Content-Type: multipart/mixed; boundary="-------------0307300717482" This is a multi-part message in MIME format. ---------------0307300717482 Content-Type: text/plain; name="03-353.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-353.keywords" Poltoratskii, Borel transform, Stieltjes transform, singular measure ---------------0307300717482 Content-Type: application/postscript; name="jalpoltf.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="jalpoltf.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: jalpoltf.dvi %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o jalpoltf.ps jalpoltf.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.07.30:1412 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false 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Fg(\026)3099 5220 y Fa(1)3144 5192 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))p 2642 5243 855 4 v 2766 5334 a(Im)16 b Fj(F)2966 5349 y Fg(\026)3019 5334 y Fn(\()p Fj(E)28 b Fn(+)22 b Fj(i\017)q Fn(\))3526 5266 y Fj(;)3645 4741 y Fn(\(2)p Fj(:)p Fn(12\))1925 5579 y(6)p eop %%Page: 7 7 7 6 bop 0 100 a Fn(where)40 b(the)f(last)f(equalit)m(y)g(is)h(just)h(m) m(ultiplying)d(and)i(dividing)f(b)m(y)h(Im)16 b Fj(F)2829 115 y Fg(\026)2882 100 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\).)60 b(Since)40 b Fj(\026)3650 115 y Fl(sing)3833 100 y Fn(is)0 279 y(singular)33 b(w.r.t.)f Fj(\026)703 294 y Fl(1)748 279 y Fn(,)h(it)f(is)i(also)e(singular)i(w.r.t.)e(\()p Fj(U)11 b(f)g Fn(\))2069 243 y Fl(2)2112 279 y Fj(\026)2172 294 y Fl(1)2250 279 y Fn(and)34 b(w)m(e)f(th)m(us)i(ha)m(v)m(e)e(b)m(y) h(Prop)s(osition)e(2.2,)1342 636 y(lim)1335 697 y Fg(\017)p Ff(!)p Fl(0)1516 561 y Fn(Im)15 b Fj(F)1715 579 y Fl(\()p Fg(U)7 b(f)h Fl(\))1885 560 y Fa(2)1925 579 y Fg(\026)1973 589 y Fa(1)2017 561 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\))p 1516 613 855 4 v 1640 704 a(Im)16 b Fj(F)1840 719 y Fg(\026)1893 704 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\))2410 636 y(=)28 b(0)1080 b(\(2)p Fj(:)p Fn(13\))0 990 y(a.e.)33 b(w.r.t.)f Fj(\026)509 1005 y Fl(sing)653 990 y Fn(.)43 b(Also,)33 b(w)m(e)h(see)g(from)e(\(2.8\))g(that)1082 1350 y(Im)16 b Fj(F)1282 1365 y Fg(\026)1330 1375 y Fa(1)1374 1350 y Fn(\()p Fj(z)t Fn(\))34 b(Im)15 b Fj(F)1735 1365 y Fg(\026)1789 1350 y Fn(\()p Fj(z)t Fn(\))28 b(=)2069 1283 y(\(Im)15 b Fj(F)2307 1298 y Fg(\026)2360 1283 y Fn(\()p Fj(z)t Fn(\)\))2527 1246 y Fl(2)p 2062 1327 518 4 v 2062 1418 a Fh(j)p Fn(1)21 b(+)i Fj(F)2325 1433 y Fg(\026)2378 1418 y Fn(\()p Fj(z)t Fn(\))p Fh(j)2534 1390 y Fl(2)2619 1350 y Fh(\024)28 b Fn(1)17 b Fj(;)826 b Fn(\(2)p Fj(:)p Fn(14\))0 1704 y(and)34 b(th)m(us)g(\(2.12\))e (implies)1346 1854 y(lim)1339 1915 y Fg(\017)p Ff(!)p Fl(0)1507 1854 y Fh(j)p Fn(Im)16 b Fj(F)1735 1872 y Fl(\()p Fg(U)7 b(f)h Fl(\))p Fg(\026)1953 1882 y Fa(1)1998 1854 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))p Fh(j)k Fn(=)i(0)1083 b(\(2)p Fj(:)p Fn(15\))0 2137 y(a.e.)35 b(w.r.t.)f Fj(\026)513 2152 y Fl(sing)657 2137 y Fn(.)52 b(By)34 b(using)j(\(2.12\))d(and)i(the)g(equalit)m(y)f(in)g(\(2.14\))f (to)i(estimate)e Fh(j)p Fn(Im)15 b Fj(F)3362 2155 y Fl(\()p Fg(U)7 b(f)h Fl(\))p Fg(\026)3580 2165 y Fa(1)3625 2137 y Fn(\()p Fj(z)t Fn(\))p Fh(j)36 b Fn(in)0 2317 y(\(2.11\),)31 b(w)m(e)j(get)168 2713 y Fh(j)p Fj(L)p Fn(\()p Fj(E)27 b Fn(+)22 b Fj(i\017)q Fn(\))p Fh(j)27 b(\024)789 2646 y(j)p Fn(Re)17 b(\(1)22 b(+)h Fj(F)1226 2661 y Fg(\026)1279 2646 y Fn(\()p Fj(E)k Fn(+)c Fj(i\017)q Fn(\)\))p Fh(j)p 789 2690 910 4 v 913 2782 a(j)p Fn(Im)16 b Fj(F)1141 2797 y Fg(\026)1194 2782 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\))p Fh(j)1774 2646 y(j)p Fn(Im)15 b Fj(F)2001 2661 y Fg(\026)2055 2646 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))p Fh(j)p 1756 2690 698 4 v 1756 2782 a(j)p Fn(1)f(+)g Fj(F)2019 2797 y Fg(\026)2073 2782 y Fn(\()p Fj(E)27 b Fn(+)22 b 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1110 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))p 835 1154 681 4 v 835 1245 a Fj(F)899 1263 y Fl(\(1+)p Fg(f)8 b Fl(\))p Fg(\026)1162 1245 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))1555 1177 y(=)35 b(lim)1660 1239 y Fg(\017)p Ff(!)p Fl(0)1841 1110 y Fj(F)1905 1125 y Fg(g)s(\027)1996 1110 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))p 1841 1154 509 4 v 1861 1245 a Fj(F)1925 1260 y Fg(\027)1975 1245 y Fn(\()p Fj(E)k Fn(+)c Fj(i\017)q Fn(\))2389 1177 y(=)28 b Fj(g)t Fn(\()p Fj(E)6 b Fn(\))25 b(=)3016 1110 y(1)p 2847 1154 388 4 v 2847 1245 a(1)d(+)g Fj(f)11 b Fn(\()p Fj(E)6 b Fn(\))3645 1177 y(\(2)p Fj(:)p Fn(20\))0 1513 y(a.e.)33 b(w.r.t.)f Fj(\026)509 1528 y Fl(sing)653 1513 y Fn(,)g(and)i(so)f(w)m(e)h(ha)m(v)m(e)g(b)m(y)f (the)h(linearit)m(y)e(of)h(the)h(Borel)f(transform,)838 1854 y(lim)831 1916 y Fg(\017)p Ff(!)p Fl(0)1012 1787 y Fj(F)1076 1802 y Fg(f)8 b(\026)1176 1787 y Fn(\()p Fj(E)28 b Fn(+)22 b Fj(i\017)q Fn(\))p 1012 1831 518 4 v 1036 1923 a Fj(F)1100 1938 y Fg(\026)1153 1923 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\))1569 1854 y(=)35 b(lim)1675 1916 y Fg(\017)p Ff(!)p Fl(0)1855 1780 y Fj(F)1919 1798 y Fl(\(1+)p Fg(f)8 b Fl(\))p Fg(\026)2183 1780 y Fn(\()p Fj(E)27 b Fn(+)22 b Fj(i\017)q Fn(\))p 1855 1831 681 4 v 1960 1923 a Fj(F)2024 1938 y Fg(\026)2078 1923 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)p Fn(\))2570 1854 y Fh(\000)g Fn(1)k(=)h Fj(f)11 b Fn(\()p Fj(E)6 b Fn(\))575 b(\(2)p Fj(:)p Fn(21\))0 2195 y(a.e.)33 b(w.r.t.)f Fj(\026)509 2210 y Fl(sing)653 2195 y Fn(.)44 b(Since)34 b(ev)m(ery)f(complex)f(v) -6 b(alued)34 b Fj(f)39 b Fh(2)28 b Fj(L)2201 2159 y Fl(1)2245 2195 y Fn(\()p Fi(R)11 b Fj(;)17 b(d\026)p Fn(\))32 b(is)h(a)g(linear)g(com)m(bination)g(of)g(four)0 2375 y(p)s(ositiv)m(e)38 b(functions)i(in)f Fj(L)999 2339 y Fl(1)1043 2375 y Fn(\()p Fi(R)11 b Fj(;)17 b(d\026)p Fn(\),)39 b(the)g(linearit)m(y)f(of)h(the)g(Borel)f(transform)g (immediately)e(implies)0 2524 y(the)25 b(result)g(for)f(an)m(y)h(suc)m (h)h Fj(f)38 b Fh(2)28 b Fj(L)1220 2488 y Fl(1)1264 2524 y Fn(\()p Fi(R)11 b Fj(;)17 b(d\026)p Fn(\).)40 b(Finally)-8 b(,)24 b(if)h Fj(\026)f Fn(is)h(complex)e(v)-6 b(alued,)27 b(then)e(w)m(e)g(ha)m(v)m(e)g Fj(\026)i Fn(=)i Fj(g)t Fh(j)p Fj(\026)p Fh(j)0 2674 y Fn(where)34 b Fh(j)p Fj(\026)p Fh(j)f Fn(is)g(p)s(ositiv)m(e)f(and)i Fj(g)d Fh(2)d Fj(L)1340 2637 y Fl(1)1385 2674 y Fn(\()p Fi(R)10 b Fj(;)17 b(d)p Fh(j)p Fj(\026)p Fh(j)p Fn(\))32 b(with)h Fh(j)p Fj(g)t Fh(j)26 b Fn(=)i(1,)33 b(so)76 3011 y(lim)69 3072 y Fg(\017)p Ff(!)p Fl(0)250 2943 y Fj(F)314 2958 y Fg(f)8 b(\026)415 2943 y Fn(\()p Fj(E)27 b Fn(+)22 b Fj(i\017)q Fn(\))p 250 2988 518 4 v 274 3079 a Fj(F)338 3094 y Fg(\026)391 3079 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q Fn(\))807 3011 y(=)36 b(lim)913 3072 y Fg(\017)p Ff(!)p Fl(0)1094 2937 y Fj(F)1158 2955 y Fg(f)8 b(g)s Ff(j)p Fg(\026)p Ff(j)1346 2937 y Fn(\()p Fj(E)28 b Fn(+)22 b Fj(i\017)q Fn(\))p 1094 2988 607 4 v 1117 3079 a Fj(F)1181 3097 y Fg(g)s Ff(j)p Fg(\026)p Ff(j)1323 3079 y Fn(\()p Fj(E)27 b Fn(+)c Fj(i\017)q 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b(TION)42 b(\(gran)m(t)g(no.)g(188/02\).)70 b(W)-8 b(e)42 b(w)m(ould)h(lik)m(e)e(to)h(thank)g(Barry)0 4215 y(Simon)32 b(for)i(motiv)-6 b(ating)31 b(this)i(w)m(ork)g(and)h (for)f(useful)i(discussions.)0 4580 y Fk(References)72 4784 y Fn(1.)49 b(L.C.)26 b(Ev)-6 b(ans)26 b(and)g(R.F.)f(Gariep)m(y)-8 b(,)27 b Fc(Me)-5 b(asur)g(e)29 b(The)-5 b(ory)29 b(and)f(Fine)g(Pr)-5 b(op)g(erties)27 b(of)i(F)-8 b(unctions)p Fn(,)27 b(CR)m(C)199 4933 y(Press,)34 b(Bo)s(ca)e(Raton,)h(1992)72 5191 y(2.)49 b(V.)g(Jak)-5 b(\024)-45 b(s)o(i)m(\023)e(c)49 b(and)h(Y.)e(Last,)53 b Fc(Sp)-5 b(e)g(ctr)g(al)50 b(structur)-5 b(e)50 b(of)g(A)m(nderson)g (typ)-5 b(e)50 b(Hamiltonians,)i Fn(In)m(v)m(en)m(t.)199 5340 y(Math.)44 b Fk(141)34 b Fn(\(2000\),)e(561{577)1925 5579 y(8)p eop %%Page: 9 9 9 8 bop 72 100 a Fn(3.)49 b(V.)25 b(Jak)-5 b(\024)-45 b(si)m(\023)e(c)25 b(and)h(Y.)f(Last,)i Fc(Simplicity)h(of)g(singular)h (sp)-5 b(e)g(ctrum)28 b(in)h(A)m(nderson)f(typ)-5 b(e)29 b(Hamiltonians)p Fn(,)199 249 y(in)k(preparation)72 518 y(4.)49 b(A.G.)41 b(P)m(oltoratskii,)g Fc(The)j(Boundary)g(Behavior)e (of)h(Pseudo)-5 b(c)g(ontinuable)43 b(F)-8 b(unctions,)43 b Fn(St.)e(P)m(e-)199 667 y(tersburg)34 b(Math.)f(J.)h Fk(5)f Fn(\(1994\),)f(389{406)72 936 y(5.)49 b(M.)26 b(Reed)i(and)e(B.)g(Simon,)g Fc(Metho)-5 b(ds)29 b(of)g(Mo)-5 b(dern)29 b(Mathematic)-5 b(al)29 b(Physics,)g Fn(I.)g Fc(F)-8 b(unctional)29 b(A)m(nal-)199 1086 y(ysis)p Fn(,)k(Academic)f (Press,)i(London,)g(San)f(Diego,)g(1980)72 1355 y(6.)49 b(W.)34 b(Rudin,)f Fc(R)-5 b(e)g(al)36 b(and)f(Complex)i(A)m(nalysis)p Fn(,)c(3rd)h(edition,)e(McGra)m(w-Hill,)h(New)g(Y)-8 b(ork,)32 b(1987)72 1624 y(7.)49 b(S.)33 b(Saks,)g Fc(The)-5 b(ory)36 b(of)f(the)h(Inte)-5 b(gr)g(al)p Fn(,)33 b(Hafner,)h(New)f(Y) -8 b(ork,)32 b(1937)72 1893 y(8.)49 b(B.)24 b(Simon,)i Fc(Sp)-5 b(e)g(ctr)g(al)27 b(analysis)h(and)g(r)-5 b(ank)27 b(one)h(p)-5 b(erturb)g(ations)27 b(and)h(applic)-5 b(ations)p Fn(,)24 b(CRM)i(Lecture)199 2042 y(Notes)31 b(V)-8 b(ol.)42 b(8)30 b(\(J.)g(F)-8 b(eldman,)30 b(R.)h(F)-8 b(ro)s(ese,)31 b(L.)f(Rosen,)i(eds.\),)f(pp.)g(109{149,)f(Amer.)f(Math.)h(So)s(c.,)199 2192 y(Pro)m(vidence,)k(RI,)f(1995.)1925 5579 y(9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0307300717482--