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Fn(N)1956 596 y Fd(\000)1882 624 y Fl(X)1893 834 y Fn(j)t Fr(=1)2042 718 y Fo(j)p Fs(E)2148 677 y Fm(\000)2142 744 y Fn(j)2229 718 y Fq(+)g(2)p Fo(j)2404 677 y Fr(3)p Fn(=)p Fr(2)2541 718 y Fs(<)28 b Fo(1)p Fs(:)555 1055 y Fe(Remark)p Fu(.)h(One)21 b(can)f(generalize)h(this)e(result)h(for)g(the)g(case)h Fs(W)14 b Fq(\(1\))27 b(=)h Fs(W)14 b Fq(\()p Fo(\000)p Fq(1\))27 b(=)h(0)p Fu(,)456 1171 y(where)37 b(instead)f(of)h(the)f (Lieb-Thirring)g(bound)g(with)g(the)h(po)n(wer)f Fq(3)p Fs(=)p Fq(2)g Fu(we)h(w)o(ould)456 1288 y(ha)n(v)o(e)27 b(the)h(one)g(with)f(the)h(po)n(wer)g Fq(5)p Fs(=)p Fq(2)p Fu(.)39 b(In)28 b(this)f(case)i(Condition)d(\(3\))j(can)f(be)g(simply) 456 1404 y(omitted.)555 1571 y(Although)36 b(this)h(theorem)g(is)h(a)g (natural)f(generalization)g(of)h(a)g(recent)g(result)f(of)456 1688 y(Killip-Simon,)32 b(it)h(has)g(some)f(disadv)n(antages.)53 b(Namely)-6 b(,)34 b(for)f(a)g(gi)n(v)o(en)f(measure)h Fs(\026)456 1804 y Fu(we)27 b(are)h(not)f(able)g(to)g(check)g(in)g(adv) n(ance)g(whether)g(the)g(condition)f(\(0.5\))h(is)g(ful\002lled)456 1920 y(for)32 b(the)g(corresponding)g(Jacobi)g(matrix)f Fs(J)9 b Fu(.)54 b(Ho)n(we)n(v)o(er)l(,)32 b(in)g(the)h(cases)f Fs(\032)p Fq(\()p Fs(e)3135 1884 y Fn(i\022)3199 1920 y Fq(\))41 b(=)h(1)456 2036 y Fu([12)o(])27 b(and)f Fs(\032)p Fq(\()p Fs(e)951 2000 y Fn(i\022)1014 2036 y Fq(\))k(=)g(cos)1319 2000 y Fr(2)1358 2036 y Fq(\()p Fs(\022)s Fq(\))c Fu([17])h(one)f(is)f (able)h(to)g(a)n(v)n(oid)g(this)f(obstacle)g(and)h(obtain)456 2153 y(stronger)e(results)g(where)h(one)g(does)g(not)f(ha)n(v)o(e)g (the)h Fe(a)g(priori)e Fu(condition)g(\(0.5\).)555 2269 y(The)29 b(main)e(idea)i(of)f(this)g(paper)h(is)f(to)g(modify)f(the)h (kno)n(wn)f(trace)i(formulae.)42 b(F)o(or)456 2385 y Fs(J)9 b Fu(')-5 b(s)31 b(with)h Fs(Q)42 b Fq(=)f Fs(J)36 b Fo(\000)29 b Fs(J)1313 2400 y Fr(0)1384 2385 y Fu(of)j(\002nite)h (rank)f(,)i(the)e(standard)g(trace)h(formulae)f(are)h(due)456 2501 y(to)k(Case)h([2],[3].)70 b(Recently)-6 b(,)40 b(Killip-Simon)c ([12])i(found)f(ho)n(w)f(to)i(e)o(xploit)e(these)456 2618 y(sum)f(rules)i(as)g(a)f(spectral)h(tool.)65 b(In)37 b(particular)l(,)i(Killip-Simon)c(ha)n(v)o(e)h(sho)n(wn)g(the)456 2734 y(importance)d(of)h(e)o(xtension)e(of)i(sum)f(rules)h(to)f(a)h (lar)n(ger)h(class)e(of)h Fs(Q)p Fu(')-5 b(s.)58 b(In)34 b(its)f(turn)456 2850 y(the)24 b(impressi)n(v)o(e)e(paper)j([12)o(])g (w)o(as)f(moti)n(v)n(ated)e(by)i(w)o(ork)g(on)g(Schr)8 b(\250)-41 b(odinger)25 b(operators)456 2966 y(by)30 b(Deift-Killip)g([6])h(and)g(Deniso)o(v)f([7],)j([8].)49 b(Finally)-6 b(,)32 b(the)e(author)h(w)o(ould)f(lik)o(e)h(to)456 3082 y(mention)23 b(the)i(paper)g([26])g(the)f(results)g(of)h(which)g (are)g(used)g(in)f(this)g(w)o(ork.)1548 3311 y(1.)52 b(P)t Ft(R)t(E)t(L)t(I)t(M)t(I)t(N)q(A)5 b(R)g(I)t(E)g(S)555 3485 y Fu(F)o(or)24 b(each)h Fs(T)42 b Fo(2)28 b Fh(S)1199 3500 y Fr(1)1238 3485 y Fu(,)c(one)g(can)h(de\002ne)g(a)f(comple)o(x-v) n(alued)e(function)i Fq(det\(1)c(+)g Fs(T)14 b Fq(\))p Fu(,)456 3601 y(so)24 b(that)1385 3728 y Fo(j)p Fq(det\(1)e(+)g Fs(T)14 b Fq(\))p Fo(j)27 b(\024)h Fq(exp)q(\()p Fo(k)p Fs(T)14 b Fo(k)2382 3743 y Fg(S)2439 3752 y Fi(1)2477 3728 y Fq(\))456 3876 y Fu(F)o(or)24 b Fs(T)42 b Fo(2)28 b Fh(S)893 3891 y Fr(3)957 3876 y Fu(one)d(de\002nes)456 4047 y(\(1.1\))532 b Fq(det)1315 4062 y Fr(3)1354 4047 y Fq(\(1)22 b(+)g Fs(T)14 b Fq(\))27 b(=)h(det\(\(1)22 b(+)g Fs(T)14 b Fq(\))p Fs(e)2335 4006 y Fm(\000)p Fn(T)c Fr(+)p Fn(T)2547 3982 y Fi(2)2581 4006 y Fn(=)p Fr(2)2656 4047 y Fq(\))p Fs(:)456 4214 y Fu(Then)456 4382 y(\(1.2\))501 b Fo(j)p Fq(det)1311 4397 y Fr(3)1351 4382 y Fq(\(1)21 b(+)h Fs(T)14 b Fq(\))p Fo(j)27 b(\024)i Fq(exp)q(\()p Fs(c)p Fo(k)p Fs(T)14 b Fo(k)2227 4341 y Fr(3)2227 4407 y Fg(S)2284 4416 y Fi(3)2322 4382 y Fq(\))p Fs(;)116 b(c)28 b(>)f Fq(0)p Fs(:)456 4563 y Fx(Lemma)c(1.1.)39 b Fe(Let)24 b Fs(A;)17 b(B)28 b Fe(be)c(of)e(\002nite)h(r)o(ank.)29 b(Ther)l(e)24 b(is)f(a)g(constant)f Fs(C)31 b Fe(suc)o(h)22 b(that)h(one)456 4679 y(of)h(the)h(conditions)456 4846 y Fu(\(1.3\))635 b Fo(j)p Fu(Im)24 b Fs(z)t Fo(j)k Fs(>)g Fq(1)50 b(or)f Fo(j)p Fu(Re)25 b Fs(z)t Fo(j)j Fs(>)g Fq(3)22 b(+)g Fo(jj)p Fs(Q)p Fo(jj)456 5014 y Fe(imply)456 5182 y Fu(\(1.4\))482 b Fo(j)17 b Fq(log)f(det)1451 5197 y Fr(3)1491 5182 y Fq(\()p Fs(I)30 b Fq(+)22 b Fs(Q)p Fq(\()p Fs(J)1869 5197 y Fr(0)1930 5182 y Fo(\000)h Fs(z)t Fq(\))2117 5141 y Fm(\000)p Fr(1)2212 5182 y Fq(\))p Fo(j)k(\024)h Fs(C)7 b Fo(jj)p Fs(Q)p Fo(jj)2676 5141 y Fr(3)2676 5206 y Fg(S)2733 5215 y Fi(3)p eop %%Page: 4 4 4 3 bop 456 251 a Fj(4)1268 b(SAFR)m(ONO)l(V)555 450 y Fe(Pr)l(oof)p Fu(.)71 b(Let)38 b Fs(z)58 b Fq(=)53 b Fs(\025)32 b Fq(+)g Fs(i\017)p Fu(,)43 b(where)c Fs(\025)f Fu(and)h Fs(\017)g Fu(are)g(real.)72 b(One)39 b(can)g(repeat)g(the)456 566 y(ar)n(guments)24 b(of)h(Proposition)e(5.2)h([13])h(to)g(sho)n(w)e (that)1832 779 y Fo(j)1889 712 y Fs(d)p 1870 756 91 4 v 1870 847 a(d\017)1986 779 y Fq(log)17 b(det)2264 794 y Fr(3)2304 779 y Fq(\()p Fs(I)30 b Fq(+)22 b Fs(Q)p Fq(\()p Fs(J)2682 794 y Fr(0)2743 779 y Fo(\000)h Fs(z)t Fq(\))2930 738 y Fm(\000)p Fr(1)3025 779 y Fq(\))p Fo(j)k Fq(=)897 969 y Fo(j)p Fu(tr)d Fq(\()p Fs(i)p Fq(\()p Fs(J)1173 984 y Fr(0)1235 969 y Fo(\000)f Fs(z)t Fq(\))1422 928 y Fm(\000)p Fr(1)1517 969 y Fs(Q)p Fq(\()p Fs(J)1686 984 y Fr(0)1747 969 y Fo(\000)g Fs(z)t Fq(\))1934 928 y Fm(\000)p Fr(1)2029 969 y Fs(Q)p Fq(\()p Fs(J)31 b Fo(\000)23 b Fs(z)t Fq(\))2416 928 y Fm(\000)p Fr(1)2511 969 y Fq(\()p Fs(J)2603 984 y Fr(0)2664 969 y Fo(\000)g Fs(z)t Fq(\))2851 928 y Fm(\000)p Fr(1)2946 969 y Fs(Q)p Fq(\))p Fo(j)28 b(\024)1782 1129 y(jj)p Fq(\()p Fs(J)1930 1144 y Fr(0)1991 1129 y Fo(\000)23 b Fs(z)t Fq(\))2178 1088 y Fm(\000)p Fr(1)2273 1129 y Fo(jj)2329 1088 y Fr(3)2368 1129 y Fo(jj)p Fq(\()p Fs(J)30 b Fo(\000)23 b Fs(z)t Fq(\))2733 1088 y Fm(\000)p Fr(1)2828 1129 y Fo(jjjj)p Fs(Q)p Fo(jj)3073 1088 y Fr(3)3073 1154 y Fg(S)3130 1163 y Fi(3)3167 1129 y Fs(:)456 930 y Fu(\(1.5\))456 1290 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Fq(\))1911 4979 y Fr(2)1973 5020 y Fq(+)g(log)17 b Fo(j)p Fq(det)2377 5035 y Fr(3)2416 5020 y Fq(\()p Fs(I)30 b Fq(+)22 b Fs(Q)p Fq(\()p Fs(J)2794 5035 y Fr(0)2856 5020 y Fo(\000)g Fs(z)t Fq(\))3042 4979 y Fm(\000)p Fr(1)3137 5020 y Fq(\))p Fo(j)28 b Fq(=:)1907 5210 y Fs(g)t Fq(\()p Fs(\022)s Fq(\))22 b(+)g(log)16 b Fo(j)p Fq(det)2507 5225 y Fr(3)2547 5210 y Fq(\()p Fs(I)30 b Fq(+)22 b Fs(Q)p Fq(\()p Fs(J)2925 5225 y Fr(0)2986 5210 y Fo(\000)h Fs(z)t Fq(\))3173 5169 y Fm(\000)p Fr(1)3268 5210 y Fq(\))p Fo(j)456 4957 y Fu(\(2.2\))p eop %%Page: 5 5 5 4 bop 1573 251 a Fj(SPECTRAL)21 b(MEASURE)1079 b(5)456 450 y Fu(The)24 b(function)g Fs(d)1043 465 y Fr(3)1082 450 y Fq(\()p Fs(k)s Fq(\))k(=)g(det)1479 465 y Fr(3)1519 450 y Fq(\()p Fs(I)h Fq(+)22 b Fs(Q)p Fq(\()p Fs(J)1896 465 y Fr(0)1958 450 y Fo(\000)g Fs(z)t Fq(\))2144 414 y Fm(\000)p Fr(1)2239 450 y Fq(\))j Fu(v)n(anishes)f(when)g Fs(k)i Fq(+)c(1)p Fs(=k)27 b Fu(is)d(an)456 566 y(eigen)l(v)n(alue)c (of)g Fs(J)9 b Fu(.)30 b(Let)21 b Fo(f)p Fp({)1393 581 y Fn(n)1440 566 y Fo(g)f Fu(be)h(the)g(zeros)g(of)g Fs(d)2154 581 y Fr(3)2193 566 y Fq(\()p Fs(k)s Fq(\))g Fu(lying)e(in)i(the)f (domain)g Fo(j)p Fs(k)s Fo(j)27 b Fs(<)h Fq(1)p Fu(.)456 683 y(W)-8 b(e)25 b(introduce)f(the)h(Blaschk)o(e)g(product)1099 893 y Fs(G)j Fq(=)1307 798 y Fl(Y)1349 1007 y Fn(n)1485 825 y Fp({)1551 840 y Fn(n)1621 825 y Fo(\000)22 b Fs(k)p 1461 870 339 4 v 1461 961 a Fq(1)g Fo(\000)g Fp({)1697 976 y Fn(n)1745 961 y Fs(k)1847 825 y Fp({)1913 840 y Fn(n)p 1819 870 169 4 v 1819 961 a Fo(j)p Fp({)1913 976 y Fn(n)1960 961 y Fo(j)1998 893 y Fs(;)315 b Fq(\()p 2378 838 114 4 v Fp({)2444 908 y Fn(n)2519 893 y Fq(=)28 b Fp({)2689 908 y Fn(n)2736 893 y Fq(\))p Fs(:)456 1151 y Fu(Clearly)d Fo(j)p Fs(G)p Fo(j)j Fq(=)g(1)d Fu(when)g Fs(k)32 b Fq(=)c Fs(e)1586 1115 y Fn(i\022)1649 1151 y Fu(.)k(Choose)25 b Fs(a)k(>)f Fq(0)d Fu(so)g(small)f(that)h(e)n(v)o (ery)f(point)g Fs(z)34 b Fq(=)456 1267 y Fs(k)11 b Fq(+)d(1)p Fs(=k)23 b Fu(with)d Fo(j)p Fs(k)s Fo(j)27 b Fq(=)h Fs(a)21 b Fu(satis\002es)g(the)g(condition)e(\(1.3\).)29 b(Ob)o(viously)19 b Fs(a)28 b(<)g Fq(min)3187 1282 y Fn(n)3251 1267 y Fo(j)p Fp({)3345 1282 y Fn(n)3392 1267 y Fo(j)p Fu(.)456 1384 y(Thus,)c(by)g(using)g(the)h(Cauchy)f(theorem)h(we)g(\002nd)1274 1528 y Fq(2)p 1269 1572 59 4 v 1269 1663 a Fs(\031)1355 1459 y Fl(Z)1454 1486 y Fn(\031)1410 1685 y Fm(\000)p Fn(\031)1529 1595 y Fq(log)16 b Fo(j)p Fs(d)1750 1610 y Fr(3)1789 1595 y Fq(\()p Fs(k)s Fq(\))p Fo(j)p Fs(\032)p Fq(\()p Fs(e)2080 1554 y Fn(i\022)2143 1595 y Fq(\))h(sin)2317 1554 y Fr(2)2374 1595 y Fs(\022)i(d\022)31 b Fq(=)1011 1893 y Fu(Re)1194 1826 y Fs(i)p 1156 1870 108 4 v 1156 1961 a Fq(2)p Fs(\031)1291 1757 y Fl(Z)1346 1983 y Fm(j)p Fn(k)r Fm(j)p Fr(=)p Fn(a)1537 1893 y Fq(log\()p Fs(d)1752 1908 y Fr(3)1791 1893 y Fq(\()p Fs(k)s Fq(\))p Fs(=G)p Fq(\))p Fs(\032)p Fq(\()p Fs(k)s Fq(\))2275 1826 y(\()p Fs(k)2367 1789 y Fr(2)2428 1826 y Fo(\000)23 b Fq(1\))2615 1789 y Fr(2)p 2275 1870 380 4 v 2418 1961 a Fs(k)2472 1932 y Fr(3)2681 1893 y Fs(dk)30 b Fq(=)456 2211 y Fu(\(2.3\))392 b(Re)1222 2144 y Fs(i)p 1185 2188 108 4 v 1185 2279 a Fq(2)p Fs(\031)1319 2075 y Fl(Z)1374 2301 y Fm(j)p Fn(k)r Fm(j)p Fr(=)p Fn(a)1565 2211 y Fq(log\()p Fs(d)1780 2226 y Fr(3)1819 2211 y Fq(\()p Fs(k)s Fq(\)\))p Fs(w)s Fq(\()p Fs(k)s Fq(\))2216 2144 y Fs(dk)p 2216 2188 105 4 v 2241 2279 a(k)2353 2211 y Fo(\000)2452 2116 y Fl(X)2503 2326 y Fn(n)2613 2211 y Fs(f)11 b Fq(\()p Fp({)2776 2226 y Fn(n)2823 2211 y Fq(\))456 2455 y Fu(where)1480 2614 y Fs(w)s Fq(\()p Fs(k)s Fq(\))27 b(=)g Fs(\032)p Fq(\()p Fs(k)s Fq(\))2003 2547 y(\()p Fs(k)2095 2511 y Fr(2)2157 2547 y Fo(\000)c Fq(1\))2344 2511 y Fr(2)p 2003 2591 380 4 v 2147 2683 a Fs(k)2201 2654 y Fr(2)2393 2614 y Fs(:)456 2782 y Fu(and)1002 2931 y Fs(f)11 b Fq(\()p Fp({)t Fq(\))28 b(=)f Fu(Re)1521 2864 y Fs(i)p 1484 2908 108 4 v 1484 2999 a Fq(2)p Fs(\031)1618 2795 y Fl(Z)1674 3021 y Fm(j)p Fn(k)r Fm(j)p Fr(=)p Fn(a)1864 2931 y Fq(log\()2063 2864 y Fp({)f Fo(\000)d Fs(k)p 2038 2908 296 4 v 2038 2999 a Fq(1)f Fo(\000)h Fp({)t Fs(k)2381 2864 y Fp({)p 2353 2908 126 4 v 2353 2999 a Fo(j)p Fp({)t Fo(j)2489 2931 y Fq(\))p Fs(w)s Fq(\()p Fs(k)s Fq(\))2756 2864 y Fs(dk)p 2756 2908 105 4 v 2781 2999 a(k)2871 2931 y(:)456 3137 y Fu(The)34 b(latter)g(inte)o(gral)f(can)i(be)g(computed)e(by)h (the)g(residue)g(calculus,)j(in)d(particular)456 3253 y(one)e(can)g(sho)n(w)f(that)h Fs(f)11 b Fq(\()p Fp({)t Fq(\))32 b Fu(is)g(a)h(\002nite)f(linear)g(combination)e(of)i(po)n (wers)g Fp({)3166 3217 y Fm(\006)p Fn(n)3301 3253 y Fu(and)456 3369 y Fq(log\()p Fo(j)p Fp({)t Fo(j)p Fq(\))p Fu(.)e(Ho)n(we)n(v)o(er) l(,)24 b(instead)g(of)h(calculating)f(it)g(e)o(xplicitly)f(we)i(pro)o (v)o(e)f(that)456 3588 y(\(2.4\))188 b Fs(f)11 b Fq(\()p Fs(t)p Fq(\))27 b(=)1146 3521 y(4)p Fs(\032)p Fq(\()p Fo(\007)p Fq(1\))p 1146 3565 302 4 v 1272 3657 a(3)1457 3588 y Fo(j)p Fs(t)22 b Fo(\006)h Fq(1)p Fo(j)1719 3547 y Fr(3)1780 3588 y Fq(+)f Fs(o)p Fq(\()p Fo(j)p Fs(t)g Fo(\006)g Fq(1)p 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Fs(sk)s Fq(\))2917 4751 y Fn(n)p 2611 4832 V 2611 4923 a Fs(n)p Fq(\(1)g Fo(\000)h Fs(k)s Fq(\))2970 4895 y Fn(n)3027 4855 y Fs(:)456 5099 y Fu(Then)35 b(one)g(can)h(inte)o(grate)f(each)h(term)f(in)g(the)h (e)o(xpansion)e(\(2.6\))h(with)g(the)g(weight)456 5216 y Fs(w)s Fq(\()p Fs(k)s Fq(\))h Fu(to)h(calculate)h(the)f(dif)n (ference)g(of)h(the)f(inte)o(grals)f(in)h(\(2.5\).)68 b(Since)37 b Fs(w)s Fq(\()p Fs(k)s Fq(\))51 b(=)p eop %%Page: 6 6 6 5 bop 456 251 a Fj(6)1268 b(SAFR)m(ONO)l(V)456 450 y Fq(4)p Fs(\032)p Fq(\(1\)\()p Fs(k)32 b Fo(\000)f Fq(1\))996 414 y Fr(2)1065 450 y Fq(+)f Fs(O)s Fq(\(\()p Fs(k)i Fo(\000)f Fq(1\))1603 414 y Fr(3)1642 450 y Fq(\))p Fu(,)38 b(as)d Fs(k)51 b Fo(!)c Fq(1)p Fu(,)38 b(we)d(come)h(to)f(\(2.4\))g (for)h(the)f(case)456 566 y Fs(t)28 b Fo(!)f Fq(1)p Fu(.)j(The)25 b(other)g(case)g(is)g(similar)-5 b(.)555 683 y(By)35 b(the)f(inequality)f(\(1.4\))i(the)f(\002rst)g(term)h(in)f(\(2.3\))g (is)g(bounded)f(by)i(a)f(constant)456 799 y Fs(C)g Fq(=)28 b Fs(C)7 b Fq(\()p Fs(a;)17 b(\032;)g Fo(jj)p Fs(Q)p Fo(jj)p Fq(\))23 b Fu(\(depending)i(on)f Fs(a;)17 b(\032)25 b Fu(and)g Fo(jj)p Fs(Q)p Fo(jj)p Fu(\))f(times)g Fo(jj)p Fs(Q)p Fo(jj)2831 763 y Fr(3)2831 825 y Fg(S)2888 834 y Fi(3)2951 799 y Fu(:)456 1020 y(\(2.7\))830 906 y Fl(\014)830 966 y(\014)830 1025 y(\014)880 885 y(Z)935 1110 y Fm(j)p Fn(k)r Fm(j)p Fr(=)p Fn(a)1126 1020 y Fq(log\()p Fs(d)1341 1035 y Fr(3)1380 1020 y Fq(\()p Fs(k)s Fq(\))p Fs(w)s Fq(\()p Fs(k)s Fq(\))1739 953 y Fs(dk)p 1739 998 105 4 v 1764 1089 a(k)1854 906 y Fl(\014)1854 966 y(\014)1854 1025 y(\014)1887 1020 y Fo(\024)k Fs(C)7 b Fo(jj)p Fs(Q)p Fo(jj)2258 979 y Fr(3)2258 1045 y Fg(S)2315 1054 y Fi(3)2353 1020 y Fs(;)116 b(C)35 b Fq(=)27 b Fs(C)7 b Fq(\()p Fs(a;)17 b(\032;)g Fo(jj)p Fs(Q)p Fo(jj)p Fq(\))p Fs(:)555 1256 y Fu(No)n(w)-6 b(,)22 b(since)g(the)g(k)o(ernel)g(of)g(the)h(resolv)o (ent)e Fs(R)q Fq(\()p Fs(z)t Fq(\))28 b(=)f(\()p Fs(H)2515 1271 y Fr(0)2567 1256 y Fo(\000)13 b Fs(z)t(I)8 b Fq(\))2795 1220 y Fm(\000)p Fr(1)2913 1256 y Fu(is)21 b(of)i(the)f(form)456 1467 y(\(2.8\))224 b Fs(G)948 1482 y Fr(2)987 1467 y Fq(\()p Fs(n;)17 b(m)p Fq(;)g Fs(z)t Fq(\))28 b(=)1590 1400 y Fs(k)p 1485 1445 265 4 v 1485 1536 a(k)1539 1507 y Fr(2)1601 1536 y Fo(\000)22 b Fq(1)1759 1467 y Fs(k)1813 1426 y Fm(j)p Fn(n)p Fm(\000)p Fn(m)p Fm(j)2016 1467 y Fs(;)117 b(z)32 b Fq(=)c Fs(k)d Fq(+)d Fs(k)2569 1426 y Fm(\000)p Fr(1)2663 1467 y Fs(;)34 b Fq(0)27 b Fs(<)h Fo(j)p Fs(k)s Fo(j)f Fs(<)g Fq(1)p Fs(;)456 1662 y Fu(we)e(obtain)456 1837 y Fx(Lemma)g(2.1.)41 b Fe(F)-10 b(or)24 b Fs(k)31 b Fq(=)c Fs(e)1392 1801 y Fn(i\022)704 2020 y Fu(Re)e(tr)g Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(QR)q Fq(\()p Fs(z)t Fq(\))p Fs(Q)30 b Fq(=)1733 1953 y Fo(\000)p Fq(1)p 1622 1997 349 4 v 1622 2092 a(4)17 b(sin)1807 2052 y Fr(2)1847 2092 y Fq(\()p Fs(\022)s Fq(\))1981 2020 y Fo(j)p Fq(2)p Fs(e)2103 1979 y Fn(i\022)2191 1995 y Fq(^)2166 2020 y Fs(A)p Fq(\(2)p Fs(\022)s Fq(\))22 b(+)2555 1995 y(^)2532 2020 y Fs(B)5 b Fq(\(2)p Fs(\022)s Fq(\))p Fo(j)2812 1979 y Fr(2)2873 2020 y Fq(+)22 b Fu(tr)j Fs(A)3130 1979 y Fr(2)3169 2020 y Fs(:)555 2252 y Fe(Pr)l(oof)o(.)k Fu(By)c(using)f(\(2.8\))h(we)g(compute)830 2455 y(Re)h(tr)f Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(B)5 b(R)q Fq(\()p Fs(z)t Fq(\))p Fs(B)34 b Fq(=)1864 2387 y Fo(\000)p Fq(1)p 1752 2432 V 1752 2527 a(4)17 b(sin)1938 2487 y Fr(2)1977 2527 y Fq(\()p Fs(\022)s Fq(\))2111 2455 y Fo(j)2162 2430 y Fq(^)2139 2455 y Fs(B)5 b Fq(\(2)p Fs(\022)s Fq(\))p Fo(j)2419 2414 y Fr(2)2458 2455 y Fs(;)116 b(z)32 b Fq(=)c(2)17 b(cos)g Fs(\022)s(;)456 2669 y Fu(Furthermore)893 2827 y(Re)25 b(tr)g Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(S)6 b(AR)q Fq(\()p Fs(z)t Fq(\))p Fs(S)g(A)29 b Fq(=)e Fu(Re)f(tr)f Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(AS)2485 2786 y Fm(\003)2525 2827 y Fs(R)q Fq(\()p Fs(z)t Fq(\))p Fs(AS)2864 2786 y Fm(\003)2932 2827 y Fq(=)1120 3035 y(=)1344 2967 y Fo(\000)p Fq(1)p 1233 3012 V 1233 3106 a(4)17 b(sin)1418 3067 y Fr(2)1458 3106 y Fq(\()p Fs(\022)s Fq(\))1592 3035 y(\()p Fo(j)1683 3010 y 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4828 a Fq(1)22 b Fo(\000)g Fs(k)1891 4799 y Fr(2)1941 4759 y Fq(\()p Fs(k)2033 4718 y Fn(n)p Fr(+)p Fn(m)p Fr(+2)2309 4759 y Fo(\000)h Fs(k)2463 4718 y Fm(j)p Fn(n)p Fm(\000)p Fn(m)p Fm(j)2667 4759 y Fq(\))k(=)1352 4962 y(=)g Fs(G)1532 4977 y Fr(1)1572 4962 y Fq(\()p Fs(n;)17 b(m)p Fq(;)g Fs(z)t Fq(\))22 b(+)g Fs(G)2125 4977 y Fr(2)2165 4962 y Fq(\()p Fs(n;)17 b(m)p Fq(;)g Fs(z)t Fq(\))p Fs(:)456 5099 y Fu(The)26 b(\002rst)h(term)f(here)i (does)e(not)g(play)h(an)o(y)f(role)g(because)h(its)f(contrib)n(ution)f (in)l(v)n(olv)o(es)456 5216 y(only)d(\002nite)g(number)h(of)g(entries)f (of)h Fs(J)9 b Fu(.)30 b(Indeed)23 b(we)g(can)h(pro)o(v)o(e)d(the)i (follo)n(wing)e(result)p eop %%Page: 7 7 7 6 bop 1573 251 a Fj(SPECTRAL)21 b(MEASURE)1079 b(7)456 450 y Fx(Pr)n(oposition)31 b(2.1.)45 b Fe(Let)32 b Fs(\036)p Fq(\()p Fs(\022)s Fq(\))g Fe(be)g(a)f(boundary)g(value)h(of)f(an)h (analytic)e(function)h(in)456 566 y(the)22 b(unit)h(disc)f(e)n(xcept)i (for)e(a)h(pole)g(of)f(or)l(der)h Fs(j)29 b 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b(the)g(inte)o(grated)f(function)g(in)g(\(2.10\))h(can) 456 1822 y(be)h(re)n(written)g(in)g(terms)g(of)h(tr)g Fs(Q)p Fq(\000)1630 1837 y Fn(k)1673 1822 y Fu(,)g(tr)h Fq(\()p Fs(Q)p Fq(\000)1984 1837 y Fn(k)2027 1822 y Fq(\))2065 1786 y Fr(2)2128 1822 y Fu(and)e(tr)i Fs(Q)p Fq(\000)2519 1837 y Fn(k)2562 1822 y Fs(QR)q Fq(\()p Fs(z)t Fq(\))p Fu(.)31 b(The)24 b(functions)456 1938 y Fq(\()p Fs(k)548 1902 y Fr(2)605 1938 y Fo(\000)18 b Fq(1\))p Fu(tr)24 b Fs(Q)p Fq(\000)1010 1953 y Fn(k)1053 1938 y Fu(,)g Fq(\()p Fs(k)1194 1902 y Fr(2)1251 1938 y Fo(\000)18 b Fq(1\))1433 1902 y Fr(2)1472 1938 y Fu(tr)25 b Fq(\()p Fs(Q)p Fq(\000)1734 1953 y Fn(k)1777 1938 y Fq(\))1815 1902 y Fr(2)1878 1938 y Fu(and)e Fq(\()p Fs(k)2137 1902 y Fr(2)2194 1938 y Fo(\000)18 b Fq(1\))2376 1902 y Fr(2)2415 1938 y Fu(tr)25 b Fs(Q)p Fq(\000)2639 1953 y Fn(k)2682 1938 y Fs(QR)q Fq(\()p Fs(z)t Fq(\))g Fu(are)f(analytic)456 2054 y(in)33 b(the)g(disc)g(and)h(the)f(coef)n(\002cients)h(of)f(the)h (T)-8 b(aylor)33 b(series)g(about)g(zero)h(depend)g(on)456 2171 y(\002nite)24 b(number)h(of)f(entries)h Fs(\013)1475 2186 y Fn(n)1547 2171 y Fu(and)g Fs(\014)1771 2186 y Fn(n)1818 2171 y Fu(.)31 b(F)o(or)24 b(e)o(xample)729 2433 y Fq(\()p Fs(k)821 2392 y Fr(2)883 2433 y Fo(\000)e Fq(1\))1069 2392 y Fr(2)1108 2433 y Fu(tr)j Fs(B)5 b Fq(\000)1334 2448 y Fn(k)1377 2433 y Fs(B)g(R)q Fq(\()p Fs(z)t Fq(\))28 b(=)g Fs(k)1842 2392 y Fr(2)1935 2309 y Fm(1)1898 2338 y Fl(X)1904 2548 y Fn(n)p Fr(=0)2059 2433 y Fs(k)2113 2392 y Fr(2)p Fn(n)2195 2352 y Fl(\000)2241 2433 y Fs(\014)2302 2392 y Fr(2)2296 2458 y Fn(n)2365 2433 y Fq(+)22 b(2)p Fs(\014)2567 2448 y Fn(n)2640 2309 y(n)p Fm(\000)p Fr(1)2635 2338 y Fl(X)2630 2548 y Fn(m)p Fr(=0)2799 2433 y Fs(\014)2854 2448 y Fn(m)2921 2352 y Fl(\001)2967 2433 y Fs(:)99 b Fc(\003)555 2745 y Fu(From)25 b(\(2.2\),)g(\(2.3\),)f(\(2.7\))h(and)g(\(2.9\))g(we)g(obtain)1151 2901 y Fq(1)p 1146 2945 59 4 v 1146 3037 a Fs(\031)1231 2833 y Fl(Z)1331 2859 y Fn(\031)1287 3058 y Fm(\000)p 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Fj(SPECTRAL)21 b(MEASURE)1079 b(9)456 450 y Fu(It)24 b(is)h(important)e(for)i(us)g(to)f(\002nd)h(conditions)e (when)456 633 y(\(3.5\))284 b Fq(\010)1001 648 y Fn(\032)1042 633 y Fq(\()p Fs(\026)p Fq(\))21 b Fo(\000)i Fq(\010)1368 648 y Fn(\032)1409 633 y Fq(\()p Fs(J)1510 592 y Fr(\()p Fn(N)7 b Fr(\))1632 633 y Fq(\))28 b(=)f Fs(P)1864 648 y Fn(N)s(;\032)1983 633 y Fq(\()p Fs(Q)p Fq(\))22 b(+)g(\003\()p Fs(Q)p Fq(\))g Fo(\000)h Fq(\003\()p Fs(Q)2782 592 y Fr(\()p Fn(N)7 b Fr(\))2904 633 y Fq(\))p Fs(:)456 812 y Fu(The)28 b(identity)e(\(3.5\))i(is)g(v)n(alid)f(if)h Fs(Q)g Fu(is)g(of)g(a)g(\002nite)g(rank.)41 b(The)28 b(case)g(when)g(the)g(func-)456 928 y(tion)34 b Fs(M)740 943 y Fn(\026)823 928 y Fu(meromorphic)h(in)g(the)g(neighborhood)f(of)i (the)g(unit)e(disc)i(follo)n(ws)e(from)456 1044 y(Proposition)23 b(4.3)h(and)h(Theorem)g(4.4)f(in)h([12)o(])g(and)g(the)g(f)o(act)g (that)1504 1300 y Fs(\032)p Fq(\()p Fs(k)s Fq(\))j(=)g(\()1930 1175 y Fn(n)1973 1184 y Fi(0)1897 1205 y Fl(X)1854 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Fu(The)36 b(follo)n(wing)e(result)h(is)g(pro)o(v)o(ed)f(in)i(the)f(paper)h(of)g (Simon)f(and)h(Killip,)h([12)o(],)456 4657 y(Corollary)24 b(5.3.)456 4850 y Fx(Lemma)38 b(3.2.)48 b Fs(S)6 b Fq(\()p Fs(\027)58 b Fo(j)52 b Fs(\026)p Fq(\))38 b Fe(is)f(weakly)i(upper)e (semi-continuous)f(in)i Fs(\026)p Fe(,)i(that)d(is,)k(if)456 4978 y Fs(\026)515 4993 y Fn(n)643 4922 y(w)589 4978 y Fo(\000)-16 b(!)27 b Fs(\026)p Fe(,)e(then)1351 5126 y Fs(S)6 b Fq(\()p Fs(\027)34 b Fo(j)27 b Fs(\026)p Fq(\))g Fo(\025)i Fq(lim)17 b(sup)1879 5206 y Fn(n)p Fm(!1)2137 5126 y Fs(S)6 b Fq(\()p Fs(\027)34 b Fo(j)27 b Fs(\026)2437 5141 y Fn(n)2484 5126 y Fq(\))p Fs(:)p eop %%Page: 11 11 11 10 bop 1573 251 a Fj(SPECTRAL)21 b(MEASURE)1042 b(11)555 450 y Fu(Let)22 b(us)f(use)g(the)g(f)o(act)h(that)f(the)g(formulae)g (\(2.12\))g(are)i(v)n(alid)d(at)h(least)g(for)h(\002nite)f(rank)456 566 y(operators)33 b Fs(A;)17 b(B)5 b Fu(.)56 b(Suppose)33 b(no)n(w)f(that)h Fs(A;)17 b(B)38 b Fu(are)c(arbitrary)g(compact)f 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1145 y Fs(A)p Fq(\(2)p Fs(\022)s Fq(\))r(+)2799 1120 y(^)2776 1145 y Fs(B)5 b Fq(\(2)p Fs(\022)s Fq(\))p Fo(j)3056 1104 y Fr(2)3094 1145 y Fs(\032)p Fq(\()p Fs(e)3227 1104 y Fn(i\022)3291 1145 y Fq(\))17 b Fs(d\022)456 1387 y Fu(as)39 b Fs(n)53 b Fo(!)h(1)p Fu(.)72 b(Let)39 b Fs(Q)1289 1402 y Fn(n)1390 1387 y Fq(=)53 b Fs(S)6 b(A)1658 1402 y Fn(n)1738 1387 y Fq(+)32 b Fs(A)1919 1402 y Fn(n)1966 1387 y Fs(S)2032 1351 y Fm(\003)2104 1387 y Fq(+)g Fs(B)2286 1402 y Fn(n)2333 1387 y Fu(,)43 b Fs(J)2455 1402 y Fn(n)2555 1387 y Fq(=)54 b Fs(S)38 b Fq(+)32 b Fs(S)2957 1351 y Fm(\003)3029 1387 y Fq(+)g Fs(Q)3214 1402 y Fn(n)3301 1387 y Fu(and)456 1504 y Fs(\026)515 1519 y Fn(n)561 1504 y Fq(\()p Fs(\016)t Fq(\))c(=)f(\()p Fs(E)925 1519 y Fn(J)964 1527 y Ff(n)1011 1504 y Fq(\()p Fs(\016)t Fq(\))p Fs(e)1179 1519 y Fr(0)1218 1504 y Fs(;)17 b(e)1307 1519 y Fr(0)1346 1504 y Fq(\))p Fs(;)25 b Fu(where)f Fs(\016)k Fu(is)c(an)g(arbitrary)g(Borel)h(set.)30 b(Since)24 b Fq(\()p Fs(J)3099 1519 y Fn(n)3166 1504 y Fo(\000)19 b Fs(z)t Fq(\))3349 1468 y Fm(\000)p Fr(1)456 1620 y Fu(con)l(v)o(er)n(ges)f(to)h Fq(\()p Fs(J)10 b Fo(\000)q Fs(z)t Fq(\))1235 1584 y Fm(\000)p Fr(1)1351 1620 y Fu(uniformly)17 b(on)i(compact)g(subsets)f(of)h(the)g(upper)h(half-plane)456 1736 y(we)25 b(obtain)f(that)g Fs(\026)1104 1751 y Fn(n)1176 1736 y Fu(is)g(weakly)h(con)l(v)o(er)n(gent)f(to)g Fs(\026)p Fu(,)1425 1930 y Fs(\026)1484 1945 y Fn(n)1612 1874 y(w)1558 1930 y Fo(\000)-16 b(!)27 b Fs(\026;)116 b Fq(as)100 b Fs(n)28 b Fo(!)f(1)p Fs(:)456 2115 y Fu(Applying)k(Lemma)h(3.2)g(we)h (obtain)f(that)g(if)h Fs(d\027)49 b Fq(=)42 b Fs(\032)p Fq(\()p Fs(e)2465 2079 y Fn(i\022)2528 2115 y Fq(\))17 b(sin)2703 2075 y Fr(2)2759 2115 y Fs(\022)j(d\022)35 b Fu(and)e Fs(\026)f Fu(is)h(the)456 2232 y(spectral)24 b(measure)h(of)g Fs(J)9 b Fu(,)25 b(then)1613 2417 y Fs(S)6 b Fq(\()p Fs(\027)34 b Fo(j)28 b Fs(\026)p Fq(\))f Fs(>)g Fo(\0001)p Fs(:)456 2602 y Fu(This)d(is)g(e)o(xactly)g(what)h (is)f(needed)h(for)g(\(3.9\).)555 2718 y(In)44 b(order)g(to)f(complete) g(the)g(proof)h(we)g(only)f(ha)n(v)o(e)g(to)g(sho)n(w)g(that)g(\(0.6\)) h(and)456 2834 y(\(0.5\))32 b(imply)f(Condition)h(\(3\))g(of)h(Theorem) f(0.1.)54 b(This)32 b(can)h(be)f(done)h(by)f(a)h(simple)456 2951 y(trick)c(\(see)g([17]\))h(whose)e(essence)i(is)f(that)f(one)i (has)f(to)g(consider)f(\002rst)i(a)f(\002nite)g(sum)456 2992 y Fl(P)561 3018 y Fn(p)561 3096 y(j)t Fr(=1)704 3067 y Fs(f)11 b Fq(\()p Fp({)867 3082 y Fn(j)904 3067 y Fq(\))p Fu(,)22 b(pro)o(v)o(e)f(that)h(this)f(sum)h(can)g(be)g (approximated)g(by)f(the)h(corresponding)456 3185 y(sum)i(for)h Fs(A)861 3200 y Fn(n)908 3185 y Fs(;)17 b(B)1026 3200 y Fn(n)1097 3185 y Fu(and)25 b(then)g(let)f Fs(p)k Fo(!)f(1)p Fu(.)1671 3464 y(R)t Ft(E)t(F)t(E)t(R)t(E)t(N)t(C)5 b(E)g(S)497 3622 y Fw([1])40 b(N.I.)18 b(Akhiezer)m(,)f Fk(The)i(classical)g (moment)e(pr)l(oblem)h(and)g(some)g(r)m(elated)g(questions)g(in)g (analysis)p Fw(,)635 3721 y(Hafner)i(Publishing)f(Co.,)h(Ne)n(w)g(Y)-9 b(ork,)19 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b(1-45)g(\(1998\).)456 2343 y([16])39 b(D.)20 b(Krutik)o(o)o(v)d(and)i(Ch.)g(Remling,)g Fk(Sc)o(hr)1816 2344 y(\250)1809 2343 y(oding)o(er)e(oper)o(ator)o(s)i(with)h(spar)o (se)f(potentials:)24 b(asymp-)635 2443 y(totics)d(of)f(the)h(F)-9 b(ourier)20 b(tr)o(ansform)g(of)g(the)h(spectr)o(al)f(measur)m(e)p Fw(,)f(preprint.)456 2542 y([17])39 b(A.)21 b(Lapte)n(v)-5 b(,)19 b(S.)h(Nabok)o(o)f(and)h(O.)g(Safrono)o(v)e Fk(On)j(ne)o(w)f(r)m (elations)g(between)g(spectr)o(al)g(pr)l(operties)635 2642 y(of)h(J)m(acobi)e(matrices)i(and)e(their)i(coef)o(\002cients)d Fw(preprint)456 2742 y([18])39 b(Y)-11 b(.)25 b(Last)f(and)f(B.)i (Simon,)e Fk(Eig)o(enfunctions,)g(tr)o(ansfer)h(matrices,)h(and)d (absolutely)h(continuous)635 2841 y(spectrum)e(of)g(one-dimensional)d (Sc)o(hr)1781 2842 y(\250)1774 2841 y(oding)o(er)h(oper)o(ator)o(s)p Fw(,)h(In)m(v)o(ent.)f(Math.)h Fb(135)g Fw(\(1999\),)f(329-)635 2941 y(367.)456 3040 y([19])39 b(S.)19 b(Molchano)o(v)-5 b(,)15 b(M.)j(No)o(vitskii)g(and)f(B.)i(V)-9 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