Content-Type: multipart/mixed; boundary="-------------0011111200941" This is a multi-part message in MIME format. ---------------0011111200941 Content-Type: text/plain; name="00-448.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-448.comments" AMS-Code: (2000 Revision) Primary 82B44; Secondary 60H25 ---------------0011111200941 Content-Type: text/plain; name="00-448.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-448.keywords" Random Schroedinger operators, surface states, spectral shift function, density of states ---------------0011111200941 Content-Type: application/postscript; name="surface_states.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="surface_states.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: c:\texmf\texfiles\surface\surface_neu.dvi %%CreationDate: Thu Nov 09 14:28:22 2000 %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Symbol Times-Italic MSBM10 %%+ EUFM10 EUSM10 MSAM10 Helvetica Courier %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: c:\texmf\miktex\bin\dvips.exe %+ c:\texmf\texfiles\surface\surface_neu %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.11.09:1428 %%BeginProcSet: texc.pro %! 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All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueXX 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 13a84eb181d2946b5d0075f71db8a991dc1a809cd3cfec573514392567a82e75 946ccf6e14f4a968313b3b6d727c83018ed58b882b96db208ebd27f7d30ffbc8 ab11e10a0f81425a62bdf7eed9f618c746103dc2d1c3e1f37d27083f4068a30e 099c92685e320266ff8047e81140658a63a6b52e4cd63e95d1b83fde2655d8b5 7dc0c494c0f8ff83dcb15fe5dff2e7e87308566e0dc35056b0a7e8b4608e51c1 f3d94ad38b8330402ffa5fe1db2d745ed34eee06e38476f5d19f8a86c4a73899 d53a42d532eb1a9913cf320d4fbac56ecc4685b2879e3f4088c450d6efea3e90 e2fa0d52ef327010c831b948d0467739137dfa71e46051e4c1931d469e2b83e2 7be839cf3f7662d8b06cddbc59ff9dee8ad1bba6362491b78ffa47286d8aec5d e584a988fcdc57b813fa27b5bc3fb7053a04a420bb26a90224a113992d2f95d0 0d04322b237bcab7e677da083aa7609fa08aa18b0bc884df1e751d0d4165f3e0 0831191e0cc47645b0072c1797d912e17fd2f78c83cee888f80795d93c0ad80c c48f1ff9368d1377f7df89d7518f9b5b1434ae5382cbd0689747a9062a7e8ab1 9da33c0437b27658b450cfdd8b4f55126d2cde66f106dd773f9de881ced96743 957d4b6b3cac832b56db0d6714a11074cad144844b5fd1c8e04a3efaa2449c5b ec40243b088079550cfa313871623a1334a46f398598f75174f9deaa4cc8e16b fc1d435631672ce22becbbfd084e14cff86a099ab54640ff2037b0903035bc23 b636def820333f92dc42e5909a634a3f00db7b8e091cf1978bdcc20a8286fb02 b9b65c70f5b59da4dcf0e1fb868bdeaae8a2873a5a4f1e267194e42248253436 6ff471479c94503db7ccdcd5a78e90f0ffdbdfa5e968f64295323664ab322de6 4805b2471c019c75467f4567cf15de3bdb9d85701b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 78 /N put dup 80 /P put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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3457 y Fx(w,L)2524 3443 y FA(,)g Fz(H)2626 3457 y Fv(0)2663 3443 y FA(\))p Fz(d)5 b FA(l)23 b(=:)38 b Fz(\265)p FA(\()p Fz(g)p FA(\))386 3645 y FC(e)o(xists)30 b(almost)f(surely)h(and)f(is)g (non-random.)47 b(The)28 b(linear)i(functional)i Fz(\265)p FA(\()p Fz(g)p FA(\))d FC(is)g(related)h(to)f(the)g(density)h(of)386 3753 y(surf)o(ace)25 b(states)g Fz(\265)940 3767 y Fw(s)970 3753 y FA(\()p Fz(g)p FA(\))f FC(\(see)g([9)q(]\))f(such)i(that)f Fz(\265)1821 3767 y Fw(s)1851 3753 y FA(\()p Fz(g)p FA(\))d(=)f Fz(\265)p FA(\()p Fz(g)2167 3720 y Fh(0)2191 3753 y FA(\))j FC(\(with)h Fz(g)2504 3720 y Fh(0)2549 3753 y FC(being)h(the)f(deri)n (v)n(ati)n(v)o(e)h(of)e Fz(g)p FC(\),)g(where)854 3949 y Fz(\265)899 3963 y Fw(s)929 3949 y FA(\()p Fz(g)p FA(\))e(=)54 b FC(lim)1125 4009 y Fx(L)1171 4019 y Fs(1)1200 4009 y Fh(!)p Fx(\245)1125 4073 y(L)1171 4083 y Fs(2)1200 4073 y Fh(!)p Fx(\245)1524 3887 y FC(1)p 1339 3928 415 4 v 1339 4011 a(meas)1526 4025 y Fx(n)1561 4035 y Fs(1)1594 4011 y FA(\(L)1686 4025 y Fv(1)1724 4011 y FA(\))1764 3949 y FC(tr)15 b Ft([)q FA(c)1910 3963 y Fx(L)1956 3973 y Fs(1)1984 3963 y Fx(\264L)2066 3973 y Fs(2)2099 3949 y FA(\()p Fz(g)p FA(\()p Fz(H)2268 3963 y Fv(0)2326 3949 y FA(+)g Fz(V)2442 3963 y Fx(w,L)2551 3973 y Fs(1)2584 3949 y FA(\))20 b(-)g Fz(g)p FA(\()p Fz(H)2843 3963 y Fv(0)2881 3949 y FA(\)\))p Ft(])c FA(,)106 b Fz(g)20 b FA(\316)15 b Fz(C)3319 3911 y Fv(2)3315 3971 y(0)3357 3949 y FA(,)386 4212 y FC(almost)23 b(surely)i(for)e(arbitrary)i (sequences)g(of)e(box)o(es)g FA(L)2140 4226 y Fv(1)2196 4212 y FA(\314)17 b Fu(R)2344 4179 y Fx(n)2379 4189 y Fs(1)2411 4212 y FC(,)22 b FA(L)2518 4226 y Fv(2)2574 4212 y FA(\314)17 b Fu(R)2722 4179 y Fx(n)2757 4189 y Fs(2)2811 4212 y FC(tending)25 b(to)e(in\002nity)-6 b(.)29 b(Englisch,)386 4320 y(Kirsch,)e(Schr)8 b(\250)-38 b(oder)l(,)29 b(and)e(Simon)f([9)q(])g(analyzed)j(the)d(surf)o(ace)j(states)e (occuring)i(at)d(the)h(boundary)i(between)386 4428 y(tw)o(o)22 b(Anderson-type)k(crystals)e(and)f(pro)o(v)o(ed)g(that)g(the)g(distrib) n(ution)j(induced)e(by)f(the)g(functional)i Fz(\265)3548 4442 y Fw(s)3578 4428 y FA(\()p Fz(g)p FA(\))e FC(\(i.e.)386 4536 y(the)i(density)i(of)d(surf)o(ace)j(states\))f(is)f(of)g(order)g (\(at)g(most\))g(3.)33 b(This)24 b(result)i(applies)h(almost)e(v)o (erbatim)h(also)f(to)386 4644 y(interactions)i(of)c(type)i(\(1\))q(,)d (so)i(we)e(ha)n(v)o(e)1389 4831 y Fz(\265)1434 4845 y Fw(s)1464 4831 y FA(\()p Fz(g)p FA(\))f(=)1661 4717 y Fq(Z)1707 4902 y Fn(R)1775 4831 y Fz(g)p FA(\(l\))p Fz(N)1997 4793 y Fh(0)1989 4853 y Fw(s)2019 4831 y FA(\(l\))p Fz(d)5 b FA(l,)197 b Fz(g)21 b FA(\316)15 b Fz(C)2658 4793 y Fv(3)2654 4853 y(0)2695 4831 y FA(\()p Fu(R)p FA(\),)386 5024 y FC(where)24 b(the)g(distrib)n(ution)j Fz(N)1264 5038 y Fw(s)1293 5024 y FA(\(l\))c FC(of)h(order)g(at)g(most)f(2)g(is)h (called)h(the)e(inte)o(grated)j(density)g(of)d(surf)o(ace)i(states.)486 5132 y(Using)30 b(a)g(slightly)j(dif)n(ferent)f(approach)g(from)f(ours) g(Chahrour)h([5])e(constructed)k(the)d(functional)i Fz(\265)p FA(\()p Fz(g)p FA(\))386 5240 y FC(for)25 b(the)f(case)h(of)g(discrete) h(Schr)8 b(\250)-38 b(odinger)27 b(operators)g(and)e(sho)n(wed)g(that)g (the)f(inte)o(grated)j(density)f(of)f(surf)o(ace)386 5348 y(states)h Fz(N)671 5362 y Fw(s)700 5348 y FA(\(l\))f FC(is)f(a)h(distrib)n(ution)j(of)d(order)h(\(at)f(most\))g(1.)32 b(In)25 b([6)q(])f(he)h(pro)o(v)o(ed)h(that)f(for)g(discrete)i(Schr)8 b(\250)-38 b(odinger)386 5456 y(operators)26 b(with)d Fz(nonr)o(andom)j FC(periodic)f(potentials)i Fz(N)2128 5470 y Fw(s)2157 5456 y FA(\(l\))c FC(is)h(a)f(measurable)i(function.) 486 5564 y(Further)c(in)g([16)q(])g(we)f(pro)o(v)o(ed)i(that)f(the)h (functional)h Fz(\265)p FA(\()p Fz(g)p FA(\))f FC(induces)h(a)d(signed) j(Borel)e(measure)h Fz(d)5 b FA(X\(l\))21 b FC(such)386 5671 y(that)j(for)g(an)o(y)f Fz(g)e FA(\316)15 b Fz(C)1031 5685 y Fv(0)1068 5671 y FA(\()p Fu(R)p FA(\))1740 5821 y Fz(\265)p FA(\()p Fz(g)p FA(\))21 b(=)1982 5707 y Fq(Z)2028 5892 y Fn(R)2095 5821 y Fz(g)p FA(\(l\))p Fz(d)5 b FA(X\(l\).)p eop %%Page: 3 3 3 2 bop 1287 259 a Fv(REGULARITY)21 b(OF)h(THE)f(DENSITY)f(OF)i(SURF)-5 b(A)m(CE)22 b(ST)-6 b(A)f(TES)866 b(3)386 459 y FC(This)23 b(result)i(implies)f(that)h(\223)p Fz(N)1344 473 y Fw(s)1373 459 y FA(\(l\))p Fz(d)5 b FA(l)p FC(\224)24 b(is)f(a)g FA(s)p FC(-\002nite)h(Borel)g(measure.)486 567 y(W)-7 b(e)22 b(will)h(no)n(w)g(e)o(xtend)i(this)f(result)h(and)f(pro)o(v)o(e) 386 701 y FJ(Theor)n(em)h(1.)43 b Fz(F)-10 b(or)25 b(continuous)k(Sc)o (hr)1619 702 y(\250)1612 701 y(oding)o(er)f(oper)o(ator)o(s)g(H)2359 715 y Fx(w)2430 701 y FA(=)21 b Fz(H)2565 715 y Fv(0)2623 701 y FA(+)c Fz(V)2741 715 y Fx(w)2816 701 y Fz(with)j(V)3043 715 y Fx(w)3117 701 y Fz(being)27 b(de\002ned)g(by)e FC(\(1\))386 809 y Fz(the)f(\(signed\))h(density)h(of)d(surface)i (states)g(measur)m(e)g(d)5 b FA(X\(l\))23 b Fz(is)g(Lebesgue)i (absolutely)i(continuous.)486 944 y FC(In)j(the)h(other)h(w)o(ords)f (Theorem)h(1)e(states)i(that)f(the)g(inte)o(grated)j(density)e(of)f (surf)o(ace)h(states)g Fz(N)3546 958 y Fw(s)3576 944 y FA(\(l\))e FC(is)h(a)386 1051 y(measurable)26 b(locally)f(inte)o (grable)h(function.)486 1159 y(Ho)n(we)n(v)o(er)l(,)18 b(it)g(remains)h(unclear)h(whether)f Fz(N)1877 1173 y Fw(s)1906 1159 y FA(\(l\))f FC(possesses)j(further)f(re)o(gularity)g (properties,)h(e.g.)d(whether)386 1267 y(it)26 b(is)g(a)g(function)j (of)d(locally)j(bounded)f(v)n(ariation)h(such)e(that)g(\223)p Fz(d)5 b(N)2506 1281 y Fw(s)2536 1267 y FA(\(l\))p FC(\224)27 b(de\002nes)g(a)f(measure.)38 b(In)27 b(f)o(act,)g(it)g(is)386 1375 y(dif)n(\002cult)e(to)f(control)i(the)f(smoothness)i(of)d Fz(N)1811 1389 y Fw(s)1840 1375 y FA(\(l\))g FC(since)i(it)e(may)g (oscillate)i(rapidly)g(due)f(to)f(the)h(presence)h(of)386 1483 y(alternating)g(surf)o(ace)g(states)e(and)g(surf)o(ace)i (\224holes\224.)486 1591 y(The)f(results)j(of)e(our)g(article)i([16)q (])d(e)o(xtend)i(almost)g(v)o(erbatim)g(\(actually)i(with)c(se)n(v)o (eral)i(simpli\002cations\))386 1699 y(to)d(the)h(case)h(of)e(discrete) j(Schr)8 b(\250)-38 b(odinger)27 b(operators.)34 b(More)25 b(precisely)i(we)d(consider)i(discrete)h(Schr)8 b(\250)-38 b(odinger)386 1807 y(operators)26 b(with)d(random)i(potentials)h(on)e (a)f(hypersurf)o(ace,)386 1974 y(\(5\))46 b FA(\()p Fz(h)612 1988 y Fx(w)663 1974 y Fz(u)p FA(\)\()p Fj(n)p FA(\))8 b(=)g(\()p Fz(h)997 1988 y Fv(0)1037 1974 y Fz(u)p FA(\)\()p Fj(n)p FA(\))g(+)1302 1954 y Fr(e)1291 1974 y Fz(V)1342 1988 y Fx(w)1394 1974 y FA(\()p Fj(n)1482 1988 y Fv(1)1520 1974 y FA(\)d\()p Fj(n)1683 1988 y Fv(2)1721 1974 y FA(\))p Fz(u)p FA(\()p Fj(n)p FA(\),)107 b(\()p Fz(h)2119 1988 y Fv(0)2157 1974 y Fz(u)p FA(\)\()p Fj(n)p FA(\))8 b(=)2437 1993 y Fp(\345)2416 2072 y Fh(j)p Fk(j)p Fh(j)p Fx(=)p Fv(1)2565 1974 y Fz(u)p FA(\()p Fj(n)g FA(-)g Fj(j)p FA(\),)107 b Fj(n)8 b FA(=)g(\()p Fj(n)3168 1988 y Fv(1)3207 1974 y FA(,)15 b Fj(n)3303 1988 y Fv(2)3340 1974 y FA(\))8 b(\316)g Fu(Z)3512 1936 y Fx(n)3547 1946 y Fs(1)3581 1974 y FA(\305)p Fu(Z)3712 1936 y Fx(n)3747 1946 y Fs(2)3779 1974 y FA(,)386 2244 y FC(where)24 b FA(d\()p Fj(n)764 2258 y Fv(2)801 2244 y FA(\))f FC(is)h(the)g(Kroneck)o(er)h(symbol)f (and)1927 2225 y Fr(e)1916 2244 y Fz(V)1967 2258 y Fx(w)2018 2244 y FA(\()p Fj(n)2106 2258 y Fv(1)2143 2244 y FA(\))f FC(a)g(metrically)j(transiti)n(v)o(e)f(random)g(\002eld)e(on)h Fu(Z)3668 2211 y Fx(n)3703 2221 y Fs(1)3735 2244 y FC(.)486 2352 y(W)-7 b(e)22 b(will)h(pro)o(v)o(e)h(the)g(follo)n(wing)h (analogue)h(of)e(Theorem)f(1)386 2487 y FJ(Theor)n(em)j(2.)43 b Fz(F)-10 b(or)25 b(discr)m(ete)j(Sc)o(hr)1506 2488 y(\250)1499 2487 y(oding)o(er)h(oper)o(ator)o(s)f(of)e(the)g(form)g FC(\(5\))f Fz(the)i(\(signed\))g(density)h(of)e(surface)386 2595 y(states)f(measur)m(e)f(d)5 b FA(X\(l\))23 b Fz(is)g(Lebesgue)j (absolutely)g(continuous.)486 2729 y FC(The)c(proofs)h(of)g(Theorems)g (1)f(and)h(2)f(will)g(be)g(gi)n(v)o(en)h(in)f(Section)i(2.)k(Section)23 b(3)f(plays)i(a)e(complementary)386 2837 y(role.)53 b(Its)32 b(aim)f(is)g(to)h(gi)n(v)o(e)g(a)f(simple)h(proof)h(of)e(the)h (Combes-Hislop-Nakamura)j(result)e(on)f(the)g(H)8 b(\250)-38 b(older)386 2945 y(continuity)37 b(of)d(the)h(inte)o(grated)i(density)f (of)e(\(b)n(ulk\))i(states)f(for)g(some)f(random)h(Schr)8 b(\250)-38 b(odinger)37 b(operators.)386 3053 y(The)d(proof)i(is)f (based)h(on)f(the)g(combination)j(of)d(the)g(Combes-Hislop-Nakamura)k Fz(L)3212 3020 y Fw(p)3249 3053 y FC(-bound)d(with)f(the)386 3161 y(formula)25 b(of)f(Birman)g(and)g(Solomyak)h([2)q(].)k(This)24 b(combination)j(is)d(a)f(generalization)28 b(of)c(Simon')-5 b(s)24 b(spectral)386 3269 y(a)n(v)o(eraging)i(method)e(which)g(w)o(as) f(used)h(to)f(pro)o(v)o(e)h(Lipshitz)g(continuity)i(of)e(the)f(inte)o (grated)j(density)f(of)e(b)n(ulk)386 3377 y(states)i(\(W)-7 b(e)o(gner')i(s)24 b(estimate\))h(for)f(some)g(random)g(Jacobi)h (matrices)g([23)q(].)486 3485 y FJ(Ackno)o(wledgements.)i FC(W)-7 b(e)20 b(are)i(indebted)h(to)e(J.M.)f(Combes)i(for)f(useful)i (discussions)h(and)e(for)f(sending)386 3593 y(us)i(the)h(preliminary)i (v)o(ersion)f(of)f(the)f(preprint)j(preprint)g([8].)1442 3820 y(2.)48 b(P)t FB(R)q(O)t(O)t(F)t(S)29 b(O)t(F)e FC(T)t FB(H)t(E)t(O)t(R)t(E)t(M)t(S)j FC(1)e FB(A)t(N)t(D)f FC(2)486 3982 y(The)f(tw)o(o)g(main)h(ingredients)i(of)e(our)g (approach)i(to)d(pro)o(v)o(e)h(Theorem)g(1)g(are)f(the)h (Banach-Alauglu)j(the-)386 4090 y(orem)24 b(\(see)g(e.g.)g([20)q(]\))g (and)g(the)g(Combes-Hislop-Nakamura)k Fz(L)2472 4057 y Fw(p)2509 4090 y FC(-bound)e(for)e(the)g(spectral)i(shift)f(function) 386 4198 y([8)q(])e(\(see)h(also)g(its)g(generalization)j(by)d (Hundertmark)h(and)f(Simon)g(in)f([13)q(]\).)486 4306 y(It)e(is)h(generally)j(kno)n(wn)e(that)f(the)h(discrete)h(case)f(is)f (much)g(easier)h(to)g(handle)g(than)g(the)g(continuous)i(case.)386 4414 y(Indeed)k(in)e(the)h(discrete)h(case)f(\(Theorem)g(2\))f(we)g (actually)i(do)f(not)f(need)h(the)g Fz(L)2973 4381 y Fw(p)3010 4414 y FC(-bound)h(for)f(the)f(spectral)386 4522 y(shift)j(function)h(and)f(will)f(use)g(instead)i(a)e(well-kno)n (wn)h(bound)h(for)e(\002nite)g(rank)h(perturbations.)49 b(W)-7 b(e)29 b(note)386 4630 y(also)24 b(that)g(in)g(the)f(case)i(of)e (\002nite)h(rank)g(perturbations)j(this)e(bound)g(is)e(implied)h(by)g (the)g Fz(L)3207 4597 y Fw(p)3244 4630 y FC(-bound.)486 4738 y(In)f(the)h(sequel)g(we)f(will)g(use)h(the)g(follo)n(wing)g (well-kno)n(wn)h(lemma,)e(which)h(is)f(a)g(direct)h(consequence)j(of) 386 4845 y(the)d(Banach-Alauglu)i(theorem.)386 4980 y FJ(Lemma)d(3.)41 b Fz(Let)23 b FC(1)e FA(<)f Fz(q)h FA(\243)f(\245)p Fz(.)27 b(Let)c(the)h(sequence)i(of)d(r)m(eal)h(valued)h(functions)40 b(f)2892 4994 y Fw(n)2950 4980 y FA(\316)20 b Fz(L)3086 4947 y Fw(q)3123 4980 y FA(\()p Fz(a)p FA(,)15 b Fz(b)p FA(\))24 b Fz(satisfy)960 5115 y Fw(b)916 5133 y Fq(Z)922 5370 y Fw(a)1015 5247 y Fl(j)14 b Fz(f)1081 5261 y Fw(n)1118 5247 y FA(\(l\))p Fl(j)1253 5209 y Fw(q)1291 5247 y Fz(d)5 b FA(l)21 b(\243)15 b Fz(C)93 b FA(\()p Fz(q)21 b FA(<)f(\245\))182 b Fz(or)240 b FC(sup)2351 5331 y Fx(l\316\()p Fw(a)p Fx(,)p Fw(b)p Fx(\))2578 5247 y Fl(j)14 b Fz(f)2644 5261 y Fw(n)2681 5247 y FA(\(l\))p Fl(j)21 b FA(\243)15 b Fz(C)93 b FA(\()p Fz(q)21 b FA(=)f(\245\))386 5509 y Fz(uniformly)28 b(in)f(n)f(for)h(some)21 b(C)k FA(<)d(\245)p Fz(.)37 b(If)26 b(the)h(sequence)43 b(f)2173 5523 y Fw(n)2211 5509 y FA(\(l\))p Fz(d)5 b FA(l)26 b Fz(con)l(ver)m(g)o(es)j(weakly)f (to)e(a)g(signed)j(measur)m(e)386 5616 y(d)5 b(\265)p FA(\(l\))24 b Fz(then)g(this)g(measur)m(e)g(is)g(absolutely)i (continuous.)486 5751 y FC(F)o(or)d(reader')-5 b(s)26 b(con)l(v)o(enience)i(we)c(recall)h(the)g(proof.)32 b(By)23 b(the)i(Banach-Alauglu)i(theorem)f(we)d(can)i(\002nd)f(a)386 5859 y(subsequence)40 b Fz(f)908 5873 y Fw(n)p Fx(\()p Fw(i)p Fx(\))1030 5859 y FC(of)d Fz(f)1169 5873 y Fw(n)1228 5859 y FC(which)23 b(con)l(v)o(er)n(ges)j(in)c(the)h(weak-)p FA(*)h FC(topology)-6 b(,)25 b(i.e.)d(there)h(e)o(xists)38 b Fz(f)3338 5873 y Fx(\245)3406 5859 y FA(\316)17 b Fz(L)3539 5826 y Fw(q)3576 5859 y FA(\()p Fz(a)p FA(,)e Fz(b)p FA(\))j(\314)p eop %%Page: 4 4 4 3 bop 386 259 a Fv(4)1168 b(V)-9 b(.)17 b(K)n(OSTR)l(YKIN)f(AND)g(R.) h(SCHRADER)386 459 y Fz(L)437 426 y Fv(1)474 459 y FA(\()p Fz(a)p FA(,)e Fz(b)p FA(\))24 b FC(such)h(that)1045 544 y Fq(Z)1158 657 y Fz(f)1185 671 y Fw(n)p Fx(\()p Fw(i)p Fx(\))1286 657 y FA(\(l\))p Fz(g)p FA(\(l\))p Fz(d)5 b FA(l)26 b Fl(!)1793 544 y Fq(Z)1906 657 y Fz(f)1933 671 y Fx(\245)1985 657 y FA(\(l\))p Fz(g)p FA(\(l\))p Fz(d)5 b FA(l,)198 b Fz(g)21 b FA(\316)f Fz(L)2773 620 y Fw(q)p Fx(/\()p Fw(q)p Fx(-)p Fv(1)p Fx(\))2976 657 y FA(\()p Fz(a)p FA(,)15 b Fz(b)p FA(\).)386 861 y FC(Thus)23 b(the)h(measure)h Fz(d)5 b(\265)p FA(\(l\))21 b(=)34 b Fz(f)1387 875 y Fx(\245)1439 861 y FA(\(l\))p Fz(d)5 b FA(l)23 b FC(is)h(absolutely)i(continuous.)486 969 y(W)-7 b(e)24 b(start)h(with)g(the)h(proof)g(of)f(Theorem)g(2)g(which)g (is)g(much)g(easier)h(than)g(the)f(one)h(for)f(Theorem)g(1.)33 b(F)o(or)386 1076 y(an)23 b(arbitrary)j(rectangular)h(box)d FA(L)c(\314)g Fu(R)1654 1043 y Fx(n)1689 1053 y Fs(1)1745 1076 y FC(with)j(inte)o(ger)n(-v)n(alued)k(v)o(ertices)e(we)e(de\002ne) 1415 1323 y Fz(V)1466 1337 y Fx(w,L)1578 1323 y FA(\()p Fj(n)p FA(\))e(=)1787 1167 y Fr(\()1866 1241 y(e)1855 1261 y Fz(V)1906 1275 y Fx(w)1957 1261 y FA(\()p Fj(n)2045 1275 y Fv(1)2082 1261 y FA(\)d\()p Fj(n)2245 1275 y Fv(2)2283 1261 y FA(\),)91 b Fj(n)2485 1275 y Fv(1)2543 1261 y FA(\316)20 b(L,)1860 1391 y FC(0)p FA(,)499 b FC(otherwise.)386 1566 y(Adopting)25 b(the)f(results)h(of)e(our)h(article)h([16)q(])e(to) h(the)g(case)g(of)f(discrete)j(Schr)8 b(\250)-38 b(odinger)26 b(operators)g(we)d(ha)n(v)o(e)386 1703 y FJ(Pr)n(oposition)i(4.)41 b Fz(F)-10 b(or)24 b(any)g(g)c FA(\316)15 b Fz(C)1478 1670 y Fv(1)1474 1729 y(0)1516 1703 y FA(\()p Fu(R)p FA(\))23 b 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FA(\(l\)x\(l)p FC(;)15 b Fz(H)2381 681 y Fv(0)2439 667 y FA(+)h Fz(V)2556 681 y Fx(w,L)2668 667 y FA(,)f Fz(H)2770 681 y Fv(0)2807 667 y FA(\))p Fz(d)5 b FA(l)386 890 y Fz(e)n(xists)31 b(and)e(de\002nes)i(a)e(linear)i(functional)h(\265)p FA(\()p Fz(g)p FA(\))e Fz(on)24 b(C)2134 857 y Fv(1)2130 916 y(0)2171 890 y FA(\()p Fu(R)p FA(\))p Fz(.)46 b(This)29 b(functional)j(e)n(xtends)f(to)e(all)h(g)25 b FA(\316)19 b Fz(C)3684 904 y Fv(0)3721 890 y FA(\()p Fu(R)p FA(\))386 997 y Fz(and)24 b(admits)g(the)g(r)m(epr)m(esentation)1759 1151 y(\265)p FA(\()p Fz(g)p FA(\))d(=)2000 1037 y Fq(Z)2099 1151 y Fz(g)p FA(\(l\))p Fz(d)5 b FA(X\(l\))386 1322 y Fz(with)23 b FA(X)g Fz(being)h(a)g(signed)h(Bor)m(el)e(measur)m(e)o (.)486 1455 y FC(Let)k Fz(s)679 1469 y Fw(j)703 1455 y FA(\()p Fz(T)12 b FA(\))27 b FC(denote)i(the)f(singular)i(v)n(alues)e (of)g(a)f(compact)i(operator)h Fz(T)11 b FC(.)40 b(F)o(or)27 b(an)o(y)h(0)23 b FA(<)31 b Fz(p)23 b FA(<)g(\245)k FC(de\002ne)h(the) 386 1563 y(functional)e 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h(relies)g(on)e(the)h(follo)n(wing)h(lemma)e(which)h(is)f(due)h(to)f (Combes,)386 2877 y(Hislop)24 b(and)g(Nakamura)g([8)q(].)k(A)23 b(generalization)k(of)d(this)g(result)h(can)f(be)f(found)i(in)e([13)r (].)386 3010 y FJ(Lemma)d(6.)38 b Fz(Let)21 b(A)f(be)h(a)g(bounded)i (self-adjoint)i(oper)o(ator)e(on)e(a)g(separ)o(able)j(Hilbert)e(space)o (.)29 b(Let)20 b(the)i(tr)o(ace)386 3118 y(class)i(oper)o(ator)d(C)k (be)e(in)h Fi(J)1259 3132 y Fv(1)p Fx(/)5 b Fw(p)1376 3118 y Fz(for)23 b(some)h FC(1)d FA(\243)27 b Fz(p)20 b FA(<)g(\245)p Fz(.)28 b(Then)1642 3294 y Fl(k)p FA(x\()p Fl(\001)p FC(;)15 b Fz(A)21 b FA(+)15 b Fz(C)r FA(,)g Fz(A)p FA(\))p Fl(k)2201 3308 y Fw(L)2242 3288 y FD(p)2296 3294 y FA(\243)20 b Fl(j)-5 b Fz(C)r Fl(j)2474 3249 y Fv(1)p Fx(/)5 b Fw(p)2474 3321 y Fv(1)p Fx(/)g Fw(p)2568 3294 y FA(.)486 3471 y FC(In)21 b(the)i(case)29 b Fz(p)14 b FA(=)g FC(1)23 b(this)f(bound)h(pro)o(vides)h(the)e(well-kno)n(wn)i Fz(L)2460 3438 y Fv(1)2497 3471 y FC(-bound)f(for)g(the)f(spectral)i (shift)e(function,)386 3579 y Fl(k)p FA(x\()p Fl(\001)p FC(;)15 b Fz(A)28 b FA(+)22 b Fz(C)r FA(,)15 b Fz(A)p FA(\))p Fl(k)959 3600 y Fw(L)996 3580 y Fs(1)1057 3579 y FA(\243)27 b Fl(j)-5 b Fz(C)r Fl(j)1242 3593 y Fv(1)1280 3579 y FC(.)56 b(The)33 b(case)41 b Fz(p)27 b FA(=)g(\245)32 b FC(is)h(rele)n(v)n(ant)i(in)e(the)g(case)h(of)f(\002nite)g(rank)h (perturbations,)386 3687 y Fl(k)p FA(x\()p Fl(\001)p FC(;)15 b Fz(A)21 b FA(+)15 b Fz(C)r FA(,)g Fz(A)p FA(\))p Fl(k)945 3701 y Fw(L)982 3681 y FH(\245)1046 3687 y FA(\243)20 b FC(Rank)11 b Fz(C)r FC(.)486 3795 y(T)-7 b(o)20 b(proceed,)j(we)d (recall)i(the)g(de\002nition)h(of)e(the)g(Birman-Solomyak)i(spaces)g Fz(l)2992 3762 y Fw(p)3029 3795 y FA(\()p Fz(L)3110 3762 y Fw(q)3147 3795 y FA(\))p FC(,)e(1)11 b FA(\243)18 b Fz(p)p FA(,)d Fz(q)c FA(\243)g(\245)p FC(.)29 b(The)o(y)386 3903 y(are)24 b(the)f(sets)i(of)e(all)h(measurable)h(functions)h (satisfying)h Fl(k)p FA(f)p Fl(k)2336 3917 y Fw(l)2361 3897 y FD(p)2392 3917 y Fx(\()p Fw(L)2451 3897 y FD(q)2480 3917 y Fx(\))2526 3903 y FA(<)21 b(\245)h FC(with)1330 4166 y Fl(k)p FA(f)p Fl(k)1467 4180 y Fw(l)1492 4161 y FD(p)1523 4180 y Fx(\()p Fw(L)1582 4161 y FD(q)1612 4180 y Fx(\))1658 4166 y FA(=)1728 4011 y Fr( )1827 4186 y Fp(\345)1800 4261 y Fk(j)p Fx(\316)p Fn(Z)1916 4242 y FH(n)1962 4038 y Fr(\024)2010 4053 y Fq(Z)2056 4238 y Fx(D)2097 4248 y Fs(0)2145 4166 y Fl(j)p FA(f\()p Fz(x)g FA(+)e Fj(j)p FA(\))p Fl(j)2466 4129 y Fw(q)2504 4166 y Fz(d)5 b(x)2594 4038 y Fr(\025)2648 4060 y Fw(p)p Fx(/)p Fw(q)2737 4011 y Fr(!)2809 4033 y Fv(1)p Fx(/)g Fw(p)386 4404 y FC(and)24 b FA(D)596 4418 y Fv(0)656 4404 y FC(being)g(the)g (unit)g(cube)h(centered)g(at)f(the)g(origin.)486 4512 y(W)-7 b(e)26 b(will)g(write)i Fz(R)1068 4526 y Fx(w,L)1179 4512 y FA(\(-)p Fz(c)p FA(\))c(=)e(\()p Fz(H)1519 4526 y Fv(0)1579 4512 y FA(+)c Fz(V)1698 4526 y Fx(w,L)1833 4512 y FA(+)23 b Fz(c)p FA(\))1976 4479 y Fx(-)p Fv(1)2077 4512 y FC(and)k Fz(R)2290 4526 y Fv(0)2327 4512 y FA(\(-)p Fz(c)p 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y FA(\243)37 b Fz(C)2787 4934 y Fv(2)2847 4920 y FC(meas)3034 4934 y Fx(n)3069 4944 y Fs(1)3102 4920 y FA(\(L\))386 5227 y FC(with)18 b Fz(C)624 5241 y Fv(2)684 5227 y FC(depending)27 b(again)d(on)f FA(k)p FC(,)37 b Fz(f)13 b FC(,)29 b Fz(p)p FC(,)23 b(and)h FA(n)f FC(only)-6 b(.)486 5335 y(T)f(o)31 b(estimate)j(the)f(norm)g(of)g Fz(R)1477 5349 y Fv(0)1514 5335 y FA(\(-)p Fz(c)p FA(\))1664 5302 y Fw(k)1699 5335 y Fl(j)-5 b Fz(V)1770 5349 y Fx(w,L)1883 5335 y Fl(j)1908 5302 y Fv(1)p Fx(/)p Fv(2)2029 5335 y FC(we)32 b(use)h(the)g (Birman-Solomyak)i(inequality)h([1)q(])c(\(see)386 5443 y(also)24 b([22)q(]\))1217 5563 y Fl(j)p Fz(w)e(g)p FA(\(-)p Fz(i)p FA(\321\))p Fl(j)1595 5577 y Fw(q)1654 5563 y FA(\243)15 b Fz(C)1778 5577 y Fw(q)1838 5563 y Fl(k)p Fz(w)p Fl(k)1989 5583 y Fw(l)2010 5564 y FD(q)2041 5583 y Fx(\()p Fw(L)2100 5564 y Fs(2)2129 5583 y Fx(\))2178 5563 y Fl(k)p Fz(g)p Fl(k)2313 5583 y Fw(l)2334 5564 y FD(q)2365 5583 y Fx(\()p Fw(L)2424 5564 y Fs(2)2453 5583 y Fx(\))2480 5563 y FA(,)196 b FC(1)21 b FA(\243)f Fz(q)h FA(\243)f FC(2)386 5701 y(and)k(the)g(Seiler)n(-Simon)g (inequality)j(\(see)d([22)q(]\))1195 5859 y Fl(j)p Fz(w)f(g)p FA(\(-)p Fz(i)p FA(\321\))p Fl(j)1574 5873 y Fw(q)1632 5859 y FA(\243)d(\()p FC(2)p FA(p\))1857 5821 y Fx(-n/)p Fw(q)2008 5859 y Fl(k)p Fz(w)p Fl(k)2159 5873 y Fw(L)2196 5853 y FD(q)2253 5859 y Fl(k)p Fz(g)p Fl(k)2388 5873 y Fw(L)2425 5853 y FD(q)2459 5859 y FA(,)197 b FC(2)21 b FA(\243)f Fz(q)g FA(<)g(\245.)p eop %%Page: 7 7 7 6 bop 1287 259 a Fv(REGULARITY)21 b(OF)h(THE)f(DENSITY)f(OF)i(SURF)-5 b(A)m(CE)22 b(ST)-6 b(A)f(TES)866 b(7)386 459 y FC(Setting)29 b Fz(w)24 b FA(=)f Fl(j)-5 b Fz(V)900 473 y Fx(w,L)1013 459 y Fl(j)1038 426 y Fv(1)p Fx(/)p Fv(2)1154 459 y FC(and)29 b Fz(g)f FC(gi)n(v)o(en)h(by)g(\(9\))f(in)h(these)g(inequalities)j(and) c(then)i(using)f(the)g(estimates)h(\(10\))386 567 y(and)24 b(\(11\))g(pro)o(v)o(es)g(the)g(lemma.)2396 b Fe(\003)386 726 y Fz(Pr)l(oof)24 b(of)f(Pr)l(oposition)j(7.)42 b FC(First)23 b(we)g(consider)j(the)e(case)g FA(n)c(\243)g FC(3.)29 b(By)23 b(the)g(resolv)o(ent)j(equation)757 875 y Fz(R)813 889 y Fx(w,L)925 875 y FA(\(-)p Fz(c)p FA(\))21 b(-)f Fz(R)1222 889 y Fv(0)1259 875 y FA(\(-)p Fz(c)p FA(\))h(=)f(-)p Fz(R)1606 889 y Fv(0)1643 875 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)1864 889 y Fx(w,L)1977 875 y Fl(j)2002 837 y Fv(1)p Fx(/)p Fv(2)2091 875 y FA(\()p Fz(I)26 b FA(+)15 b Fz(V)2309 837 y Fv(1)p Fx(/)p Fv(2)2293 898 y Fx(w,L)2405 875 y Fz(R)2461 889 y Fv(0)2498 875 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)2719 889 y Fx(w,L)2832 875 y Fl(j)2857 837 y Fv(1)p Fx(/)p Fv(2)2947 875 y FA(\))2977 837 y Fx(-)p Fv(1)3046 875 y Fz(V)3113 837 y Fv(1)p Fx(/)p Fv(2)3097 898 y Fx(w,L)3209 875 y Fz(R)3265 889 y Fv(0)3302 875 y FA(\(-)p Fz(c)p FA(\).)386 1033 y FC(Since)22 b(the)g(operator)i (norm)e(of)g FA(\()p Fz(I)e FA(+)9 b Fz(V)1581 1000 y Fv(1)p Fx(/)p Fv(2)1565 1060 y Fx(w,L)1677 1033 y Fz(R)1733 1047 y Fv(0)1770 1033 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)1991 1047 y Fx(w,L)2104 1033 y Fl(j)2129 1000 y Fv(1)p Fx(/)p Fv(2)2219 1033 y FA(\))2249 1000 y Fx(-)p Fv(1)2344 1033 y FC(is)22 b(uniformly)h(bounded,)i(we)c(obtain)i(from)f (\(7\))386 1141 y(and)i(Lemma)e(8)717 1288 y Fl(j)p Fz(R)798 1302 y Fx(w,L)909 1288 y FA(\(-)p Fz(c)p FA(\))f(-)f Fz(R)1206 1302 y Fv(0)1243 1288 y FA(\(-)p Fz(c)p FA(\))p Fl(j)1423 1302 y Fw(p)1482 1288 y FA(\243)15 b Fz(C)25 b Fl(j)p Fz(R)1714 1302 y Fv(0)1751 1288 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)1972 1302 y Fx(w,L)2085 1288 y Fl(j)2110 1250 y Fv(1)p Fx(/)p Fv(2)2199 1288 y Fl(j)2224 1302 y Fv(2)5 b Fw(p)2323 1288 y Fl(j)-5 b Fz(V)2411 1250 y Fv(1)p Fx(/)p Fv(2)2394 1311 y Fx(w,L)2506 1288 y Fz(R)2562 1302 y Fv(0)2599 1288 y FA(\(-)p Fz(c)p FA(\))p Fl(j)2774 1302 y Fv(2)5 b Fw(p)2871 1288 y FA(\243)15 b Fz(C)25 b FC(meas)3209 1302 y Fx(n)3244 1312 y Fs(1)3277 1288 y FA(\(L\))3399 1250 y Fv(1)p Fx(/)5 b Fw(p)3494 1288 y FA(.)486 1435 y FC(W)-7 b(e)22 b(turn)i(to)g(the)g(case)g FA(n)c(\263)g FC(4.)29 b(F)o(or)22 b(an)o(y)i Fz(k)e FA(\316)e Fu(N)j FC(we)f(ha)n(v)o(e)705 1642 y Fz(R)761 1656 y Fx(w,L)872 1642 y FA(\(-)p Fz(c)p FA(\))1022 1604 y Fw(k)1078 1642 y FA(-)e Fz(R)1204 1656 y Fv(0)1241 1642 y FA(\(-)p Fz(c)p FA(\))1391 1604 y Fw(k)1510 1642 y FA(=)1657 1580 y(\(-)p FC(1)p FA(\))1812 1547 y Fw(k)q Fx(-)p Fv(1)p 1653 1621 269 4 v 1653 1704 a FA(\()p Fz(k)i FA(-)e FC(1)p FA(\))p FC(!)1961 1580 y Fz(d)2011 1547 y Fw(k)q Fx(-)p Fv(1)p 1941 1621 195 4 v 1941 1704 a Fz(d)5 b(c)2031 1678 y Fw(k)q Fx(-)p Fv(1)2161 1642 y Ft([)p Fz(R)2242 1656 y Fx(w,L)2354 1642 y FA(\(-)p Fz(c)p FA(\))21 b(-)f Fz(R)2651 1656 y Fv(0)2688 1642 y FA(\(-)p Fz(c)p FA(\))p Ft(])1510 1840 y FA(=)1784 1860 y Fp(\345)1674 1935 y Fw(l)s Fx(+)p Fw(m)p Fx(+)p Fw(n)p Fx(=)p Fw(k)q Fx(-)p Fv(1)1643 1993 y Fw(l)s Fx(,)p Fw(m)p Fx(,)p Fw(n)p Fx(\316)p Fn(N)p Fh([f)p Fv(0)p Fh(g)2033 1840 y Fz(c)2073 1854 y Fw(l)s(nm)2203 1840 y Fz(R)2259 1854 y Fv(0)2296 1840 y FA(\(-)p Fz(c)p FA(\))2446 1803 y Fw(l)s Fx(+)p Fv(1)2543 1840 y Fl(j)-5 b Fz(V)2614 1854 y Fx(w,L)2726 1840 y Fl(j)2751 1803 y Fv(1)p Fx(/)p Fv(2)2863 1840 y Fz(K)2922 1854 y Fw(m)2991 1840 y Fz(V)3059 1803 y Fv(1)p Fx(/)p Fv(2)3042 1864 y Fx(w,L)3155 1840 y Fz(R)3211 1854 y Fv(0)3248 1840 y FA(\(-)p Fz(c)p FA(\))3398 1803 y Fw(n)p Fx(+)p Fv(1)3506 1840 y FA(,)386 2128 y FC(with)23 b(some)h(coef)n(\002cients)h Fz(c)1264 2142 y Fw(l)s(nm)1372 2128 y FC(.)j(The)23 b(operators)j Fz(K)2007 2142 y Fw(m)2081 2128 y FC(are)e(gi)n(v)o(en)g(by)386 2319 y(\(12\))607 b Fz(K)1202 2333 y Fw(m)1273 2319 y FA(=)1373 2257 y Fz(d)1423 2224 y Fw(m)p 1353 2298 143 4 v 1353 2381 a Fz(d)5 b(c)1443 2355 y Fw(m)1506 2319 y FA(\()p Fz(I)25 b FA(+)15 b Fz(V)1724 2281 y Fv(1)p Fx(/)p Fv(2)1707 2342 y Fx(w,L)1820 2319 y Fz(R)1876 2333 y Fv(0)1913 2319 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)2134 2333 y Fx(w,L)2247 2319 y Fl(j)2272 2281 y Fv(1)p Fx(/)p Fv(2)2361 2319 y FA(\))2391 2281 y Fx(-)p Fv(1)2465 2319 y FA(,)106 b Fz(m)20 b FA(\316)g Fu(N)g Fl([)f(f)p FC(0)p Fl(g)p FA(.)386 2491 y FC(Applying)25 b(Lemma)e(8)g(we)f(obtain)1205 2666 y Fz(R)1261 2680 y Fv(0)1298 2666 y FA(\(-)p Fz(c)p FA(\))1448 2628 y Fw(l)s Fx(+)p Fv(1)1544 2666 y Fl(j)-5 b Fz(V)1615 2680 y Fx(w,L)1728 2666 y Fl(j)1753 2628 y Fv(1)p Fx(/)p Fv(2)1862 2666 y FA(\316)20 b Fi(J)1998 2680 y Fw(p)2031 2690 y Fs(1)2239 2666 y FC(for)182 b Fz(p)2571 2680 y Fv(1)2628 2666 y FA(>)2821 2604 y(n)p 2709 2645 272 4 v 2709 2728 a FC(2)p FA(\()p Fz(l)25 b FA(+)20 b FC(1)p FA(\))2990 2666 y(,)1328 2889 y Fz(V)1396 2851 y Fv(1)p Fx(/)p Fv(2)1379 2912 y Fx(w,L)1491 2889 y Fz(R)1547 2903 y Fv(0)1584 2889 y FA(\(-)p Fz(c)p FA(\))1734 2851 y Fw(n)p Fx(+)p Fv(1)1862 2889 y FA(\316)g Fi(J)1998 2903 y Fw(p)2031 2913 y Fs(2)2239 2889 y FC(for)182 b Fz(p)2571 2903 y Fv(2)2628 2889 y FA(>)2828 2827 y(n)p 2709 2868 288 4 v 2709 2951 a FC(2)p FA(\()p Fz(n)21 b FA(+)f FC(1)p FA(\))386 3078 y FC(with)1319 3225 y Fl(j)p Fz(R)1400 3239 y Fv(0)1438 3225 y FA(\(-)p Fz(c)p FA(\))1588 3187 y Fw(l)s Fx(+)p Fv(1)1684 3225 y Fl(j)-5 b Fz(V)1755 3239 y Fx(w,L)1868 3225 y Fl(j)1893 3187 y Fv(1)p Fx(/)p Fv(2)1982 3225 y Fl(j)2012 3239 y Fw(p)2045 3249 y Fs(1)2161 3225 y FA(\243)78 b Fz(C)25 b FC(meas)2562 3239 y Fx(n)2597 3249 y Fs(1)2630 3225 y FA(\(L\))2752 3187 y Fv(1)p Fx(/)5 b Fw(p)2841 3197 y Fs(1)2876 3225 y FA(,)1448 3495 y Fl(j)-5 b Fz(V)1535 3457 y Fv(1)p Fx(/)p Fv(2)1519 3518 y Fx(w,L)1631 3495 y Fz(R)1687 3509 y Fv(0)1724 3495 y FA(\(-)p Fz(c)p FA(\))1874 3457 y Fw(n)p Fx(+)p Fv(1)1982 3495 y Fl(j)2012 3509 y Fw(p)2045 3519 y Fs(2)2161 3495 y FA(\243)78 b Fz(C)25 b FC(meas)2562 3509 y Fx(n)2597 3519 y Fs(1)2630 3495 y FA(\(L\))2752 3457 y Fv(1)p Fx(/)5 b Fw(p)2841 3467 y Fs(2)2876 3495 y FA(.)486 3642 y FC(W)-7 b(e)22 b(turn)i(to)g(the)g(discussion)i(of)e (the)f(operators)j Fz(K)2077 3656 y Fw(m)2129 3642 y FC(.)i(Ob)o(viously)d(we)e(ha)n(v)o(e)739 3771 y Fz(d)p 719 3812 91 4 v 719 3895 a(d)5 b(c)820 3833 y FA(\()p Fz(I)25 b FA(+)15 b Fz(V)1038 3795 y Fv(1)p Fx(/)p Fv(2)1021 3856 y Fx(w,L)1134 3833 y Fz(R)1190 3847 y Fv(0)1227 3833 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)1448 3847 y Fx(w,L)1561 3833 y Fl(j)1586 3795 y Fv(1)p Fx(/)p Fv(2)1675 3833 y FA(\))1705 3795 y Fx(-)p Fv(1)730 4013 y FA(=)20 b(\()p Fz(I)25 b FA(+)15 b Fz(V)1018 3976 y Fv(1)p Fx(/)p Fv(2)1001 4036 y Fx(w,L)1113 4013 y Fz(R)1169 4027 y Fv(0)1206 4013 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)1427 4027 y Fx(w,L)1541 4013 y Fl(j)1566 3976 y Fv(1)p Fx(/)p Fv(2)1655 4013 y FA(\))1685 3976 y Fx(-)p Fv(1)1777 4013 y Fz(V)1844 3976 y Fv(1)p Fx(/)p Fv(2)1828 4036 y Fx(w,L)1940 4013 y Fz(R)1996 4027 y Fv(0)2033 4013 y FA(\(-)p Fz(c)p FA(\))2183 3976 y Fv(2)2221 4013 y Fl(j)g Fz(V)2292 4027 y Fx(w,L)2405 4013 y Fl(j)2430 3976 y Fv(1)p Fx(/)p Fv(2)2542 4013 y FA(\()p Fz(I)25 b FA(+)15 b Fz(V)2760 3976 y Fv(1)p Fx(/)p Fv(2)2743 4036 y Fx(w,L)2855 4013 y Fz(R)2911 4027 y Fv(0)2948 4013 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)3169 4027 y Fx(w,L)3283 4013 y Fl(j)3308 3976 y Fv(1)p Fx(/)p Fv(2)3397 4013 y FA(\))3427 3976 y Fx(-)p Fv(1)3501 4013 y FA(.)386 3902 y FC(\(13\))386 4159 y(Let)22 b Fz(m)p FA(,)15 b Fz(i)i FA(\316)g Fu(N)p FC(,)22 b Fz(i)17 b FA(\243)g Fz(m)22 b FC(be)h(gi)n(v)o(en.)29 b(By)22 b Fz(M)1626 4173 y Fw(m)p Fx(,)p Fw(i)1735 4159 y FC(we)g(denote)i(the)f(set)g(of)g(all)f(multiindices)k Fz(m)p 3065 4175 66 4 v 17 w FA(=)17 b(\()p Fz(m)3311 4173 y Fv(1)3348 4159 y FA(,)e Fy(:)g(:)g(:)h FA(,)f Fz(m)3611 4173 y Fw(i)3633 4159 y FA(\))23 b FC(with)386 4267 y Fz(m)452 4281 y Fv(1)489 4267 y FA(,)15 b Fy(:)g(:)g(:)h FA(,)f Fz(m)752 4281 y Fw(i)794 4267 y FA(\316)20 b Fu(N)j FC(satisfying)j(the)e(follo)n(wing)h(conditions)1768 4414 y Fz(m)1834 4428 y Fv(1)1871 4414 y FA(,)38 b Fy(:)15 b(:)g(:)h FA(,)37 b Fz(m)2179 4428 y Fw(i)2244 4414 y FA(\263)21 b FC(2)p FA(,)1768 4549 y Fz(m)1834 4563 y Fv(1)1914 4549 y FA(+)43 b Fy(:)15 b(:)g(:)44 b FA(+)e Fz(m)2314 4563 y Fw(i)2357 4549 y FA(=)20 b Fz(m)g FA(+)g Fz(i)p FA(.)386 4696 y FC(Applying)26 b(the)g(formula)g(\(13\))f(to)g (\(12\))g(recursi)n(v)o(ely)j(we)c(obtain)i(that)g(for)f(an)o(y)g Fz(m)c FA(\263)g FC(1)j(the)h(operator)i Fz(K)3645 4710 y Fw(m)3721 4696 y FC(can)386 4804 y(be)c(represented)k(in)d(the)f (form)973 5006 y Fz(K)1032 5020 y Fw(m)1104 5006 y FA(=)1197 4909 y Fw(m)1174 5025 y Fp(\345)1176 5099 y Fw(i)p Fx(=)p Fv(1)1338 5025 y Fp(\345)1282 5098 y Fw(m)p 1282 5109 48 3 v Fx(\316)p Fw(M)1432 5108 y FD(mi)1502 5006 y Fz(c)1542 5020 y Fw(mim)p 1608 5031 V 1684 5006 a FA(\()p Fz(I)i FA(+)15 b Fz(V)1902 4968 y Fv(1)p Fx(/)p Fv(2)1885 5029 y Fx(w,L)1997 5006 y Fz(R)2053 5020 y Fv(0)2090 5006 y FA(\(-)p Fz(c)p FA(\))p Fl(j)-5 b Fz(V)2311 5020 y Fx(w,L)2425 5006 y Fl(j)2450 4968 y Fv(1)p Fx(/)p Fv(2)2539 5006 y FA(\))2569 4968 y Fx(-)p Fv(1)1286 5305 y Fl(\001)1376 5208 y Fw(i)1331 5325 y Fp(\325)1345 5397 y Fw(j)r Fx(=)p Fv(1)1449 5305 y Fz(V)1516 5267 y Fv(1)p Fx(/)p Fv(2)1500 5328 y Fx(w,L)1635 5305 y Fz(R)1691 5319 y Fv(0)1728 5305 y FA(\(-)p Fz(c)p FA(\))1878 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y(From)f(Lemma)h(7)g(it)g(no)n(w)g(follo)n(ws)h(that)1659 2854 y Fz(R)1715 2868 y Fx(w,L)1826 2854 y FA(\(-)p Fz(c)p FA(\))1976 2817 y Fw(k)2032 2854 y FA(-)c Fz(R)2158 2868 y Fv(0)2195 2854 y FA(\(-)p Fz(c)p FA(\))2345 2817 y Fw(k)2401 2854 y FA(\316)g Fi(J)2537 2868 y Fw(p)386 3030 y FC(with)30 b Fz(p)23 b FC(satisfying)1049 3189 y(1)p 1045 3230 53 4 v 1052 3313 a Fz(p)1128 3250 y FA(=)1230 3189 y FC(1)p 1208 3230 90 4 v 1215 3313 a Fz(p)1260 3327 y Fv(1)1328 3250 y FA(+)1430 3189 y FC(1)p 1408 3230 V 1415 3313 a Fz(p)1460 3327 y Fv(2)1528 3250 y FA(+)1630 3189 y FC(1)p 1608 3230 V 1615 3313 a Fz(p)1660 3327 y Fv(3)1727 3250 y FA(<)1807 3189 y FC(2)p FA(\()p Fz(l)j FA(+)20 b FC(1)p FA(\))h(+)f FC(2)p FA(\()p Fz(n)i FA(+)e FC(1)p FA(\))h(+)f FC(2)p Fz(m)g FA(+)g FC(2)p 1807 3230 987 4 v 2277 3313 a FA(n)2824 3250 y(=)2904 3189 y FC(2)p FA(\()p Fz(k)j FA(+)d FC(2)p FA(\))p 2904 3230 284 4 v 3022 3313 a(n)386 3470 y FC(and)g(the)g(estimate)h(\(8\))f (holds.)29 b(Since)20 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y(a)975 4196 y Fr(\014)975 4250 y(\014)975 4305 y(\014)975 4359 y(\014)1015 4266 y FA(x\(l)p FC(;)15 b Fz(H)1244 4280 y Fv(0)1301 4266 y FA(+)g Fz(V)1417 4280 y Fx(w,L)1530 4266 y FA(,)g Fz(H)1632 4280 y Fv(0)1668 4266 y FA(\))p 1015 4307 684 4 v 1168 4390 a FC(meas)1355 4404 y Fx(n)1390 4414 y Fs(1)1422 4390 y FA(\(L\))1708 4196 y Fr(\014)1708 4250 y(\014)1708 4305 y(\014)1708 4359 y(\014)1739 4221 y Fv(1)p Fx(/)5 b Fw(p)1848 4328 y Fz(d)g FA(l)921 4583 y(\243)83 b Ft(\()p FC(meas)1276 4597 y Fx(n)1311 4607 y Fs(1)1344 4583 y FA(\(L\))p Ft(\))1502 4540 y Fx(-)p Fv(1)p Fx(/)5 b Fw(p)1648 4469 y Fq(Z)1732 4495 y Fw(b)1694 4654 y(a)1784 4506 y Fr(\014)1784 4560 y(\014)1814 4583 y FA(x\(\(l)21 b(+)f Fz(c)p FA(\))2130 4545 y Fx(-)p Fw(k)2202 4583 y FC(;)15 b Fz(R)2298 4597 y Fx(w,L)2410 4583 y FA(\(-)p Fz(c)p FA(\))2560 4545 y Fw(k)2595 4583 y FA(,)g Fz(R)2689 4597 y Fv(0)2726 4583 y FA(\(-)p Fz(c)p FA(\))2876 4545 y Fw(k)2912 4583 y FA(\))2942 4506 y Fr(\014)2942 4560 y(\014)2972 4525 y Fv(1)p Fx(/)5 b Fw(p)3081 4583 y Fz(d)g FA(l)921 4837 y(=)83 b Ft(\()p FC(meas)1276 4851 y Fx(n)1311 4861 y Fs(1)1344 4837 y FA(\(L\))p Ft(\))1502 4794 y Fx(-)p Fv(1)p Fx(/)5 b Fw(p)1648 4837 y Fz(k)1690 4799 y Fx(-)p Fv(1)1779 4723 y Fq(Z)1863 4749 y Fx(\()p Fw(a)p Fx(+)p Fw(c)p Fx(\))2005 4725 y FH(-)p FD(k)1825 4908 y Fx(\()p Fw(b)p Fx(+)p Fw(c)p Fx(\))1967 4889 y FH(-)p FD(k)2080 4760 y Fr(\014)2080 4814 y(\014)2110 4837 y FA(x\()n Fz(t)h FC(;)15 b Fz(R)2310 4851 y Fx(w,L)2422 4837 y FA(\(-)p Fz(c)p FA(\))2572 4799 y Fw(k)2608 4837 y FA(,)g Fz(R)2702 4851 y Fv(0)2739 4837 y FA(\(-)p Fz(c)p FA(\))2889 4799 y Fw(k)2924 4837 y FA(\))2954 4760 y Fr(\014)2954 4814 y(\014)2985 4779 y Fv(1)p Fx(/)5 b Fw(p)3092 4837 y Fz(t)3123 4799 y Fx(-)p Fv(1)p Fx(/)p Fw(k)q Fx(-)p Fv(1)3316 4837 y Fz(d)s(t)921 5070 y FA(\243)83 b Ft(\()p FC(meas)1276 5084 y Fx(n)1311 5094 y Fs(1)1344 5070 y FA(\(L\))p Ft(\))1502 5027 y Fx(-)p Fv(1)p Fx(/)5 b Fw(p)1648 5070 y Fz(k)1690 5032 y Fx(-)p Fv(1)1764 5070 y FA(\()p Fz(b)20 b FA(+)h Fz(c)p FA(\))2000 5032 y Fw(k)q Fx(+)p Fv(1)2120 4956 y Fq(Z)2166 5141 y Fn(R)2234 4993 y Fr(\014)2234 5047 y(\014)2264 5070 y FA(x\()n Fz(t)6 b FC(;)15 b Fz(R)2464 5084 y Fx(w,L)2576 5070 y FA(\(-)p Fz(c)p FA(\))2726 5032 y Fw(k)2762 5070 y FA(,)g Fz(R)2856 5084 y Fv(0)2892 5070 y FA(\(-)p Fz(c)p FA(\))3042 5032 y Fw(k)3078 5070 y FA(\))3108 4993 y Fr(\014)3108 5047 y(\014)3139 5012 y Fv(1)p Fx(/)5 b Fw(p)3248 5070 y Fz(d)s(t)h FA(.)386 5280 y FC(No)n(w)22 b(applying)k(Lemma)c(6)i(we)e(get)716 5413 y Fq(Z)800 5438 y Fw(b)763 5598 y(a)853 5398 y Fr(\022)930 5465 y FA(x\(l)p FC(;)15 b Fz(H)1159 5479 y Fv(0)1216 5465 y FA(+)g Fz(V)1332 5479 y Fx(w,L)1444 5465 y FA(,)g Fz(H)1546 5479 y Fv(0)1583 5465 y FA(\))p 930 5505 V 1083 5589 a FC(meas)1270 5603 y Fx(n)1305 5613 y Fs(1)1337 5589 y FA(\(L\))1623 5398 y Fr(\023)1690 5420 y Fv(1)p Fx(/)5 b Fw(p)1799 5526 y Fz(d)g FA(l)21 b(\243)15 b Fz(C)40 b Ft(\()p FC(meas)2308 5540 y Fx(n)2343 5550 y Fs(1)2376 5526 y FA(\(L\))p Ft(\))2534 5484 y Fx(-)p Fv(1)p Fx(/)5 b Fw(p)2679 5526 y Fl(j)p Fz(R)2760 5540 y Fx(w,L)2872 5526 y FA(\(-)p Fz(c)p FA(\))3022 5489 y Fw(k)3078 5526 y FA(-)20 b Fz(R)3204 5540 y Fv(0)3241 5526 y FA(\(-)p Fz(c)p FA(\))3391 5489 y Fw(k)3426 5526 y Fl(j)3456 5540 y Fw(p)3494 5526 y FA(.)386 5751 y FC(By)g(Proposition)k(7)d(the)g(r)-5 b(.h.s.)20 b(of)h(this)h(inequality)i(is)d(bounded)i(uniformly)g(in)e FA(L)g FC(and)g FA(w)11 b(\316)g(W)p FC(.)27 b(Thus)21 b(Lemma)386 5859 y(3)i(with)g Fz(q)e FA(=)27 b Fz(p)c FC(pro)o(v)o(es)i(the)e(absolute)j(continuity)h(of)c Fz(d)5 b FA(X)p FC(.)p eop %%Page: 9 9 9 8 bop 1287 259 a Fv(REGULARITY)21 b(OF)h(THE)f(DENSITY)f(OF)i(SURF)-5 b(A)m(CE)22 b(ST)-6 b(A)f(TES)866 b(9)682 459 y FC(3.)47 b(H)878 451 y(\250)867 459 y FB(O)t(L)t(D)t(E)t(R)29 b FC(C)t FB(O)t(N)t(T)t(I)t(N)t(U)t(I)t(T)t(Y)h(O)t(F)e(T)t(H)t(E)g FC(I)t FB(N)t(T)t(E)t(G)t(R)t(A)l(T)t(E)t(D)j FC(D)t FB(E)t(N)t(S)t(I)t(T)t(Y)g(O)t(F)c FC(B)s FB(U)t(L)t(K)h FC(S)t 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y(Thus)e(from)h(Lemma)f(6)g(it)h (follo)n(ws)g(that)h(the)f(l.h.s.)f(of)h(\(17\))g(is)g(bounded)i(by)e (a)f(constant)e Fz(C)d FA(>)e FC(0)22 b(uniformly)386 1808 y(in)h FA(w)e(\316)f(W)p FC(.)28 b(Finally)c(this)g(leads)h(to)e (the)h(estimate)1554 1998 y Fz(N)6 b FA(\(l)1701 2012 y Fv(2)1738 1998 y FA(\))20 b(-)g Fz(N)6 b FA(\(l)2005 2012 y Fv(1)2042 1998 y FA(\))20 b(\243)c Fz(C)2222 2012 y Fw(p)2259 1998 y Fl(j)p FA(l)2334 2012 y Fv(2)2391 1998 y FA(-)21 b(l)2512 2012 y Fv(1)2549 1998 y Fl(j)2588 1924 y FD(p)p FH(-)p Fs(1)p 2584 1940 81 3 v 2614 1978 a FD(p)386 2161 y FC(for)30 b(an)o(y)37 b Fz(p)25 b FA(>)g FC(1,)30 b(which)g(pro)o(v)o(es)h(the)f(H)8 b(\250)-38 b(older)31 b(continuity)h(of)e(the)g(inte)o(grated)i(density)g(of)d (states.)49 b(W)-7 b(e)29 b(can)386 2269 y(apply)23 b(similar)h(ar)n (guments)g(to)e(models)i(more)e(general)i(than)f(the)g (Holden-Martinelli)j(model.)j(W)-7 b(e)21 b(will)h(not)386 2377 y(dwell)i(on)f(this)h(here.)1862 2609 y(R)t FB(E)t(F)t(E)t(R)t(E)t (N)t(C)t(E)t(S)423 2754 y FI([1])38 b(M.S.)19 b(Birman)g(and)i(M.Z.)e (Solomyak,)h FE(Estimates)f(of)h(singular)h(number)o(s)g(of)e(inte)m (gr)o(al)i(oper)o(ator)o(s.)f(III)p FI(,)f(V)-8 b(estnik)20 b(Leningrad)548 2845 y(Uni)n(v)-5 b(.)19 b Fb(24)g FI(\(1969\))h(35)g (\226)f(48)g(\(Russian\).)423 2937 y([2])38 b(M.S.)18 b(Birman)h(and)g(M.Z.)f(Solomyak,)i FE(Remarks)f(on)g(the)g(spectr)o (al)h(shift)e(function)p FI(,)h(J.)g(So)o(v)-5 b(.)18 b(Math.)h Fb(3)g FI(\(1975\))h(408)g(\226)f(419.)423 3028 y([3])38 b(M.Sh.)26 b(Birman)h(and)g(M.Z.)f(Solomyak,)k FE(Spectr)o(al)d(Theory)g(of)g(Self-Adjoint)g(Oper)o(ator)o(s)g(in)g (Hilbert)g(Space)p FI(,)i(D.)d(Reidel,)548 3119 y(Dordrecht,)19 b(1987.)423 3211 y([4])38 b(M.Sh.)23 b(Birman)g(and)h(D.R.)e(Y)-7 b(af)o(ae)n(v)i(,)25 b FE(The)e(spectr)o(al)h(shift)f(function.)g(W)-7 b(ork)25 b(by)e(M.G.Kr)m(ein)g(and)i(its)d(further)i(de)o(velopment)p FI(,)548 3302 y(St.)17 b(Petersb)o(ur)o(g)i(Math.)g(J.)f Fb(4)h FI(\(1993\))h(833)g(\226)f(870.)423 3393 y([5])38 b(A.)21 b(Chahrour)m(,)i FE(Densit)t(\264)-29 b(e)22 b(int)t(\264)-29 b(egr)t(\264)g(ee)23 b(d')t(\264)-29 b(etats)22 b(surfaciques)h(et)f(la)f(fonction)i(g)t(\264)-29 b(en)t(\264)g(er)o(alis)t(\264)g(ee)23 b(de)f(d)t(\264)-29 b(eplacement)23 b(spectr)o(al)g(pour)f(un)548 3484 y(op)t(\264)-29 b(er)o(ateur)20 b(de)f(Sc)o(hr)6 b(\250)-31 b(oding)o(er)21 b(surfacique)g(er)m(godicue)p FI(,)f(Helv)-5 b(.)19 b(Phys.)f(Acta)h Fb(72)g FI(\(1999\))h(93)f(\226)h(122.)423 3576 y([6])38 b(A.)18 b(Chahrour)m(,)i FE(On)f(the)g(spectrum)g(of)g(Sc)o(hr)6 b(\250)-31 b(oding)o(er)21 b(oper)o(ator)f(with)f(periodic)g(surface)h (potential)p FI(,)f(preprint)g(\(2000\).)423 3667 y([7])38 b(J.M.)16 b(Combes)i(and)g(P)-8 b(.)16 b(Hislop,)h FE(Localization)g (for)g(some)h(continuous,)h(r)o(andom)f(Hamiltonians)g(in)e(d)22 b(dimensions)p FI(,)c(J.)e(Funct.)548 3758 y(Anal.)i Fb(124)i FI(\(1994\))g(149)g(\226)f(180.)423 3850 y([8])38 b(J.M.)22 b(Combes,)i(P)-8 b(.)22 b(Hislop,)h(S.)f(Nakamura,)j FE(The)d(L)1923 3823 y FD(p)1952 3850 y FE(-theory)h(of)g(the)g(spectr) o(al)g(shift)f(function,)j(the)d(W)-7 b(e)m(gner)25 b(estimate)o(,)e (and)548 3941 y(the)c(inte)m(gr)o(ated)g(density)h(of)f(states)g(for)f (some)i(r)o(andom)g(oper)o(ator)o(s)p FI(,)g(Commun.)f(Math.)g(Phys.)g (\(to)f(appear\).)423 4032 y([9])38 b(H.)23 b(Englisch,)i(W)-7 b(.)22 b(Kirsch,)j(M.)e(Schr)6 b(\250)-31 b(oder)m(,)25 b(and)g(B.)e(Simon,)h FE(Random)h(Hamiltonians)f(er)m(godic)h(in)e(all) h(b)o(ut)f(one)h(dir)m(ection)p FI(,)548 4124 y(Commun.)19 b(Math.)g(Phys.)g Fb(128)g FI(\(1990\))h(613)g(\226)f(625.)386 4215 y([10])38 b(F)-6 b(.)16 b(Gesztesy)i(and)g(K.A.)e(Makaro)o(v)-5 b(,)18 b FE(The)g FF(X)f FE(oper)o(ator)h(and)g(its)f(r)m(elation)g(to) h(Kr)m(ein')m(s)f(spectr)o(al)h(shift)e(function)p FI(,)i(J.)f(Anal.)g (Math.)548 4306 y Fb(81)i FI(\(2000\))h(139)g(\226)f(183.)386 4398 y([11])38 b(F)-6 b(.)23 b(Gesztesy)-5 b(,)25 b(K.A.)e(Makaro)o(v) -5 b(,)26 b(and)f(A.K.)e(Moto)o(vilo)o(v)-5 b(,)26 b FE(Monotonicity)f(and)g(concavity)g(pr)m(operties)f(of)g(the)g(spectr)o (al)g(shift)548 4489 y(function)p FI(,)f(in)g(F)-6 b(.)21 b(Gesztesy)-5 b(,)24 b(H.)e(Holden,)i(J.)e(Jost,)h(S.)e(P)o(aycha,)j (M.)e(R)6 b(\250)-31 b(ockner)m(,)24 b(and)f(S.)e(Scarlatti)h (\(Eds.\),)g(Canadian)h(Mathe-)548 4580 y(matical)18 b(Society')l(s)h(Conference)g(Proceedings)h(Series,)e(Canadian)h(Math.) g(Soc.)f(and)i(Amer)l(.)e(Math.)h(Soc.,)f(Pro)o(vidence,)h(RI.)548 4672 y(\(to)f(appear\);)i(a)o(v)n(ailable)f(from)g Fa (http://www.ma.utexas.edu/mp)p 2581 4672 23 4 v 22 w(arc/)f FI(as)h(99-339.)386 4763 y([12])38 b(I.C.)17 b(Gohber)o(g)i(and)f(M.G.) g(Krein,)g FE(Intr)m(oduction)h(to)f(the)g(Theory)g(of)g(Linear)g (Nonselfadjoint)h(Oper)o(ator)o(s)g(in)f(Hilbert)g(Space)p FI(,)548 4854 y(Amer)l(.)g(Math.)h(Soc.)g(T)m(ransl.)f(of)h(Math.)g (Monographs)j(18,)d(Pro)o(vidence,)g(RI,)f(1969.)386 4946 y([13])38 b(D.)23 b(Hundertmark)i(and)g(B.)e(Simon,)h FE(An)g(optimal)g(L)1933 4919 y FD(p)1962 4946 y FE(-bound)h(on)f(the)g (Kr)m(ein)g(spectr)o(al)g(shift)g(function)p FI(,)h(J.)f(Anal.)f(Math.) h(\(to)548 5037 y(appear\);)19 b(a)o(v)n(ailable)g(from)g Fa(http://www.ma.utexas.edu/mp)p 2479 5037 V 23 w(arc/)f FI(as)g(00-370.)386 5128 y([14])38 b(V)-10 b(.)23 b(K)m(ostrykin,)i FE(Concavity)f(of)f(eig)o(en)m(value)i(sums)f(and)g(the)f(spectr)o(al)h (shift)f(function)p FI(,)h(J.)f(Funct.)g(Anal.)g Fb(176)h FI(\(2000\))h(100)f(\226)548 5220 y(114.)386 5311 y([15])38 b(V)-10 b(.)19 b(K)m(ostrykin)h(and)g(R.)e(Schrader)m(,)h FE(Scattering)h(theory)g(appr)m(oac)o(h)h(to)e(r)o(andom)h(Sc)o(hr)6 b(\250)-31 b(oding)o(er)22 b(oper)o(ator)o(s)e(in)f(one)h(dimension)p FI(,)548 5402 y(Re)n(v)-5 b(.)19 b(Math.)g(Phys.)f Fb(11)i FI(\(1999\))f(187)h(\226)f(242.)386 5494 y([16])38 b(V)-10 b(.)19 b(K)m(ostrykin)i(and)f(R.)f(Schrader)m(,)h FE(The)f(density)h (of)g(states)f(and)i(the)e(spectr)o(al)h(shift)g(density)g(of)f(r)o (andom)i(Sc)o(hr)6 b(\250)-31 b(oding)o(er)22 b(oper)o(-)548 5585 y(ator)o(s)p FI(,)d(Re)n(v)-5 b(.)19 b(Math.)g(Phys.)f Fb(12)i FI(\(2000\))g(807)f(\226)g(847.)386 5676 y([17])38 b(V)-10 b(.)18 b(K)m(ostrykin)i(and)g(R.)e(Schrader)m(,)h FE(Global)g(bounds)h(for)f(the)g(L)n(yapuno)o(v)h(e)o(xponent)g(and)g (the)f(inte)m(gr)o(ated)g(density)h(of)e(states)h(of)548 5768 y(r)o(andom)h(Sc)o(hr)6 b(\250)-31 b(oding)o(er)21 b(oper)o(ator)o(s)g(in)e(one)g(dimension)p FI(,)h(J.)f(Phys.)f(A:)h (Math.)g(Gen.)g Fb(33)g FI(\(2000\))h(\(to)f(appear\);)h(a)o(v)n (ailable)f(from)548 5859 y Fa(http://www.ma.utexas.edu/mp)p 1767 5859 V 22 w(arc/)f FI(as)h(00-226.)p eop %%Page: 11 11 11 10 bop 1287 259 a Fv(REGULARITY)21 b(OF)h(THE)f(DENSITY)f(OF)i(SURF) -5 b(A)m(CE)22 b(ST)-6 b(A)f(TES)833 b(11)386 459 y FI([18])38 b(S.)28 b(Nakamura,)k FE(A)c(r)m(emark)i(on)g(the)f(Diric)o (hlet-Neuman)g(decoupling)i(and)f(the)f(inte)m(gr)o(ated)h(density)f (of)g(states)p FI(,)j(preprint)548 550 y(\(2000\);)19 b(a)o(v)n(ailable)g(from)g Fa(http://www.ma.utexas.edu/mp)p 2454 550 23 4 v 23 w(arc/)f FI(as)h(00-121.)386 641 y([19])38 b(L.)18 b(P)o(astur)g(and)i(A.)e(Figotin,)g FE(Spectr)o(a)i(of)f (Random)h(and)f(Almost-P)-6 b(eriodic)19 b(Oper)o(ator)o(s)p FI(,)h(Springer)m(,)e(Berlin,)g(1992.)386 733 y([20])38 b(M.)23 b(Reed)h(and)g(B.)e(Simon,)j FE(Methods)f(of)f(Modern)i (Mathematical)f(Physics,)g(I:)f(Functional)i(Analysis)p FI(,)f(Academic)g(Press,)548 824 y(Ne)n(w)19 b(Y)-8 b(ork,)18 b(1972.)386 915 y([21])38 b(M.)24 b(Reed)g(and)h(B.)f(Simon,)h FE(Methods)g(of)g(Modern)g(Mathematical)g(Physics,)g(III:)f(Scattering) h(Theory)p FI(,)h(Academic)f(Press,)548 1007 y(Ne)n(w)19 b(Y)-8 b(ork,)18 b(1979.)386 1098 y([22])38 b(B.)18 b(Simon,)g FE(T)l(r)o(ace)h(Ideals)h(and)f(Their)g(Applications)p FI(,)g(Cambridge)h(Uni)n(v)o(ersity)f(Press,)f(Cambridge,)h(1979.)386 1189 y([23])38 b(B.)16 b(Simon,)h FE(Spectr)o(al)g(analysis)h(of)e(r)o (ank)i(one)f(perturbations)i(and)f(applications)p FI(,)g(in)e(J.)h (Feldman,)g(R.)f(Froese,)h(and)g(L.)f(Rosen)548 1281 y(\(eds.\))k FE(Mathematical)h(Quantum)g(Theory)f(II:)g(Sc)o(hr)m (oeding)o(er)i(Oper)o(ator)o(s)p FI(,)f(Proceedings)g(of)f(the)g (Canadian)h(Mathematical)548 1372 y(Society)c(annual)h(seminar)g(on)g (mathematical)f(quantum)i(theory)f(held)g(in)f(V)-8 b(ancouv)o(er)m(,) 18 b(Canada,)h(August)e(4-14,)i(1993,)f(CRM)548 1463 y(Proceedings)i(Lecture)f(Notes,)f(V)-10 b(ol.)19 b(8,)g(Amer)l(.)f (Math.)h(Soc.,)f(Pro)o(vidence,)i(R.I.,)d(1995,)j(p.)f(109)g(\226)g (149.)386 1555 y([24])38 b(B.)18 b(Simon,)g FE(Spectr)o(al)i(aver)o(a)o (ging)h(and)f(the)f(Kr)m(ein)g(spectr)o(al)g(shift)p FI(,)f(Proc.)g(Amer)l(.)h(Math.)g(Soc.)f Fb(126)i FI(\(1998\))g(1409)g (\226)f(1413.)386 1646 y([25])38 b(D.R.)14 b(Y)-7 b(af)o(ae)n(v)i(,)17 b FE(Mathematical)f(Scattering)h(Theory)l(.)e(Gener)o(al)h(Theory)p FI(,)h(Amer)l(.)e(Math.)h(Soc.)f(T)m(ransl.)g(of)h(Math.)f(Monographs) 548 1737 y(V)-10 b(ol.)19 b(105,)g(Pro)o(vidence,)g(RI,)f(1992.)488 1908 y(V)-5 b FG(A)t(D)t(I)t(M)35 b FI(K)r FG(O)t(S)t(T)t(RY)t(K)t(I)t (N)t FI(,)i(F)t FG(R)t(A)q(U)t(N)t(H)t(O)t(F)t(E)t(R)t FI(-)t(I)t FG(N)t(S)t(T)t(I)t(T)t(U)t(T)30 b(F)2037 1902 y FI(\250)2028 1908 y FG(U)t(R)36 b FI(L)t FG(A)t(S)t(E)t(RT)t(E)t(C)t (H)t(N)t(I)t(K)t FI(,)f(S)t FG(T)t(E)t(I)t(N)t(B)r(A)r(C)t(H)t(S)t(T)t (R)t(A)t(SS)t(E)30 b FI(1)t(5)t(,)40 b(D)t(-)t(5)t(2)t(0)t(7)t(4)t(,) 388 2000 y(A)t FG(A)r(C)t(H)t(E)t(N)t FI(,)20 b(G)t FG(E)t(R)t(M)t(A)t (N)t(Y)486 2091 y FE(E-mail)e(addr)m(ess)p FI(:)24 b Fa(kostrykin@t-online.de,)41 b(kostrykin@ilt.fhg.de)488 2246 y FI(R)q FG(O)t(B)t(E)t(RT)21 b FI(S)t FG(C)t(H)t(R)t(A)t(D)t(E)t (R)t FI(,)f(I)t FG(N)t(S)t(T)t(I)t(T)t(U)t(T)g(F)1537 2240 y FI(\250)1528 2246 y FG(U)t(R)j FI(T)t FG(H)t(E)t(O)t(R)t(E)t(T)t (I)t(S)t(C)t(H)t(E)18 b FI(P)t FG(H)t(Y)t(S)t(I)t(K)t FI(,)j(F)t FG(R)t(E)t(I)t(E)g FI(U)t FG(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T) 3029 2240 y FI(\250)3020 2246 y FG(A)t(T)e FI(B)t FG(E)t(R)t(L)t(I)t(N) t FI(,)h(A)t FG(R)t(N)t(I)t(M)t(A)t(L)t(L)t(E)t(E)388 2338 y FI(1)t(4)t(,)i(D)t(-)t(1)t(4)t(1)t(9)t(5)h(B)t FG(E)t(R)t(L)t(I)t(N)t FI(,)c(G)t FG(E)t(R)t(M)t(A)t(N)t(Y)486 2429 y FE(E-mail)f(addr)m(ess)p FI(:)24 b Fa (schrader@physik.fu-berlin.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0011111200941--