 4742 y Fx(2.)h Fq(Here)33 b(w)m(e)h(consider)f(the)g (functionals)1192 4792 y Ff(8)1192 4867 y(<)1192 5016 y(:)1308 4904 y Fq(\001)1389 4868 y Fj(\(0\))1389 4929 y Fh(p)1483 4904 y Fq(\()p Fo(T)14 b Fq(\))28 b(=)f(lim)17 b(sup)2060 4927 y Fh(s)p Fk(!)p Fj(0)2219 4904 y Fo(s)2265 4868 y Fh(p)2305 4904 y Fo(n)2363 4919 y Fj(0)2403 4904 y Fq(\()p Fo(s;)g(T)d Fq(\))p Fo(;)1339 5027 y(\016)1386 4991 y Fj(\(0\))1382 5052 y Fh(p)1481 5027 y Fq(\()p Fo(T)g Fq(\))27 b(=)g(lim)17 b(inf)2029 5042 y Fh(s)p Fk(!)p Fj(0)2188 5027 y Fo(s)2234 4991 y Fh(p)2273 5027 y Fo(n)2331 5042 y Fj(0)2371 5027 y Fq(\()p Fo(s;)g(T)d Fq(\))p Fo(;)456 5216 y Fq(whic)m(h)33 b(are)f(\014nite)h(on)f(\006) 1352 5231 y Fh(p)1425 5216 y Fq(\(see)h([3]\).)p eop %%Page: 8 8 8 7 bop 456 236 a Fi(8)1180 b(O.)34 b(SAFR)n(ONO)n(V)555 425 y Fq(Assume)g(that)e Fo(T)41 b Fl(2)29 b Fq(\006)1391 440 y Fh(p)1431 425 y Fo(;)49 b(p)27 b(>)h Fq(0)p Fo(;)k Fq(and)h(for)f(ev)m(ery)j Fo(s)27 b(>)h Fq(0)1405 593 y Fo(n)1463 608 y Fj(0)1503 593 y Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b Fl(\025)h Fo(n)p Fq(\()p Fo(s;)17 b(T)d Fq(\))22 b Fl(\000)g Fo(\036)p Fq(\()p Fo(s)p Fq(\))p Fo(;)776 b Fq(\(20\))456 767 y(where)33 b Fo(\036)28 b Fl(2)g Fo(L)983 782 y Fk(1)p Fh(;l)q(oc)1164 767 y Fq(\(0)p Fo(;)17 b Fl(1)p Fq(\))32 b(is)g(a)g(real-v)-5 b(alued)31 b(function,)h(suc)m(h)i(that)1646 935 y(lim)1645 992 y Fh(s)p Fk(!)p Fj(0)1800 935 y Fo(s)1846 894 y Fh(p)1885 935 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))28 b(=)f(0)p Fo(:)999 b Fq(\(21\))555 1135 y(Let)40 b(the)g(functionals)e(\001)1496 1150 y Fh(p)1536 1135 y Fo(;)56 b(\016)1662 1150 y Fh(p)1742 1135 y Fq(b)s(e)40 b(de\014ned)h(in)d(\(7\).)64 b(As)40 b(an)g(application)d(of)456 1251 y(the)c(inequalit)m(y)f(\(17\))g(w)m (e)i(presen)m(t)g(the)f(follo)m(wing)d(result.)594 1431 y Fx(Prop)s(osition)35 b(2.3.)49 b Fm(L)-5 b(et)37 b Fo(T)45 b Fl(2)32 b Fq(\006)1862 1446 y Fh(p)1902 1431 y Fo(;)54 b(p)31 b(>)g Fq(0)p Fo(:)37 b Fm(Assume)f(that)i(the)f(c)-5 b(onditions)456 1548 y Fq(\(20\))34 b Fm(and)45 b Fq(\(21\))34 b Fm(ar)-5 b(e)34 b(ful\014l)5 b(le)-5 b(d.)45 b(Then)1576 1724 y Fq(\001)1657 1682 y Fj(\(0\))1657 1748 y Fh(p)1752 1724 y Fq(\()p Fo(T)14 b Fq(\))27 b(=)g(\001)2110 1739 y Fh(p)2150 1724 y Fq(\()p Fo(T)14 b Fq(\))p Fo(;)947 b Fq(\(22\))1612 1965 y Fo(\016)1659 1924 y Fj(\(0\))1655 1990 y Fh(p)1753 1965 y Fq(\()p Fo(T)14 b Fq(\))28 b(=)f Fo(\016)2074 1980 y Fh(p)2114 1965 y Fq(\()p Fo(T)14 b Fq(\))p Fo(:)983 b Fq(\(23\))555 2163 y Fm(Pr)-5 b(o)g(of.)42 b Fq(T)-8 b(o)29 b(b)s(e)f(de\014nite)h(w)m(e)h(consider)f(the)g (functional)e(\001)2680 2127 y Fj(\(0\))2680 2188 y Fh(p)2775 2163 y Fq(\()p Fo(T)14 b Fq(\).)41 b(The)30 b(equal-)456 2284 y(it)m(y)j(\(23\))f(can)h(b)s(e)g(c)m(hec)m(k)m(ed)j(similarly)-8 b(.)39 b(It)33 b(follo)m(ws)e(from)g(\(20\),)h(\(21\))g(that)1575 2460 y(\001)1656 2418 y Fj(\(0\))1656 2484 y Fh(p)1751 2460 y Fq(\()p Fo(T)14 b Fq(\))27 b Fl(\025)h Fq(\001)2111 2475 y Fh(p)2151 2460 y Fq(\()p Fo(T)14 b Fq(\))p Fo(:)456 2632 y Fq(Let)36 b(us)i(no)m(w)f(pro)m(v)m(e)h(the)f(inequalit)m(y)e Fl(\024)i Fq(in)f(\(22\).)55 b(There)38 b(exists)f Fo(s)2959 2647 y Fj(0)3033 2632 y Fo(>)d Fq(0,)k(suc)m(h)456 2748 y(that)28 b Fo(n)721 2763 y Fj(0)761 2748 y Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b(=)g Fo(n)p Fq(\()p Fo(s;)17 b(T)d Fq(\))27 b(=)h(0)h(for)f Fo(s)f(>)h(s)2000 2763 y Fj(0)2039 2748 y Fq(.)43 b(Therefore,)30 b(m)m(ultiplying)c Fo(\036)i Fq(b)m(y)i(the)456 2864 y(indicator)g(function)j(of)f(\(0)p Fo(;)17 b(s)1538 2879 y Fj(0)1576 2864 y Fq(])33 b(w)m(e)h(can)f (assume)g(that)1560 3032 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))27 b(=)g(0)p Fo(;)49 b Fl(8)p Fo(s)28 b(>)g(s)2274 3047 y Fj(0)2313 3032 y Fo(:)456 3200 y Fq(Then)33 b(for)f(ev)m(ery)j Fo(t)28 b(>)f Fq(0,)33 b(according)f(to)g(\(20\))o(,)h(w)m(e)g(ha)m(v)m (e)1114 3322 y Fk(1)1089 3339 y Ff(Z)1100 3578 y Fh(t)1189 3456 y Fo(n)1247 3471 y Fj(0)1286 3456 y Fq(\()p Fo(s;)17 b(T)d Fq(\))1533 3388 y Fo(ds)p 1533 3433 97 4 v 1558 3524 a(s)1667 3456 y Fl(\025)1797 3322 y Fk(1)1772 3339 y Ff(Z)1782 3578 y Fh(t)1872 3456 y Fo(n)p Fq(\()p Fo(s;)j(T)d Fq(\))2177 3388 y Fo(ds)p 2177 3433 V 2202 3524 a(s)2305 3456 y Fl(\000)2429 3322 y Fk(1)2405 3339 y Ff(Z)2415 3578 y Fh(t)2504 3456 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))2694 3388 y Fo(ds)p 2694 3433 V 2719 3524 a(s)2800 3456 y(:)444 b Fq(\(24\))456 3715 y(Note)32 b(that)1406 3812 y Fk(1)1381 3829 y Ff(Z)1391 4068 y Fh(t)1481 3946 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))1671 3879 y Fo(ds)p 1671 3923 V 1696 4014 a(s)1805 3946 y Fq(=)27 b Fo(o)p Fq(\()p Fo(t)2028 3905 y Fk(\000)p Fh(p)2123 3946 y Fq(\))p Fo(;)81 b(t)28 b Fl(!)g Fq(0)p Fo(:)735 b Fq(\(25\))456 4211 y(Subtracting)33 b(\(24\))f(from)f(\(17\),)h(w)m(e)i(obtain)1129 4345 y Fh(t)1082 4362 y Ff(Z)1086 4596 y Fh(\021)1181 4479 y Fo(n)1239 4494 y Fj(0)1279 4479 y Fq(\()p Fo(s;)17 b(T)d Fq(\))1526 4412 y Fo(ds)p 1526 4456 V 1551 4547 a(s)1659 4479 y Fl(\024)1812 4345 y Fh(t)1765 4362 y Ff(Z)1769 4596 y Fh(\021)1864 4479 y Fo(n)p Fq(\()p Fo(s;)j(T)d Fq(\))2169 4412 y Fo(ds)p 2169 4456 V 2194 4547 a(s)2297 4479 y Fq(+)2420 4345 y Fk(1)2395 4362 y Ff(Z)2406 4601 y Fh(t)2495 4479 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))2685 4412 y Fo(ds)p 2685 4456 V 2710 4547 a(s)2791 4479 y(:)456 4752 y Fq(Putting)24 b Fo(\021)31 b Fq(=)d(\(1)6 b Fl(\000)g Fo(")p Fq(\))p Fo(t;)40 b Fq(0)28 b Fo(<)f(")h(<)f Fq(1)p Fo(;)d Fq(and)h(estimating)d(the)j(left-)f(and)g(righ)m(t-hand)456 4868 y(side,)32 b(w)m(e)i(\014nd)f(that)860 5118 y Fo("n)964 5133 y Fj(0)1003 5118 y Fq(\()p Fo(t;)17 b(T)d Fq(\))27 b Fl(\024)1457 5051 y Fo(")p 1371 5095 217 4 v 1371 5187 a Fq(1)22 b Fl(\000)h Fo(")1598 5118 y(n)p Fq(\(\(1)f Fl(\000)g Fo(")p Fq(\))p Fo(t;)17 b(T)d Fq(\))21 b(+)2318 4985 y Fk(1)2293 5001 y Ff(Z)2304 5240 y Fh(t)2393 5118 y Fo(\036)p Fq(\()p Fo(s)p Fq(\))2583 5051 y Fo(ds)p 2583 5095 97 4 v 2608 5187 a(s)2689 5118 y(;)82 b(t)28 b(>)f Fq(0)p Fo(:)p eop %%Page: 9 9 9 8 bop 905 236 a Fi(THE)34 b(DISCRETE)f(SPECTR)n(UM)h(IN)e(THE)i (SPECTRAL)f(GAPS)411 b(9)456 425 y Fq(Multiplying)20 b(the)i(last)g(inequalit)m(y)f(b)m(y)j Fo(t)1912 389 y Fh(p)1952 425 y Fo(=")d Fq(and)i(taking)e(upp)s(er)i(limits)d(as)i Fo(t)28 b Fl(!)g Fq(0,)456 541 y(in)j(view)i(of)39 b(\(25\))32 b(w)m(e)i(obtain)1337 731 y(\001)1418 690 y Fj(\(0\))1418 756 y Fh(p)1512 731 y Fq(\()p Fo(T)14 b Fq(\))28 b Fl(\024)g Fq(\(1)22 b Fl(\000)g Fo(")p Fq(\))2084 690 y Fk(\000)p Fj(1)p Fk(\000)p Fh(p)2269 731 y Fq(\001)2350 746 y Fh(p)2390 731 y Fq(\()p Fo(T)14 b Fq(\))p Fo(:)456 913 y Fq(The)33 b(passage)g(to)g(the)g(limit)c(as)k Fo(")27 b Fl(!)g Fq(0)32 b(giv)m(es)h(the)g(estimate)1575 1098 y(\001)1656 1057 y Fj(\(0\))1656 1122 y Fh(p)1751 1098 y Fq(\()p Fo(T)14 b Fq(\))27 b Fl(\024)h Fq(\001)2111 1113 y Fh(p)2151 1098 y Fq(\()p Fo(T)14 b Fq(\))p Fo(:)456 1280 y Fq(The)33 b(pro)s(of)f(is)g(completed.)43 b Fd(2)748 1549 y Fq(3.)55 b Fp(The)37 b(pr)n(oof)h(of)h(the)e(main)h(resul)-7 b(t)37 b(\(Theorem)i(1.1\))555 1723 y Fq(It)33 b(is)f(su\016cien)m(t)i(to)e (pro)m(v)m(e)i(that)1053 1898 y Fo(N)1131 1913 y Fk(\000)1190 1898 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)1750 1913 y Fj(0)1789 1898 y Fq(\()p Fo(V)g Fq(;)17 b Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)2349 1913 y Fk(\000)2408 1898 y Fq(\()p Fl(\000)p Fo(V)g Fq(;)17 b Fo(\025;)g(\013)q Fq(\))1229 2014 y(+)p Fo(N)1383 2029 y Fj(0)1423 2014 y Fq(\()p Fl(\000)p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)i Fo(o)p Fq(\()p Fo(\013)2141 1978 y Fh(p)2180 2014 y Fq(\))p Fo(;)82 b(\013)28 b Fl(!)f(1)p Fo(:)3271 1958 y Fq(\(26\))456 2193 y(It)32 b(follo)m(ws)g(from)f(the)i (Prop)s(osition)e(1.1)h(that)1040 2367 y Fo(N)1118 2382 y Fj(+)1177 2367 y Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)1737 2382 y Fk(\000)1796 2367 y Fq(\()p Fo(V)f Fq(;)c Fo(\025;)g(\013)q Fq(\))k(+)h Fo(N)2355 2382 y Fj(0)2395 2367 y Fq(\()p Fo(V)f Fq(;)c Fo(\025;)g(\013)q Fq(\))27 b(=)1306 2417 y Ff(P)1394 2504 y Fj(0)p Fh()g Fq(0)p Fo(:)k Fq(The)i(cub)s(es)f Fg(Q)2607 1082 y Fh(n)2689 1067 y Fq(=)29 b Fg(Q)40 b Fq(+)22 b Fo(n;)51 b(n)29 b Fl(2)g Fg(Z)3379 1031 y Fh(d)3417 1067 y Fo(;)456 1183 y Fq(form)44 b(a)h(partition)e(of)h Fg(R)1401 1147 y Fh(d)1448 1183 y Fo(:)h Fq(With)g(this)f(partition)g(w)m(e)i(asso)s (ciate)f(the)g(classes)456 1299 y Fo(`)497 1314 y Fh(r)534 1299 y Fq(\()p Fg(Z)641 1263 y Fh(d)679 1299 y Fq(;)17 b Fo(L)789 1314 y Fh(q)827 1299 y Fq(\()p Fg(Q)12 b Fq(\)\))p Fo(;)59 b Fq(0)33 b Fo(<)h(r)m(;)17 b(q)38 b(<)33 b Fl(1)p Fo(;)j Fq(of)g(all)e(functions)i(on)g Fg(R)2612 1263 y Fh(d)2659 1299 y Fq(,)h(for)f(whic)m(h)g(the)h(se-)456 1418 y(quence)31 b(of)e Fo(L)949 1433 y Fh(q)988 1418 y Fq(\()p Fg(Q)1097 1433 y Fh(n)1150 1418 y Fq(\)-norms)g(b)s(elongs)g (to)h Fo(`)2016 1433 y Fh(r)2054 1418 y Fq(\()p Fg(Z)2161 1381 y Fh(d)2198 1418 y Fq(\))p Fo(:)g Fq(Let)g(\010)g(b)s(e)g(the)g(F) -8 b(ourier)29 b(trans-)456 1536 y(form)i(on)h Fg(R)887 1500 y Fh(d)966 1536 y Fq(de\014ned)i(on)f(the)g(Sc)m(h)m(w)m(artz)h (class)f(b)m(y)h(the)f(form)m(ula)1229 1728 y(\010)p Fo(u)p Fq(\()p Fo(\030)5 b Fq(\))27 b(=)g(\(2)p Fo(\031)t Fq(\))1793 1687 y Fk(\000)p Fh(d=)p Fj(2)1975 1611 y Ff(Z)2075 1728 y Fo(e)2120 1687 y Fk(\000)p Fh(i)p Fk(h)p Fh(\030)s(;x)p Fk(i)2351 1728 y Fo(u)p Fq(\()p Fo(x)p Fq(\))p Fo(dx:)456 1921 y Fq(W)-8 b(e)33 b(write)38 b(^)-55 b Fo(u)27 b Fq(:=)h(\010)p Fo(u:)555 2037 y Fq(In)33 b(our)g(example)f Fn(H)27 b Fq(=)h Fo(L)1502 2052 y Fj(2)1541 2037 y Fq(\()p Fg(R)1645 2001 y Fh(d)1692 2037 y Fq(\))p Fo(;)49 b(d)p Fq([)p Fo(a)p Fq(])28 b(=)f Fo(H)2182 2001 y Fh(`)2215 2037 y Fq(\()p Fg(R)2319 2001 y Fh(d)2365 2037 y Fq(\))p Fo(;)49 b(`)28 b Fl(2)g Fg(N)47 b Fq(and)829 2235 y Fo(a)p Fq([)p Fo(u;)17 b(u)p Fq(])27 b(=)1220 2118 y Ff(Z)1320 2235 y Fl(j)p Fo(\030)5 b Fl(j)1424 2194 y Fj(2)p Fh(`)1491 2235 y Fl(j)h Fq(^)-55 b Fo(u)o Fq(\()p Fo(\030)5 b Fq(\))p Fl(j)1726 2194 y Fj(2)1765 2235 y Fo(d\030)26 b Fq(+)2023 2152 y Ff(X)1983 2340 y Fk(j)p Fh(\014)s Fk(j\024)p Fh(m)2199 2118 y Ff(Z)2299 2235 y Fo(f)2347 2250 y Fh(\014)2394 2235 y Fq(\()p Fo(x)p Fq(\))p Fl(j)p Fo(D)2637 2194 y Fh(\014)2684 2235 y Fo(u)p Fq(\()p Fo(x)p Fq(\))p Fl(j)2899 2194 y Fj(2)2938 2235 y Fo(dx;)200 b Fq(\(30\))1415 2503 y Fl(j)p Fo(\014)6 b Fl(j)27 b(\024)h Fo(m)g Fl(2)g Fg(N)9 b Fo(;)55 b Fq(0)27 b Fl(\024)i Fo(m)f(<)f(`:)456 2646 y Fq(As)36 b Fo(V)57 b Fq(w)m(e)37 b(tak)m(e)g(an)f(op)s(erator)f(of)h(m)m(ultiplication)31 b(b)m(y)37 b(a)f(measurable)f(function)456 2762 y Fo(V)49 b Fq(:)28 b Fg(R)683 2726 y Fh(d)757 2762 y Fl(\000)-16 b(!)27 b Fg(R)5 b Fq(.)49 b(Let)33 b(us)g(put)1752 2885 y Fo(p)27 b Fq(=)h Fo(d=)p Fq(2)p Fo(`:)456 3029 y Fq(W)-8 b(e)33 b(b)s(orro)m(w)f(from)f([2])i(the)g(follo)m(wing)d(assumptions:) 1099 3197 y Fo(f)1147 3212 y Fh(\014)1221 3197 y Fq(:)e Fg(R)1342 3156 y Fh(d)1416 3197 y Fl(\000)-16 b(!)27 b Fg(R)5 b Fo(;)88 b(f)1833 3212 y Fh(\014)1908 3197 y Fl(2)28 b Fo(L)2068 3212 y Fk(1)2143 3197 y Fq(\()p Fg(R)2247 3156 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Fk(\000)p Fh(d)2334 4860 y Fo(!)2395 4875 y Fh(d)2452 4743 y Ff(Z)2551 4860 y Fo(V)2630 4813 y Fh(p)2608 4879 y Fj(+)2670 4860 y Fo(dx;)468 b Fq(\(34\))1100 5148 y(lim)1075 5198 y Fh(\013)p Fk(!1)1278 5148 y Fo(\013)1341 5107 y Fk(\000)p Fh(p)1435 5148 y Fo(N)10 b Fq(\()p Fl(\000)p Fo(V)23 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b(=)i(\(2)p Fo(\031)t Fq(\))2278 5107 y Fk(\000)p Fh(d)2373 5148 y Fo(!)2434 5163 y Fh(d)2491 5031 y Ff(Z)2590 5148 y Fo(V)2669 5101 y Fh(p)2647 5167 y Fk(\000)2708 5148 y Fo(dx:)430 b Fq(\(35\))p eop %%Page: 11 11 11 10 bop 905 236 a Fi(THE)34 b(DISCRETE)f(SPECTR)n(UM)h(IN)e(THE)i (SPECTRAL)f(GAPS)373 b(11)594 425 y Fx(Theorem)37 b(4.1.)49 b Fm(L)-5 b(et)41 b Fo(a;)17 b(v)44 b Fm(b)-5 b(e)40 b(the)g(forms)g(de\014ne)-5 b(d)39 b(as)h(in)48 b Fq(\(30\))40 b Fm(and)50 b Fq(\(33\))p Fm(,)456 544 y(and)35 b(let)h(the)g(c)-5 b(onditions)43 b Fq(\(31\))36 b Fm(and)45 b Fq(\(32\))36 b Fm(b)-5 b(e)35 b(ful\014l)5 b(le)-5 b(d.)48 b(Then)35 b(for)h(al)5 b(l)35 b 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b(\(2)p Fo(\031)t Fq(\))1179 1375 y Fk(\000)p Fh(d)1274 1412 y Fo(!)1335 1427 y Fh(d)1392 1341 y Ff(R)1464 1412 y Fo(V)1543 1364 y Fh(p)1521 1431 y Fj(+)1582 1412 y Fo(dx)50 b Fq(and)f Fo(J)1998 1427 y Fk(\000)2113 1412 y Fq(=)56 b(\(2)p Fo(\031)t Fq(\))2429 1375 y Fk(\000)p Fh(d)2524 1412 y Fo(!)2585 1427 y Fh(d)2642 1341 y Ff(R)2714 1412 y Fo(V)2792 1364 y Fh(p)2771 1431 y Fk(\000)2832 1412 y Fo(dx:)49 b Fq(As)h(it)e(w)m(as)456 1528 y(noted,)31 b(the)h(condition)e(\(32\))g(implies)f(\(8\))o(.)43 b(The)32 b(asymptotics)f(\(34\))o(,\(35\))g(mean)456 1644 y Fo(N)10 b Fq(\()p Fo(V)22 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))26 b Fl(\030)i Fo(\013)1101 1608 y Fh(p)1141 1644 y Fo(J)1195 1659 y Fj(+)1254 1644 y Fo(;)17 b(\013)28 b Fl(!)f(1)p Fo(;)d Fq(and)g Fo(N)10 b Fq(\()p Fl(\000)p Fo(V)23 b Fq(;)17 b Fo(\025;)g(\013)q Fq(\))27 b Fl(\030)h Fo(\013)2571 1608 y Fh(p)2610 1644 y Fo(J)2664 1659 y Fk(\000)2723 1644 y Fo(;)17 b(\013)28 b Fl(!)f(1)p Fq(.)41 b(T)-8 b(aking)456 1760 y(in)m(to)31 b(accoun)m(t)j(\(12\))o(,)f(it)e (remains)h(to)g(note)h(that)f(\(see)i([2]\))1294 1932 y(lim)1292 1989 y Fh(s)p Fk(!)p Fj(0)1447 1932 y Fo(s)1493 1891 y Fh(p)1533 1932 y Fo(n)1591 1947 y Fj(+)1650 1932 y Fq(\()p Fo(s;)17 b(X)8 b Fq(\()p Fo(\025)p Fq(\))p Fo(S)e Fq(\))27 b(=)g Fo(J)2288 1947 y Fj(+)2370 1932 y Fq(+)22 b Fo(J)2522 1947 y Fk(\000)2581 1932 y Fo(;)1443 2150 y Fq(lim)1441 2207 y Fh(s)p Fk(!)p Fj(0)1596 2150 y Fo(s)1642 2109 y Fh(p)1682 2150 y Fo(n)1740 2165 y Fk(\000)1799 2150 y Fq(\()p Fo(s;)17 b(X)8 b Fq(\()p Fo(\025)p Fq(\))p Fo(S)e Fq(\))27 b(=)g(0)p Fo(:)456 2328 y Fq(The)33 b(theorem)f(is)h(pro)m(v)m(ed.)45 b Fd(2)1660 2567 y Fp(References)497 2725 y Fw([1])c(Alama,)34 b(S.,)h(Deift,)h(P)-7 b(.A.,)35 b(Hemp)r(el,)g(R.:)48 b(Eigen)n(v)-5 b(alue)32 b(brunc)n(hes)h(of)g(the)g(Sc)n(hr\177)-42 b(odinger)626 2825 y(op)r(erator)20 b Fu(H)12 b Fs(\000)5 b Fu(\025W)33 b Fw(in)22 b(a)e(gap)h(of)g Fu(\033)s Fw(\()p Fu(H)7 b Fw(\).)22 b(Comm)n(un.)f(Math.)g(Ph)n(ys.)f Fb(121)p Fw(,)i(291-321)c(\(1989\))497 2924 y([2])41 b(Birman,)c(M.Sh.:)53 b(Discrete)35 b(sp)r(ectrum)h(in)f(the)h(gaps)e (of)i(a)f(con)n(tin)n(uous)f(one)h(for)g(p)r(er-)626 3024 y(turbations)d(with)g(large)f(coupling)h(constan)n(ts.)f(Adv)-5 b(ances)32 b(in)h(So)n(viet)e(Math.)h Fb(7)p Fw(,)i(57-73)626 3123 y(\(1991\))497 3223 y([3])41 b(Birman,)g(M.S.,)g(Solomjak,)g (M.Z.:)58 b(Sp)r(ectral)39 b(theory)e(of)i(self-adjoin)n(t)f(op)r (erators)e(in)626 3323 y(Hilb)r(ert)31 b(space.)e(Mathematics)h(and)f (Its)h(Applications,)h(D.)f(Reidel)h(Publishing)e(Com-)626 3422 y(pan)n(y)-7 b(,)27 b(1987)497 3522 y([4])41 b(Birman,)28 b(M.Sh.,)i(Solom)n(y)n(ak,)c(M.Z.:)39 b(Estimates)28 b(for)g(the)g(n)n(um)n(b)r(er)h(of)f(negativ)n(e)f(eigen-)626 3622 y(v)-5 b(alues)27 b(of)g(the)g(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erator)g(and)i(its)g(generalizations.)d(Adv)-5 b(ances)27 b(in)g(So)n(viet)626 3721 y(Math.)h Fb(7)p Fw(,)f(1-55)f(\(1991\))497 3821 y([5])41 b(Deift,)c(P)-7 b(.A.,)36 b(Hemp)r(el,)h(R.:)50 b(On)34 b(the)g(existence)g(of)g(eigen) n(v)-5 b(alues)33 b(of)h(the)g(Sc)n(hr\177)-42 b(odinger)626 3920 y(op)r(erator)20 b Fu(H)12 b Fs(\000)5 b Fu(\025W)33 b Fw(in)22 b(a)e(gap)h(of)g Fu(\033)s Fw(\()p Fu(H)7 b Fw(\).)22 b(Comm)n(un.)f(Math.)g(Ph)n(ys.)f Fb(103)p Fw(,)i(461-490)c(\(1986\))497 4020 y([6])41 b(Gesztesy)-7 b(,)29 b(F.,)g(Gurarie,)f(D.,)h(Holden,)g(H.,)h(Klaus,)e(M.,)h(Sadun,)g (L.,)g(Simon,)g(B.,)g(V)-7 b(ogl,)626 4120 y(P)g(.:)45 b(T)-7 b(rapping)31 b(and)h(Cascading)e(of)i(eigen)n(v)-5 b(alues)31 b(in)h(the)g(large)e(coupling)h(limit.)i(Com-)626 4219 y(m)n(un.)28 b(Math.)g(Ph)n(ys.)f Fb(118)p Fw(,)f(597-634)f (\(1988\))497 4319 y([7])41 b(Gesztesy)-7 b(,)25 b(F.,)g(Simon,)f(B.:) 35 b(On)24 b(a)g(Theorem)f(of)h(Deift)h(and)f(Hemp)r(el.)g(Comm)n(un.)g (Math.)626 4419 y(Ph)n(ys.)j Fb(116)p Fw(,)g(503-505)d(\(1988\))497 4518 y([8])41 b(Hemp)r(el,)36 b(R.:)49 b(On)33 b(the)h(asymptotic)e (distribution)i(of)f(the)h(eigen)n(v)-5 b(alue)32 b(branc)n(hes)h(of)g (a)626 4618 y(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erator)g Fu(H)f Fs(\000)16 b Fu(\025W)40 b Fw(in)27 b(a)f(sp)r(ectral)g(gap)g (of)h Fu(H)7 b Fw(.)27 b(J.)g(Reine)g(Angew.)f(Math.)626 4717 y Fb(399)p Fw(,)h(38-59)f(\(1989\))497 4817 y([9])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 4917 y(ditions.)g(J.)f(Math.)h(Anal.)g(Appl.)g Fb(169)p Fw(,)f(no.)g(1,)h(229-259)c(\(1992\))456 5016 y([10])40 b(Hemp)r(el,)22 b(R.:)33 b(On)18 b(the)h(asymptotic)g(distribution)g (of)g(eigen)n(v)-5 b(alues)17 b(in)i(gaps.)f(Pro)r(ceedings)626 5116 y(of)30 b(the)g(IMA-w)n(orkshop)e(on)h(quasiclassical)f(metho)r (ds,)j(Ma)n(y)d(1995;)h(The)h(IMA)g(V)-7 b(ol.)30 b(in)626 5216 y(Math.)e(and)f(its)h(Appl.,)h(Springer-V)-7 b(erlag,)25 b(to)i(app)r(ear)p eop %%Page: 12 12 12 11 bop 456 236 a Fi(12)1142 b(O.)34 b(SAFR)n(ONO)n(V)456 425 y Fw([11])40 b(Klaus,)33 b(M.:)47 b(On)33 b(the)f(p)r(oin)n(t)h(sp) r(ectrum)g(of)f(Dirac)g(op)r(erators.)f(Helv.)i(Ph)n(ys.)e(Acta)i Fb(53)p Fw(,)626 525 y(453-462)25 b(\(1980\))456 624 y([12])40 b(Klaus,)28 b(M.:)38 b(Some)28 b(applications)g(of)g(the)h (Birman-Sc)n(h)n(winger)d(principle.)i(Helv.)g(Ph)n(ys.)626 724 y(Acta)g Fb(55)p Fw(,)f(49-68)f(\(1982\))456 824 y([13])40 b(Konno,)30 b(R.,)i(Kuro)r(da,)e(S.T.:)43 b(On)30 b(the)h(\014niteness)f(of)g(p)r(erturb)r(ed)h(eigen)n(v)-5 b(alues.)29 b(J.)i(F)-7 b(ac.)626 923 y(Sci.)28 b(Univ.)g(T)-7 b(oky)n(o)26 b(Sect.)i(I,)g Fb(13)p Fw(,)f(55-63)e(\(1966\))456 1023 y([14])40 b(Lev)n(endorski)-9 b(\025)-32 b(\020,)36 b(S.Z.:)55 b(Lo)n(w)n(er)34 b(b)r(ound)j(for)f(the)h(n)n(um)n(b)r(er)f (of)g(eigen)n(v)-5 b(alue)36 b(branc)n(hes)f(for)626 1123 y(the)h(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erator)g Fu(H)d Fs(\000)23 b Fu(\025W)47 b Fw(in)35 b(a)g(gap)f(of)h Fu(H)7 b Fw(:)51 b(the)36 b(case)e(of)h(inde\014nite)g Fu(W)n(:)626 1222 y Fw(Comm)n(un.)28 b(P)-7 b(.D.E.)27 b Fb(20)p Fw(,)g(827-854)e(\(1995\))456 1322 y([15])40 b(Safrono)n(v,)d(O.L.:)54 b(Discrete)36 b(sp)r(ectrum)g(in)h(a)e(gap)h (of)g(the)g(con)n(tin)n(uous)g(sp)r(ectrum)g(for)626 1421 y(v)-5 b(ariable)23 b(sign)g(p)r(erturbations)g(with)i(large)d (coupling)h(constan)n(t.)g(St.)i(P)n(etersburg)c(Math.)626 1521 y(Journal)26 b Fb(8)p Fw(,)i(No.)f(2)h(\(1997\))456 1621 y([16])40 b(Safrono)n(v,)31 b(O.L.:)45 b(The)32 b(discrete)g(sp)r(ectrum)g(in)g(the)g(gaps)f(of)h(the)g(con)n(tin)n (uous)f(one)g(for)626 1720 y(non-signde\014nite)21 b(p)r(erturbations)f (with)h(a)g(large)f(coupling)g(constan)n(t.)g(Comm)n(un.)h(Math.)626 1820 y(Ph)n(ys)27 b Fb(193)g Fw(1,)g(233-243)d(\(1998\).)456 1920 y([17])40 b(Simon,)29 b(B.:)40 b(Bro)n(wnian)27 b(motion,)i Fu(L)1788 1889 y Fa(p)1854 1920 y Fw(prop)r(erties)f(of)h (Sc)n(hr\177)-42 b(odinger)27 b(op)r(erators)g(and)h(the)626 2019 y(lo)r(calization)f(of)g(binding.)h(Journal)e(of)i(F)-7 b(unctional)28 b(Analysis)f Fb(35)p Fw(,)g(215-229)d(\(1980\))555 2206 y Fv(O.L.)50 b(Safr)n(ono)n(v,)55 b(Dep)-6 b(ar)g(tment)52 b(of)f(Ma)-6 b(thema)g(tics,)57 b(R)n(o)n(y)-6 b(al)50 b(Institute)h(of)456 2306 y(Technology,)24 b(10044)e(Stockholm,)h (Sweden.)38 b(e-mail:)f(safr)n(ono)n(v@ma)-6 b(th.kth.se)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0008251206635--