This is a multi-part message in MIME format. ---------------9909100535267 Content-Type: text/plain; name="99-332.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-332.keywords" spectral shift function, asymptotics, coupling constant ---------------9909100535267 Content-Type: application/postscript; name="preprint408.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint408.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: xi.dvi %%CreationDate: Fri Sep 10 10:15:53 1999 %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\emtex\bin\dvips32.exe xi -o xi.ps -D 300 -O %+ 0mm,0mm %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.09.09:2005 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 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b(is)e(monotone)g(decreasing)g(in)g Fr(\013)h Ft(\(for)228 621 y(a)e(\014xed)g Fr(\025)e Ft(=)p 480 581 29 2 v 14 w Fr(\025)p Ft(\))i(and)h(coincides)d(with)i (a)g(certain)g(in)o(tegral)f(of)h(the)g(distribution)228 679 y(function)j(of)h(a)g(compact)e(selfadjoin)o(t)h(op)q(erator.)32 b(A)19 b(suitable)g(v)o(ersion)g(of)g(the)228 737 y(represen)o(tation)f (form)o(ula)f(can)h(b)q(e)h(found)g(in)f([6)o(],)g([7].)28 b(In)18 b([7],)g(the)g(said)g(ab)q(o)o(v)o(e)228 795 y(forms)d(a)i(basis)g(of)f(the)g(pro)q(of)i(of)e(a)h(stabilit)o(y)e (theorem)f(for)j Fr(V)25 b(<)14 b Ft(0)p Fr(:)278 853 y Ft(The)g(study)h(of)g(the)f(p)q(erturbations)i(\(1\))f(requires,)e (\014st)i(of)g(all,)f(a)h(new)g(v)o(ersion)228 911 y(of)f(represen)o (tation)g(form)o(ula)f(for)h Fr(\030)r Ft(\()p Fr(\025;)8 b(\013)p Ft(\).)22 b(Suc)o(h)14 b(a)h(form)o(ula)e(is)h(obtained)g(in)g ([4].)228 969 y(Moreo)o(v)o(er,)g(the)h(tec)o(hnique)f(used)i(in)g ([8],)e([9])i(can)g(b)q(e)f(also)i(simpli\014ed)c(with)i(the)228 1028 y(help)g(of)i(this)f(represen)o(tation.)278 1086 y FA(Ac)n(kno)n(wledgmen)n(ts.)33 b Ft(The)20 b(author)i(is)e(grateful) h(to)f(A.)g(Pushnitskii)g(for)228 1144 y(his)c(suggestions)h(and)g (remarks)e(to)i(the)f(text.)601 1277 y(1.)27 b Fs(Main)18 b(resul)m(t)g(\(Theorem)g(1.1\))278 1364 y FA(1.)24 b(Notations)14 b Ft(Throughout)j(the)e(pap)q(er)h(form)o(ulae)e(and)i(statemen)o(ts)d (with)228 1422 y(double)j(indices)f(\()p Fo(\006)h Ft(or)g Fo(\007)p Ft(\))g(are)g(understo)q(o)q(d)i(indep)q(enden)o(tly)l(,)c (as)j(pairs)f(of)h(for-)228 1481 y(m)o(ulae.)h(Belo)o(w)13 b Fo(H)h Ft(is)f(a)h(separable)g(Hilb)q(ert)e(space.)21 b(By)13 b Fr(\032)p Ft(\()p Fr(T)7 b Ft(\),)13 b Fr(\033)r Ft(\()p Fr(T)7 b Ft(\))12 b(w)o(e)h(denote)228 1539 y(resp)q(ectiv)o (ely)f(the)i(resolv)o(en)o(t)g(set,)g(the)g(sp)q(ectrum)g(of)g(a)h (linear)f(op)q(erator)i Fr(T)7 b Ft(.)20 b(F)l(or)228 1597 y(a)f(selfadjoin)o(t)f(op)q(erator)h Fr(T)25 b Ft(let)18 b Fr(E)869 1604 y Fk(T)896 1597 y Ft(\()p Fr(\016)r 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624 y Ft(is)22 b(understo)q(o)q(d)i(in)d(the)h(sense)g(of)g(quadratic)g (forms)g(\(see)f([10)q(]\).)38 b(F)l(or)22 b(a)h(real)228 682 y(v)m(alued)16 b(function)g Fr(f)5 b Ft(\()p Fo(\001)p Ft(\))16 b(\(de\014ned)g(on)h Fm(R)p Ft(\))c(suc)o(h)j(that)839 770 y Fo(\006)p Fr(f)5 b Ft(\()p Fo(\006)p Ft(1\))14 b Fr(>)f Ft(0)p Fr(;)228 858 y Ft(w)o(e)j(in)o(tro)q(duce)f(the)h(op)q (erator)834 946 y Fr(V)862 953 y Fk(f)899 946 y Ft(=)d Fr(f)5 b Ft(\()p Fr(J)g Ft(\))p Fo(j)p Fr(V)11 b Fo(j)228 1034 y Ft(and)17 b(de\014ne)f(the)g(family)e(of)i(p)q(erturb)q(ed)h(op) q(erators)688 1122 y Fr(H)728 1129 y Fk(f)751 1122 y Ft(\()p Fr(\013)p Ft(\))d(=)g Fr(H)926 1129 y Fq(0)957 1122 y Ft(+)d Fr(\013V)1065 1129 y Fk(f)1089 1122 y Fr(;)24 b(\013)14 b(>)g Ft(0)p Fr(:)228 1210 y Ft(The)k(main)f(question)h(of)h (our)g(pap)q(er)g(is)f(ho)o(w)h(to)g(describ)q(e)e(the)h(motion)g(of)g (the)228 1268 y(sp)q(ectrum)h(of)i Fr(H)t Ft(\()p Fr(\013)p Ft(\))g(as)g Fr(\013)g Ft(gro)o(ws.)34 b(The)21 b(family)d(of)j(op)q 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Fr(tB)1147 1851 y Fk(\025)1169 1844 y Ft(\))p Fr(;)d(f)d Ft(\()p Fr(J)g Ft(\))1309 1824 y Fp(\000)p Fq(1)1356 1844 y Ft(\))p Fr(dw)q Ft(\()p Fr(t)p Ft(\))p Fr(;)116 b Ft(\(2.8\))714 1957 y Fr(dw)q Ft(\()p Fr(t)p Ft(\))14 b(=)f Fr(\031)926 1936 y Fp(\000)p Fq(1)973 1957 y Ft(\(1)e(+)g Fr(t)1094 1936 y Fq(2)1114 1957 y Ft(\))1133 1936 y Fp(\000)p Fq(1)1180 1957 y Fr(dt:)278 2046 y Fg(Pr)n(o)n(of.)32 b Ft(F)l(or)20 b(ev)o(ery)f(pair)h(of)h (selfadjoin)o(t)e(b)q(ounded)i(op)q(erators)h Fr(R;)8 b(K)t Ft(,)21 b(suc)o(h)228 2104 y(that)16 b(0)f Fo(2)f Fr(\032)p Ft(\()p Fr(R)p Ft(\))j(and)f Fr(K)i Fo(2)c FA(S)767 2111 y Fp(1)821 2104 y Ft(w)o(e)i(in)o(tro)q(duce)g(the)g(op)q (erators)707 2185 y Fr(R)744 2192 y Fk(s)762 2185 y Fr(:)d Ft(=)h Fr(S)s(RS;)41 b(K)1037 2192 y Fk(s)1070 2185 y Ft(=)14 b Fr(S)s(K)t(S;)228 2267 y Ft(where)23 b Fr(S)29 b Ft(=)d Fr(S)532 2249 y Fp(\003)575 2267 y Ft(is)d(a)h(b)q(ounded)h (in)o(v)o(ertible)20 b(op)q(erator.)45 b(Then)23 b(in)g(order)h(to)228 2325 y(establish)16 b(\(2.8\))g(it)g(is)g(su\016cien)o(t)f(to)i(pro)o (v)o(e)e(that)599 2407 y Fr(\021)r Ft(\(0;)8 b Fr(R)727 2414 y Fk(s)757 2407 y Ft(+)j Fr(K)847 2414 y Fk(s)866 2407 y Fr(;)d(R)925 2414 y Fk(s)943 2407 y Ft(\))14 b(=)g Fr(\021)r Ft(\(0;)8 b Fr(R)j Ft(+)g Fr(K)q(;)d(R)p Ft(\))p Fr(:)272 b Ft(\(2.9\))228 2492 y(Moreo)o(v)o(er)11 b(the)i(substan)o (tiation)h(of)i(\(2.9\))d(can)g(b)q(e)g(reduced)f(to)i(the)e(cases)h Fr(K)18 b Fo(\025)c Ft(0)228 2550 y(and)19 b Fr(K)j Fo(\024)c Ft(0.)29 b(But)19 b(if)f Fr(K)23 b Ft(is)18 b(of)h(de\014nite)f(sign,)h (the)g(quan)o(tit)o(y)e Fr(\021)r Ft(\(0;)8 b Fr(R)14 b Ft(+)e Fr(K)q(;)c(R)p Ft(\))228 2608 y(coincides)13 b(with)h(the)g(n)o(um)o(b)q(er)f(of)h(eigen)o(v)m(alues)f(of)i(the)f (op)q(erator)h Fr(R)7 b Ft(+)g Fr(tK)19 b Ft(whic)o(h)p eop %%Page: 6 6 6 5 bop 228 118 a Fl(6)590 b(O.)16 b(SAFR)o(ONO)o(V)228 213 y Ft(pass)25 b(p)q(oin)o(t)f(0)h(as)f Fr(t)g Ft(gro)o(ws)h(from)e (0)i(to)f(1.)45 b(Therefore)24 b(\(2.9\))g(follo)o(ws)g(from)228 271 y(equiv)m(alence)14 b(of)j(the)f(t)o(w)o(o)g(statemen)o(ts:)600 391 y(0)e Fo(2)g Fr(\033)r Ft(\()p Fr(R)d Ft(+)g Fr(tK)t Ft(\))j Fo(,)g Ft(0)g Fo(2)g Fr(\033)r Ft(\()p Fr(R)1162 398 y Fk(s)1191 391 y Ft(+)d Fr(tK)1299 398 y Fk(s)1317 391 y Ft(\))p Fr(:)228 465 y Ft(No)o(w)18 b(to)h(complete)d(the)i(pro)q (of)i(let)e(us)h(tak)o(e)f Fr(S)i Ft(=)e Fo(j)p Fr(f)5 b Ft(\()p Fr(J)g Ft(\))p Fo(j)1306 447 y Fp(\000)p Fq(1)p Fk(=)p Fq(2)1388 465 y Fr(;)26 b(R)18 b Ft(=)g(sign)p Fr(f)5 b Ft(\()p Fr(J)g Ft(\))228 524 y(and)17 b Fr(K)364 531 y Fk(s)396 524 y Ft(=)d Fr(\013)p Ft(\()p Fr(A)535 531 y Fk(\025)569 524 y Ft(+)d Fr(tB)673 531 y Fk(\025)695 524 y Ft(\))16 b(in)g(\(2.9\).)103 b Ff(2)278 582 y FA(2.)29 b Ft(The)20 b(rest)e(of)i(the)f(section)f(is)h(dev)o(oted)g(to)g(the)g (pro)q(of)h(of)g(Theorem)d(1.1.)228 640 y(Consider)f(the)g(function)708 723 y Fr(\021)r Ft(\(0;)8 b Fr(J)15 b Ft(+)c Fr(\013)p Ft(\()p Fr(A)977 730 y Fk(\025)1011 723 y Ft(+)g 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Fr(\013)i Fo(2)g Fm(R)737 1113 y Fq(+)783 1106 y Ft(:)38 b(0)20 b Fo(2)g Fr(\032)p Ft(\()p Fr(J)e Ft(+)13 b Fr(\013X)1143 1113 y Fk(\026)1167 1106 y Ft(\))p Fo(g)19 b Ft(and)i(using)e(the)h(Birman-)228 1164 y(Sc)o(h)o(winger)15 b(principle)g(\(see)g(Prop)q(osition)j(1.1\)) e(w)o(e)g(obtain)441 1251 y Fo(\000)p Fr(n)509 1258 y Fp(\000)538 1251 y Ft(\(1)c Fo(\000)f Ft(0)p Fr(;)d(T)725 1231 y Fq(\(+\))781 1251 y Ft(\))14 b Fo(\024)g Fr(\020)t Ft(\()p Fr(\013)p Ft(\))g Fo(\024)f Fr(n)1056 1258 y Fq(+)1086 1251 y Ft(\(1)p Fr(;)8 b(T)1187 1231 y Fq(\()p Fp(\000)p Fq(\))1244 1251 y Ft(\))p Fr(;)89 b(\013)14 b Fo(2)g(Y)t Fr(;)228 1334 y Ft(where)644 1398 y Fr(T)680 1377 y Fq(\()p Fp(\006)p Fq(\))750 1398 y Ft(:=)815 1346 y Fn(q)p 857 1346 39 2 v 52 x Fr(Q)895 1421 y Fp(\006)925 1398 y Ft(\()p Fr(J)h Ft(+)c Fr(\013X)1106 1405 y Fk(\026)1131 1398 y Ft(\))1150 1377 y Fp(\000)p Fq(1)1197 1346 y Fn(q)p 1238 1346 V 1238 1398 a Fr(Q)1277 1421 y Fp(\006)228 1478 y Ft(and)641 1540 y Fr(Q)i Ft(=)h Fr(Q)p Ft(\()p Fr(t)p Ft(\))f(=)h Fr(\013)p Ft(\()p Fr(A)992 1547 y Fk(\025)1026 1540 y Ft(+)d Fr(tB)1130 1547 y Fk(\025)1163 1540 y Fo(\000)g Fr(X)1253 1547 y Fk(\026)1277 1540 y Ft(\))p Fr(:)228 1613 y Ft(By)k(the)h(Horn)h(inequalit)o(y)e(\(1.1\),) 488 1700 y Fr(n)517 1707 y Fp(\006)546 1700 y Ft(\(1)p Fr(;)8 b(T)647 1679 y Fq(\()p Fp(\007)p Fq(\))704 1700 y Ft(\))14 b Fo(\024)f Ft(2)p Fr(n)p Ft(\()p Fr(\017;)8 b(Q)p Ft(\))j(+)g Fr(n)p Ft(\()p Fr(\017)1089 1679 y Fp(\000)p Fq(1)1136 1700 y Fr(;)d Ft(\()p Fr(J)15 b Ft(+)c Fr(\013X)1339 1707 y Fk(\026)1364 1700 y Ft(\))1383 1679 y Fp(\000)p Fq(1)1430 1700 y Ft(\))p Fr(:)228 1782 y Ft(Since)489 1798 y Fn(Z)531 1811 y Fq(+)p Fp(1)512 1892 y(\0001)604 1856 y Fr(n)p Ft(\()p Fr(\017;)d(Q)p Ft(\()p Fr(t)p Ft(\)\)\(1)i(+)h Fr(t)928 1836 y Fq(2)947 1856 y Ft(\))966 1836 y Fp(\000)p Fq(1)1013 1856 y Fr(dt)j Ft(=)g Fr(o)p Ft(\()p Fr(\013)1195 1836 y Fk(p)1215 1856 y Ft(\))p Fr(;)41 b(\013)14 b Fo(!)g(1)p Fr(;)228 1949 y Ft(it)i(is)g(su\016cien)o(t)e(to)j(establish)f(that)562 2032 y(lim)8 b(sup)590 2066 y Fk(\013)p Fp(!1)720 2032 y Fr(\013)751 2011 y Fp(\000)p Fk(p)798 2032 y Fr(n)p Ft(\()p Fr(\017)866 2011 y Fp(\000)p Fq(1)913 2032 y Fr(;)g Ft(\()p Fr(J)16 b Ft(+)11 b Fr(\013X)1117 2039 y Fk(\026)1141 2032 y Ft(\))1160 2011 y Fp(\000)p Fq(1)1207 2032 y Ft(\))j Fo(\024)f Fr(C)t Ft(\()p Fr(\017)p Ft(\))228 2134 y(and)837 2196 y(lim)837 2225 y Fk(\017)p Fp(!)p Fq(0)913 2196 y Fr(C)t Ft(\()p Fr(\017)p Ft(\))g(=)h(0)p Fr(:)228 2282 y Ft(First)i(of)g(all)g(note)g(that)491 2365 y Fr(n)p Ft(\()p Fr(\017)559 2344 y Fp(\000)p Fq(1)606 2365 y Fr(;)8 b Ft(\()p Fr(J)15 b Ft(+)c Fr(\013X)809 2372 y Fk(\026)834 2365 y Ft(\))853 2344 y Fp(\000)p Fq(1)900 2365 y Ft(\))j(=)f(rank)q Fr(E)1117 2373 y Fq(\()p Fk(J)s Fq(+)p Fk(\013X)1232 2377 y Fe(\026)1253 2373 y Fq(\))1268 2365 y Ft(\()p Fo(\000)p Fr(\017;)8 b(\017)p Ft(\))13 b Fo(\024)451 2452 y(\024)h Fr(\021)r Ft(\()p Fr(\017)p Ft(;)8 b Fr(J)15 b Ft(+)c Fr(\013X)753 2459 y Fk(\026)777 2452 y Fr(;)d(J)d Ft(\))10 b Fo(\000)h Fr(\021)r Ft(\()p Fo(\000)p Fr(\017)p Ft(;)d Fr(J)15 b Ft(+)c Fr(\013X)1198 2459 y Fk(\026)1222 2452 y Fr(;)d(J)d Ft(\))13 b(=:)g Fr(m)p Ft(\()p Fr(\013)p Ft(\))p Fr(:)228 2525 y Ft(It)j(is)g(easy)g(to)h(see)e(from)h(the)g(de\014nition)f(of)i (the)f(function)g Fr(\021)r Ft(,)g(that)278 2608 y Fr(m)p Ft(\()p Fr(\013)p Ft(\))e(=)f Fr(\021)r Ft(\(0)p Fr(;)8 b Ft(\()p Fr(f)589 2615 y Fq(+)619 2608 y Ft(\()p Fr(J)d Ft(\))11 b(+)g Fr(\013X)820 2615 y Fk(\026)844 2608 y Ft(\))p Fr(;)d(f)909 2615 y Fq(+)938 2608 y Ft(\()p Fr(J)d Ft(\)\))11 b Fo(\000)f Fr(\021)r Ft(\(0)p Fr(;)e Ft(\()p Fr(f)1221 2615 y Fp(\000)1251 2608 y Ft(\()p Fr(J)d Ft(\))11 b(+)g Fr(\013X)1452 2615 y Fk(\026)1476 2608 y Ft(\))p Fr(;)d(f)1541 2615 y Fp(\000)1570 2608 y Ft(\()p Fr(J)d Ft(\)\))p Fr(;)p eop %%Page: 7 7 7 6 bop 690 118 a Fl(SPECTRAL)16 b(SHIFT)h(FUNCTION)441 b(7)228 213 y Ft(where)805 273 y Fr(f)829 280 y Fp(\006)859 273 y Ft(\()p Fr(J)5 b Ft(\))13 b(=)h Fr(J)h Fo(\007)c Fr(\017I)t(:)228 344 y Ft(Denoting)676 404 y Fr(H)720 384 y Fp(\006)750 404 y Ft(\()p Fr(\013)p Ft(\))j(=)g Fr(H)925 411 y Fq(0)956 404 y Ft(+)d Fr(\013f)1060 411 y Fp(\006)1090 404 y Ft(\()p Fr(J)5 b Ft(\))1160 384 y Fp(\000)p Fq(1)1207 404 y Fo(j)p Fr(V)11 b Fo(j)228 475 y Ft(and)17 b(using)f(the)g(represen)o(tation)g(for)g Fr(\030)r Ft(\()p Fr(\026;)8 b(H)1064 457 y Fp(\006)1095 475 y Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)1226 482 y Fq(0)1246 475 y Ft(\))17 b(\(see)e(\(2.8\))q(\))h(w)o(e)g(obtain)511 556 y Fr(m)p Ft(\()p Fr(\013)p Ft(\))e(=)g Fr(\030)r Ft(\()p Fr(\026;)8 b(H)826 535 y Fq(+)856 556 y Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)987 563 y Fq(0)1008 556 y Ft(\))j Fo(\000)f Fr(\030)r Ft(\()p Fr(\026;)e(H)1224 535 y Fp(\000)1255 556 y Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)1386 563 y Fq(0)1406 556 y Ft(\))p Fr(:)228 636 y Ft(Therefore)400 717 y Fr(m)p Ft(\()p Fr(\013)p Ft(\))14 b Fo(\030)f Ft(\()p Fr(C)632 724 y Fq(+)673 717 y Ft(+)e Fr(C)757 724 y Fp(\000)786 717 y Ft(\)[\(1)g Fo(\000)g Fr(\017)p Ft(\))962 697 y Fp(\000)p Fk(p)1020 717 y Fo(\000)g Ft(\(1)g(+)g Fr(\017)p Ft(\))1212 697 y Fp(\000)p Fk(p)1259 717 y Ft(])p Fr(\013)1304 697 y Fk(p)1324 717 y Fr(;)40 b(\013)14 b Fo(!)g(1)p Fr(;)228 798 y Ft(and)711 859 y(lim)711 887 y Fk(\017)p Fp(!)p Fq(0)787 859 y Ft(lim)8 b(sup)815 893 y Fk(\013)p Fp(!1)945 859 y Fr(\013)976 838 y Fp(\000)p Fk(p)1023 859 y Fr(m)p Ft(\()p Fr(\013)p Ft(\))14 b(=)g(0)p Fr(:)228 949 y Ft(The)i(pro)q(of)h(is)f(complete.)573 1054 y(3.)28 b Fs(Limits)19 b(of)f(comp)m(a)o(ct)h(opera)m(tors)278 1141 y Ft(Here)g(w)o(e)i(presen)o(t)f(some)g(conditions)g(on)i Fr(V)32 b Ft(and)21 b Fr(H)1284 1148 y Fq(0)1325 1141 y Ft(whic)o(h)f(ensure)h(exis-)228 1199 y(tence)13 b(of)18 b(\(1.3\))c(and)h(guaran)o(tee)f(\(1.4\).)21 b(W)l(e)14 b(b)q(egin)g(with)g(the)g(statemen)o(t)e(whic)o(h)228 1258 y(imme)o(diately)g(follo)o(ws)k(from)f(the)g(results)h(of)g([5].) 21 b(F)l(or)16 b(0)e Fr(<)g(q)h Fo(\024)f Ft(1)i(w)o(e)g(de\014ne)f Fr(q)1703 1239 y Fp(\003)737 1338 y Fr(q)761 1318 y Fp(\003)794 1338 y Ft(=)f Fr(q)r(;)154 b Ft(if)19 b Fr(q)c(<)f Ft(1;)649 1422 y Fr(q)673 1401 y Fp(\003)706 1422 y Fr(>)f Ft(1)p Fr(;)8 b Fo(\000)p Ft(an)o(y)17 b(n)o(um)o(b)q(er)p Fr(;)d Ft(if)19 b Fr(q)d Ft(=)d(1)p Fr(:)297 1510 y FA(Prop)r(osition)18 b(3.1.)23 b Fg(Assume)13 b(that)h(for)e(every)h(b)n(ounde)n(d)g(op)n (en)g(interval)h Fr(\016)h Fo(\032)228 1568 y Fm(R)f Fg(the)k(fol)r(lowing)i(inclusion)f(holds:)627 1650 y Fo(j)p Fr(V)11 b Fo(j)694 1630 y Fq(1)p Fk(=)p Fq(2)749 1650 y Fr(E)785 1657 y Fk(H)814 1662 y Fd(0)834 1650 y Ft(\()p Fr(\016)r Ft(\))i Fo(2)h FA(S)987 1657 y Fi(2q)1031 1650 y Fr(;)60 b Ft(0)14 b Fr(<)g(q)h Fo(\024)f Ft(1)p Fr(:)275 b Ft(\(3.10\))228 1731 y Fg(Then)18 b(for)f(a.e.)22 b Fr(\025)15 b Fo(2)f Fm(R)g Fg(the)k(limit)478 1823 y Fr(X)t Ft(\()p Fr(\025;)8 b(\016)r Ft(\))14 b(:=)45 b(lim)713 1853 y Fk(z)q Fp(!)p Fk(\025)p Fq(+)p Fk(i)p Fq(0)853 1771 y Fn(q)p 894 1771 67 2 v 894 1823 a Fo(j)p Fr(V)11 b Fo(j)p Fr(E)997 1830 y Fk(H)1026 1835 y Fd(0)1045 1823 y Ft(\()p Fr(\016)r Ft(\)\()p Fr(H)1166 1830 y Fq(0)1197 1823 y Fo(\000)g Fr(z)r(I)t Ft(\))1317 1803 y Fp(\000)p Fq(1)1363 1771 y Fn(q)p 1405 1771 V 52 x Fo(j)p Fr(V)g Fo(j)228 1923 y Fg(exists)18 b(in)g(the)g FA(S)533 1930 y Fi(q)554 1921 y Fj(\003)574 1923 y Fg(-norm)g(and)803 2003 y Ft(Im)o Fr(X)t Ft(\()p Fr(\025;)8 b(\016)r Ft(\))13 b Fo(2)i FA(S)1110 2010 y Fi(q)1133 2003 y Fr(:)278 2092 y Ft(The)h(follo)o(wing)g(statemen)o(t)e(is)i(a)h(direct)e(consequence) h(of)g(Prop)q(osition)i(3.1.)297 2180 y FA(Prop)r(osition)g(3.2.)23 b Ft([7])33 b Fg(Assume)17 b(that)g(for)g(every)h(b)n(ounde)n(d)f (interval)h Fr(\016)d Fo(\032)228 2238 y Fm(R)777 2319 y Fo(j)p Fr(V)c Fo(j)844 2299 y Fq(1)p Fk(=)p Fq(2)899 2319 y Fr(E)935 2326 y Fk(H)964 2331 y Fd(0)984 2319 y Ft(\()p Fr(\016)r Ft(\))i Fo(2)h FA(S)1137 2326 y Fi(2)1159 2319 y Fr(:)425 b Ft(\(3.11\))228 2403 y Fg(Then)21 b(for)e(a.e.)31 b Fr(\025)20 b Fo(2)f Fm(R)e Fg(the)k(limit)26 b Ft(\(1.3\))20 b Fg(exists)h(and)g(the)f(c)n(ondition)25 b Ft(\(1.4\))20 b Fg(is)228 2461 y(ful\014l)r(le)n(d.)278 2550 y Ft(The)15 b(pro)q(of)g(of)h(the)e(follo)o(wing)h(statemen)o(t)e(is)i(a)g(rep)q (eat)g(of)g(the)g(pro)q(of)h(of)f(The-)228 2608 y(orem)g(4.1)h(in)g ([7].)p eop %%Page: 8 8 8 7 bop 228 118 a Fl(8)590 b(O.)16 b(SAFR)o(ONO)o(V)297 213 y FA(Prop)r(osition)i(3.3.)23 b Fg(L)n(et)17 b(the)h(c)n(ondition)k Ft(\(3.10\))c Fg(b)n(e)g(satis\014e)n(d)f(and)810 293 y Fr(p)d Fo(\025)g Fr(q)r(;)25 b Ft(if)20 b Fr(q)15 b(<)f Ft(1;)810 376 y Fr(p)g(>)g Ft(1)p Fr(;)26 b Ft(if)20 b Fr(q)15 b Ft(=)f(1)p Fr(:)228 446 y Fg(Assume)k(that)711 527 y Fo(j)p Fr(V)11 b Fo(j)778 506 y Fq(1)p Fk(=)p Fq(2)833 527 y Fo(j)p Fr(H)887 534 y Fq(0)918 527 y Ft(+)g Fr(iI)t Fo(j)1024 506 y Fp(\000)p Fq(1)p Fk(=)p Fq(2)1119 527 y Fo(2)j Ft(\006)1201 534 y Fq(2)p Fk(p)1598 527 y Ft(\(3.12\))228 611 y Fg(Then)k(for)f(a.e.)22 b Fr(\025)15 b Fo(2)f Fm(R)g Fg(the)k(c)n(ondition)k Ft(\(1.7\))17 b Fg(holds.)668 716 y Ft(4.)28 b Fs(Regular)17 b(points)h(of)g Fr(H)1262 723 y Fq(0)278 803 y Ft(Here)g(w)o(e)h(presen)o(t)g(some)g(su\016cien)o (t)f(for)i(\(1.6\))g(conditions.)31 b(F)l(or)20 b(a)g(regular)228 861 y(p)q(oin)o(t)c Fr(\026)e Fo(2)g Fr(\032)p Ft(\()p Fr(H)529 868 y Fq(0)550 861 y Ft(\))i(w)o(e)g(in)o(tro)q(duce)f(the)h (op)q(erators)669 955 y Fr(X)713 934 y Fp(\006)709 967 y Fk(\026)756 955 y Ft(:=)822 903 y Fn(q)p 863 903 58 2 v 863 955 a Fr(V)891 962 y Fp(\006)921 955 y Ft(\()p Fr(H)980 962 y Fq(0)1011 955 y Fo(\000)11 b Fr(\026I)t Ft(\))1135 934 y Fp(\000)p Fq(1)1182 903 y Fn(q)p 1223 903 V 1223 955 a Fr(V)1251 962 y Fp(\006)228 1038 y Ft(and)667 1099 y Fr(X)711 1078 y Fq(0)707 1111 y Fk(\026)745 1099 y Ft(:=)810 1047 y Fn(q)p 851 1047 V 851 1099 a Fr(V)879 1106 y Fq(+)909 1099 y Ft(\()p Fr(H)968 1106 y Fq(0)1000 1099 y Fo(\000)f Fr(\026I)t Ft(\))1123 1078 y Fp(\000)p Fq(1)1170 1047 y Fn(q)p 1212 1047 V 52 x Fr(V)1240 1106 y Fp(\000)1270 1099 y Fr(:)297 1192 y FA(Theorem)16 b(4.1.)24 b Fg(L)n(et)c Fr(\026)f Ft(=)p 837 1166 30 2 v 18 w Fr(\026)g Fo(2)g Fr(\032)p Ft(\()p Fr(H)1021 1199 y Fq(0)1042 1192 y Ft(\))p Fg(.)30 b(Assume)20 b(that)h(ther)n(e)f(exist)h(c)n(on-)228 1250 y(stants)d Fr(C)409 1232 y Fq(0)405 1262 y Fp(\000)434 1250 y Fr(;)8 b(C)495 1232 y Fq(0)491 1262 y(+)538 1250 y Fg(such)18 b(that)592 1335 y Fr(n)621 1342 y Fp(\007)651 1335 y Ft(\()p Fr(s;)8 b(X)759 1314 y Fp(\006)755 1347 y Fk(\026)789 1335 y Ft(\))14 b Fo(\030)f Fr(C)913 1314 y Fq(0)909 1347 y Fp(\006)939 1335 y Fr(s)962 1314 y Fp(\000)p Fk(p)1009 1335 y Fr(;)42 b(s)14 b Fo(!)g Ft(0)p Fr(;)26 b(p)14 b(>)g Ft(0)p Fr(;)240 b Ft(\(4.13\))228 1415 y Fg(and)691 1496 y Fr(n)p Ft(\()p Fr(s;)8 b(X)828 1475 y Fq(0)824 1508 y Fk(\026)848 1496 y Ft(\))14 b(=)f Fr(o)p Ft(\()p Fr(s)997 1475 y Fp(\000)p Fk(p)1045 1496 y Ft(\))p Fr(;)43 b(s)14 b Fo(!)f Ft(0)p Fr(:)339 b Ft(\(4.14\))228 1582 y Fg(Then)22 b Ft(\(1.6\))17 b Fg(is)h(ful\014l)r(le)n(d)h(with)f Fr(C)840 1589 y Fp(\006)883 1582 y Ft(=)c Fr(C)974 1563 y Fq(0)970 1594 y Fp(\006)1017 1582 y Fg(,)j(in)h(p)n(articular)516 1666 y Fr(\030)r Ft(\()p Fr(\026)p Ft(;)8 b Fr(H)t Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)784 1673 y Fq(0)806 1666 y Ft(\))14 b Fo(\030)f Ft(\()p Fr(C)949 1645 y Fq(0)945 1678 y(+)986 1666 y Fo(\000)d Fr(C)1074 1645 y Fq(0)1070 1678 y Fp(\000)1100 1666 y Ft(\))p Fr(\013)1150 1645 y Fk(p)1170 1666 y Fr(;)78 b(\013)14 b Fo(!)f(1)p Fr(:)164 b Ft(\(4.15\))278 1757 y Fg(Pr)n(o)n(of.)20 b Ft(It)15 b(is)h(su\016cien)o(t)f(to)i(establish)f(\(4.15\))q(.)21 b(In)16 b(our)h(case)606 1838 y Fr(\030)r Ft(\()p Fr(\026)p Ft(;)8 b Fr(H)t Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)874 1845 y Fq(0)896 1838 y Ft(\))13 b(=)h Fr(\021)r Ft(\(0;)8 b Fr(J)16 b Ft(+)11 b Fr(\013X)1234 1845 y Fk(\026)1258 1838 y Fr(;)d(J)d Ft(\))p Fr(:)228 1918 y Ft(W)l(e)16 b(are)g(going)h(to)g(compare)e(this)h(function)g(with)621 1999 y Fr(\021)r Ft(\(0;)8 b Fr(J)16 b Ft(+)11 b Fr(\013L;)d(J)d Ft(\))p Fr(;)40 b(L)14 b Ft(:=)f Fr(X)1151 1978 y Fq(+)1147 2011 y Fk(\026)1192 1999 y Ft(+)e Fr(X)1285 1978 y Fp(\000)1281 2011 y Fk(\026)1315 1999 y Fr(:)228 2080 y Ft(W)l(e)16 b(use)g(its)g(additivit)o(y)f(:)489 2160 y Fr(\020)510 2167 y Fq(0)530 2160 y Ft(\()p Fr(\013)p Ft(\))f(:=)f Fr(\021)r Ft(\(0;)8 b Fr(J)16 b Ft(+)11 b Fr(\013X)932 2167 y Fk(\026)956 2160 y Fr(;)d(J)d Ft(\))11 b Fo(\000)g Fr(\021)r Ft(\(0;)d Fr(J)15 b Ft(+)c Fr(\013L;)d(J)d Ft(\))14 b(=)705 2243 y(=)g Fr(\021)r Ft(\(0;)8 b Fr(J)15 b Ft(+)c Fr(\013X)1010 2250 y Fk(\026)1035 2243 y Fr(;)d(J)15 b Ft(+)c Fr(\013L)p Ft(\))p Fr(:)228 2314 y Ft(Denoting)20 b Fo(Y)475 2321 y Fq(0)516 2314 y Ft(:=)g Fo(f)p Fr(\013)h Fo(2)g Fm(R)758 2321 y Fq(+)805 2314 y Ft(:)41 b(0)21 b Fo(2)f Fr(\032)p Ft(\()p Fr(J)f Ft(+)14 b Fr(\013L)p Ft(\))p Fo(g)20 b Ft(and)h(using)g(the)f(Birman-)228 2372 y(Sc)o(h)o(winger)15 b(principle)g(\(see)g(Prop)q(osition)j(1.1\)) e(w)o(e)g(obtain)425 2461 y Fo(\000)p Fr(n)493 2468 y Fp(\000)523 2461 y Ft(\(1)11 b Fo(\000)g Ft(0)p Fr(;)d(T)709 2436 y Fq(\(+\))702 2472 y(0)766 2461 y Ft(\))13 b Fo(\024)h Fr(\020)872 2468 y Fq(0)892 2461 y Ft(\()p Fr(\013)p Ft(\))g Fo(\024)g Fr(n)1057 2468 y Fq(+)1086 2461 y Ft(\(1)p Fr(;)8 b(T)1187 2436 y Fq(\()p Fp(\000)p Fq(\))1180 2472 y(0)1244 2461 y Ft(\))p Fr(;)89 b(\013)14 b Fo(2)g(Y)1491 2468 y Fq(0)1511 2461 y Fr(;)228 2542 y Ft(where)659 2604 y Fr(T)695 2579 y Fq(\()p Fp(\006)p Fq(\))688 2615 y(0)765 2604 y Ft(:=)830 2551 y Fn(q)p 872 2551 69 2 v 53 x Fr(Q)911 2587 y Fq(0)911 2614 y Fp(\006)940 2604 y Ft(\()p Fr(J)i Ft(+)11 b Fr(\013L)p Ft(\))1134 2584 y Fp(\000)p Fq(1)1181 2551 y Fn(q)p 1223 2551 V 53 x Fr(Q)1262 2587 y Fq(0)1262 2614 y Fp(\006)p eop %%Page: 9 9 9 8 bop 690 118 a Fl(SPECTRAL)16 b(SHIFT)h(FUNCTION)441 b(9)228 213 y Ft(and)786 300 y Fr(Q)825 279 y Fq(0)858 300 y Ft(:=)13 b Fr(\013)p Ft(\()p Fr(X)1013 307 y Fk(\026)1049 300 y Fo(\000)d Fr(L)p Ft(\))p Fr(:)228 397 y Ft(By)15 b(the)h(Horn)h(inequalit)o(y)e(\(1.1\),)493 508 y Fr(n)522 515 y Fp(\006)552 508 y Ft(\(1)p Fr(;)8 b(T)653 488 y Fq(\()p Fp(\007)p Fq(\))709 508 y Ft(\))14 b Fo(\024)g Ft(2)p Fr(n)p Ft(\()p Fr(\017;)8 b(Q)948 488 y Fq(0)967 508 y Ft(\))j(+)g Fr(n)p Ft(\()p Fr(\017)1114 488 y Fp(\000)p Fq(1)1161 508 y Fr(;)d Ft(\()p Fr(J)16 b Ft(+)11 b Fr(\013L)p Ft(\))1377 488 y Fp(\000)p Fq(1)1424 508 y Ft(\))p Fr(:)228 616 y Ft(Since)k(b)o(y)i(\(4.14\))689 703 y Fr(n)p Ft(\()p Fr(\017;)8 b(Q)818 683 y Fq(0)837 703 y Ft(\))14 b(=)g Fr(o)p Ft(\()p Fr(\013)995 683 y Fk(p)1015 703 y Ft(\))p Fr(;)41 b(\013)14 b Fo(!)f(1)p Fr(;)228 801 y Ft(it)j(is)g(su\016cien)o (t)e(to)j(establish)f(that)577 908 y(lim)8 b(sup)605 943 y Fk(\013)p Fp(!1)735 908 y Fr(\013)766 888 y Fp(\000)p Fk(p)813 908 y Fr(n)p Ft(\()p Fr(\017)881 888 y Fp(\000)p Fq(1)928 908 y Fr(;)g Ft(\()p Fr(J)16 b Ft(+)11 b Fr(\013L)p Ft(\))1144 888 y Fp(\000)p Fq(1)1191 908 y Ft(\))j Fo(\024)g Fr(C)t Ft(\()p Fr(\017)p Ft(\))228 1035 y(and)837 1122 y(lim)837 1151 y Fk(\017)p Fp(!)p Fq(0)913 1122 y Fr(C)t Ft(\()p Fr(\017)p Ft(\))f(=)h(0)p Fr(:)228 1233 y Ft(First)i(of)g(all)g (note)g(that)519 1340 y Fr(n)p Ft(\()p Fr(\017)587 1320 y Fp(\000)p Fq(1)634 1340 y Fr(;)8 b Ft(\()p Fr(J)16 b Ft(+)11 b Fr(\013L)p Ft(\))850 1320 y Fp(\000)p Fq(1)897 1340 y Ft(\))j(=)g(rank)p Fr(E)1114 1348 y Fq(\()p Fk(J)s Fq(+)p Fk(\013L)p Fq(\))1240 1340 y Ft(\()p Fo(\000)p Fr(\017;)8 b(\017)p Ft(\))13 b Fo(\024)472 1477 y(\024)g Fr(\021)r Ft(\()p Fr(\017)p Ft(;)8 b Fr(J)15 b Ft(+)c Fr(\013L;)d(J)d Ft(\))11 b Fo(\000)g Fr(\021)r Ft(\()p Fo(\000)p Fr(\017)p Ft(;)d Fr(J)15 b Ft(+)c Fr(\013L;)d(J)d Ft(\))14 b(=:)f Fr(m)1376 1484 y Fq(0)1395 1477 y Ft(\()p Fr(\013)p Ft(\))p Fr(:)228 1574 y Ft(It)j(is)g(easy)g(to)h(see)e(from)h (the)g(de\014nition)f(of)i(the)f(function)g Fr(\021)r Ft(,)g(that)299 1681 y Fr(m)342 1688 y Fq(0)361 1681 y Ft(\()p Fr(\013)p Ft(\))e(=)g Fr(\021)r Ft(\(0)p Fr(;)8 b Ft(\()p Fr(f)630 1688 y Fq(+)659 1681 y Ft(\()p Fr(J)d Ft(\))11 b(+)g Fr(\013L)p Ft(\))p Fr(;)d(f)918 1688 y Fq(+)948 1681 y Ft(\()p Fr(J)d Ft(\)\))10 b Fo(\000)h Fr(\021)r 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y(for)k(non-semib)q(ounded)g(op)q(erators)i(is)e(giv)o(en)g(in)f([10)q (].)278 1876 y(The)e(sp)q(ectrum)f(of)h(the)g(op)q(erator)i Fr(H)952 1883 y Fq(0)985 1876 y Ft(is)e(absolutely)g(con)o(tin)o(uous)g (and)h(co)o(v)o(ers)228 1934 y(the)19 b(complemen)o(t)d(of)21 b(the)e(in)o(terv)m(al)g(\003)h(=)g(\()p Fo(\000)p Ft(1)p Fr(;)8 b Ft(1\).)33 b(The)20 b(essen)o(tial)e(sp)q(ectrum)228 1993 y(of)g(the)f(op)q(erator)i Fr(A)p Ft(\()p Fr(\013)p Ft(\))e(coincides)f(with)i(the)f(sp)q(ectrum)f(of)i Fr(H)1407 2000 y Fq(0)1427 1993 y Ft(.)25 b(Besides,)16 b(the)228 2051 y(op)q(erator)k Fr(H)t Ft(\()p Fr(\013)p Ft(\))f(has)h(a)f (discrete)f(sp)q(ectrum)f(in)i(the)f(gap)i(\003.)29 b(It)18 b(is)h(clear)f(that)228 2109 y(the)g(generalized)e(sp)q(ectral)i(shift) g(function)g(of)h(the)e(pair)i Fr(H)t Ft(\()p Fr(\013)p Ft(\))p Fr(;)26 b(H)1496 2116 y Fq(0)1534 2109 y Ft(exists)18 b(on)228 2167 y(the)e(in)o(terv)m(al)f(\003)h(in)g(the)g(sense)g(of)k (\(1.5\).)297 2259 y FA(Theorem)c(5.1.)24 b Fg(L)n(et)15 b Fr(H)768 2266 y Fq(0)803 2259 y Fg(b)n(e)g(the)g(Dir)n(ac)f(op)n(er)n (ator)g(and)h Fr(\026)f Fo(2)g Ft(\003)p Fg(.)21 b(Under)16 b(the)228 2317 y(c)n(ondition)21 b Ft(\(5.18\))e Fg(the)f(fol)r(lowing) h(asymptotics)e(holds)795 2403 y Fr(\030)r Ft(\()p Fr(\026)p Ft(;)8 b Fr(H)t Ft(\()p Fr(\013)p Ft(\))p Fr(;)g(H)1063 2410 y Fq(0)1084 2403 y Ft(\))14 b Fo(\030)442 b Ft(\(5.19\))636 2522 y Fo(\030)13 b Ft(\(3)p Fr(\031)r Ft(\))780 2502 y Fp(\000)p Fq(2)827 2522 y Fr(\013)858 2502 y Fq(3)878 2522 y Ft(\()897 2464 y Fn(Z)947 2522 y Fr(V)986 2502 y Fq(3)975 2534 y(+)1006 2522 y Fr(dx)e Fo(\000)1120 2464 y Fn(Z)1170 2522 y Fr(V)1209 2502 y Fq(3)1198 2534 y Fp(\000)1229 2522 y Fr(dx)p Ft(\))p Fr(;)889 2608 y(\013)j Fo(!)g(1)p Fr(:)p eop %%Page: 11 11 11 10 bop 690 118 a Fl(SPECTRAL)16 b(SHIFT)h(FUNCTION)422 b(11)278 213 y Fg(Pr)n(o)n(of.)32 b Ft(It)20 b(is)h(su\016cien)o(t)e (to)h(note)h(that)g(the)f(condition)g(\(4.13\))q(,)h(\(4.14\))g(are)228 271 y(ful\014lled)15 b(and)749 335 y Fr(C)788 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Fo(!)g(1)p Fr(:)830 1483 y Fs(References)249 1562 y Fz([1])19 b(M.)13 b(Sh.)g(Birman:)j(Discrete)f(sp)q(ectrum)f(in)f(the)h(gaps)f (of)g(a)g(con)o(tin)o(uous)g(one)h(for)f(p)q(ertur-)313 1612 y(bations)h(with)f(large)h(coupling)f(constan)o(ts.)h Fb(A)n(dv.)h(Sov.)g(Math.)f Fa(7)g Fz(\(1991\),)e(57-73.)249 1661 y([2])19 b(M.Sh.)f(Birman,)g(M.Z.)h(Solom)o(y)o(ak:)25 b(Sp)q(ectral)19 b(theory)h(of)e(self-adjoin)o(t)g(op)q(erators)i(in) 313 1711 y(Hilb)q(ert)15 b(space.)h(Mathematics)e(and)h(Its)h (Applications,)e(D.)g(Reidel)h(Publishing)f(Com-)313 1761 y(pan)o(y)m(,)f(1987.)249 1811 y([3])19 b(M.Sh.)14 b(Birman,)f(A.A.)h(Laptev:)20 b(Discrete)c(sp)q(ectrum)g(of)e(the)h(p)q (erturb)q(ed)i(Dirac)e(op)q(er-)313 1861 y(ator.)e(Ark.)h(Mat.,)f Fa(32)h Fz(13-32,)e(\(1994\).)249 1910 y([4])19 b(F.)10 b(Gesztesy)h(K.A.)e(Mak)n(aro)o(v,)g(:)16 b(The)11 b(\004)e(op)q (erator)h(and)g(its)g(relation)f(to)h(Krein's)g(sp)q(ectral)313 1960 y(shift)k(function,)f Fb(pr)n(eprint)p Fz(.)249 2010 y([5])19 b(S.)d(N.)f(Nab)q(ok)o(o,)h(On)g(b)q(oundary)g(problems)f (for)h(analytic)f(op)q(erator)i(v)n(alued)e(functions)313 2060 y(with)f(p)q(ositiv)o(e)f(imaginary)f(part,)27 b Fb(Zap.)16 b(Nauchn.)f(Seminar)n(ov)h(LOMI)e Fz(\(Russian\))g Fa(157)313 2110 y Fz(\(1987\),)f(55{69.)249 2159 y([6])19 b(A.)13 b(Pushnitski,)g(Represen)o(tation)h(for)f(the)g(sp)q(ectral)i (shift)e(function)f(for)h(p)q(erturbations)313 2209 y(of)g(a)h (de\014nite)h(sign,)e Fb(St.Petersbur)n(g)h(Math.)h(J.)e Fa(9)p Fz(,)g(No.)h(6)f(\(1998\).)249 2259 y([7])19 b(A.)13 b(Pushnitski:)k(Sp)q(ectral)d(shift)f(function)g(of)f(the)h(Sc)o(hro)q (dinger)h(op)q(erator)f(in)g(the)g(large)313 2309 y(coupling)g(constan) o(t)i(limit,)10 b(to)k(app)q(ear)g(in)g Fb(Comm.)g(in)h(PDE)p Fz(.)249 2359 y([8])k(O.)f(Safrono)o(v:)25 b(The)19 b(discrete)g(sp)q (ectrum)g(in)f(gaps)g(of)f(the)i(con)o(tin)o(uous)f(one)g(for)f(non-) 313 2409 y(signde\014nite)e(p)q(erturbations)g(with)f(a)f(large)h (coupling)g(constan)o(t)g Fb(St)h(Petersbur)n(g)g(Math.)313 2458 y(Journal)f Fa(8)p Fz(,)f(No)h(2)f(\(1997\).)249 2508 y([9])19 b(O.)25 b(Safrono)o(v:)40 b(The)26 b(discrete)h(sp)q (ectrum)f(of)f(selfadjoin)o(t)f(op)q(erators)i(under)g(non-)313 2558 y(signde\014nite)21 b(p)q(erturbations)h(with)e(a)g(large)g (coupling)g(constan)o(t,)i(Sfb)e(288)g Fb(pr)n(eprint)313 2608 y(No)15 b(387)p Fz(,)f(German)o(y)m(.)p eop %%Page: 12 12 12 11 bop 228 118 a Fl(12)571 b(O.)16 b(SAFR)o(ONO)o(V)228 213 y Fz([10])j(D.)h(Y)m(afaev:)30 b(Mathematical)18 b(Scattering)j(Theory)m(,)g Fb(T)m(r)n(ansl.)f(of)g(Math.)h(Mono)n(gr)n (aphs)313 262 y Fa(105)29 b Fz(,)14 b(AMS.)278 356 y Fy(Universit)481 353 y(\177)480 356 y(at)h(Potsd)o(am,)h(Institut)f(f) 968 353 y(\177)967 356 y(ur)i(Ma)m(thema)m(tik,)g(PF)e(60)h(15)g(53,)g (144)f(15)228 406 y(Potsd)o(am,)h(Germany,)i(e-mail:)j(safr)o(ono)o (v@rz.uni-potsd)o(am.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9909100535267--