Content-Type: multipart/mixed; boundary="-------------0402271119941" This is a multi-part message in MIME format. ---------------0402271119941 Content-Type: text/plain; name="04-50.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-50.keywords" Random Schroedinger operators, Integrated density of states, Lifshitz tails ---------------0402271119941 Content-Type: application/postscript; name="LifInh.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="LifInh.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: LifInh.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman CMMI8 CMR8 CMMI6 CMSY8 CMR6 CMR5 %%+ CMSY6 Times-Italic CMMI10 CMR10 CMSY10 CMR7 CMMI5 CMSY7 CMMI7 CMSY5 %%+ CMEX10 MSAM10 Helvetica TeX-cmex7 Courier %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /usr/bin/dvips -o LifInh.ps LifInh.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.02.25:1036 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % File 8r.enc as of 2002-03-12 for PSNFSS 9 % % This is the encoding vector for Type1 and TrueType fonts to be used % with TeX. This file is part of the PSNFSS bundle, version 9 % % Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % % (6) /Euro is assigned to 128, as in Windows ANSI % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /Euro /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: aae443f0.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: f7b6d320.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: bbad153f.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 % /TeXbbad153fEncoding [ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: 0ef0afca.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmr5 % /TeX0ef0afcaEncoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: TeX-cmex7 %!PS-AdobeFont-1.0: TeX-cmex7 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex7) def /FullName (TeX\040cmex7\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex7 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 101 /e put readonly def /FontBBox {-14 -2954 1627 771} readonly def /UniqueID 4314415 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % 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 (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 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 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 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 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 0 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 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 0 /.notdef put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC27E4A7E07D09FB08355F6EA74E530B0743143F2A871732D62D80F35B19FD2 C7FDF08105847F13D50934419AC647CBA71DF74F4531DC02BBDA22AEEA3FBBBB 407E0ACC52BDC60D01A29407CC4F93EB8BF6D4813E9BA858D54F38918AC82720 4956D50291F0546E50FCAFA6DBD0099123F5ECD4AB338DB310DB4CAE11337A89 8ED99B6F483940C97544F888EAF0CBEB11094A13C073D0061808662A04A82BA0 AD35E8782F854AF66C20C0FEF18D0ECDD1646321B93D327E53D88CA0E825FA95 05AA57BD77B4E4C5AE617269154084CCE82545C6478B4F02A0BB093613384B08 360CE3632C03E70CCB358E0E06269A6485F9E705E06A5CC124531442B94C9E37 8109055A64A4F8DF3064F1F66C9E28558911EDC33CCF4FF27CF4E1A1C7ECA0E7 7893DF6AA3807B8115533945445BB9EB9F97DB960E2C7052BC27C9526B461B7C CC10251152A294DD6671C3B12CE0F4D06540C27FC1B2AE97CC1C489E105FE258 98DCCD2FCC8A5BEE7BC3E8D53039640142BB42AB054A9689CD803D771D3910A9 52968CF8AAFEBA136E0739C605EDC4B23D88DB3E76A9427C61961BEB10177B56 ECA827ACFD99CA700A56F9781409982FFA8F79779612DDF3C9B2CFD01B514E0D C43FD0BE188D603F31102D7CE9FC712485B6C2A53D102CD56729DAD404C5A615 FCCC0AED9876A35C031EC6942116B17B2455210BC56AFB7F3AAE5FCB7EF26D58 B92584E8EC364A31BC2A82CCB25331F0887CE0BF4687F54669E788F573ACF410 A3E891FC38E521AF3F8DDF2959EACADC1213484EEB185FF8A2AE1CBE1B05CFF6 391D0C8E17439908D03A9845F5BB52725D2F0A6A132F031C35EA8EF39A4E0131 CF7B4ED8E0AC33CC8F22D2650CD890670A00DD8E0DA3527636F7C1ACD38711EB F0E277800C2BF3C2E95328CEAEA677BFD5AE6F294719CEA8B253AB8D747E3B64 3159DA6E74F80136DE4495FA93D7419B7BE4FF0939E604C9D3C44DCC7DDB6E89 7CC155E08A2FC5AB8B900A265D4CC1C46C66F8729A8541640730F24F2A229726 E15CA1E4E4756E11BC7D133CF25FD1BEEBF23D65617DE830C29ECA7682A56952 E303EC53526E2BB5590D23130C7D5E9A637C2A30E6F13C3E235349898DFB0060 2CBCD4E59A8CB391A04C6FBAEAFBCB4D89BA8005C3932CC2D8F806A990BA2FE2 739B4428D10F19D307F7ECD033AE54D8962924DC07828CE7A9BA108256D6CA8A 279915B6CA59653F61DED137AD092ED3490B50CC397DFA8CE74288B4B6074A3A 07191D7CB6940EA8CF29FB078B37E0AE5CDDA10D52840CB889BB47AC3F6D99A9 BD9F5C10F86B11DC3982883DF00FA4C1F34089481391CF109B4E4B3B0F45D596 23459285968D48B02A196F2EF584B78FBDB3BAABBDDD7BAF003B12151417BD56 D630A8A3ACD2C20994209DD890C5E506A0DB7B456BD4FE32ED23CAE96C94D3D3 6DF85AD0BF93AFD189D98C6090500527B97A3DF54273D5DDD2C5F623E1BD084A 4566B8E3FA91FFC162608193FE4990D5B16AEA909E50F44776FA7ABA2C0250EC CFFC8823539BBF0F2455F11A23AEB7AC0B1E88AFE50374FFCA6F5413DE8CBFA9 F6EC3B7EEB9A948EA82DC414366511D4D11FC644B0AB6ABE79674A523E4CDEBC 3D05B08FF2038340B3D045360F3CEFAA3D948E969BFC0D67DBC5F76AD6DD391B 8410882A81FCBC1A0B25AE9477B088257F16AFBC154FEC8F52DA837687877E13 CFADFAFFA3F55FFCB484832B0A419D7BDB1C859FFA3A0F7E93B72A12C96856A0 FFA38941E48AF7AD114182F58D55445F56C971B09B71C4A9F85EC4D7FDBC226E 837549A06077AF38E3463E3C2D2A5653740C46784AD4E308EF4439BC194E8C43 E08047B45DC992766B59651883E17C18A4802D82EB377E9F42094F40D92249E7 3E9EC0C99F865B17CF84E8AFFC78BBEADCCEF73047DEB121F5A9E4D9010C6FA9 68F7B8BF15EA34157DD3EE52D7FE29C39DB361989FB2498ECAA06A01802ED913 BAF9D940283CD889104AF900D61940E7604932F92C4FAC5AA6D18363A803C28C 878D90B29A0FEE523248578A77E6DBAF5615D5395805E5D88E5E048E37ACD259 0A679030053487E5A153456C6CBFB8AD04E9074BB2EE38F34AC0FE7143A8448D 9DBDACFE2116E12EBB09BFADD154672FA981B3AF42B38E73929636727513C4F3 074C90D4440E259124D1AFB0FBD5E21305AFD5B0BE7CE36A4E2E286386DC8ACA 1A46126E8B95B2D1689E934A5A878EF35C7483B5D9038BB0DEDAEE89423ACA47 C968AF1A960ABC2536705E27D05C4C513E650E3EE3983AF7C389AA2B6FFE4772 44190D8C945BAD22EBB510AB60BD6F39AE86E757D4B3B970C4009367DF4EA8C2 EA9B6694DC2D578199A205B380138D591709215FB782B0277EE06F4C46C032D3 0FEFB0042F64BADC1A74FA387AF920BB22B47C969C62222CDA68B72A080E3FF0 93413844BD3143823E572B26F0230666494BB80D4B7396E7C954BCB93D2C2784 32ACEF5771E6B4FA59878D688CBF2BF9FC503815316CA036F9E84BADCCF0DDF9 09F7E8135E0FC461D26C9792DEDE90015AEABA461EAF0804C83F0A06D12B4A69 15EDA6034B288BA5D902A723253CD1C6981E538D098FFDC1FB0D291A05F58B53 CDB3D4173B006279DB7BF073D2BDAB7FB8D3CE97CDEFC5F4A2A90B2FFCFD3934 576CE82DADB78DC726BFDE2E4F9B6A21573C5931BA0DD3F9E09C21817CECC8EE 0B7FDCAFF89CFECD171D1A340673DB6EC3DB7080296C00114DE0B7DF42B525B7 B2B2F7FA81E237A9BEAA430FEFCF9830F8DA8E924B285D0ACD5B46C76F102183 25057F8CBA8452BB4C41B669260B328873F224CDF0A5E3CC3267B2BE2947C42D FF568DAC6D413BC85C844AD06FE5BBA298A7538128CF9FE610A4CC7BC8BF0CC2 BAF4AEB094EC4E131A98F8A1DA63A62C483A40A0EB4B274A914F3FA18C509306 50836D6A70B510B667D9568043FAA523B9B0707DE2731A95242088BF1167EB6B AB71ECEBC67F2C7057B06AFD1EEFD6FFBCD7BD1C728935CCABCE446D24A14A26 0C0A0482EBA4EC0DB671AC75AED3969F6244F4FDBC103AA4ECFF5AFC61CAFA10 4F08AAA098C81867AFE1D0FB1C749B2DB727A0BE23F1F6CC5326B3CD5BB8D080 BAE815117BA8473B55EC31DD429A12C09D79AFC058B8D03CD9E4A99D56565378 588A007471842F1559B9F29A8BBDF9E2FE53C9A7DBF8360B5314B0DB4CE2D5B4 FEF2965C236720DB136F5C509D45B72947316510CCFDCE915796AA144EF4148D C7371D3F49B2FCBAD865839CDA9423F9B356745BD6480CFA4CFA5DC22081BE53 DB45DFD790E2997C30C180C37AC2C8DDFF3071FA9D676324AC0A2269933BD99C B754A20CD75075164CC494138C2A29E5E28A9C3BC43C80596E85360B216670FC 9CE1A32ADE9865E61E98650E9968A3084D0BCDC49D63840073E6A286E0106793 455DCA963EA1E8A1F99E3CF6EF004DF7B783AB04E9C03D3942F67404D96C08E8 F312B5623F3320C8DC141BF12C93A6D9E2CCBD2575FDA9EF2A532D206FD84679 605555CB30D2A4F1F9AEAE866855273C75DDC5EB78AA97E42E7CE9AAC82EC8B7 4F2CCB6097351A9EEFB202F590941E1B51226D1B9F36E2B03FB20AB3ACBE57B9 F6969DA017ADF6D780EABEF95FC0D02F6C61C1E5AC1C7199003F18FAD0458F8C 58FBC5D98B6D8F68310C044F57BB8EBC8F8F2D317417248C3AFD1C1B8AAB2E3F 82E4AC6305032FCA3142FEF6BD5DD440ABB0CC38CCFD6CA4F69AB5EEF006DA8D 1E5FF4F352FA8E02D58DEF5A6185DC856C7C6FEE63657E271587CD02F060E021 3D0EF57EB3EA14CFCBDCC6972EBA4F3DE76F89746F9E81D68132D0CCD913AA0B B66616E6CBE80DE46373529091567914104C177A77193BF7FD674A0D376E281A 5CDCF7F5B47EB9EF53096A86FD112845A6D34E0E39F035B0405BD7B8C2388DF2 C0CC0A57D6023FD65528728F4C64439760CC4CE11628C758CD171A10F36A6F0E 12167E6C1966649D2A2EAAD5A6902E631DC70608EF55C90DD3689FE571085572 123DF949F0F11D8EC8DBB49124B51E101BD18AF1144F203E8C76E83D71314D8D BC72DDA6756DD27D8B2BABE7333E3A44854720626FC88A0F5420CBFE54A55EE9 B164F53494D7AEE4E73E2C915AB7ED608D02293A4E602D71896AC1E30FD22D9F 2853D2B8730AF2D5F5251A029A438740AC2311763A92BE2D39A823F7DD316DA8 86ADF393D9CF68073B924535BEDD6E0C8EB66ACD264D9D8A7E01233D978C4735 8605DB9D2B1E79F026C5AA351B0AF7A26A555BE94ED65279652F1C353D74A8E6 FBC66519E55166F3E4E931E8CCEDCE9869BC078725E8B721ADC263812002CA07 2AC32F2D4775CEE2823A824AE28BA9DBF3CF8C4FA7664BE87A3F883DC6AE86C9 B4A2A316C49776788B549E4AC7B7AB61BD4BA46846D457ECE580477DAC5D4B0E 06BAA8E0069CB6BAAA5DCA91463BAAD6E8C2607659E7D788772FA3A0A762333D AD0D76FD221B6570B7B8C2610AA48D4087F066BD0298E7713DCD84122046A5BF 39FD2FD267A61BD81E7D3A3C3C6E50986C5A4771922AE09FE86DCB89C8C23484 A0AAAE986DD6636501153AA7E159501623BED6B9505485C1D54EC956D93F1F3A 5B6597B71E6FF59D753CF7FE2D7D77865FF21E9211B81B33B0502D61C7836A47 D9BE48E631915F5FD66A347FDA9237B9A577647C9B6AB0D4D5D81A82785DFDD1 12A800A95BCD213686B4AAE62F362C1BC0CFB5F2BB760E177CF0A97F4804130B 7EB231BE3D56C3494E10EB92C8B78B91CFAC59E42D763CCA1D810AE5FEC1080F D3729C16D4A82AACEC9FE0E34F6B2E9791D4DCB072DC93F37DE5B2A3DD3D2801 6F1C34012D32543510988DF9636C4AD83532581C55CBCC2069A665378A883A39 9FF961860B722A2B254D715EC6CEE7E654FCC64288270B301AF2DFE757C33E9C 66BD3F584157CAEFF1F18B6F717519CD29866E8C2292767B19E8BFC9445F1AB5 6F2525C331BAE60CB424713BC96B92FC7E9E3BF1C1A79CE9446FB18B26BBC64D 741002B59AC57E84D8832028BD7C0C65E81DF92E16A043A590C34B2B5508708C 017A7AFF3FF69C876CC17261FFF2DC7C109EDAC97E43CACB3B51229B0BC05D8C 2390CC00D72BFBD5B490390B595FE96807C12EDE3A6A4F84059A20FCF18EF31C 7DAC0912200C36E044E73432476EF7D07444B4BE08976E656B6DB634CC4C18C8 0137BA9D786774F83F88874DA32FB90F0D582E52C27C9027B94A3E63B0017034 0A236C9CF06C80462419D27998359BD00D703668D494B589B82EB2CEADD1BD84 B41D301612C9782CCB5483A47C549FA44331C520606742B8B2CF187139113A3A DF300908B150C99EFD4B564219C091C737696A9572C198C26DD727B0EEAE0868 F8AA577E32006F5EF9037B2209B5778BD95C1E0BCB6C3565A2AD443461C432C5 3E74502760950D30BD7A7695463F6E5A317202B7F017948971E78A756D64EDE0 D6F84DA89435EB677E8B1CB69DE567B3BC6877AA64E69E711D711EE01A9F1A1B 16C15A05160840339D6433DE2B50B974082930868EFC47B762F868777DEDC3A1 E795DB49A728118188462E8FD25C939ADC5D6CE3BA28657A1A9A9B65CA21A37C 6EF3D77AA1D427D09C066626499404B9F7A3D57A14495B2DC8EE396A1D084ABB 8983BA17BAFC65A4B850073976C259EC1EB5D508CCA12E03966747D6C4430A69 2FE3053DF84BA6E6924793172D8320195189597BD4CA5CD7FEDDC3DEC3452A97 589DC740E107D9B4F021CFD05C4D052594A3C3770568B4FA561AD430BD61707E 59189E0612EC06A234B2CAC16D7AE6BD9F408E92301F3174543E9D7ACBC65FEF E4D2B442EBF6A7F130260794D5DAA981C246B0B8616DB07810F860DFD40D414B F5E7D48143C3263FD13567D59A7EDAEE9FF948A04B55099D1E3782B02D1AEB2F D0D86B37D0B3F168C7A1ED7B0C645ABBDD762D80124E29CC8AB4FB6ADE8DF615 CAB5C2F5096BAA40A6BAB16A3C01EA4A09C9D34A2206F0EEA39AF69A005D3A87 895378F09F0D9375297EBD1EEBCD462963556F88FBD8459EB3FDB72203CA5CAD 527053BBEC06736C0E834624AEFD20AA37868EFCBBA08027BC97C1CEA1DC71E6 26A1AAB78B6A9775655ED261D41299FB7839B432053DDF1037209A893F1734DD B9A6603F20B5283560A0CEC3006579B7231C66C079F6736A64D05C17975818D7 54ECDB3AA9779F55EC88EBABD0D0D684E69AD2ECC1307CEC602DE9F9C26C6365 2DBE3881F714B860B35F785E4F4A510B85D61D74C430A71E90CD0B3D41ABC9AD DC8EAB8BD64A9D9C800FF902FEBC2586EA4380D79356FB8D246C763D5E910F24 F0706E891DBF30547778BECC3DA4475B9608752F99588AE69D68844385F0FB5E FC88E295C5B69A2B2D0E1C55FB83BC2B53ABC01852CF267D896D48FB1F5374E5 2F870D3BDC96098CB4726CE5E9EFB5711B85A36603E7BEF49FD75FE39E868154 59A7493941770C085C2E9747061C101F75B4C2FE2554121FF12B40CFC4BEF980 4D4CC3974BEF12CE48E4C617077448985C86C24A9848C22DDBAAEC0B6CF5E9B5 20C08C37A2412FBC6BD4C0B1797E35EF90026071E0E3C54D2804F1944F7168F7 0EF323538D976988D2697B35A735458ACD654E930D8648921869B5A69AC43011 F503B09481A1D5541D1CC12A61C7C39DED49F8F1D283344001827324D3AB7800 BBDE6C05F24E799EA1C4F41A3117D8E1C4881AC1C5A4D5E18441C3A7E5650EE5 366E0BEFCABC34B238DA540CF0657C641E7A0EC81F10309A50D636BD9BEECE97 FF52C8FEA68329448B9FC4703B3EADD2DE590FF1B9871C98DE7724F1DA064AC8 A2A104DDB18C45266130F8AFF94C3DB65D194E313874882FDFED9B852FCA44BC 62986EC4F6C01B7588BC1D3511F5A4C03E7EAB15796D0AECF005C7D54CFD509B B58D91A3BCB915780449914F42D36D432F92A9EC907B0336C663A663C34624A0 BC285E19CBE8E8D1B8F84B62A85CD38B196D406D9C946986E797A42E12D0435E F51C5F3A188F23AF8DF24C3F594BC6406FF54722C0185BAB76AD46053BA83216 8739BC7CACC32E04FB8C5B5EDB81ED82B0CB63A99698F5FFF9BF9B8AFED29EAA 65BB34ED7C435953001680340D6637FB9638E67E1CD18A6B54306979AECE63D0 F1F3A85D6AF8F23525CA1867230934E3249E2FD12627C86360DF57B77460956B C7F584CD846630048D536AA92B6425316EBE52C3E65F93B1487DD528A460FEAD 8F4A227F6DFD05E154A1344B9AAA80EE744B29341BD1167983835EFBA6A1FEF1 1E29DF84BB3056BAFAEA74DDB78282986A67ECCEF659D864AB1E4E726FB47C2E 8D2AE85EA444C450C6CF1847A708D426C3BF09D51707067117E12BBA18BDBF0A DD5213366E0E860EC124A2E2D28DE56C22C39D541B91B9C26042F74747F72B1F 77738A94874DD92867FACBC0236C9E00255A5C8057475FD7E2DF733484D2CC5F 7E9C590B8EC1B0E9F07248BB77282FED794C669C716B794FAA16734CEE4C982C 6E1362CD29E10D7615D056A0A5B849DC61C39BE6EFDFB4622394E0F1AB3B748B 6388FAC9D77F1C4350AAEE95E04B2FCD2F5EFA16CCE7799636546DD4A726086A BA302C6B8319725CCB20CA389D4734BA8D47DA7196D1EDD51063B0D42C00FEB3 202300ECE8E2D7C6079B5F9B0A67415EA6C1529D24EA77AE3D681BF63615F043 6D173FEC87789E814E1B0A8A846B293E33C18865353C4B6C8110FD67DA530CEA 5DB61E77CCD5C4CF580212C6EB6EC7F6F54CEF856CC76B53D342B67E7995E9F7 9113EF63DCECD7D89DDC913AEBF6E3592F1C9B52DC26F8B6F030B3529642B4B6 69E3236C9FAB233EED4EF5EEC39F13DD5519730909E3C12DB6999727821B18F3 1F607EDA17B966AAD7628F368AA6F3621CA7A53FED6A3FD9E4ACD89D734ADF0F 116C3E91B749C790ADBDAEB477BF780077A68C11EF0943E279C3D00304C7134E 283EC6493EC4DA1948629518227B4C097866E764B2B338564E1F2EDF4B8B1143 6C033FF3CAA3BB4DEECE8A192B2802E0AC1452C9711127497C1C28612C504817 E0FB94CA5714B5001B750F7FBCF2BC7214794E48ACEEC7BBB32D69BEB0AE4975 04E396B3368C7E709C90DA62749FFC04687028F5433568C41F8F41A4DB5713D2 B74A1396C1E70F0E8B1D47ABC9441E8A6592FA16906D23D6AE41DFB1C5C5D01B 1511A9C528A1FD3E86B6111B68B421E455A950A15B8D5EF05E158C1159918D2E 6179A224F876E5966AEF083C592034892899D6D81BB780B5E40C2915C60C8CDC 58AF9DADFC0792C5091D0F9DD7FC9173D1B717D7E86CE2748299D632CAC37AF1 651B1FBA88ADBF656BE062C25315BBC3939B84F136B6F0CAE0A37554CF933E07 7FDE511ADED3CBEE4C55E4E20D4A75EEE82501C2EDC6F9D654F7C3DCA2FA1A62 FAF83892B9F6D8FB1A13475E4D227172A3C92277546DEA068E0CC7409FAEBAB1 623C6429ADF97663ECFC2BF4187953E4C1E6B95F3F81D5D009904E6F7FB05A96 100A1BA40C441AA20913CA0618A525D8A6689BCCB3C7A7F647A52A2E19A894C1 2FFF9E2F8111A0866266D404F52C8FB5FA537FC699EB8154E9AFC5C99C868656 0A62F487EA133C7B3D9317E057F18E701C748D2C255BB3F2225FA4708C86DBE3 D04F1CB6C6F47701F8735152CBC89C4FA9EB7379A1A9961F6DDCEBC0A189211C CD0E88047B815A68043168144224F3739448E70DDC82EE192937CB64F41B69AD 4316B44366D5ABEECA6E32BCDC63D472F700BE413FAA9DA5159EAE6CCFBD54E2 38FCAE2DE7BA39F45DE94592D85FB24076E81F7D0ACFDB7B9195052EA3230F9D 0A6A819DA2D2EDCECA486EAD6B71566D4EDE04960FF08D7BDC67A8D34959AFD1 5D220294C46B99A96217C9CB028F0976E0C2BEA81F9BF6B598E15A6629588B25 790F676816044841999D3F1C7A12C2BFF3378116ED0599485FAD7B78CFD42D24 4C5EF8F9ECD74E7574690F339AD324E8E95AA47A8817A9CF3066A7EEE411EEE4 4E8E902B9315C43EFC730677CD019BE6D44EB5EF5460F341754185E85B0E162F 4F2367264D331BD19E8B48923A813AA9E2CF90BFD6F409C46B86D864E019297A 7859ACD851A4CC9D2B679886F7DAF4E2DFD233FA32A3D76145C84974165E6A63 FF24C5F7A3CEF45DA79D94F802A3A290E95EB94952599355B76B402C96287BB4 6AD16E436CB68F7B7267C95EABCBAB1C28F54956FBC2115202551CC0A7F87F01 773A7BE50EA3A478B38EB866A89D8595B614D5C88BC392CD67B0512692726E95 E79EE5349CB96BDE1AF539B406ED2297C8E38CE1409049DB6644779DED7A32F0 126F0A942C76129F6B2E037E94BD2E7708ED076A27F13890B32BEDD1B508AAD9 693664D2A16DC7136BD8B1EE0ACFCDF4FA460E85CCAF6DE831E38C5E7856F38A CA0846CBBF305022395F1054948B0D6F3E22861DC56DEC46C247EC0A8FB4A7B9 189C42B5F2407B0733C03079CDBCDC9E8A724619ADC0B2141171605322BEDF0B 95BD362D916C5448B9E71B7111CBCC641FEE3520EA9691541298475C3C8D32A8 ECC5706C644F22B2EC0C1C03324EB621F8DA8F707367C380B26B73C75173B103 3F78B21CFB1BC7AA37BD6C86EE5D49B3BAA62F30B86B59C80A127112AE592E01 396352C829156A02116202213304FB1BFBAA9F484C875B3BE129670A86B9D3EA 8703B9CD3A757352D1B7DA588BA807FBA3CAB083B8ECC8BF244A650222516096 9405C999331F1A699C8E07DC2A9AB26CC00197648742A49EA728373DE8385927 3F8F7D2BB9CA46E7DECBE62C529B0B371DF86623AD26D338B955DD2335497080 2EA166270131198F75233A93E9CC8C54294510C91B90C17F531CBD6F9AEE96CC 7D76B1C2E3D21F9CBCD27812F04332AA770E534D0F9E78860E0C3D3BF2ED205D 90DE51D5B97A950D48EA1C8E308FFB4CE64430C19223DDA7617443F605978220 CF8A1CFE969B8ACDF5400F559AEB45A337B6AA53D8AC877D993E36D26EE263EF 358A390BBA072593F999F7F24430AA64368F144794BE583FB6507CBBF2831F54 A00D915A3D1DE9F8086A1CD3CCF1BDAC4407050EF73FF2BDC90F8C7BF42CA2E6 A4D6737B06DDA931637683BD2B2C2227D66C047B027441B4B708977F4E5DE974 BC70A0884EFE0A6D50C6F95CCE37DB668BF757348321FA047F17B20CC2989567 478C9CC029563A8FACCA60D7DF7E67C0CB152BC6BD1C7AE99D6D40CDFDE89613 FE5DBE64C6833F45610C48C45F3DDF9111EAA7C7C042E803767403682ABF0BA2 C4106BC8F87C7868390470FEF651F8A968D4C74371387CDA8C4263CFA7D571BB 5455E8C3152D3E5C6ED56B6555965A44770EBA908BE866F073F42A3FA60C2E5E C577FA552FBB46ECF420ECB78371B71BAEF3D1B668ACF439988485E9A9FC1A3E 87E74E90266717F2D9C84827B39B54298E4E20F693AE9CE1465641F294B1E90A 8AC087B63EB97BF98976209CAC1CFB51C47AA0639E380FD995D4979DD3AFFBE2 DAE145F74BB9C21EEF7F82C96B96293886053B353702AA1045A9C85AADCC1696 317DB4009A3707146B161F9115160BFC089BB947C0AD7E514D2B25001D508D26 62B5E57D58BDBD9F6157C5D3BBB02ED5FDDDAA7D2F811B96413843CE225AB93D D324A1310D2EB1448B88786340173DBD9C6BCEED7A4ADC241A1030519F4FED7B A43C74284762461B716F21A80662226DA73DEBBDEC368E8737D5F1FE777A0F4F FEC3EDE2016F8505067EDD7CDD0339DFEF1ADBFCBDD070014D7D34B56F2A8D01 926C2656DE0EEE94C270AB6B066F0301CE30685C7A136C667D0E468B19028BDE D05BBE78109EEED9FF4B0B30ABE2959FEC3141190851C0D63DCDEED7758F8A1A 8803294C115ACEC18DE831A09EA9EFF3A2A72D8BBD55CB20E131B11B2B54DD41 A87011B71BE2F68966640299D839C9F9B2AA6B6CA949B8F0DE50267936A08046 0E4098EEEE362BEC87F24234E21386A95DF0814869AD23267EF667E2F6E105CE A42B00A087E860247F526D5CC0DDA9D480087B10A490FB224E963C5EF85249EB 14A17C1801CA1492E5FA12CF0DEB4A97FDCB3D84DBD47933189DF884DF6B662A 6AA9C26B997D7DFC712E2BF2FBBF71A3DC0E829A59034C8D45349C68347881FE 7BB9809F3505B3EC60CCDB3D4F518B113EAAB35924140A95F850EF6DDA52BD53 4CB39568EDAD22E308F9C907214BC7E6E4BDF6742CECEB0ED192D77FFEE90EF5 49E1150778AF243E81D2975E887121D576CD787CD6C0CBA51DC3329094CCFAA6 4A33913B472FBDE1AA708CFAB9FF9ABC1EC676248B1A8892FC2FF716C9A00920 4F97D9F6F63CC0F6A96C12FA0565F99DE4F560E179D3C9BB21A5DD9EB7775025 C30CE129F13A66462902FB7B41462FFD63E9E3F16535B658CD7AE6C45EBE16EB F7785416FED5082085B42A7DB13548392AD485921B118C9F47E462650FB77413 0AD1F85EA1D31D36F91AF67FE8E87C6AB8B4181543AC8372B9A6AAFC128884E9 6DCA6AC175AE5D31D4E97A6E10386C4C64941973CDD1DA0D6BD46B647B065EBC 12097AAC0CDCAA8E4A77402C61ECD53B3C2FBDE97837BA4789484B6EC934EEE7 41E8449D396391AAA662CC2D7AA76717DF98BCF6CBEB36F4E65BBC092FCC1B28 31EF6ED2CE244DE585474B1C7BE8C758A756B71B457E1C6B1A939D5B3EA89BD4 7A26339E2C94885932A2C2D30864F7E354C64D5A2E54F92CA27365F6279C6AF5 515134864FBEB0498B662AB845F7F21D4E66186C170CB0251642E1B404A6EF5F 5D8BA722D21C1018A2854CF323A3D0551A456D1245FAC2C5EF467DAAD2A3FB2A A87298D0B557F8E3D074D1A09EBD247B503351143F6D5B10DB4A84C56D3E867D 7C8962FB1123DC8F1C7B89A6D301C691A60E551E22D7466B16168E1740C92BD4 7FC5BF1FA24A5FF5182ED83134677AEFF30EAD6EF62C54ABEBA0088F0B2530E7 92B6F6E086C4BF099823C6F4DB19CBDF7D51185EFD7B1FC0506782A9C616485F F951138131CD707657554AA64571348CD5C3567B51DF6EADF6A0E40998514FBE A8D9D09B762221BE427C0793ED77CE0A6F3157310783C5DE893AD917E31F7926 1ACC190CE65973F80DDBC8C470A234BBA37CD794B32042D936D0DBCAE6FD9A40 6F0BFB82B1A24547093B625A10A9487ADE6DABC1BE5B7E5A212C062850D34A57 D038C87F568B83E7EB2BB07796F285ECFFD761E7AC65FA7260B1604B6798A395 7C75A0E72CD47430852618C838401A55F5B2A0B328E505836CA892CA36C6B395 DA883664E5F9AD3F6F4A4B0B6025EB95E80BF38A3BF68C3139C2F9850C114322 9491EF32A2F99AF6848C04DF8BAFBA306C4CA1E2B1D9AD7BDD4F558D872367DB 714471706595949DF1594F6DA658B90EC2EFD41743C3DDF2211BB7FFCFD66031 97CFCDB3782CD2F682BA33D7B8FD22507ADCBA808AE19878C4D47BF8099D7E57 549376DE27C9574DEC5A3B24198A429A03019D8D5ADD027FC142CEC42E4AF55E CFB2C8EE4A7060FC2080587D2CA57B0294A3B56F08EB0D482D1343733905A3CE A5EB6CA69E2F7B82B8CF1C5925E41C20FBAE76D8561F51B8F9C7A369D3A153C0 F7B5E3B7037521AD96A84255356E825699F5FC384C5B994788FEE84A9E9526F1 046B66F140DCB72AF3735965FBA44AACB445A7EE6D905CA8B04F48573BF9C3DD E6087E9B440187262007F0ECE2BA9A62F72107A3097BBC0B7CA373A056372005 FE27BDD929BB4AAEB68011F616E97D1BF27E235ADA8B629C8A64C35D5BE0B885 353AF23A840EAF28226FAAC12043C23C868AF9511B09C87B5B134352E3F70FDF 7131D0900B0FA7CC06F838193E7EEB43F274D7971FC04E7778334B648E788DA3 055BECDD7AD2B82EB5267E6AC4F24C01732ECF4417C7E4C99212934206DF9707 2DE20E0FEDA62FE48C98C083B5AAA812239CFC93B384A342FC32E5B23ECE52C8 801326C7ABB268601C2CCEEE35FC8F1466B460A1DB76A43D4C38C071B5ECCEBE 68BB66172674F7A8398B2882FD4FBEF20E974CEB783668DC727A918E7E579683 34F7D114CE9248A392D74FB83D4667D2BC93397AC057400EC4A9C702FB76701A A319B3D9A1E2946E4C5066CB2ED9FC045E9F0CBB2BB49054431F0C34CBCA4C4E 62D0FD8B5F25D9E2B57A578D08AB5FFC868F37913F35E66B49C4F80B7F8792F1 056A41D80727D878BC70503E65B609DE3E820012077B989E83759B01D22FA32B E293C71DF5E456F0EDCCD37891C1E153E1D20F0AB669C428CE864F9CD0D1CE01 A3BCD03052C1BBD6004A89E620BFB84F99529C31A8A155C2ABF99B5DCFCB7A11 ED76C88C6B80A3F961C5FF5B830E48C906A5A5C714ED5CC22584C01C1F429C2D EB9AB23DC64B63865C6D69C5E445C60C47E763C335549B2277125A8280A062E1 D717A1C943AF7792C0974B67FC1202144D1379B1921F6D84C2554C90BEFD73AA 09BC0F09BB2B41A6EBF28E92E00CBB33F92287E2F7E587F104BFD9AE66227918 B5CDDA70A160035C573D05DC247808B48A20CBF62C79341E6778BA6F93250901 D39DF8AC345F1C11B13F621868B341B8B1E13FFE98E2830D35BA3A21EB2EDFE5 3EA0350906DE859C92E587DF6FB4F1FD0B9F627E3DD7009DB73F733AF68DFF20 939D3A2EB2ED312B5E8E39097BAC36326652BF9A451CC3F9F50D04A6D74301CB CDD3384E6B4D1B56F0086DDA56AADE5590760E72F711DC981294B88AA85A3827 768D51200BBD5F9BCFF67AF1B5EA3340E7F4CF79204F732E638B729E30796D8C 9F01BEFD30FCE2616F85272FB6EDFEFD410C9E06A6369717AF861BBFECA65501 09314615092C27D07CA9C9634F22A2B9FD6F88891DC714A3DE5AD365D1DFBF36 718542BCF37B4F209FFEF2A9D584C95D9E909E0B748BDDE2B299E3259F04D51B 7791780218CE05B8EFD668107853BA9298DBCCA4C6336AA203E0C108C9B89FB0 3EC2140D9F3522EDDB25411FD850DF99DF2BE4791746FB3F7300E958ADCCFF7C 1A8AF0A4E8F590A9F93600D4A3F9A8DA312B8F97D1BA45B4795DAF00835C88F0 D9051EF1387037986C2572FB04AB622C19D18A39BA99379B29F90B3358CA0D3F 588836FE78B9DE57ECBAB1176F539B275213AE83B8AAF10DAFCBF23766D72EF5 4018CEC1A361AF50B2D924F5580B7457F9CA6399912EEF332E9C1D1C2781BD56 A76D0A986C310E829C25000BB4A4281DE50B0829E5E8D1F4157E925C2683FDAC 9A304DDBC32473C252FEA0B484B4743966A98D70935E919801C007DC4CCAC226 2EC08B53F3E0DA6AD6725B0C084BD788392102566F1EB4D1A7D4708DC7AF0AC0 3469C9C56D283239EEA4FEBB8BBEB9A0A3531D64B52C5D99B0AF038030500660 39F3D1DDA7576EA5563B89FA730E044E5FFF5AD650373DFD695E842AA7B8EC52 1237E3E2C95C29FBF65CD8D38105B3748694572E45641BFE340E0135FBFA81B1 04A66B2CFC510EEBE92A828CE2F3C11C7E15EBD5942F0F02E6BE0777755C1155 6EA5C953F017AB4A6B0A12397A1907FAACE877FC472B0550CE829558D1BEF3A7 7A692C7016550078FA04C75E3D748E2DBC7D4C9DA552D62F93EC7CD5108B6785 7BA08839587DAF42D7C465DD0D5FDE7C661EF43AA69187F2AE13D8C0EEFE97FB AD7388407F2649E94FD360182CCD306441492CB7D3C2437E73C1A080E961FA3E C7FE1E1BB7FE59C4BB380F5B7CBDB122A3ABBB46D2786EDED9B4657C134E676D 5E12ED94E476E22CA86CC297314EA4A3697C0A8A570DA70D00151E81083CA1F8 D0F0AD5EBB6A46267F135E91056FF91C7731294372A1C5C61D911B894B6970C8 D476C179455E53A9C3A578CB96BB5C196A49FB6B78E1F09846557AE23FC21AD1 821D9416AF09DE19F81265942FE81403F4201629CF0417627A1C9E23ECE0FCD1 7C75AE457E0B32DCA67FC55C3B298AD252FA20A7F3B96B6192D003BEB252711C FAE2668B47DFFA736655EEFA4FE445058F506CBA968F9B9DD9C83CDA6E7478C2 150A1D4C0CA9F6482DF3D25D31351FC0EED9B142F2EFA2D73C6472A0BD581362 3BE8C86F46F053C6C556E18DC327E38B1CFE24E76A25DAA7EB2B45C9CCFCDEDE 58B3056DCDDAD9691A796BC9C99DCACFC759F01F62B6B77D9A4CC180E0D0C616 828E92920860D62B839445B1BEF9F8827FBC2BB6BD4CF7FB55EFCC65FF5ED020 59B68B6CCE75742D95DBE480088A909219AFEFA16B4A062A080712A13D098866 A0CE62220D4CF84B3FCE378176B5ABB98D3D989DFA8DDBFE6D86A299737E7A27 67CD54494866F7F6B8CE3426972621C8FE53A1020EC9C30D57AD5AA0A90434E3 29A1C33EB3E8294257641A10B1ADD1382C7C92BF30562AB05E1DCBF1BDD44453 6BF510325C34A7F24E1A1D8F888B12AD327FB5C0D12C873FC799E0D2FF484B32 00D5EAB721FF3BFDE625D61094C2165B1E834FB3A7E0059E62044BB0D2F0AA51 16C2190AC75C57EC2700B35D46E87A0FBBF63080106156901884BB65DD192C09 454A48FC137D6574071AAB3B90CD9882E74263EE89EA5430FB4AAA123C2BBEE2 F412ED2EEBA556D21F17FEB90DD62D5C688EE48EE779582034612FE1D11AD198 6D7AE460632EBD42ECCEED98DA0E1609BF6297BB774B8337D6D39A121F43EA7F 2EBC3E091ED42FA334A57CE00DE4B0BCDFFD3B616F5944436D856BFEDFEC131C 4CBB2FA8DD432C165283D86F0D5D8314899DCF3FF69A4B6B4952B90D83ECCBD5 B3EC9A254316CB175A0FF2428CC47F24CC345AB1CF9DA406D0847930A9BB531F A8D51D93C34481FF0E6C7D3444C85CE22D29E93BF6914EE6DF46FFAC1019CE98 DE93029138E1B72A4A8F551C6AF9527D40E85189EB9FEF30952FED344C8599A7 281AA8EF07A645C4EF7B286D9A7D5752A1CA9F09228907B949C47C65A04342BA 1E56C4E2DA1F3F1B61C2259470C51D5E62BBA0DB710BA6C8D07445CC6B251AC8 C07AC2FC1424AC55F9CEF192BDA64A6973A5FFF5035D4D74E5924689AAD8C633 BFA81AF5F13D6CF5CE3A3F6B89BDC8660476D2B9963D22647DA93710D3B650B2 068A876F2580AAF51C1771F0B38C97DA473C77477B9D09A078AF8D9218F19D3B EBA4D97FF3CD0C2BC47C655D9B11242C6F9113E3EC7E6E5F56C15212B317BA03 BD7F082448416321A123C16C70D17A6C835A9BAF76ED2F1CCA7517E0A49896CB E6B55A32F794169A6117E36FF82221354047104D05530BA4E711C5E3E6CB0E69 CDC135475DB9AE50BD57D6DF86C1088FF048609CF2A4F6148C62E7134E6060D8 0FB7B0EF5677F2C04BA6D4CAF95A213CD0546FD224B4A5A8CA877D3F3F032FCE E93EB1E9F1EF440730E094B9B584FFED6229D00AC0A80A3F047E57100B7DE517 954A2033F48C3BED602F0030DE0BE0D403F13924039726807F73CB495258F75A 53C79F93EB23E873CD4494455B9CE86BB6AE10DA66C02C983BE70D4F0B6AC281 FBBF212477A0CEB5FDCDA248ED3B023ACA12295BD0D6AE9EE96348A80E5083A7 8F143D4CA73FB30F9836C03CF8D828C36ABDD82E6EBDDA9B81F9454D6EC8D6E8 BEC3E609FE383D2BDAE794B51D3B0B985DDB19F5A2C35B1B8B83D4B8F1273CBB ABD3972E16993A462B88DAF708A5B04740AD939446ED2DC9B6061683D79439E4 C2D2332F75959C63A982BB9D7BAC7B543F5AD8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 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 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 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 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 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 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 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 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 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 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (LifInh.dvi) @start /Fa 133[40 40 1[40 1[40 40 40 40 1[40 40 40 40 40 40 1[40 40 2[40 40 40 40 40 32[40 17[40 40 45[{ TeXBase1Encoding ReEncodeFont }23 66.4176 /Courier rf /Fb 154[38 101[{}1 58.1154 /TeX-cmex7 rf /Fc 142[46 3[69 18 8[42 51[23 5[28 28 40[{ TeXBase1Encoding ReEncodeFont }7 83.022 /Helvetica rf /Fd 252[65 3[{}1 83.022 /MSAM10 rf %DVIPSBitmapFont: Fe bbm7 7 3 /Fe 3 91 df78 D82 D<007FB612F0A29038E007809039000F01E0007C010E13C00078EB1E030070011C138000 F0EB3C07489038780F00A248EBF01EECE01C0101133CC648485AA249485A4A5AEB0F0101 0E5BEB1E0349485A013890C7FCEB780FEB700ED9F01E131848485AA248485AD980701338 000713F0380F01E0A2261E03C01370001C4913F0EA3C070038EB0001D8780F1307D8F01E 133FB7FCA225287CA72E>90 D E %EndDVIPSBitmapFont /Ff 87[28 17[42 27[32 37 37 55 37 42 23 32 32 42 42 42 42 60 23 37 1[23 42 42 23 37 42 37 42 42 12[46 42 51 1[51 2[69 46 9[51 51 6[28 1[42 42 1[42 42 42 42 42 2[21 28 21 2[28 28 37[42 2[{ TeXBase1Encoding ReEncodeFont }49 83.022 /Times-Italic rf /Fg 144[55 55 4[39 39 2[46 9[92 46 1[120 4[69 39 1[88 17[74 1[74 1[74 1[74 14[67 67 4[48 48 66 66 4[62 62 4[44 44 61 61 50 50 4[39 39 48 48 4[35 35 38 38{}37 83.022 /CMEX10 rf /Fh 255[45{ TeXbbad153fEncoding ReEncodeFont }1 41.511 /CMSY5 rf /Fi 133[32 34 38 2[39 4[34 1[41 59 1[35 27 23 39 3[35 30 16[50 5[46 1[37 3[42 49 54 48 2[35 1[52 34 52 20 20 22[40 1[42 1[41 6[30 3[39 2[34 3[35 38 43 11[{ TeXaae443f0Encoding ReEncodeFont }35 58.1154 /CMMI7 rf /Fj 145[34 3[20 2[34 34 9[45 41[45 66 19 12[34 1[66 11[52 52 19[52{ TeXbbad153fEncoding ReEncodeFont }13 58.1154 /CMSY7 rf /Fk 146[51 1[31 24 5[31 40[19 41[31 4[33 12[{ TeXaae443f0Encoding ReEncodeFont }7 41.511 /CMMI5 rf /Fl 139[26 1[26 1[37 33 37 55 19 2[19 3[30 1[30 37[51 4[33 33 33 33 33 33 33 33 33 4[51 1[26 26 29[47 6[45 3[{ TeXf7b6d320Encoding ReEncodeFont }25 58.1154 /CMR7 rf /Fm 141[69 3[42 2[42 23 32 32 42 42 9[55 9[70 1[58 2[100 9[44 55 66 5[42 3[0 0 2[55 55 83 13[42 1[83 6[65 1[65 2[65 65 17[65 23 65{ TeXbbad153fEncoding ReEncodeFont }30 83.022 /CMSY10 rf %DVIPSBitmapFont: Fn bbm10 10 5 /Fn 5 91 df69 D78 D 80 D82 D<003FB8FCA39139000F001E01F049133E01C0011E133C90C7003E137C003E023C137800 3C027C13F803785B48ECF8014A48485A12704A48485A03C05B00600107130F038090C7FC 020F5BED001EC748133E021E133C023E137C4A5BA24A485A02F05B0101130302E05B0103 130702C05B0107130F028090C8FC010F5B90381F003EA2013E49EB0180013C1378017C13 F801785BEBF80101F04913030001130301E05B00031307484848481307A2484848C71300 D9001E5C48013E5C001E133C003E017C147F003C01785C007C01F81303007849133FB9FC A331397BB83B>90 D E %EndDVIPSBitmapFont /Fo 135[44 2[46 1[33 2[46 42 46 69 23 2[23 1[42 25 37 3[42 3[23 1[23 14[52 14[65 1[23 23 1[42 2[42 42 42 42 42 42 4[65 1[32 32 4[69 1[23 22[60 6[58 1[69 1[{ TeXf7b6d320Encoding ReEncodeFont }34 83.022 /CMR10 rf /Fp 133[39 41 47 5[37 37 42 1[50 73 1[43 34 1[48 40 41 39 43 36 11[78 48 57 49 1[63 3[67 81 57 1[46 1[69 1[53 61 1[59 2[44 1[65 42 65 23 23 18[54 1[43 1[49 39 52 54 52 49 2[47 2[36 41 50 48 48 1[39 41 2[37 43 47 53 11[{ TeXaae443f0Encoding ReEncodeFont }55 83.022 /CMMI10 rf /Fq 87[28 17[42 27[37 42 42 60 42 46 28 32 37 46 46 42 46 69 23 46 1[23 46 42 28 37 46 37 46 42 6[55 2[83 60 60 55 46 60 65 51 1[60 78 55 65 42 32 65 1[51 55 60 60 55 60 6[28 42 42 42 42 42 42 42 42 42 42 1[21 28 21 41[46 2[{ TeXBase1Encoding ReEncodeFont }64 83.022 /Times-Bold rf /Fr 165[36 42 1[55 1[42 36 32 39 1[32 42 42 52 36 42 1[19 42 1[32 36 42 39 39 42 7[29 29 29 29 29 29 29 29 29 29 48[{ TeXBase1Encoding ReEncodeFont }31 58.1154 /Times-Roman rf /Fs 87[22 45[26 29 29 44 29 33 18 26 26 33 33 33 33 48 18 29 1[18 33 33 18 29 33 29 33 33 6[37 3[41 48 37 33 41 1[41 48 44 55 37 44 29 22 48 48 41 41 48 44 41 41 6[22 2[33 33 2[33 33 1[33 18 17 22 17 2[22 22 37[33 2[{ TeXBase1Encoding ReEncodeFont }61 66.4176 /Times-Italic rf /Ft 87[28 17[42 1[37 37 24[37 42 42 60 42 42 23 32 28 42 42 42 42 65 23 42 23 23 42 42 28 37 42 37 42 37 3[28 1[28 2[60 78 60 60 51 46 55 60 46 60 60 74 51 60 1[28 60 60 46 51 60 55 55 60 5[23 23 42 42 42 42 42 42 42 42 42 42 1[21 28 21 2[28 28 28 35[46 46 2[{ TeXBase1Encoding ReEncodeFont }75 83.022 /Times-Roman rf /Fu 255[48{ TeXbbad153fEncoding ReEncodeFont }1 49.8132 /CMSY6 rf /Fv 139[22 5[31 4[17 54[28 28 28 4[43 1[22 22 40[{ TeX0ef0afcaEncoding ReEncodeFont }9 41.511 /CMR5 rf /Fw 205[30 30 49[{ TeXf7b6d320Encoding ReEncodeFont }2 49.8132 /CMR6 rf /Fx 149[20 56[71 15[71 8[55 24[{ TeXbbad153fEncoding ReEncodeFont }4 66.4176 /CMSY8 rf /Fy 155[32 88[40 11[{ TeXaae443f0Encoding ReEncodeFont }2 49.8132 /CMMI6 rf %DVIPSBitmapFont: Fz bbm8 8 1 /Fz 1 83 df82 D E %EndDVIPSBitmapFont /FA 164[20 32[20 7[35 1[35 4[55 1[27 27 40[{ TeXf7b6d320Encoding ReEncodeFont }7 66.4176 /CMR8 rf /FB 135[40 3[25 13[34 1[36 38[35 1[20 47[45 11[{ TeXaae443f0Encoding ReEncodeFont }7 66.4176 /CMMI8 rf /FC 75[22 11[22 17[33 27[29 33 33 48 33 33 18 26 22 33 33 33 33 52 18 33 18 18 33 33 22 29 33 29 33 29 3[22 1[22 41 48 1[63 48 48 41 37 44 48 37 48 48 59 41 48 26 22 48 48 37 41 48 44 44 48 6[18 33 33 33 33 33 33 33 33 33 33 1[17 22 17 2[22 22 34[22 2[37 2[{ TeXBase1Encoding ReEncodeFont }74 66.4176 /Times-Roman rf /FD 166[38 1[50 38 38 32 30 35 1[30 38 38 47 32 38 1[18 38 38 30 32 38 35 35 38 65[{ TeXBase1Encoding ReEncodeFont }22 53.134 /Times-Roman rf /FE 133[41 6[36 30 2[46 46 71 25 2[25 46 2[41 46 41 1[41 9[86 3[51 7[66 75[{ TeXBase1Encoding ReEncodeFont }16 91.3242 /Times-Roman rf /FF 134[50 3[55 33 39 44 55 55 50 55 83 28 2[28 55 50 33 44 55 44 55 50 20[66 17[33 12[33 45[{ TeXBase1Encoding ReEncodeFont }23 99.6264 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 1067 1164 a FF(Lifshits)25 b(tails)f(caused)h(by) h(anisotr)n(opic)f(decay:)998 1280 y(the)h(emer)o(gence)g(of)f(a)g (quantum-classical)g(r)n(egime)1316 1496 y FE(W)-7 b(erner)22 b(Kirsch)g(and)g(Simone)g(W)-7 b(arzel)756 1722 y FC(A)t FD(B)t(S)t(T)t(R)t(A)r(C)t(T)p FC(.)39 b(W)-5 b(e)20 b(in)m(v)o(estigate)k(Lifshits-tail)g(beha)o(viour)e(of)f(the)h(inte)o (grated)i(density)e(of)f(states)h(for)754 1805 y(a)k(wide)g(class)g(of) f(Schr)6 b(\250)-28 b(odinger)27 b(operators)g(with)f(positi)n(v)o(e)i (random)d(potentials.)49 b(The)25 b(setting)i(in-)754 1888 y(cludes)e(allo)o(y-type)i(and)d(Poissonian)h(random)f (potentials.)42 b(The)24 b(considered)i(\(single-site\))g(impu-)754 1971 y(rity)f(potentials)i FB(f)40 b FA(:)33 b Fz(R)1338 1948 y Fy(d)1407 1971 y Fx(!)g FA([0)p FB(;)11 b Fx(1)p FA([)23 b FC(decay)i(at)g(in\002nity)g(in)g(an)f(anisotropic)i(w)o(ay)l (,)g(for)e(e)o(xample,)754 2054 y FB(f)7 b FA(\()p FB(x)862 2063 y Fw(1)897 2054 y FB(;)12 b(x)969 2063 y Fw(2)1003 2054 y FA(\))20 b Fx(\030)g FA(\()p Fx(j)p FB(x)1212 2063 y Fw(1)1246 2054 y Fx(j)1266 2031 y Fy(\013)1306 2041 y Fv(1)1347 2054 y FA(+)5 b Fx(j)p FB(x)1467 2063 y Fw(2)1501 2054 y Fx(j)1521 2031 y Fy(\013)1561 2041 y Fv(2)1597 2054 y FA(\))1624 2031 y Fu(\000)p Fw(1)1721 2054 y FC(as)14 b Fx(j)p FA(\()p FB(x)1877 2063 y Fw(1)1911 2054 y FB(;)e(x)1983 2063 y Fw(2)2017 2054 y FA(\))p Fx(j)19 b(!)h(1)p FC(.)f(As)13 b(is)h(e)o(xpected)j(from)d(the)h (isotropic)754 2137 y(situation,)24 b(there)d(is)f(a)h(so-called)h (quantum)f(re)o(gime)h(with)f(Lifshits)f(e)o(xponent)j FB(d=)p FA(2)e FC(if)f(both)h FB(\013)2995 2146 y Fw(1)3050 2137 y FC(and)754 2220 y FB(\013)799 2229 y Fw(2)858 2220 y FC(are)k(big)g(enough,)h(and)f(there)g(is)f(a)h(so-called)h (classical)h(re)o(gime)e(with)g(Lifshits)f(e)o(xponent)j(de-)754 2303 y(pending)g(on)e FB(\013)1129 2312 y Fw(1)1188 2303 y FC(and)h FB(\013)1354 2312 y Fw(2)1413 2303 y FC(if)f(both)h(are)f (small.)45 b(In)25 b(addition)i(to)e(this)h(we)f(\002nd)f(tw)o(o)i(ne)n (w)f(re)o(gimes)754 2386 y(where)18 b(the)g(Lifshits)g(e)o(xponent)h(e) o(xhibits)g(a)e(mixture)i(of)e(quantum)h(and)g(classical)h(beha)o (viour)l(.)24 b(More-)754 2469 y(o)o(v)o(er)m(,)c(the)g(transition)h (lines)f(between)g(these)g(re)o(gimes)f(depend)i(in)e(a)f(nontri)n (vial)k(w)o(ay)e(on)e FB(\013)2898 2478 y Fw(1)2951 2469 y FC(and)i FB(\013)3111 2478 y Fw(2)754 2552 y FC(simultaneously)l(.) 897 2884 y Ft(Dedicated)g(to)g(the)g(memory)f(of)h(G.)g(A.)h (Mezincescu)e(\()h(1943)f(\226)h(2001)f(\).)1757 3102 y(C)t FC(O)t(N)t(T)t(E)t(N)t(T)t(S)555 3252 y Ft(1.)83 b(Introduction)2288 b(2)555 3352 y(2.)83 b(Basic)21 b(quantities)e(and) h(main)g(result)1641 b(3)555 3451 y(2.1.)82 b(Random)19 b(potentials)2014 b(3)555 3551 y(2.2.)82 b(Examples)2311 b(5)555 3650 y(2.3.)82 b(Random)19 b(Sch)7 b(\250)-35 b(odinger)18 b(operators)g(and)i(their)g(inte)o(grated)f(density)g(of)h (states)410 b(6)555 3750 y(2.4.)82 b(Lifshits)20 b(tails)2228 b(7)555 3850 y(3.)83 b(Basic)21 b(inequalities)e(and)h(auxiliary)f (results)1420 b(8)555 3949 y(3.1.)82 b(Mezincescu)19 b(boundary)e(conditions)i(and)h(basic)g(inequalities)806 b(8)555 4049 y(3.2.)82 b(Elementary)18 b(f)o(acts)j(about)e(mar)o (ginal)g(impurity)f(potentials)868 b(10)555 4149 y(4.)83 b(Upper)19 b(bound)2222 b(10)555 4248 y(4.1.)82 b(Re)o(gularisation)19 b(of)g(random)g(Borel)h(measure)1247 b(11)555 4348 y(4.2.)82 b(Quantum)19 b(re)o(gime)2037 b(11)555 4447 y(4.3.)82 b(Quantum-classical)18 b(re)o(gime)1729 b(13)555 4547 y(4.4.)82 b(Classical)21 b(re)o(gime)2047 b(17)555 4647 y(5.)83 b(Lo)n(wer)19 b(bound)2215 b(19)555 4746 y(5.1.)82 b(Upper)19 b(bound)f(on)i(lo)n(west)h(Dirichlet)e(eigen)m(v)n(alue)1130 b(19)555 4846 y(5.2.)82 b(Proof)19 b(of)h(Theorem)e(2.8)i(\226)g (\002nal)h(parts)1465 b(21)555 4946 y(Appendix)18 b(A.)83 b(Proof)19 b(of)h(mixing)f(of)h(random)f(Borel)h(measure)895 b(22)555 5045 y(References)2436 b(22)p 456 5122 499 4 v 605 5216 a Fs(K)n(e)n(y)18 b(wor)n(ds)e(and)h(phr)o(ases.)34 b FC(Random)18 b(Schr)6 b(\250)-28 b(odinger)19 b(operators,)f(Inte)o (grated)i(density)e(of)f(states,)h(Lifshits)g(tails.)1935 5315 y Fr(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 456 251 a Fr(2)915 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)1673 450 y Fq(1.)41 b(Intr)o(oduction)605 600 y Ft(The)21 b(inte)o(grated)e(density)i(of)g (states)h Fp(N)34 b Fo(:)25 b Fn(R)g Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ft(is)h(an)g(important)d(basic)j(quantity)e(in)h (the)456 699 y(theory)28 b(of)h(disordered)e(electronic)i(systems)h([)p Fq(Kir89)n(,)g(CL90)o(,)g(Lan91)o(,)g(PF92)n(,)g(Sto01)n(,)g(LMW03,)456 799 y(V)-8 b(es03)n Ft(].)48 b(Roughly)26 b(speaking,)j Fp(N)9 b Fo(\()p Fp(E)c Fo(\))28 b Ft(describes)f(the)h(number)e(of)i (ener)o(gy)d(le)n(v)o(els)j(belo)n(w)f(a)h(gi)n(v)o(en)456 898 y(ener)o(gy)23 b Fp(E)31 b Ft(per)24 b(unit)h(v)n(olume)g(\(see)g (\(15\))f(belo)n(w)h(for)f(a)i(precise)f(de\002nition\).)38 b(A)26 b(characteristic)e(fea-)456 998 y(ture)f(of)g(disordered)f (systems)i(is)h(the)f(beha)n(viour)d(of)j Fp(N)33 b Ft(near)23 b(band)f(edges.)35 b(It)24 b(w)o(as)h(\002rst)f(studied)f(by)456 1098 y(Lifshits)d([)p Fq(Lif63)o Ft(].)25 b(He)c(ga)n(v)o(e)e(con)m (vincing)e(physical)i(ar)o(guments)f(that)j(the)f(polynomial)e (decrease)1188 1267 y Fo(log)c Fp(N)9 b Fo(\()p Fp(E)c Fo(\))23 b Fm(\030)g Fo(log)g(\()p Fp(E)h Fm(\000)18 b Fp(E)2017 1279 y Fl(0)2055 1267 y Fo(\))2097 1210 y Fk(d)p 2097 1219 31 3 v 2098 1253 a Fv(2)2308 1267 y Ft(as)83 b Fp(E)28 b Fm(#)23 b Fp(E)2675 1279 y Fl(0)3348 1267 y Ft(\(1\))456 1424 y(kno)n(wn)e(as)j(v)n(an-Ho)o(v)o(e)c (singularity)h(\(see)i([)p Fq(KS87)o Ft(])g(for)f(a)i(rigorous)d (proof\))g(near)h(a)h(band)f(edge)g Fp(E)3314 1436 y Fl(0)3375 1424 y Ft(of)456 1524 y(an)d(ideal)h(periodic)e(system)i(in)g Fp(d)h Ft(space)e(dimensions)g(is)i(replaced)d(by)h(an)h(e)o (xponential)e(decrease)h(in)h(a)456 1624 y(disordered)e(system.)27 b(In)20 b(his)h(honour)m(,)d(this)k(decrease)e(is)h(kno)n(wn)e(as)j (Lifshits)f(singularity)e(or)i(Lifshits)456 1723 y(tail)f(and)g (typically)f(gi)n(v)o(en)g(by)1143 1894 y Fo(log)14 b Fp(N)9 b Fo(\()p Fp(E)c Fo(\))24 b Fm(\030)f Fo(log)14 b Fp(e)1742 1860 y Fj(\000)p Fi(C)22 b Fl(\()p Fi(E)s Fj(\000)p Fi(E)2043 1868 y Fv(0)2075 1860 y Fl(\))2101 1835 y Fh(\000)p Fk(\021)2352 1894 y Ft(as)83 b Fp(E)29 b Fm(#)22 b Fp(E)2719 1906 y Fl(0)3348 1894 y Ft(\(2\))456 2052 y(where)d Fp(\021)26 b(>)d Fo(0)d Ft(is)h(called)f(the)h(Lifshits) f(e)o(xponent)e(and)i Fp(C)29 b(>)23 b Fo(0)d Ft(is)h(some)f(constant.) 605 2151 y(The)25 b(\002rst)i(rigorous)d(proof)g([)p Fq(D)m(V75)n Ft(])i(\(see)f([)p Fq(Nak77)o Ft(]\))g(of)g(Lifshits)h (tails)h(\(in)e(the)h(sense)g(that)g(\(2\))456 2251 y(holds\))c (concerns)g(the)h(bottom)f Fp(E)1451 2263 y Fl(0)1512 2251 y Ft(of)h(the)g(ener)o(gy)f(spectrum)g(of)h(a)g(continuum)e(model) h(in)m(v)n(olving)g(a)456 2351 y(Poissonian)d(random)g(potential)1492 2522 y Fp(V)1540 2534 y Fi(!)1589 2522 y Fo(\()p Fp(x)p Fo(\))24 b(:=)1834 2443 y Fg(X)1879 2620 y Fi(j)1987 2522 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)g Fp(\030)2253 2534 y Fi(!)r(;j)2352 2522 y Fo(\))p Fp(;)941 b Ft(\(3\))456 2776 y(where)25 b Fp(\030)721 2788 y Fi(!)r(;j)852 2776 y Fm(2)34 b Fn(R)1011 2746 y Fi(d)1075 2776 y Ft(are)26 b(Poisson)f(distrib)n(uted)g(points)g(and)g Fp(f)41 b Fo(:)33 b Fn(R)2450 2746 y Fi(d)2522 2776 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)25 b Ft(is)i(a)f(non-ne)o(gati)n(v)o(e)456 2875 y(impurity)g(potential.)49 b(Donsk)o(er)27 b(and)g(V)-9 b(aradhan)27 b([)p Fq(D)m(V75)n Ft(])h(particularly)f(sho)n(wed)g(that) h(the)h(Lifshits)456 2975 y(e)o(xponent)17 b(is)22 b(uni)n(v)o(ersally) c(gi)n(v)o(en)h(by)h Fp(\021)26 b Fo(=)d Fp(d=)p Fo(2)d Ft(in)g(case)775 3132 y Fo(0)i Fm(\024)h Fp(f)9 b Fo(\()p Fp(x)p Fo(\))38 b Fm(\024)e Fp(f)1268 3144 y Fl(0)1319 3132 y Fo(\(1)18 b(+)g Fm(j)p Fp(x)p Fm(j)p Fo(\))1619 3098 y Fj(\000)p Fi(\013)1802 3132 y Ft(with)j(some)f Fp(\013)j(>)g(d)18 b Fo(+)g(2)j Ft(and)e(some)h Fp(f)2915 3144 y Fl(0)2975 3132 y Fp(>)j Fo(0)p Ft(.)222 b(\(4\))456 3290 y(It)20 b(w)o(as)h(P)o(astur)f([)p Fq(P)o(as77)o Ft(])g(who)g(pro)o(v)o(ed)d(that)k(the)f(Lifshits)g(e)o(xponent)e (changes)h(to)i Fp(\021)26 b Fo(=)d Fp(d=)p Fo(\()p Fp(\013)c Fm(\000)f Fp(d)p Fo(\))j Ft(if)746 3497 y Fp(f)787 3509 y Fi(u)844 3497 y Fo(\(1)d(+)g Fm(j)p Fp(x)p Fm(j)p Fo(\))1144 3463 y Fj(\000)p Fi(\013)1267 3497 y Fm(\024)23 b Fp(f)9 b Fo(\()p Fp(x)p Fo(\))24 b Fm(\024)e Fp(f)1668 3509 y Fl(0)1719 3497 y Fo(\(1)c(+)g Fm(j)p Fp(x)p Fm(j)p Fo(\))2019 3463 y Fj(\000)p Fi(\013)2244 3446 y Ft(with)i(some)g Fp(d)k(<)e(\013)i(<)e(d)d Fo(+)f(2)2322 3546 y Ft(and)h(some)h Fp(f)2699 3558 y Fi(u)2742 3546 y Ft(,)h Fp(f)2825 3558 y Fl(0)2885 3546 y Fp(>)h Fo(0)p Ft(.)3348 3497 y(\(5\))456 3698 y(This)g(change)f(from)g(a)i(uni)n(v)o(ersal)e(Lifshits)h(e)o (xponent)e(to)i(a)h(non-uni)n(v)o(ersal)c(one,)j(which)g(depends)e(on) 456 3798 y(the)27 b(decay)g(e)o(xponent)e Fp(\013)k Ft(of)e Fp(f)9 b Ft(,)29 b(may)e(be)h(heuristically)e(e)o(xplained)g(in)i (terms)f(of)g(a)h(competition)e(of)456 3898 y(the)g(kinetic)h(and)f (the)g(potential)g(ener)o(gy)f(of)h(the)h(underlying)d(one-particle)h (Schr)7 b(\250)-35 b(odinger)24 b(operator)-5 b(.)456 3997 y(In)22 b(the)h(\002rst)g(case)g(\()p Fp(\021)31 b Fo(=)c Fp(d=)p Fo(2)p Ft(\))22 b(the)h(quantum)e(mechanical)g (kinetic)h(ener)o(gy)f(has)i(a)g(crucial)f(in\003uence)456 4097 y(on)29 b(the)h(\(\002rst)h(order\))d(asymptotics)i(of)g Fp(N)9 b Ft(.)55 b(The)29 b(Lifshits)i(tail)f(is)h(then)f(said)h(to)f (ha)n(v)o(e)f(a)i Ff(quantum)456 4197 y Ft(character)-5 b(.)25 b(In)c(the)g(other)f(case)h(it)g(is)h(said)f(to)g(ha)n(v)o(e)f (a)h Ff(classical)g Ft(character)f(since)h(then)f(the)h(\(classical\)) 456 4296 y(potential)k(ener)o(gy)f(determines)h(the)h(asymptotics)f(of) h Fp(N)9 b Ft(.)42 b(F)o(or)26 b(details,)h(see)g(for)e(e)o(xample)g([) p Fq(Lan91)o(,)456 4396 y(PF92)n(,)c(L)-8 b(W04)o Ft(].)605 4495 y(Analogous)25 b(results)i(ha)n(v)o(e)g(been)f(obtained)f(for)i (other)f(random)f(potentials.)44 b(F)o(or)27 b(e)o(xample,)g(in)456 4595 y(case)20 b(of)g(an)g(allo)o(y-type)f(random)f(potential)1468 4766 y Fp(V)1516 4778 y Fi(!)1564 4766 y Fo(\()p Fp(x)p Fo(\))24 b(:=)1828 4687 y Fg(X)1810 4871 y Fi(j)s Fj(2)p Fe(Z)1931 4855 y Fk(d)1981 4766 y Fp(q)2018 4778 y Fi(!)r(;j)2130 4766 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(j)5 b Fo(\))916 b Ft(\(6\))456 5013 y(which)17 b(is)h(gi)n(v)o(en)f(in)g (terms)h(of)f(independent)e(identically)i(distrib)n(uted)g(random)e(v)n (ariables)i Fp(q)3111 5025 y Fi(!)r(;j)3228 5013 y Ft(and)g(an)456 5116 y(impurity)e(potential)i Fp(f)31 b Fo(:)23 b Fn(R)1256 5086 y Fi(d)1318 5116 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)p Ft(,)j(the)h(Lifshits)f(tails)h(at)g(the)f(lo)n(west)g(band)f (edge)h Fp(E)3061 5128 y Fl(0)3116 5116 y Ft(ha)n(v)o(e)g(been)456 5216 y(in)m(v)o(estigated)k(by)i([)p Fq(KM83a)n(,)h(KS86)o(,)g(Mez87)o Ft(].)36 b(Similarly)23 b(to)h(the)g(Poissonian)f(case)h(the)g(authors) e(of)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)771 b(3)456 450 y Ft([)p Fq(KS86)n(,)24 b(Mez87)o Ft(])h(consider)d Fp(f)34 b Ft(as)24 b(in)g(\(4\))g(and)f(\(5\))h(and)f (detect)h(a)g(quantum)e(and)i(a)g(classical)h(re)o(gime)456 550 y(for)19 b(which)h(the)g(Lifshits)g(e)o(xponent)e(equals)743 769 y Fp(\021)27 b Fo(=)898 627 y Fg(\()1064 664 y Fi(d)p 1064 678 35 4 v 1065 725 a Fl(2)1239 697 y Ft(in)20 b(case)14 b Fo(\(4\))23 b(:)106 b Fp(d)19 b Fo(+)f(2)23 b Fp(<)f(\013)1064 804 y Fi(d)p 1017 818 130 4 v 1017 866 a(\013)p Fj(\000)p Fi(d)1239 837 y Ft(in)e(case)14 b Fo(\(5\))23 b(:)106 b Fp(d)24 b(<)e(\013)i(<)e(d)d Fo(+)f(2)2285 627 y Fg(\))2375 769 y Fo(=)23 b(max)2631 652 y Fg(\032)2703 713 y Fp(d)p 2703 750 44 4 v 2704 826 a Fo(2)2757 769 y Fp(;)2875 713 y(d=\013)p 2803 750 282 4 v 2803 826 a Fo(1)18 b Fm(\000)g Fp(d=\013)3094 652 y Fg(\033)3348 769 y Ft(\(7\))456 985 y(In)31 b(f)o(act)i(the)o(y)e(do)g(not)h(obtain)f(the)h (asymptotics)g(\(2\))f(on)h(a)g(logarithmic)e(scale)j(b)n(ut)f(only)f (double-)456 1085 y(logarithmic)19 b(asymptotics)i(\(confer)f(\(16\))h (belo)n(w\).)29 b(\(See)22 b(also)g([)p Fq(Sto99)n Ft(])g(for)f(an)h (alternati)n(v)o(e)e(proof)g(of)456 1184 y(this)27 b (double-logarithmic)c(asymptotics)k(in)g(case)h(of)f(allo)o(y-type)e (and)i(Poissonian)g(random)e(poten-)456 1284 y(tials.\))605 1384 y(Our)e(main)f(point)g(is)i(to)f(generalise)f(these)h(results)g (on)f(the)h(Lifshits)g(e)o(xponent)d(to)j(impurity)f(po-)456 1483 y(tentials)c Fp(f)27 b Ft(that)18 b(decay)f(in)h(an)g Ff(anisotr)l(opic)f Ft(w)o(ay)g(at)i(in\002nity)e(\(confer)f(\(8\))h (belo)n(w\).)24 b(In)17 b(addition)g(we)h(are)456 1583 y(able)i(to)g(handle)f(a)i(wide)f(class)i(of)e(random)e(potentials)i (gi)n(v)o(en)f(in)h(terms)h(of)f(random)e(Borel)i(measures)456 1682 y(which)26 b(include)h(among)f(further)f(interesting)i(e)o (xamples)f(both)h(the)g(case)h(of)f(allo)o(y-type)f(potentials)456 1782 y(and)19 b(Poisson)h(potential.)k(Thus)c(the)g Ff(same)h Ft(proof)d(w)o(orks)i(for)g(these)g(tw)o(o)h(most)f(important)e(cases.) 605 1882 y(In)h(our)g(opinion)e(it)j(is)g(interesting)f(to)g(e)o (xplore)f(the)h(transition)g(between)f(quantum)f(and)i(classical)456 1981 y(Lifshits)i(beha)n(viour)f(in)h(such)g(models)g(from)f(both)h(a)h (mathematical)e(and)h(a)h(physical)e(point)h(of)g(vie)n(w)-5 b(.)456 2081 y(The)24 b(interesting)g(cases)i(are)e(those)h(for)f (which)g Fp(f)34 b Ft(decays)25 b(f)o(ast)g(enough)e(in)i(some)g (directions)e(to)i(en-)456 2181 y(sure)20 b(a)h(quantum)e(character)g (while)i(it)g(decays)f(slo)n(wly)h(in)g(the)f(other)g(direction)f(so)i (that)g(the)g(e)o(xpected)456 2280 y(character)c(there)h(is)i(the)e (classical)i(one.)k(In)18 b(the)h(follo)n(wing)e(we)i(gi)n(v)o(e)e(a)i (complete)f(picture)g(of)g(the)h(clas-)456 2380 y(sical)i(and)f(the)g (quantum)f(re)o(gime)g(of)h(the)h(inte)o(grated)e(density)h(of)g (states)h(as)g(well)g(as)h(of)e(the)g(emer)o(ging)456 2479 y(mix)o(ed)g Ff(quantum-classical)g Ft(re)o(gime.)29 b(W)-7 b(e)23 b(found)d(it)j(remarkable)d(that)i(the)g(borderline)e (between)h(the)456 2579 y(quantum)e(and)i(classical)i(beha)n(viour)c (caused)i(by)g(the)h(decay)f(of)g Fp(f)31 b Ft(in)21 b(a)h(certain)f(direction)g(is)h(not)f(de-)456 2679 y(termined)e(by)i (the)g(corresponding)d(decay)i(e)o(xponent)f(of)h(these)i(directions)e (alone,)g(b)n(ut)h(depends)f(also)456 2778 y(in)g(a)h(nontri)n(vial)d (w)o(ay)i(on)g(the)g(decay)g(in)g(the)g(other)g(directions.)605 2878 y(A)k(second)e(moti)n(v)n(ation)g(for)h(this)h(paper)e(came)h (from)f(in)m(v)o(estigations)g(of)h(the)g(Lifshits)h(tails)g(in)g(a)456 2978 y(constant)15 b(magnetic)f(\002eld)i(in)g(three)g(space)g (dimensions)e([)p Fq(W)-5 b(ar01)n(,)16 b(HKW03)n(,)h(L)-8 b(W04)o Ft(].)24 b(In)16 b(contrast)f(to)456 3077 y(the)21 b(tw)o(o-dimensional)f(situation)h([)p Fq(BHKL95)o(,)h(Erd98)o(,)g(HL) -8 b(W99,)22 b(HL)-8 b(W00)o(,)22 b(Erd01,)g(W)-5 b(ar01)n Ft(],)22 b(the)456 3177 y(magnetic)14 b(\002eld)h(introduces)f(an)h (anisotrop)o(y)f(in)h Fn(R)1911 3147 y Fl(3)1948 3177 y Ft(,)i(such)e(that)g(it)i(is)f(quite)f(natural)f(to)i(look)e(at)i Fp(f)24 b Ft(which)456 3276 y(are)17 b(anisotropic)f(as)i(well.)24 b(In)18 b(f)o(act,)f(in)h(the)f(three-dimensional)e(magnetic)h(case)i (a)g(quantum-classical)456 3376 y(re)o(gime)c(has)i(already)f(been)g (sho)n(wn)h(to)g(occur)f(for)g(certain)g Fp(f)25 b Ft(with)16 b(isotropic)f(decay)g([)p Fq(W)-5 b(ar01)n(,)16 b(L)-8 b(W04)p Ft(].)456 3476 y(The)19 b(present)h(paper)f(will)i(contrib)n (ute)e(to)h(a)h(better)f(understanding)d(of)j(these)g(results.)605 3575 y(The)i(results)h(mentioned)e(abo)o(v)o(e)g(as)i(well)g(as)g(the)g (results)g(in)g(this)g(paper)e(concern)g(Lifshits)i(tails)456 3675 y(at)f(the)h(bottom)e(of)h(the)g(spectrum.)30 b(In)22 b(accordance)e(with)j(Lifshits')f(heuristics,)g(the)g(inte)o(grated)f (den-)456 3775 y(sity)g(of)g(states)i(should)d(beha)n(v)o(e)g(in)i(a)f (similar)h(w)o(ay)f(at)h(other)e(edges)h(of)g(the)h(spectrum.)27 b(Such)21 b(internal)456 3874 y(Lifshits)f(tails)h(were)f(pro)o(v)o(en) e(in)i([)p Fq(Mez86)o(,)h(Sim87)o(,)g(Mez93)o(,)f(Klo99)n(,)h(KW02)o(,) f(Klo02)n Ft(].)456 4073 y Fq(Ackno)o(wledgement:)26 b Ft(W)-7 b(e)22 b(are)f(grateful)f(to)h(Hajo)g(Leschk)o(e)f(for)g (helpful)g(remarks.)27 b(This)21 b(w)o(ork)f(w)o(as)456 4173 y(partially)f(supported)f(by)i(the)g(DFG)h(within)f(the)g(SFB)i (TR)f(12.)1329 4379 y Fq(2.)41 b(Basic)21 b(quantities)f(and)h(main)f (r)o(esult)605 4528 y(2.1.)40 b(Random)20 b(potentials.)40 b Ft(W)-7 b(e)22 b(consider)d(random)f(potentials)808 4715 y Fp(V)42 b Fo(:)23 b(\012)c Fm(\002)f Fn(R)1176 4680 y Fi(d)1237 4715 y Fm(!)23 b Fo([0)p Fp(;)14 b Fm(1)p Fo([)p Fp(;)97 b Fo(\()p Fp(!)s(;)14 b(x)p Fo(\))23 b Fm(7!)g Fp(V)2051 4727 y Fi(!)2100 4715 y Fo(\()p Fp(x)p Fo(\))h(:=)2346 4602 y Fg(Z)2392 4790 y Fe(R)2446 4774 y Fk(d)2499 4715 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)g Fp(y)s Fo(\))c Fp(\026)2869 4727 y Fi(!)2917 4715 y Fo(\()p Fp(dy)s Fo(\))p Fp(;)257 b Ft(\(8\))456 4917 y(which)26 b(are)h(gi)n(v)o(en)f(in)h(terms)g(of)g(a)g(random)e(Borel)i(measure)f Fp(\026)36 b Fo(:)g(\012)f Fm(!)h(M)p Fo(\()p Fn(R)2847 4887 y Fi(d)2885 4917 y Fo(\))p Ft(,)29 b Fp(!)39 b Fm(7!)c Fp(\026)3226 4929 y Fi(!)3275 4917 y Ft(,)29 b(and)456 5016 y(an)24 b(impurity)e(potential)h Fp(f)39 b Fo(:)30 b Fn(R)1387 4986 y Fi(d)1456 5016 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)p Ft(.)36 b(W)-7 b(e)26 b(recall)d(from)g([)p Fq(Kal83)n(,)i(SKM87)o(,)f(D)m(VJ88)n Ft(])g(that)h(a)456 5116 y(random)13 b(Borel)i(measure)g(is)h(a)g(measurable)e(mapping)f (from)h(a)i(probability)d(space)i Fo(\()q(\012)p Fp(;)f Fm(A)p Fp(;)g Fn(P)p Fo(\))i Ft(into)f(the)456 5216 y(set)k(of)f(Borel) h(measures)1186 5148 y Fg(\000)1224 5216 y Fm(M)p Fo(\()p Fn(R)1426 5185 y Fi(d)1465 5216 y Fo(\))p Fp(;)14 b Fm(B)s Fo(\()p Fm(M)p Fo(\))1756 5148 y Fg(\001)1793 5216 y Ft(,)20 b(that)e(is,)i(the)e(set)h(of)f(positi)n(v)o(e,)g (locally-\002nite)f(measures)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 456 251 a Fr(4)915 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)456 450 y Ft(on)18 b Fn(R)628 420 y Fi(d)666 450 y Ft(.)25 b(Here)18 b Fm(B)s Fo(\()p Fm(M)p Fo(\))h Ft(denotes)e(the)i(Borel)f Fp(\033)s Ft(-algebra)f(of)i Fm(M)p Fo(\()p Fn(R)2361 420 y Fi(d)2399 450 y Fo(\))p Ft(,)g(that)g(is,)g(the)g(smallest)g Fp(\033)s Ft(-algebra)456 550 y(rendering)29 b(the)j(mappings)e Fm(M)p Fo(\()p Fn(R)1497 520 y Fi(d)1535 550 y Fo(\))45 b Fm(3)g Fp(\027)k Fm(7!)c Fp(\027)5 b Fo(\(\003\))32 b Ft(measurable)f(for)g(all)h(bounded)d(Borel)j(sets)456 649 y Fo(\003)22 b Fm(2)i(B)s Fo(\()p Fn(R)775 619 y Fi(d)812 649 y Fo(\))p Ft(.)605 749 y(The)c(follo)n(wing)e(assumptions) i(on)g Fp(\026)g Ft(are)g(supposed)f(to)i(be)f(v)n(alid)f(throughout)f (the)i(paper)-5 b(.)605 906 y Fq(Assumption)24 b(2.1.)42 b Ft(The)23 b(random)e(Borel)i(measure)f Fp(\026)28 b Fo(:)g(\012)h Fm(!)f(M)p Fo(\()p Fn(R)2690 875 y Fi(d)2728 906 y Fo(\))p Ft(,)c Fp(!)31 b Fm(7!)d Fp(\026)3049 918 y Fi(!)3121 906 y Ft(is)c(de\002ned)456 1005 y(on)19 b(some)h(complete)g(probability)e(space)i Fo(\(\012)p Fp(;)14 b Fm(A)p Fp(;)g Fn(P)p Fo(\))p Ft(.)26 b(W)-7 b(e)21 b(suppose)f(that:)543 1142 y(\(i\))83 b Fp(\026)20 b Ft(is)i Fn(Z)911 1112 y Fi(d)950 1142 y Ft(-stationary)-5 b(.)520 1301 y(\(ii\))83 b(there)25 b(e)o(xists)h(a)g(partition)e(of)h Fn(R)1646 1271 y Fi(d)1717 1301 y Fo(=)1815 1239 y Fg(S)1884 1326 y Fi(j)s Fj(2)p Fe(Z)2005 1309 y Fk(d)2059 1301 y Fo(\003)2117 1313 y Fi(j)2177 1301 y Ft(into)h(disjoint)f(unit)g (cubes)g Fo(\003)3036 1313 y Fi(j)3104 1301 y Fo(=)32 b(\003)3259 1313 y Fl(0)3319 1301 y Fo(+)21 b Fp(j)705 1419 y Ft(centred)c(at)h(the)g(sites)h(of)f(the)g(lattice)g Fn(Z)1820 1389 y Fi(d)1878 1419 y Ft(such)f(that)h(the)g(random)e(v)n (ariables)2900 1352 y Fg(\000)2938 1419 y Fp(\026)p Fo(\(\003)3078 1389 y Fl(\()p Fi(j)s Fl(\))3165 1352 y Fg(\001)3203 1452 y Fi(j)s Fj(2)p Fi(J)3343 1419 y Ft(are)705 1546 y(stochastically)f(independent)d(for)j(an)o(y)f(\002nite)i(collection)e Fp(J)31 b Fm(\032)23 b Fn(Z)2595 1515 y Fi(d)2650 1546 y Ft(of)15 b(Borel)g(sets)i Fo(\003)3134 1515 y Fl(\()p Fi(j)s Fl(\))3243 1546 y Fm(\032)23 b Fo(\003)3389 1558 y Fi(j)3424 1546 y Ft(.)497 1707 y(\(iii\))83 b(the)20 b(intensity)g(measure)p 1431 1661 51 4 v 19 w Fp(\026)j Fo(:)g Fm(B)s Fo(\()p Fn(R)1710 1677 y Fi(d)1748 1707 y Fo(\))g Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)p Ft(,)20 b(which)g(is)h(gi)n(v)o(en)e(by)p 1644 1808 V 1644 1854 a Fp(\026)p Fo(\(\003\))k(:=)g Fn(E)2015 1787 y Fg(\002)2049 1854 y Fp(\026)p Fo(\(\003\))2221 1787 y Fg(\003)3348 1854 y Ft(\(9\))705 2006 y(in)h(terms)g(of)g(the)g(probabilistic)f(e)o (xpectation)f Fn(E)p Fo([)p Fm(\001)p Fo(])30 b(:=)2351 1939 y Fg(R)2390 2035 y Fl(\012)2442 2006 y Fo(\()p Fm(\001)p Fo(\))14 b Fn(P)p Fo(\()p Fp(d!)s Fo(\))p Ft(,)26 b(is)f(a)g(Borel)f (measure)705 2105 y(which)19 b(does)h(not)g(v)n(anish)g(identically)p 1835 2060 V 19 w Fp(\026)j Fm(6)p Fo(=)g(0)p Ft(.)504 2264 y(\(i)n(v\))82 b(there)29 b(is)i(some)f(constant)f Fp(\024)41 b(>)g Fo(0)30 b Ft(such)g(that)g Fn(P)14 b Fm(f)o Fp(!)26 b Fm(2)e Fo(\012)f(:)36 b Fp(\026)2574 2276 y Fi(!)2623 2264 y Fo(\(\003)2713 2276 y Fl(0)2750 2264 y Fo(\))23 b Fm(2)h Fo([0)p Fp(;)14 b(")p Fo([)p Fm(g)39 b(\025)i Fp(")3274 2234 y Fi(\024)3348 2264 y Ft(for)705 2364 y(small)20 b(enough)f Fp(")j(>)h Fo(0)p Ft(.)605 2520 y Fq(Remark)h(2.2.)42 b Ft(Assumption)22 b(2.1\(i\))g(implies)i(that)f(the)h(intensity)f(measure)p 2873 2474 V 23 w Fp(\026)h Ft(is)g Fn(Z)3085 2490 y Fi(d)3124 2520 y Ft(-periodic.)456 2620 y(Assumption)19 b(2.1\(iii\))g(is)i(thus) f(equi)n(v)n(alent)f(to)h(the)g(e)o(xistence)g(of)g(the)g(\002rst)h (moment)e Fn(E)14 b Fo([)p Fp(\026)p Fo(\(\003)3158 2632 y Fl(0)3195 2620 y Fo(\)])24 b Fp(<)e Fm(1)456 2719 y Ft(of)f(the)h(random)e(v)n(ariable)h Fp(\026)p Fo(\(\003)1375 2731 y Fl(0)1412 2719 y Fo(\))26 b(:)g Fp(!)j Fm(7!)d Fp(\026)1759 2731 y Fi(!)1807 2719 y Fo(\(\003)1897 2731 y Fl(0)1934 2719 y Fo(\))p Ft(.)31 b(Moreo)o(v)o(er)m(,)19 b(we)j(emphasis)f(that)h(the)g(unit)f(cubes)456 2819 y Fo(\(\003)546 2831 y Fi(j)581 2819 y Fo(\))g Ft(introduced)c(in)k (Assumption)e(2.1\(ii\))g(are)h(neither)f(open)g(nor)h(closed.)605 2979 y(W)-7 b(e)22 b(recall)f(from)f([)p Fq(Kal83)n(,)h(SKM87)o(,)g(D)m (VJ88)n Ft(])g(that)g Fn(Z)2215 2948 y Fi(d)2254 2979 y Ft(-stationarity)f(of)h Fp(\026)g Ft(requires)f(the)h(group)456 3078 y Fo(\()p Fp(T)537 3090 y Fi(j)571 3078 y Fo(\))603 3095 y Fi(j)s Fj(2)p Fe(Z)724 3078 y Fk(d)29 b Ft(of)19 b(lattice)g(translations,)g(which)f(is)i(de\002ned)e(on)h Fm(M)p Fo(\()p Fn(R)2385 3048 y Fi(d)2423 3078 y Fo(\))h Ft(by)e Fo(\()p Fp(T)2658 3090 y Fi(j)2693 3078 y Fp(\027)5 b Fo(\)\(\003\))24 b(:=)f Fp(\027)5 b Fo(\(\003)14 b(+)g Fp(j)5 b Fo(\))20 b Ft(for)456 3185 y(all)g Fo(\003)j Fm(2)h(B)s Fo(\()p Fn(R)879 3155 y Fi(d)916 3185 y Fo(\))d Ft(and)f(all)h Fp(j)28 b Fm(2)23 b Fn(Z)1413 3155 y Fi(d)1452 3185 y Ft(,)e(to)f(be)g(probability)e(preserving)g(in)j(the)f(sense)g (that)1602 3332 y Fm(P)g(f)p Fp(T)1771 3344 y Fi(j)1805 3332 y Fp(M)9 b Fm(g)22 b Fo(=)h Fm(P)e(f)o Fp(M)9 b Fm(g)1007 b Ft(\(10\))456 3482 y(for)27 b(all)h Fp(M)46 b Fm(2)38 b(B)s Fo(\()p Fm(M)p Fo(\))27 b Ft(and)h(all)g Fp(j)42 b Fm(2)c Fn(Z)1649 3452 y Fi(d)1688 3482 y Ft(.)48 b(Here)28 b(we)g(ha)n(v)o(e)g(introduced)d(the)j(notation)e Fm(P)21 b(f)o Fp(M)9 b Fm(g)37 b Fo(:=)456 3582 y Fn(P)14 b Fm(f)o Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\026)939 3594 y Fi(!)1010 3582 y Fm(2)23 b Fp(M)9 b Fm(g)23 b Ft(for)f(the)h(induced)e(probability)g(measure)h(on)2568 3514 y Fg(\000)2606 3582 y Fm(M)p Fo(\()p Fn(R)2808 3551 y Fi(d)2847 3582 y Fo(\))p Fp(;)14 b Fm(B)s Fo(\()p Fm(M)p Fo(\))3138 3514 y Fg(\001)3175 3582 y Ft(.)33 b(T)-7 b(o)23 b(en-)456 3689 y(sure)30 b(the)g(\()p Fn(Z)844 3659 y Fi(d)883 3689 y Ft(-\)er)o(godicity)d(of)j(the)g(random)e (potential)i Fp(V)19 b Ft(,)33 b(it)d(is)i(useful)d(to)i(kno)n(w)e (that)h(under)f(the)456 3789 y(assumptions)17 b(made)g(abo)o(v)o(e,)g Fo(\()p Fp(T)1400 3801 y Fi(j)1434 3789 y Fo(\))i Ft(is)g(a)f(group)f (of)g(mixing)g(\(hence)g(er)o(godic\))f(transformations)f(on)j(the)456 3892 y(probability)g(space)1046 3824 y Fg(\000)1084 3892 y Fm(M)p Fo(\()p Fn(R)1286 3862 y Fi(d)1324 3892 y Fo(\))p Fp(;)c Fm(B)s Fo(\()p Fm(M)p Fo(\))p Fp(;)g Fm(P)1717 3824 y Fg(\001)1754 3892 y Ft(.)605 4048 y Fq(Lemma)21 b(2.3.)40 b Ff(Assumption)19 b(2.1\(i\))g(and)g(2.1\(ii\))g(imply)i (that)e Fp(\026)i Ff(is)g(mixing)f(in)g(the)h(sense)f(that)1291 4195 y Fo(lim)1247 4253 y Fj(j)p Fi(j)s Fj(j!1)1463 4195 y Fm(P)h(f)o Fp(T)1632 4207 y Fi(j)1667 4195 y Fp(M)27 b Fm(\\)19 b Fp(M)1939 4161 y Fj(0)1962 4195 y Fm(g)j Fo(=)h Fm(P)d(f)p Fp(M)9 b Fm(g)k(P)20 b(f)p Fp(M)2589 4161 y Fj(0)2611 4195 y Fm(g)653 b Ft(\(11\))456 4394 y Ff(for)20 b(all)h Fp(M)t(;)14 b(M)894 4363 y Fj(0)939 4394 y Fm(2)24 b(B)s Fo(\()p Fm(M)p Fo(\))p Ff(.)607 4550 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(See)20 b(Appendix)e(A.)1904 b Fd(\003)605 4717 y Ft(The)35 b(considered)f(impurity)h(potentials)g Fp(f)60 b Fo(:)52 b Fn(R)2094 4687 y Fi(d)2184 4717 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)36 b Ft(comprise)f(a)h(lar)o(ge)f (class)h(of)456 4817 y(functions)22 b(with)j(anisotropic)d(decay)-5 b(.)36 b(More)23 b(precisely)-5 b(,)24 b(we)g(decompose)e(the)j (con\002guration)c(space)456 4917 y Fn(R)526 4887 y Fi(d)595 4917 y Fo(=)30 b Fn(R)760 4887 y Fi(d)795 4895 y Fv(1)852 4917 y Fm(\002)21 b(\001)14 b(\001)g(\001)21 b(\002)h Fn(R)1213 4887 y Fi(d)1248 4895 y Fk(m)1331 4917 y Ft(into)i Fp(m)30 b Fm(2)h Fn(N)25 b Ft(subspaces)f(with)g(dimensions)f Fp(d)2745 4929 y Fl(1)2783 4917 y Fp(;)14 b(:)g(:)g(:)g(;)g(d)3011 4929 y Fi(m)3104 4917 y Fm(2)31 b Fn(N)p Ft(.)38 b(Ac-)456 5016 y(cordingly)-5 b(,)20 b(we)j(will)g(write)f Fp(x)28 b Fo(=)f(\()p Fp(x)1524 5028 y Fl(1)1562 5016 y Fp(;)14 b(:)g(:)g(:)f(;)h(x)1793 5028 y Fi(m)1857 5016 y Fo(\))27 b Fm(2)h Fn(R)2069 4986 y Fi(d)2107 5016 y Ft(,)c(where)d Fp(x)2424 5028 y Fi(k)2493 5016 y Fm(2)28 b Fn(R)2646 4986 y Fi(d)2681 4995 y Fk(k)2743 5016 y Ft(and)22 b Fp(k)30 b Fm(2)e(f)p Fo(1)p Fp(;)14 b(:)g(:)g(:)e(;)i(m)p Fm(g)p Ft(.)456 5116 y(Denoting)24 b(by)h Fm(j)p Fp(x)970 5128 y Fi(k)1012 5116 y Fm(j)33 b Fo(:=)g(max)1344 5131 y Fi(i)p Fj(2f)p Fl(1)p Fi(;:::)o(;d)1613 5140 y Fk(k)1649 5131 y Fj(g)1701 5116 y Fm(j)p Fo(\()p Fp(x)1803 5128 y Fi(k)1844 5116 y Fo(\))1876 5128 y Fi(i)1904 5116 y Fm(j)27 b Ft(the)f(maximum)e(norm)g(on)i Fn(R)2826 5086 y Fi(d)2861 5095 y Fk(k)2900 5116 y Ft(,)i(our)d(precise)h(as-)456 5216 y(sumptions)19 b(on)h Fp(f)29 b Ft(are)20 b(as)h(follo)n(ws.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)771 b(5)605 450 y Fq(Assumption)20 b(2.4.)38 b Ft(The)18 b(impurity)g(potential)g Fp(f)31 b Fo(:)23 b Fn(R)2165 420 y Fi(d)2227 450 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)19 b Ft(is)h(positi)n(v)o(e,)e(strictly)h(positi)n(v)o(e)456 550 y(on)g(some)h(non-empty)e(open)h(set)i(and)f(satis\002es:)543 707 y(\(i\))83 b(the)20 b(Birman-Solomyak)d(condition)1796 645 y Fg(P)1884 732 y Fi(j)s Fj(2)p Fe(Z)2005 715 y Fk(d)2058 640 y Fg(\000)d(R)2149 737 y Fl(\003)2194 745 y Fv(0)2245 707 y Fm(j)p Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(j)5 b Fo(\))p Fm(j)2593 677 y Fi(p)2632 707 y Fp(dx)2722 640 y Fg(\001)2761 657 y Fl(1)p Fi(=p)2889 707 y Fp(<)23 b Fm(1)e Ft(with)g Fp(p)i Fo(=)g(2)705 815 y Ft(if)d Fp(d)j Fm(2)h(f)p Fo(1)p Fp(;)14 b Fo(2)p Fp(;)g Fo(3)p Fm(g)k Ft(and)i Fp(p)i(>)h(d=)p Fo(2)d Ft(if)h Fp(d)i Fm(\025)f Fo(4)p Ft(.)520 978 y(\(ii\))83 b(there)19 b(e)o(xist)i(constants)e Fp(\013)1450 990 y Fl(1)1488 978 y Fp(;)14 b(:)g(:)g(:)f(;)h(\013)1725 990 y Fi(m)1812 978 y Fm(2)23 b Fo([0)p Fp(;)14 b Fm(1)p Fo(])20 b Ft(and)g Fo(0)i Fp(<)h(f)2452 990 y Fi(u)2495 978 y Ft(,)e Fp(f)2578 990 y Fl(0)2638 978 y Fp(<)h Fm(1)f Ft(such)f(that)1115 1129 y Fp(f)1156 1141 y Fi(u)p 935 1166 445 4 v 935 1181 a Fg(P)1022 1201 y Fi(m)1022 1268 y(k)q Fl(=1)1161 1243 y Fm(j)p Fp(x)1231 1255 y Fi(k)1273 1243 y Fm(j)1296 1219 y Fi(\013)1339 1228 y Fk(k)1412 1185 y Fm(\024)1500 1072 y Fg(Z)1546 1261 y Fl(\003)1591 1269 y Fv(0)1628 1185 y Fp(f)9 b Fo(\()p Fp(y)21 b Fm(\000)d Fp(x)p Fo(\))c Fp(dy)s(;)180 b(f)9 b Fo(\()p Fp(x)p Fo(\))24 b Fm(\024)2704 1129 y Fp(f)2745 1141 y Fl(0)p 2521 1166 V 2521 1181 a Fg(P)2608 1201 y Fi(m)2608 1268 y(k)q Fl(=1)2747 1243 y Fm(j)p Fp(x)2817 1255 y Fi(k)2858 1243 y Fm(j)2881 1219 y Fi(\013)2924 1228 y Fk(k)3306 1185 y Ft(\(12\))705 1413 y(for)e(all)h Fp(x)k Fo(=)g(\()p Fp(x)1176 1425 y Fl(1)1214 1413 y Fp(;)14 b(:)g(:)g(:)g(;)g(x)1446 1425 y Fi(m)1509 1413 y Fo(\))28 b Fm(2)g Fn(R)1722 1383 y Fi(d)1783 1413 y Ft(with)23 b(lar)o(ge)e(enough)g(v)n(alues)h(of)g (their)g(maximum)f(norm)705 1512 y Fm(j)p Fp(x)p Fm(j)i Fo(=)g(max)o Fm(fj)p Fp(x)1175 1524 y Fl(1)1213 1512 y Fm(j)p Fp(;)14 b(:)g(:)g(:)f(;)h Fm(j)p Fp(x)1490 1524 y Fi(m)1554 1512 y Fm(jg)p Ft(.)605 1679 y Fq(Remark)30 b(2.5.)47 b Ft(In)30 b(order)f(to)h(simultaneously)e(treat)j(the)f (case)g Fp(\013)2570 1691 y Fi(k)2653 1679 y Fo(=)41 b Fm(1)31 b Ft(for)e(some)h(\(or)g(all\))456 1778 y Fp(k)g Fm(2)d(f)p Fo(1)p Fp(;)14 b(:)g(:)g(:)f(;)h(m)p Fm(g)p Ft(,)22 b(we)h(adopt)e(the)i(con)m(v)o(entions)c Fm(j)p Fp(x)1978 1790 y Fi(k)2019 1778 y Fm(j)2042 1748 y Fj(1)2140 1778 y Fo(:=)27 b Fm(1)c Ft(for)e Fm(j)p Fp(x)2550 1790 y Fi(k)2592 1778 y Fm(j)27 b Fp(>)f Fo(0)d Ft(and)f Fo(1)p Fp(=)p Fm(1)k Fo(:=)g(0)p Ft(.)32 b(An)456 1878 y(e)o(xample)18 b(for)i(such)g(a)g(situation)g(is)h(gi)n(v)o(en)e(by)h Fp(f)29 b Ft(with)21 b(compact)e(support)g(in)h(the)g Fp(x)2858 1890 y Fi(k)2899 1878 y Ft(-direction.)605 2044 y Fq(2.2.)40 b(Examples.)h Ft(The)33 b(setting)h(in)g(Subsection)e (2.1)h(co)o(v)o(ers)f(a)i(huge)e(class)j(of)e(random)f(po-)456 2144 y(tentials)26 b(which)g(are)g(widely)g(encountered)e(in)i(the)h (literature)e(on)h(random)f(Schr)7 b(\250)-35 b(odinger)23 b(operators)456 2244 y([)p Fq(Kir89)n(,)30 b(CL90)o(,)g(PF92)o(,)g (Sto01)n Ft(].)54 b(In)30 b(this)g(Subsection)f(we)h(list)h(prominent)d (e)o(xamples,)j(some)e(of)456 2343 y(which)19 b(ha)n(v)o(e)h(already)f (been)g(\(informally\))f(introduced)g(in)i(the)g(Introduction.)605 2542 y(From)26 b(the)h(physical)f(point)g(of)h(vie)n(w)-5 b(,)28 b(it)g(natural)e(to)h(consider)f(inte)o(ger)n(-v)n(alued)e (random)h(Borel)456 2642 y(measures)k Fp(\027)45 b Fo(=)986 2580 y Fg(P)1073 2667 y Fi(j)1122 2642 y Fp(k)1165 2654 y Fi(j)1200 2642 y Fp(\016)1237 2654 y Fi(x)1275 2662 y Fk(j)1310 2642 y Ft(,)32 b(also)e(kno)n(wn)e(as)i(point)e(processes)i ([)p Fq(D)m(VJ88)m Ft(].)53 b(Here)29 b(each)g Fp(k)3215 2654 y Fi(j)3281 2642 y Ft(is)h(an)456 2749 y(inte)o(ger)n(-v)n(alued) 25 b(random)i(v)n(ariable)h(and)g(the)h(distinct)g(points)f Fo(\()p Fp(x)2402 2761 y Fi(j)2438 2749 y Fo(\))i Ft(inde)o(xing)c(the) j(atoms,)i(equi)n(v)n(a-)456 2848 y(lently)19 b(the)g(Dirac)g(measure)g Fp(\016)s Ft(,)h(form)e(a)i(countable)e(\(random\))f(set)j(with)g(at)g (most)f(\002nitely)g(man)o(y)f Fp(x)3324 2860 y Fi(j)3380 2848 y Ft(in)456 2948 y(an)o(y)g(bounded)g(Borel)h(set.)26 b(In)19 b(f)o(act,)h(interpreting)e Fo(\()p Fp(x)1985 2960 y Fi(j)2020 2948 y Fo(\))j Ft(as)f(the)g(\(random\))d(positions)i (of)h(impurities)f(in)456 3048 y(a)h(disordered)e(solid)j(justi\002es)g (the)f(name)g('impurity)e(potential')h(for)g Fp(f)30 b Ft(in)20 b(\(8\).)605 3147 y(T)-7 b(w)o(o)21 b(e)o(xamples)e(of)h (point)f(processes)h(satisfying)f(Assumptions)h(2.1\(i\))n (\2262.1\(iii\))f(are:)562 3284 y(\(P\))82 b(the)29 b Ff(g)o(ener)o(alised)e(P)-7 b(oisson)29 b(measur)m(e)f Fp(\027)45 b Fo(=)2073 3222 y Fg(P)2161 3309 y Fi(j)2209 3284 y Fp(\016)2246 3296 y Fi(\030)2276 3304 y Fk(j)2341 3284 y Ft(with)29 b(some)f(non-zero)f Fn(Z)3106 3254 y Fi(d)3145 3284 y Ft(-periodic)746 3385 y(Borel)32 b(intensity)g (measure)p 1591 3340 47 4 v 31 w Fp(\027)5 b Ft(.)61 b(The)31 b(Poisson)h(measure)f(is)i(uniquely)d(characterised)g(by)746 3485 y(requiring)21 b(that)h(the)h(random)e(v)n(ariables)g Fp(\027)5 b Fo(\(\003)2079 3455 y Fl(\(1\))2169 3485 y Fo(\))p Fp(;)14 b(:)g(:)g(:)g(;)g(\027)5 b Fo(\(\003)2522 3455 y Fl(\()p Fi(n)p Fl(\))2619 3485 y Fo(\))23 b Ft(are)g (stochastically)f(inde-)746 3588 y(pendent)f(for)h(an)o(y)g(collection) f(of)i(disjoint)f(Borel)h(sets)g Fo(\003)2422 3558 y Fl(\(1\))2511 3588 y Fp(;)14 b(:)g(:)g(:)g Fo(\003)2717 3558 y Fl(\()p Fi(n)p Fl(\))2841 3588 y Fm(2)28 b(B)s Fo(\()p Fn(R)3084 3558 y Fi(d)3121 3588 y Fo(\))c Ft(and)e(that)746 3688 y(each)e Fp(\027)5 b Fo(\(\003\))21 b Ft(is)g(distrib)n(uted)f (according)e(to)i(Poisson')-5 b(s)21 b(la)o(w)886 3933 y Fn(P)951 3866 y Fg(\010)1000 3933 y Fp(!)k Fm(2)f Fo(\012)37 b(:)g Fp(\027)1354 3945 y Fi(!)1402 3933 y Fo(\(\003\))23 b(=)g Fp(k)1681 3866 y Fg(\011)1752 3933 y Fo(=)1849 3804 y Fg(\000)p 1888 3826 V 1888 3872 a Fp(\027)5 b Fo(\(\003\))2056 3804 y Fg(\001)2094 3822 y Fi(k)p 1849 3914 286 4 v 1958 3990 a Fp(k)s Fo(!)2159 3933 y(exp)2299 3866 y Fg(\002)2352 3933 y Fm(\000)p 2435 3887 47 4 v 18 w Fp(\027)h Fo(\(\003\))2604 3866 y Fg(\003)2639 3933 y Fp(;)97 b(k)25 b Fm(2)f Fn(N)2977 3945 y Fl(0)3306 3933 y Ft(\(13\))746 4133 y(for)j(an)o(y)f(bounded)f Fo(\003)35 b Fm(2)h(B)s Fo(\()p Fn(R)1675 4103 y Fi(d)1713 4133 y Fo(\))p Ft(.)46 b(The)27 b(case)p 2140 4088 V 28 w Fp(\027)5 b Fo(\(\003\))36 b(=)g Fp(\045)p Fm(j)p Fo(\003)p Fm(j)27 b Ft(corresponds)e(to)i(the)g(usual)746 4233 y Ff(P)-7 b(oisson)20 b(pr)l(ocess)h Ft(with)f(parameter)f Fp(\045)k(>)g Fo(0)p Ft(.)548 4370 y(\(D\))82 b(the)19 b Ff(displacement)f(measur)m(e)h Fp(\027)28 b Fo(=)1780 4308 y Fg(P)1868 4395 y Fi(j)s Fj(2)p Fe(Z)1989 4378 y Fk(d)2042 4370 y Fp(\016)2079 4382 y Fi(j)s Fl(+)p Fi(d)2195 4390 y Fk(j)2231 4370 y Ft(.)d(Here)19 b(the)g(random)e(v)n (ariables)i Fp(d)3213 4382 y Fi(j)3271 4370 y Fm(2)k Fo(\003)3407 4382 y Fl(0)746 4472 y Ft(are)i(independent)d(and)j (identically)f(distrib)n(uted)f(o)o(v)o(er)h(the)h(unit)f(cube.)39 b(The)24 b(case)i Fp(d)3240 4484 y Fi(j)3307 4472 y Fo(=)31 b(0)746 4572 y Ft(corresponds)18 b(to)i(the)h(\(non-random\))16 b Ff(periodic)j(point)g(measur)m(e)h Fp(\027)29 b Fo(=)2802 4510 y Fg(P)2890 4597 y Fi(j)s Fj(2)p Fe(Z)3011 4580 y Fk(d)3065 4572 y Fp(\016)3102 4584 y Fi(j)3137 4572 y Ft(.)605 4710 y(An)o(y)20 b(\(generalised\))f(Poisson)i(measure)g (\(P\))g(also)g(satis\002es)i(Assumption)d(2.1\(i)n(v\))m(.)29 b(It)21 b(gi)n(v)o(es)g(rise)456 4810 y(to)26 b(the)g(\(generalized\))e (Poissonian)h(random)g(potential)g(\(3\).)42 b(Unfortunately)-5 b(,)25 b(Assumption)g(2.1\(i)n(v\))456 4910 y(is)c(ne)n(v)o(er)f (satis\002ed)h(for)g(an)o(y)f(displacement)f(measure)h(\(D\).)g(Ho)n (we)n(v)o(er)m(,)f(a)j(corresponding)17 b(compound)456 5009 y(point)k(process)g Fp(\027)31 b Fo(=)1082 4947 y Fg(P)1170 5034 y Fi(j)s Fj(2)p Fe(Z)1291 5017 y Fk(d)1345 5009 y Fp(q)1382 5021 y Fi(j)1417 5009 y Fp(\016)1454 5021 y Fi(x)1492 5029 y Fk(j)1549 5009 y Ft(will)22 b(satisfy)g (Assumption)f(2.1\(i)n(v\))f(under)g(suitable)h(conditions)456 5116 y(on)27 b(the)g(random)f(v)n(ariables)h Fo(\()p Fp(q)1373 5128 y Fi(j)1408 5116 y Fo(\))p Ft(.)48 b(In)28 b(order)e(to)i(satisfy)g(Assumption)e(2.1\(iii\))o(,)j(we)f(tak)o(e)g Fo(\()p Fp(q)3216 5128 y Fi(j)3251 5116 y Fo(\))3283 5133 y Fi(j)s Fj(2)p Fe(Z)3404 5116 y Fk(d)456 5216 y Ft(independent)17 b(and)j(identically)f(distrib)n(uted,)g(positi)n(v)o (e)g(random)g(v)n(ariables)g(with)i Fo(0)h Fp(<)h Fn(E)p Fo([)p Fp(q)3108 5228 y Fl(0)3145 5216 y Fo(])g Fp(<)g Fm(1)p Ft(.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 456 251 a Fr(6)915 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)605 450 y Ft(T)g(w)o(o)34 b(e)o(xamples)e(of)h(such)h(compound)c(point)j(processes,)j(for)d (which)g(Assumptions)g(2.1\(i\))n(\226)456 550 y(2.1\(i)n(v\))18 b(hold,)h(are:)534 687 y(\(P'\))82 b(the)20 b Ff(compound)e(\(g)o(ener) o(alised\))g(P)-7 b(oisson)20 b(measur)m(e)g Fp(\027)29 b Fo(=)2429 624 y Fg(P)2517 712 y Fi(j)2566 687 y Fp(q)2603 699 y Fi(j)2638 687 y Fp(\016)2675 699 y Fi(\030)2705 707 y Fk(j)2761 687 y Ft(with)20 b Fo(\()p Fp(\030)2997 699 y Fi(j)3033 687 y Fo(\))h Ft(as)g(in)f(\(P\).)520 831 y(\(D'\))82 b(the)23 b Ff(compound)d(displacement)h(measur)m(e)h Fp(\027)33 b Fo(=)2168 769 y Fg(P)2255 856 y Fi(j)s Fj(2)p Fe(Z)2376 839 y Fk(d)2430 831 y Fp(q)2467 843 y Fi(j)2502 831 y Fp(\016)2539 843 y Fi(j)s Fl(+)p Fi(d)2655 851 y Fk(j)2714 831 y Ft(with)22 b Fp(d)2927 843 y Fi(j)2986 831 y Ft(as)h(in)f(\(D\).)h(As-)746 946 y(sumption)17 b(2.1\(i)n(v\))g(requires)g Fn(P)1666 879 y Fg(\010)1714 946 y Fp(!)26 b Fm(2)d Fo(\012)g(:)37 b Fp(q)2050 958 y Fi(!)r(;)p Fl(0)2174 946 y Fm(2)24 b Fo([0)p Fp(;)14 b(")p Fo([)2417 879 y Fg(\011)2488 946 y Fm(\025)22 b Fp(")2614 916 y Fi(\024)2676 946 y Ft(for)17 b(small)i(enough)d Fp(")23 b(>)g Fo(0)746 1046 y Ft(and)e(some)g Fp(\024)j(>)g Fo(0)p Ft(.)k(The)21 b(case)g Fp(d)1695 1058 y Fi(j)1755 1046 y Fo(=)k(0)c Ft(gi)n(v)o(es)f(the)h Ff(alloy-type)f(measur)m(e)h Fp(\027)30 b Fo(=)3038 983 y Fg(P)3126 1070 y Fi(j)s Fj(2)p Fe(Z)3247 1054 y Fk(d)3300 1046 y Fp(q)3337 1058 y Fi(j)3373 1046 y Fp(\016)3410 1058 y Fi(j)746 1147 y Ft(associated)20 b(with)h(the)f(allo)o(y-type)e(random)h(potential)g (\(6\).)605 1318 y Fq(Remark)31 b(2.6.)48 b Ft(W)-7 b(e)32 b(note)f(that)g(in)g(case)h(\(P'\))f(there)g(are)g(no)f(further)g (requirements)f(on)i Fo(\()p Fp(q)3356 1330 y Fi(j)3391 1318 y Fo(\))p Ft(.)456 1417 y(Moreo)o(v)o(er)m(,)h(our)g(results)h(in) g(Subsection)e(2.4)h(belo)n(w)g(also)h(apply)f(to)g(allo)o(y-type)f (random)g(poten-)456 1517 y(tials)24 b(\(6\))g(with)f(bounded)f(belo)n (w)h(random)f(v)n(ariables)h Fo(\()p Fp(q)2116 1529 y Fi(j)2151 1517 y Fo(\))p Ft(,)j(not)d(only)g(positi)n(v)o(e)g(ones.)35 b(This)24 b(follo)n(ws)456 1617 y(from)k(the)h(f)o(act)g(that)g(one)g (may)g(add)f Fp(x)40 b Fm(7!)1767 1554 y Fg(P)1854 1642 y Fi(j)s Fj(2)p Fe(Z)1975 1625 y Fk(d)2029 1617 y Fp(q)2066 1629 y Fl(min)2180 1617 y Fp(f)9 b Fo(\()p Fp(x)26 b Fm(\000)e Fp(j)5 b Fo(\))30 b Ft(to)g(the)f(periodic)e(background)456 1718 y(potential)19 b Fp(U)824 1730 y Fl(p)r(er)942 1718 y Ft(\(confer)g(\(14\))g(and)h(Assumption)f(2.7)g(belo)n(w\).)605 1889 y Fq(2.3.)40 b(Random)32 b(Sch)1244 1888 y(\250)1237 1889 y(odinger)g(operators)f(and)h(their)g(integrated)f(density)h(of)g (states.)40 b Ft(F)o(or)456 1989 y(an)o(y)19 b(of)g(the)h(abo)o(v)o(e)f (de\002ned)g(random)f(potentials)h Fp(V)g Ft(,)i(we)f(study)f(the)h (corresponding)d(random)h(Schr)7 b(\250)-35 b(o-)456 2088 y(dinger)18 b(operator)m(,)g(which)i(is)h(informally)d(gi)n(v)o (en)h(by)h(the)g(second)g(order)e(dif)n(ferential)h(operator)1471 2260 y Fp(H)7 b Fo(\()p Fp(V)1627 2272 y Fi(!)1675 2260 y Fo(\))24 b(:=)e Fm(\000)p Fo(\001)d(+)f Fp(U)2134 2272 y Fl(p)r(er)2250 2260 y Fo(+)g Fp(V)2381 2272 y Fi(!)3306 2260 y Ft(\(14\))456 2436 y(on)c(the)i(Hilbert)f(space)g Fo(L)1175 2406 y Fl(2)1212 2436 y Fo(\()p Fn(R)1314 2406 y Fi(d)1353 2436 y Fo(\))h Ft(of)f(comple)o(x-v)n(alued,)d(square-inte) o(grable)g(functions)h(on)i Fn(R)3139 2406 y Fi(d)3177 2436 y Ft(.)24 b(Thereby)456 2535 y(the)g(periodic)e(background)f (potential)i Fp(U)1667 2547 y Fl(p)r(er)1790 2535 y Ft(\(acting)g(in)h (\(14\))f(as)i(a)g(multiplication)d(operator\))g(is)j(re-)456 2635 y(quired)18 b(to)j(satisfy)f(the)h(follo)n(wing)605 2806 y Fq(Assumption)34 b(2.7.)48 b Ft(The)32 b(background)e(potential) h Fp(U)2205 2818 y Fl(p)r(er)2349 2806 y Fo(:)46 b Fn(R)2488 2776 y Fi(d)2573 2806 y Fm(!)g Fn(R)33 b Ft(is)h Fn(Z)2953 2776 y Fi(d)2992 2806 y Ft(-periodic)d(and)456 2909 y Fp(U)513 2921 y Fl(p)r(er)633 2909 y Fm(2)24 b Fo(L)764 2869 y Fi(p)764 2934 y Fl(lo)r(c)865 2841 y Fg(\000)903 2909 y Fn(R)973 2878 y Fi(d)1011 2841 y Fg(\001)1070 2909 y Ft(for)c(some)g Fp(p)j(>)f(d)p Ft(.)605 3088 y(Assumptions)i (2.1)h(and)g(2.4)f(particularly)g(imply)h([)p Fq(CL90)o Ft(,)h(Cor)-5 b(.)26 b(V)-11 b(.3.4])24 b(that)i Fp(V)2964 3100 y Fi(!)3044 3088 y Fm(2)33 b Fo(L)3184 3048 y Fi(p)3184 3113 y Fl(lo)r(c)3271 3088 y Fo(\()p Fn(R)3373 3058 y Fi(d)3412 3088 y Fo(\))456 3187 y Ft(for)20 b Fn(P)p Ft(-almost)h(all)h Fp(!)27 b Fm(2)f Fo(\012)c Ft(with)f(the)g(same)h Fp(p)f Ft(as)h(in)f(Assumption)g(2.4\(ii\))n(.)28 b(T)-7 b(ogether)20 b(with)i(Assump-)456 3287 y(tion)c(2.7)h(this)g(ensures)f ([)p Fq(KM83b)o Ft(])h(that)g Fp(H)7 b Fo(\()p Fp(V)1782 3299 y Fi(!)1831 3287 y Fo(\))20 b Ft(is)g(essentially)f(self-adjoint)e (on)i(the)g(space)g Fm(C)3202 3257 y Fj(1)3197 3308 y Fi(c)3271 3287 y Fo(\()p Fn(R)3373 3257 y Fi(d)3412 3287 y Fo(\))456 3387 y Ft(of)k(comple)o(x-v)n(alued,)d(arbitrarily)j(often) f(dif)n(ferentiable)g(functions)g(with)i(compact)e(support)h(for)g Fn(P)p Ft(-)456 3486 y(almost)k(all)h Fp(!)39 b Fm(2)d Fo(\012)p Ft(.)47 b(Since)28 b Fp(V)47 b Ft(is)28 b Fn(Z)1576 3456 y Fi(d)1615 3486 y Ft(-er)o(godic)d(\(confer)g(Lemma)i(2.3\),)h (the)f(spectrum)g(of)g Fp(H)7 b Fo(\()p Fp(V)3364 3498 y Fi(!)3412 3486 y Fo(\))456 3586 y Ft(coincides)19 b(with)h(a)h (non-random)16 b(set)21 b(for)f Fn(P)p Ft(-almost)g(all)h Fp(!)k Fm(2)f Fo(\012)c Ft([)p Fq(KM82)o Ft(,)h(Thm.)e(1].)605 3785 y(F)o(or)h(an)o(y)g Fp(d)p Ft(-dimensional)g(open)f(cuboid)h Fo(\003)j Fm(\032)h Fn(R)2049 3755 y Fi(d)2088 3785 y Ft(,)d(the)g(restriction)f(of)g(\(14\))g(to)h Fm(C)2996 3755 y Fj(1)2991 3806 y Fi(c)3066 3785 y Fo(\(\003\))g Ft(de\002nes)456 3885 y(a)c(self-adjoint)f(operator)f Fp(H)1276 3855 y Fi(D)1269 3908 y Fl(\003)1336 3885 y Fo(\()p Fp(V)1416 3897 y Fi(!)1465 3885 y Fo(\))j Ft(on)f Fp(L)1673 3855 y Fl(2)1710 3885 y Fo(\(\003\))p Ft(,)h(which)f (corresponds)d(to)k(taking)e(Dirichlet)h(boundary)456 3984 y(conditions)h([)p Fq(RS78)n Ft(].)25 b(It)20 b(is)g(bounded)d (belo)n(w)i(and)g(has)h(purely)e(discrete)h(spectrum)f(with)i(eigen)m (v)n(alues)456 4084 y Fp(\025)504 4096 y Fl(0)541 4084 y Fo(\()p Fp(H)649 4054 y Fi(D)642 4107 y Fl(\003)710 4084 y Fo(\()p Fp(V)790 4096 y Fi(!)838 4084 y Fo(\))38 b Fp(<)f(\025)1058 4096 y Fl(1)1095 4084 y Fo(\()p Fp(H)1203 4054 y Fi(D)1196 4107 y Fl(\003)1263 4084 y Fo(\()p Fp(V)1343 4096 y Fi(!)1392 4084 y Fo(\))h Fm(\024)f Fp(\025)1612 4096 y Fl(2)1649 4084 y Fo(\()p Fp(H)1757 4054 y Fi(D)1750 4107 y Fl(\003)1817 4084 y Fo(\()p Fp(V)1897 4096 y Fi(!)1946 4084 y Fo(\))h Fm(\024)e Fp(:)14 b(:)g(:)29 b Ft(ordered)d(by)h (magnitude)f(and)h(repeated)456 4184 y(according)h(to)i(their)g (multiplicity)-5 b(.)55 b(Our)30 b(main)g(quantity)f(of)h(interest,)i (the)f(inte)o(grated)d(density)i(of)456 4283 y(states,)21 b(is)g(then)e(de\002ned)h(as)h(the)f(in\002nite-v)n(olume)e(limit)999 4486 y Fp(N)9 b Fo(\()p Fp(E)c Fo(\))23 b(:=)74 b(lim)1339 4543 y Fj(j)p Fl(\003)p Fj(j!1)1611 4430 y Fo(1)p 1580 4467 104 4 v 1580 4543 a Fm(j)p Fo(\003)p Fm(j)1708 4486 y Fo(#)1777 4394 y Fg(n)1832 4486 y Fp(n)23 b Fm(2)g Fn(N)2054 4498 y Fl(0)2128 4486 y Fo(:)37 b Fp(\025)2236 4498 y Fi(n)2296 4418 y Fg(\000)2334 4486 y Fp(H)2410 4451 y Fi(D)2403 4506 y Fl(\003)2470 4486 y Fo(\()p Fp(V)2550 4498 y Fi(!)2599 4486 y Fo(\))2631 4418 y Fg(\001)2692 4486 y Fp(<)23 b(E)2846 4394 y Fg(o)3306 4486 y Ft(\(15\))456 4717 y(More)e(precisely)-5 b(,)21 b(thanks)g(to)h(the)g Fn(Z)1505 4687 y Fi(d)1544 4717 y Ft(-er)o(godicity)d(of)i(the)h (random)e(potential)h(there)h(is)g(a)h(set)f Fo(\012)3233 4729 y Fl(0)3297 4717 y Fm(2)k(A)456 4817 y Ft(of)g(full)h(probability) -5 b(,)26 b Fn(P)p Fo(\(\012)1261 4829 y Fl(0)1298 4817 y Fo(\))36 b(=)e(1)p Ft(,)29 b(and)d(a)h(non-random)c(unbounded)h (distrib)n(ution)h(function)g Fp(N)44 b Fo(:)456 4917 y Fn(R)31 b Fm(!)h Fo([0)p Fp(;)14 b Fm(1)p Fo([)25 b Ft(such)g(that)g(\(15\))e(holds)i(for)f(all)h Fp(!)35 b Fm(2)d Fo(\012)2066 4929 y Fl(0)2129 4917 y Ft(and)24 b(all)i(continuity)d(points)h Fp(E)37 b Fm(2)32 b Fn(R)25 b Ft(of)g Fp(N)9 b Ft(.)456 5016 y(The)22 b(set)h(of)f(gro)n(wth)g (points)g(of)g Fp(N)32 b Ft(coincides)22 b(with)g(the)h(almost-sure)e (spectrum)h(of)g Fp(H)7 b Fo(\()p Fp(V)3107 5028 y Fi(!)3156 5016 y Fo(\))p Ft(,)23 b(confer)456 5116 y([)p Fq(Kir89)n(,)d(CL90,)g (PF92)o Ft(].)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)771 b(7)605 450 y Fq(2.4.)40 b(Lifshits)26 b(tails.)41 b Ft(The)24 b(main)g(result)g(of)h(the)f(present)g(paper)f(generalises) h(the)h(result)f(\(7\))g(of)456 550 y([)p Fq(KS86)n(,)30 b(Mez87)o Ft(])f(on)g(the)g(Lifshits)h(e)o(xponent)c(for)j(allo)o (y-type)f(random)f(potentials)i(with)g(isotrop-)456 649 y(ically)h(decaying)f(impurity)g(potential)g Fp(f)40 b Ft(to)31 b(the)f(case)h(of)f(anisotropic)f(decay)h(and)f(more)h (general)456 749 y(random)22 b(potentials)i(\(8\).)36 b(W)-7 b(e)26 b(note)e(that)g(isotropic)g(decay)f(corresponds)f(to)i (taking)g Fp(m)30 b Fo(=)h(1)24 b Ft(in)h(As-)456 849 y(sumption)18 b(2.4)i(or)m(,)f(what)h(is)i(the)e(same,)g Fp(\013)j Fo(:=)g Fp(\013)1850 861 y Fi(k)1912 849 y Ft(for)c(all)i Fp(k)26 b Fm(2)d(f)p Fo(1)p Fp(;)14 b(:)g(:)g(:)f(;)h(m) p Fm(g)p Ft(.)605 1002 y Fq(Theor)o(em)j(2.8.)34 b Ff(Let)17 b Fp(H)7 b Fo(\()p Fp(V)1383 1014 y Fi(!)1432 1002 y Fo(\))17 b Ff(be)g(a)g(r)o(andom)e(Sc)o(hr)2069 1003 y(\250)2062 1002 y(oding)o(er)g(oper)o(ator)g(\(14\))h(with)h(r)o (andom)e(poten-)456 1102 y(tial)20 b(\(8\))f(satisfying)g(Assumptions)g (2.1)g(and)g(2.4,)g(and)g(a)g(periodic)g(bac)n(kgr)l(ound)f(potential)g (satisfying)456 1201 y(Assumption)k(2.7.)32 b(Then)23 b(its)h(inte)m(gr)o(ated)d(density)i(of)g(states)g Fp(N)33 b Ff(dr)l(ops)22 b(down)h(to)g(zer)l(o)h(e)n(xponentially)456 1301 y(near)c Fp(E)690 1313 y Fl(0)750 1301 y Fo(:=)j Ft(inf)d(spec)13 b Fp(H)7 b Fo(\(0\))21 b Ff(with)g(Lifshits)g(e)n (xponent)e(given)g(by)1043 1509 y Fp(\021)26 b Fo(:=)49 b(lim)1221 1563 y Fi(E)s Fj(#)p Fi(E)1356 1571 y Fv(0)1412 1453 y Fo(log)14 b Fm(j)g Fo(log)g Fp(N)9 b Fo(\()p Fp(E)c Fo(\))p Fm(j)p 1412 1490 509 4 v 1417 1566 a(j)14 b Fo(log\()p Fp(E)24 b Fm(\000)18 b Fp(E)1822 1578 y Fl(0)1860 1566 y Fo(\))p Fm(j)1953 1509 y Fo(=)2072 1405 y Fi(m)2042 1430 y Fg(X)2041 1609 y Fi(k)q Fl(=1)2176 1509 y Fo(max)2344 1392 y Fg(\032)2416 1453 y Fp(d)2459 1465 y Fi(k)p 2416 1490 85 4 v 2438 1566 a Fo(2)2510 1509 y Fp(;)2611 1453 y(\015)2654 1465 y Fi(k)p 2557 1490 191 4 v 2557 1566 a Fo(1)g Fm(\000)g Fp(\015)2758 1392 y Fg(\033)2834 1509 y Fp(;)449 b Ft(\(16\))456 1734 y Ff(wher)m(e)20 b Fp(\015)719 1746 y Fi(k)783 1734 y Fo(:=)j Fp(d)937 1746 y Fi(k)978 1734 y Fp(=\013)1073 1746 y Fi(k)1134 1734 y Ff(and)c Fp(\015)28 b Fo(:=)1461 1672 y Fg(P)1548 1693 y Fi(m)1548 1759 y(k)q Fl(=1)1687 1734 y Fp(\015)1730 1746 y Fi(k)1771 1734 y Ff(.)605 1888 y Fq(Remarks)21 b(2.9.)169 b Ft(\(i\))82 b(As)26 b(a)f(by-product,)d(it)k(turns)e(out)h(that)g(the)f(in\002mum)g (of)h(the)f(almost-)456 1987 y(sure)c(spectrum)f(of)h Fp(H)7 b Fo(\()p Fp(V)1186 1999 y Fi(!)1234 1987 y Fo(\))21 b Ft(coincides)f(with)g(that)g(of)g Fp(H)7 b Fo(\(0\))23 b(=)g Fm(\000)p Fo(\001)18 b(+)g Fp(U)2610 1999 y Fl(p)r(er)2708 1987 y Ft(.)562 2106 y(\(ii\))82 b(Thanks)21 b(to)g(the)h(con)m(v)o (ention)c Fo(0)25 b(=)g Fp(d)1809 2118 y Fi(k)1850 2106 y Fp(=)p Fm(1)14 b Fo(\(=)24 b Fp(\015)2153 2118 y Fi(k)2194 2106 y Fo(\))p Ft(,)f(Theorem)c(2.8)i(remains)g(v)n(alid)g(if)h Fp(\013)3314 2118 y Fi(k)3380 2106 y Fo(=)456 2205 y Fm(1)e Ft(for)g(some)g(\(or)g(all\))g Fp(k)26 b Fm(2)d(f)p Fo(1)p Fp(;)14 b(:)g(:)g(:)f(;)h(m)p Fm(g)p Ft(,)20 b(confer)e(Remark)i (2.5.)539 2324 y(\(iii\))82 b(Assumption)21 b(2.7)g(on)h(the)g(local)g (singularities)f(of)h Fp(U)2287 2336 y Fl(p)r(er)2407 2324 y Ft(is)h(slightly)f(more)f(restricti)n(v)o(e)g(than)456 2423 y(the)32 b(one)f(in)i([)p Fq(KS86)n(,)f(Mez87)o Ft(].)61 b(It)33 b(is)g(tailored)e(to)h(ensure)g(certain)f(re)o (gularity)f(properties)h(of)h(the)456 2523 y(ground-state)24 b(eigenfunction)h(of)i Fp(H)7 b Fo(\(0\))p Ft(.)46 b(As)28 b(can)f(be)g(inferred)f(from)g(Subsection)g(3.1)h(belo)n(w)-5 b(,)28 b(we)456 2623 y(may)19 b(relax)h(Assumption)f(2.7)g(and)h (require)e(only)i Fp(p)j(>)f(d=)p Fo(2)e Ft(\(as)h(in)f([)p Fq(KS86)n(,)h(Mez87)o Ft(]\))f(in)g(the)g(interior)456 2722 y(of)g(the)g(unit)g(cube)f(and)h(thus)g(allo)n(w)g(for)g(Coulomb)f (singularities)g(there.)545 2840 y(\(i)n(v\))82 b(Ev)o(en)25 b(in)g(the)h(isotropic)f(situation)g Fp(m)32 b Fo(=)h(1)26 b Ft(Assumption)e(2.4)h(co)o(v)o(ers)f(slightly)h(more)g(im-)456 2940 y(purity)20 b(potentials)h(than)g(in)h([)p Fq(KS86)n(,)g(Mez87)o Ft(],)g(since)f(we)h(allo)n(w)g Fp(f)30 b Ft(to)22 b(ha)n(v)o(e)f (zeros)g(at)h(arbitrary)e(lar)o(ge)456 3040 y(distance)f(from)h(the)g (origin.)566 3158 y(\(v\))82 b(An)26 b(inspection)e(of)h(the)h(proof)e (belo)n(w)h(sho)n(ws)g(that)h(we)g(pro)o(v)o(e)d(a)j(slightly)g(better) f(estimate)456 3258 y(than)19 b(the)h(double)f(logarithmic)e (asymptotics)j(gi)n(v)o(en)e(in)i(\(16\).)k(In)c(particular)m(,)e(if)i (the)g(measure)f Fp(\026)3265 3270 y Fi(!)3334 3258 y Ft(has)456 3357 y(an)h(atom)g(at)g(zero,)g(more)f(e)o(xactly)g(if)i Fn(P)14 b Fm(f)o Fp(!)26 b Fm(2)d Fo(\012)g(:)g Fp(\026)1984 3369 y Fi(!)2032 3357 y Fo(\(\003)2122 3369 y Fl(0)2160 3357 y Fo(\))g(=)g(0)p Fm(g)f Fp(>)g Fo(0)p Ft(,)e(then)g(we)h (actually)e(pro)o(v)o(e)1135 3499 y Fm(\000)p Fp(C)34 b Fo(\()p Fp(E)24 b Fm(\000)18 b Fp(E)1554 3511 y Fl(0)1591 3499 y Fo(\))1623 3457 y Fi(\021)1687 3499 y Fm(\024)23 b Fo(log)14 b Fp(N)9 b Fo(\()p Fp(E)c Fo(\))23 b Fm(\024)g(\000)p Fp(C)2343 3465 y Fj(0)2394 3499 y Fo(\()p Fp(E)g Fm(\000)18 b Fp(E)2654 3511 y Fl(0)2692 3499 y Fo(\))2724 3457 y Fi(\021)3306 3499 y Ft(\(17\))456 3640 y(for)28 b(small)h Fp(E)h Fm(\000)24 b Fp(E)1032 3652 y Fl(0)1070 3640 y Ft(.)51 b(This)29 b(is)h(not)e(quite)g(the)h(logarithmic)e(beha)n (viour)f(\(2\))j(of)f Fp(N)38 b Ft(since)29 b(the)g(con-)456 3740 y(stants)g Fp(C)44 b(>)38 b Fo(0)28 b Ft(and)g Fp(C)1164 3710 y Fj(0)1226 3740 y Fp(>)37 b Fo(0)29 b Ft(do)f(not)g(agree.)49 b(Note)28 b(that)g Fp(\026)2290 3752 y Fi(!)2367 3740 y Ft(has)h(an)f(atom)g(at)h(zero)f(for)f(the)i(an)o(y)456 3840 y(generalized)20 b(Poisson)i(measure)g(\(P\))g(as)h(well)f(as)h (for)f(a)g(compound)e(displacement)h(measure)g(\(D'\))h(if)456 3939 y Fn(P)14 b Fm(f)o Fp(!)26 b Fm(2)d Fo(\012)g(:)g Fp(q)898 3951 y Fi(!)r(;)p Fl(0)999 3939 y Fo(\()p Fp(!)s Fo(\))g(=)g(0)p Fm(g)f Fp(>)h Fo(0)p Ft(.)605 4093 y(F)o(or)32 b(an)g(illustration)g(and)g(interpretation)e(of)i(Theorem)e(2.8)i(we)h (consider)e(the)h(special)h(case)456 4192 y Fp(m)23 b Fo(=)f(2)p Ft(.)j(The)20 b(right-hand)e(side)i(of)g(\(16\))f(then)h (suggests)g(to)g(distinguish)g(the)g(follo)n(wing)e(three)i(cases:)497 4414 y Fq(Quantum)g(r)o(egime:)1541 4358 y Fp(d)1584 4370 y Fl(1)p 1541 4395 81 4 v 1560 4471 a Fo(2)1654 4414 y Fm(\025)1807 4358 y Fp(\015)1850 4370 y Fl(1)p 1752 4395 191 4 v 1752 4471 a Fo(1)e Fm(\000)g Fp(\015)2036 4414 y Ft(and)2248 4358 y Fp(d)2291 4370 y Fl(2)p 2248 4395 81 4 v 2268 4471 a Fo(2)2362 4414 y Fm(\025)2515 4358 y Fp(\015)2558 4370 y Fl(2)p 2460 4395 191 4 v 2460 4471 a Fo(1)f Fm(\000)i Fp(\015)2660 4414 y Ft(.)150 b Fc(\(qm\))497 4633 y Fq(Quantum-classical)19 b(r)o(egime:)1541 4576 y Fp(d)1584 4588 y Fl(1)p 1541 4613 81 4 v 1560 4690 a Fo(2)1654 4633 y Fm(\025)1807 4576 y Fp(\015)1850 4588 y Fl(1)p 1752 4613 191 4 v 1752 4690 a Fo(1)f Fm(\000)g Fp(\015)2036 4633 y Ft(and)2274 4576 y Fp(d)2317 4588 y Fl(2)p 2274 4613 81 4 v 2294 4690 a Fo(2)2388 4633 y Fp(<)2540 4576 y(\015)2583 4588 y Fl(2)p 2485 4613 191 4 v 2485 4690 a Fo(1)g Fm(\000)g Fp(\015)2790 4633 y Fc(\(qm/cl\))1356 4851 y Ft(or:)1541 4795 y Fp(d)1584 4807 y Fl(1)p 1541 4832 81 4 v 1560 4908 a Fo(2)1654 4851 y Fp(<)1807 4795 y(\015)1850 4807 y Fl(1)p 1752 4832 191 4 v 1752 4908 a Fo(1)g Fm(\000)g Fp(\015)2036 4851 y Ft(and)2274 4795 y Fp(d)2317 4807 y Fl(2)p 2274 4832 81 4 v 2294 4908 a Fo(2)2388 4851 y Fm(\025)2540 4795 y Fp(\015)2583 4807 y Fl(2)p 2485 4832 191 4 v 2485 4908 a Fo(1)g Fm(\000)g Fp(\015)2790 4851 y Fc(\(cl/qm\))497 5069 y Fq(Classical)j(r)o(egime:)1541 5013 y Fp(d)1584 5025 y Fl(1)p 1541 5050 81 4 v 1560 5126 a Fo(2)1654 5069 y Fp(<)1807 5013 y(\015)1850 5025 y Fl(1)p 1752 5050 191 4 v 1752 5126 a Fo(1)d Fm(\000)g Fp(\015)2036 5069 y Ft(and)2274 5013 y Fp(d)2317 5025 y Fl(2)p 2274 5050 81 4 v 2294 5126 a Fo(2)2388 5069 y Fp(<)2540 5013 y(\015)2583 5025 y Fl(2)p 2485 5050 191 4 v 2485 5126 a Fo(1)g Fm(\000)g Fp(\015)2859 5069 y Fc(\(cl\))p eop end %%Page: 8 8 TeXDict begin 8 7 bop 456 251 a Fr(8)915 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)605 450 y Ft(In)34 b(comparison)f(to)i(the)f(result)h(\(7\))f(for)g Fp(m)49 b Fo(=)h(1)34 b Ft(the)h(main)f(\002nding)f(of)i(this)g(paper)e(is)j (the)456 550 y(emer)o(gence)29 b(of)i(a)g(re)o(gime)g(corresponding)d (to)j(mix)o(ed)f(quantum)g(and)h(classical)h(character)e(of)h(the)456 649 y(Lifshits)i(tail.)62 b(A)34 b(remarkable)c(f)o(act)j(about)f(the)g (Lifshits)h(e)o(xponent)e(\(16\))g(is)j(that)e(the)h(directions)456 749 y Fp(k)53 b Fm(2)e(f)p Fo(1)p Fp(;)14 b Fo(2)p Fm(g)33 b Ft(related)i(to)g(the)g(anisotrop)o(y)e(do)h(not)h(sho)n(w)g(up)f (separately)h(as)g(one)g(might)f(e)o(xpect)456 849 y(nai)n(v)o(ely)-5 b(.)60 b(In)32 b(particular)m(,)i(the)f(transition)f(from)f(a)i (quantum)e(to)i(a)g(classical)g(re)o(gime)f(for)g(the)g Fp(x)3375 861 y Fi(k)3417 849 y Ft(-)456 948 y(direction)23 b(does)h(not)g(occur)g(if)g Fp(d)1412 960 y Fi(k)1453 948 y Fp(=)p Fo(2)30 b(=)h Fp(\015)1706 960 y Fi(k)1747 948 y Fp(=)p Fo(\(1)21 b Fm(\000)g Fp(\015)2013 960 y Fi(k)2054 948 y Fo(\))p Ft(,)26 b(b)n(ut)e(rather)g(if)h Fp(d)2600 960 y Fi(k)2641 948 y Fp(=)p Fo(2)30 b(=)h Fp(\015)2894 960 y Fi(k)2934 948 y Fp(=)p Fo(\(1)21 b Fm(\000)h Fp(\015)5 b Fo(\))p Ft(.)38 b(This)456 1048 y(intriguing)17 b(intertwining)h(of)h(directions)f(through)f Fp(\015)25 b Ft(may)19 b(be)g(interpreted)f(in)h(terms)g(of)g(the)h (mar)o(ginal)456 1148 y(impurity)26 b(potentials)i Fp(f)1172 1118 y Fl(\(1\))1290 1148 y Ft(and)g Fp(f)1489 1118 y Fl(\(2\))1606 1148 y Ft(de\002ned)g(in)g(\(24\))f(and)h(\(25\))f(belo)n (w)-5 b(.)48 b(In)29 b(f)o(act,)h(when)e(writing)456 1247 y Fp(\015)499 1259 y Fl(2)536 1247 y Fp(=)p Fo(\(1)21 b Fm(\000)h Fp(\015)5 b Fo(\))32 b(=)f Fp(d)1011 1259 y Fl(2)1049 1247 y Fp(=)14 b Fo(\()o Fp(\013)1189 1259 y Fl(2)1227 1247 y Fo(\(1)k Fm(\000)g Fp(\015)1445 1259 y Fl(1)1482 1247 y Fo(\))h Fm(\000)f Fp(d)1659 1259 y Fl(2)1696 1247 y Fo(\))26 b Ft(and)f(identifying)e Fp(\013)2343 1259 y Fl(2)2380 1247 y Fo(\(1)f Fm(\000)f Fp(\015)2605 1259 y Fl(1)2643 1247 y Fo(\))k Ft(as)h(the)f(decay)f(e)o(xponent)456 1350 y(of)i Fp(f)602 1320 y Fl(\(2\))719 1350 y Ft(by)g(Lemma)g(3.4)h (belo)n(w)-5 b(,)27 b(it)h(is)g(clear)f(that)g Fp(f)2035 1320 y Fl(\(2\))2151 1350 y Ft(serv)o(es)g(as)h(an)f(ef)n(fecti)n(v)o (e)e(potential)i(for)f(the)456 1450 y Fp(x)503 1462 y Fl(2)540 1450 y Ft(-direction)16 b(as)j(f)o(ar)f(as)g(the)g (quantum-classical)e(transition)h(is)i(concerned.)i(In)d(analogy)-5 b(,)16 b Fp(f)3131 1420 y Fl(\(1\))3238 1450 y Ft(serv)o(es)456 1550 y(as)26 b(the)f(ef)n(fecti)n(v)o(e)f(potential)h(for)f(the)i Fp(x)1603 1562 y Fl(1)1640 1550 y Ft(-direction.)39 b(Heuristic)25 b(ar)o(guments)f(for)g(the)i(importance)d(of)456 1649 y(the)d(mar)o(ginal)e(potentials)i(in)g(the)g(presence)g(of)f(an)i (anisotrop)o(y)d(can)i(be)g(found)f(in)h([)p Fq(L)-8 b(W04)o Ft(].)1216 1849 y Fq(3.)41 b(Basic)21 b(inequalities)g(and)f (auxiliary)f(r)o(esults)605 1999 y Ft(In)j(order)g(to)g(k)o(eep)g(our)g (notation)g(as)h(transparent)e(as)i(possible,)g(we)g(will)g (additionally)e(suppose)456 2098 y(that)1486 2209 y Fp(E)1547 2221 y Fl(0)1608 2209 y Fo(=)h(0)166 b Ft(and)f Fp(m)23 b Fo(=)f(2)892 b Ft(\(18\))456 2336 y(throughout)19 b(the)k(subsequent) e(proof)h(of)g(Theorem)f(2.8.)32 b(In)23 b(f)o(act,)g(the)g(\002rst)g (assumption)f(can)h(al)o(w)o(ays)456 2436 y(be)d(achie)n(v)o(ed)e(by)i (adding)f(a)h(constant)g(to)g Fp(H)7 b Fo(\(0\))p Ft(.)605 2536 y(The)24 b(strate)o(gy)f(of)h(the)g(proof)f(is)i(roughly)d(the)i (same)g(as)h(in)g([)p Fq(KS86)n(,)f(Mez87)p Ft(],)h(which)e(in)i(turn)e (is)456 2635 y(based)d(on)g([)p Fq(KM83a)n(,)h(Sim85)p Ft(].)26 b(W)-7 b(e)22 b(use)f(bounds)e(on)h(the)h(inte)o(grated)e (density)h(of)h(states)g Fp(N)30 b Ft(and)20 b(sub-)456 2735 y(sequently)g(emplo)o(y)g(the)h(Rayleigh-Ritz)g(principle)f(and)h (T)-6 b(emple')h(s)21 b(inequality)f([)p Fq(RS78)o Ft(])h(to)h (estimate)456 2834 y(the)29 b(occurring)e(ground-state)f(ener)o(gies)i (from)g(abo)o(v)o(e)g(and)g(belo)n(w)-5 b(.)51 b(The)29 b(basic)g(idea)g(to)g(construct)456 2934 y(the)e(bounds)e(on)i Fp(N)37 b Ft(is)28 b(to)f(partition)f(the)h(con\002guration)e(space)i Fn(R)2433 2904 y Fi(d)2499 2934 y Ft(into)g(congruent)d(domains)j(and) 456 3034 y(emplo)o(y)f(some)i(brack)o(eting)e(technique)g(for)h Fp(H)7 b Fo(\()p Fp(V)1946 3046 y Fi(!)1994 3034 y Fo(\))p Ft(.)49 b(The)27 b(most)h(straightforw)o(ard)d(of)j(these)f(tech-)456 3133 y(niques)22 b(is)i(Dirichlet)e(or)h(Neumann)e(brack)o(eting.)31 b(Ho)n(we)n(v)o(er)m(,)22 b(to)h(apply)f(T)-6 b(emple')h(s)22 b(inequality)g(to)h(the)456 3233 y(arising)g(Neumann)g(ground-state)f (ener)o(gy)-5 b(,)22 b(the)i(authors)f(of)h([)p Fq(KS86)o Ft(])g(required)e(that)j Fp(U)3030 3245 y Fl(p)r(er)3152 3233 y Ft(is)g(re\003ec-)456 3333 y(tion)20 b(in)m(v)n(ariant.)k(T)-7 b(o)20 b(get)h(rid)f(of)g(this)h(additional)e(assumption,)g(Mezincescu) g([)p Fq(Mez87)o Ft(])i(suggested)e(an)456 3432 y(alternati)n(v)o(e)i (upper)h(bound)f(on)i Fp(N)32 b Ft(which)23 b(is)h(based)e(on)h(a)h (brack)o(eting)d(technique)g(corresponding)f(to)456 3532 y(certain)e(Robin)h(\(mix)o(ed\))f(boundary)e(conditions.)23 b(In)c(his)h(honour)m(,)c(we)k(will)g(refer)e(to)i(these)f(particular) 456 3631 y(Robin)g(boundary)f(conditions)g(as)j(Mezincescu)f(boundary)d (conditions.)605 3802 y Fq(3.1.)40 b(Mezincescu)27 b(boundary)e (conditions)g(and)h(basic)g(inequalities.)41 b Ft(Assumption)25 b(2.7)f(on)456 3902 y Fp(U)513 3914 y Fl(p)r(er)630 3902 y Ft(implies)19 b([)p Fq(Sim82)o Ft(,)h(Thm.)e(C.2.4])g(that)h(there)f (is)i(a)g(continuously)c(dif)n(ferentiable)h(representati)n(v)o(e)456 4004 y Fp( )26 b Fo(:)d Fn(R)652 3974 y Fi(d)713 4004 y Fm(!)14 b Fo(]0)p Fp(;)g Fm(1)p Fo([)k Ft(of)f(the)g(strictly)g (positi)n(v)o(e)g(ground-state)e(eigenfunction)f(of)j Fp(H)7 b Fo(\(0\))23 b(=)g Fm(\000)p Fo(\001)8 b(+)g Fp(U)3327 4016 y Fl(p)r(er)3424 4004 y Ft(,)456 4104 y(which)19 b(is)i Fo(L)807 4074 y Fl(2)845 4104 y Ft(-normalised)d(on)i (the)g(unit)g(cube)f Fo(\003)1873 4116 y Fl(0)1910 4104 y Ft(,)1643 4170 y Fg(Z)1689 4359 y Fl(\003)1734 4367 y Fv(0)1785 4283 y Fp( )s Fo(\()p Fp(x)p Fo(\))1953 4249 y Fl(2)1991 4283 y Fp(dx)24 b Fo(=)f(1)p Fp(:)1048 b Ft(\(19\))456 4491 y(The)29 b(function)f Fp( )34 b Ft(is)c Fn(Z)1154 4461 y Fi(d)1193 4491 y Ft(-periodic,)h(bounded)c(from)i (belo)n(w)g(by)g(a)h(strictly)g(positi)n(v)o(e)f(constant)h(and)456 4591 y(obe)o(ys)19 b Fp(H)7 b Fo(\(0\))p Fp( )26 b Fo(=)c Fp(E)1079 4603 y Fl(0)1117 4591 y Fp( )k Fo(=)d(0)p Ft(.)605 4691 y(Subsequently)-5 b(,)19 b(we)j(denote)e(by)h Fo(\003)k Fm(\032)g Fn(R)1800 4661 y Fi(d)1860 4691 y Ft(a)d Fp(d)p Ft(-dimensional,)e(open)g(cuboid)g(which)h(is)h(compat-)456 4790 y(ible)g(with)h(the)f(lattice)h Fn(Z)1183 4760 y Fi(d)1222 4790 y Ft(,)g(that)g(is,)g(we)g(suppose)e(that)i(it)g (coincides)e(with)i(the)f(interior)g(of)g(the)g(union)456 4890 y(of)d Fn(Z)604 4860 y Fi(d)643 4890 y Ft(-translates)h(of)g(the)g (closed)g(unit)g(cube.)k(On)c(the)h(boundary)c Fp(@)5 b Fo(\003)20 b Ft(of)g Fo(\003)g Ft(we)h(de\002ne)e Fp(\037)k Fo(:)g Fp(@)5 b Fo(\003)23 b Fm(!)g Fn(R)456 4990 y Ft(as)e(the)f(ne)o (gati)n(v)o(e)e(of)i(the)g(outer)f(normal)g(deri)n(v)n(ati)n(v)o(e)f (of)i Fo(log)14 b Fp( )s Ft(,)1248 5171 y Fp(\037)p Fo(\()p Fp(x)p Fo(\))24 b(:=)f Fm(\000)1684 5115 y Fo(1)p 1621 5152 170 4 v 1621 5228 a Fp( )s Fo(\()p Fp(x)p Fo(\))1827 5171 y(\()p Fp(n)c Fm(\001)f(r)p Fo(\))d Fp( )s Fo(\()p Fp(x)p Fo(\))p Fp(;)98 b(x)23 b Fm(2)g Fp(@)5 b Fo(\003)p Fp(:)654 b Ft(\(20\))p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)771 b(9)456 450 y Ft(Since)20 b Fp(\037)j Fm(2)g Fp(L)871 420 y Fj(1)941 450 y Fo(\()p Fp(@)5 b Fo(\003\))21 b Ft(is)g(bounded,)c(the)k(sesquilinear)e(form)852 627 y Fo(\()p Fp(')938 639 y Fl(1)976 627 y Fp(;)14 b(')1067 639 y Fl(2)1104 627 y Fo(\))24 b Fm(7!)1266 514 y Fg(Z)1312 703 y Fl(\003)p 1375 555 273 4 v 1375 627 a Fm(r)p Fp(')1498 639 y Fl(1)1536 627 y Fo(\()p Fp(x)p Fo(\))c Fm(\001)e(r)p Fp(')1831 639 y Fl(2)1869 627 y Fo(\()p Fp(x)p Fo(\))c Fp(dx)20 b Fo(+)2187 514 y Fg(Z)2233 703 y Fi(@)t Fl(\003)2335 627 y Fp(\037)p Fo(\()p Fp(x)p Fo(\))p 2512 555 204 4 v 14 w Fp(')2566 639 y Fl(1)2605 627 y Fo(\()p Fp(x)p Fo(\))q Fp(')2771 639 y Fl(2)2808 627 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx;)260 b Ft(\(21\))456 816 y(with)20 b(domain)f Fp(')948 828 y Fl(1)985 816 y Ft(,)i Fp(')1081 828 y Fl(2)1142 816 y Fm(2)i Fp(W)1310 786 y Fl(1)p Fi(;)p Fl(2)1400 816 y Fo(\(\003\))g(:=)1656 749 y Fg(\010)1705 816 y Fp(')g Fm(2)h Fp(L)1918 786 y Fl(2)1954 816 y Fo(\(\003\))37 b(:)24 b Fm(r)2229 828 y Fi(j)2264 816 y Fp(')f Fm(2)h Fp(L)2477 786 y Fl(2)2513 816 y Fo(\(\003\))g Ft(for)19 b(all)k Fp(j)29 b Fm(2)23 b(f)p Fo(1)p Fp(;)14 b(:)g(:)g(:)f(;)h(d)p Fm(g)3376 749 y Fg(\011)3424 816 y Ft(,)456 916 y(is)22 b(symmetric,)e(closed)h(and)f(lo)n(wer)h(bounded,)e(and)h(thus)h (uniquely)e(de\002nes)i(a)h(self-adjoint)e(operator)456 1016 y Fm(\000)p Fo(\001)590 976 y Fi(\037)590 1040 y Fl(\003)673 1016 y Fo(=:)33 b Fp(H)870 976 y Fi(\037)863 1040 y Fl(\003)916 1016 y Fo(\(0\))22 b Fm(\000)h Fp(U)1189 1028 y Fl(p)r(er)1313 1016 y Ft(on)j Fp(L)1480 986 y Fl(2)1516 1016 y Fo(\(\003\))p Ft(.)44 b(In)25 b(f)o(act,)j(the)e (condition)e Fp(\037)34 b Fm(2)g Fp(L)2671 986 y Fj(1)2741 1016 y Fo(\()p Fp(@)5 b Fo(\003\))26 b Ft(guarantees)f(that)456 1115 y(boundary)17 b(term)k(in)g(\(21\))e(is)j(form-bounded)16 b(with)21 b(bound)e(zero)h(relati)n(v)o(e)g(to)h(the)g(\002rst)g(term,) g(which)f(is)456 1215 y(just)f(the)f(quadratic)f(form)h(corresponding)d (to)j(the)h(\(ne)o(gati)n(v)o(e\))d(Neumann)g(Laplacian.)24 b(Consequently)456 1315 y([)p Fq(RS78)n Ft(,)29 b(Thm.)44 b(XIII.68],)26 b(both)g(the)h(Robin)g(Laplacian)e Fm(\000)p Fo(\001)2292 1275 y Fi(\037)2292 1339 y Fl(\003)2369 1315 y Ft(as)j(well)f(as)g Fp(H)2808 1275 y Fi(\037)2801 1339 y Fl(\003)2854 1315 y Fo(\()p Fp(V)2934 1327 y Fi(!)2983 1315 y Fo(\))35 b(:=)g Fm(\000)p Fo(\001)3307 1275 y Fi(\037)3307 1339 y Fl(\003)3380 1315 y Fo(+)456 1415 y Fp(U)513 1427 y Fl(p)r(er)630 1415 y Fo(+)19 b Fp(V)762 1427 y Fi(!)810 1415 y Ft(,)j(de\002ned)f(as)h(a)g(form)e(sum)i(on)f Fp(W)1809 1385 y Fl(1)p Fi(;)p Fl(2)1899 1415 y Fo(\(\003\))k Fm(\032)g Fp(L)2193 1385 y Fl(2)2230 1415 y Fo(\(\003\))p Ft(,)d(ha)n(v)o(e)f(compact)f(resolv)o(ents.)28 b(Since)456 1514 y Fp(H)532 1475 y Fi(\037)525 1539 y Fl(\003)577 1514 y Fo(\()p Fp(V)657 1526 y Fi(!)706 1514 y Fo(\))20 b Ft(generates)f(a)g(positi)n(vity)g(preserving)f(semigroup,)f(its)k (ground-state)c(is)j(simple)f(and)g(comes)456 1614 y(with)h(a)h (strictly)f(positi)n(v)o(e)f(eigenfunction)f([)p Fq(RS78)n Ft(,)j(Thm.)e(XIII.43].)605 1750 y Fq(Remarks)i(3.1.)169 b Ft(\(i\))82 b(In)20 b(the)g(boundary)e(term)i(in)g(\(21\))f(we)i (took)e(the)i(liberty)e(to)i(denote)e(the)456 1849 y(trace)h(of)g Fp(')782 1861 y Fi(j)840 1849 y Fm(2)j Fp(W)1008 1819 y Fl(1)p Fi(;)p Fl(2)1098 1849 y Fo(\(\003\))e Ft(on)f Fp(@)5 b Fo(\003)20 b Ft(again)f(by)h Fp(')1830 1861 y Fi(j)1865 1849 y Ft(.)562 1968 y(\(ii\))82 b(P)o(artial)31 b(inte)o(gration)d(sho)n(ws)i(that)h(the)f(quadratic)f(form)g(\(21\))h (corresponds)e(to)i(imposing)456 2067 y(Robin)e(boundary)f(conditions)g Fo(\()p Fp(n)19 b Fm(\001)f(r)h Fo(+)f Fp(\037)p Fo(\))c Fp( )s Fm(j)1898 2079 y Fi(@)t Fl(\003)2026 2067 y Fo(=)40 b(0)29 b Ft(on)f(functions)g Fp( )33 b Ft(in)c(the)g(domain)f(of)h(the) 456 2167 y(Laplacian)21 b(on)h Fo(L)964 2137 y Fl(2)1001 2167 y Fo(\(\003\))p Ft(.)32 b(Ob)o(viously)-5 b(,)20 b(Neumann)h(boundary)f(conditions)g(correspond)g(to)j(the)f(special)456 2267 y(case)h Fp(\037)28 b Fo(=)f(0)p Ft(.)33 b(W)m(ith)23 b(the)g(present)f(choice)g(\(20\))f(of)i Fp(\037)g Ft(the)o(y)f(arise)h (if)g Fp(U)2509 2279 y Fl(p)r(er)2635 2267 y Fo(=)k(0)c Ft(such)g(that)g Fp( )31 b Fo(=)c(1)c Ft(or)m(,)456 2366 y(more)c(generally)-5 b(,)18 b(if)i Fp(U)1120 2378 y Fl(p)r(er)1239 2366 y Ft(is)h(re\003ection)e(in)m(v)n(ariant)g(\(as)i (w)o(as)g(supposed)d(in)j([)p Fq(KS86)n Ft(]\).)539 2485 y(\(iii\))82 b(Denoting)18 b(by)g Fp(\025)1225 2497 y Fl(0)1263 2485 y Fo(\()p Fp(H)1371 2445 y Fi(\037)1364 2509 y Fl(\003)1416 2485 y Fo(\()p Fp(V)1496 2497 y Fi(!)1545 2485 y Fo(\)\))24 b Fp(<)e(\025)1768 2497 y Fl(1)1806 2485 y Fo(\()p Fp(H)1914 2445 y Fi(\037)1907 2509 y Fl(\003)1960 2485 y Fo(\()p Fp(V)2040 2497 y Fi(!)2088 2485 y Fo(\)\))i Fm(\024)f Fp(\025)2312 2497 y Fl(2)2349 2485 y Fo(\()p Fp(H)2457 2445 y Fi(\037)2450 2509 y Fl(\003)2503 2485 y Fo(\()p Fp(V)2583 2497 y Fi(!)2632 2485 y Fo(\)\))g Fm(\024)g Fp(:)14 b(:)g(:)33 b Ft(the)19 b(eigen)m(v)n(alues)456 2584 y(of)h Fp(H)622 2544 y Fi(\037)615 2609 y Fl(\003)667 2584 y Fo(\()p Fp(V)747 2596 y Fi(!)796 2584 y Fo(\))p Ft(,)g(the)h(eigen)m(v)n(alue-counting)15 b(function)1037 2719 y Fp(N)22 b Fo(\()p Fp(E)5 b Fo(;)14 b Fp(H)1337 2679 y Fi(\037)1330 2743 y Fl(\003)1383 2719 y Fo(\()p Fp(V)1463 2731 y Fi(!)1512 2719 y Fo(\)\))23 b(:=)g(#)14 b Fm(f)o Fp(n)23 b Fm(2)h Fn(N)2057 2731 y Fl(0)2131 2719 y Fo(:)37 b Fp(\025)2239 2731 y Fi(n)2298 2719 y Fo(\()p Fp(H)2406 2679 y Fi(\037)2399 2743 y Fl(\003)2452 2719 y Fo(\()p Fp(V)2532 2731 y Fi(!)2581 2719 y Fo(\)\))23 b Fp(<)g(E)5 b Fm(g)442 b Ft(\(22\))456 2853 y(is)25 b(well-de\002ned)d(for)i(all)g Fp(!)33 b Fm(2)e Fo(\012)24 b Ft(and)g(all)g(ener)o(gies)f Fp(E)35 b Fm(2)c Fn(R)p Ft(.)37 b(If)24 b Fp(U)2458 2865 y Fl(p)r(er)2580 2853 y Ft(is)h(bounded)c(from)i(belo)n(w)-5 b(,)24 b(it)456 2953 y(follo)n(ws)g(from)f([)p Fq(Min02)o Ft(,)j(Thm.)e(1.3])f(and)h (\(15\))f(that)i Fp(N)9 b Fo(\()p Fp(E)c Fo(\))31 b(=)f(lim)2484 2968 y Fj(j)p Fl(\003)p Fj(j!1)2705 2953 y Fm(j)p Fo(\003)p Fm(j)2809 2923 y Fj(\000)p Fl(1)2898 2953 y Fp(N)9 b Fo(\()p Fp(E)c Fo(;)14 b Fp(H)3185 2913 y Fi(\037)3178 2977 y Fl(\003)3230 2953 y Fo(\()p Fp(V)3310 2965 y Fi(!)3359 2953 y Fo(\)\))p Ft(.)456 3052 y(W)-7 b(e)32 b(also)f(refer)f(to)h([)p Fq(Min02)o Ft(])g(for)f(proofs)f(of)i(some)g(of)f(the)h(abo)o(v)o (e-mentioned)26 b(properties)j(of)i(the)456 3152 y(Robin)19 b(Laplacian.)605 3288 y(One)e(important)e(point)i(about)f(the)h (Mezincescu)f(boundary)e(conditions)h(\(20\))h(is)i(that)f(the)g (restric-)456 3388 y(tion)26 b(of)g Fp( )k Ft(to)d Fo(\003)f Ft(continues)g(to)g(be)h(the)f(ground-state)e(eigenfunction)g(of)i Fp(H)2737 3348 y Fi(\037)2730 3412 y Fl(\003)2782 3388 y Fo(\(0\))i Ft(with)e(eigen)m(v)n(alue)456 3487 y Fp(\025)504 3499 y Fl(0)541 3487 y Fo(\()p Fp(H)649 3447 y Fi(\037)642 3512 y Fl(\003)695 3487 y Fo(\(0\)\))e(=)f Fp(E)1006 3499 y Fl(0)1068 3487 y Fo(=)g(0)p Ft(.)j(This)21 b(follo)n(ws)f(from)g (the)g(f)o(act)h(that)g Fp( )j Ft(satis\002es)e(the)e(eigen)m(v)n(alue) f(equation,)456 3587 y(the)h(boundary)d(conditions)i(and)g(that)i Fp( )j Ft(is)d(strictly)f(positi)n(v)o(e.)605 3786 y(Our)k(proof)e(of)i (Theorem)e(2.8)i(is)h(based)e(on)h(the)g(follo)n(wing)e(sandwiching)g (bound)h(on)g(the)h(inte-)456 3886 y(grated)19 b(density)g(of)h (states.)605 4021 y Fq(Pr)o(oposition)i(3.2.)42 b Ff(Let)24 b Fo(\003)29 b Fm(\032)g Fn(R)1587 3991 y Fi(d)1649 4021 y Ff(be)24 b(a)f Fp(d)p Ff(-dimensional)f(open)g(cuboid,)h(whic)o(h)g (is)h(compatible)456 4121 y(with)c(the)h(lattice)f Fn(Z)1029 4091 y Fi(d)1068 4121 y Ff(.)25 b(Then)20 b(the)g(inte)m(gr)o(ated)f (density)h(of)g(states)h Fp(N)30 b Ff(obe)n(ys)857 4277 y Fm(j)p Fo(\003)p Fm(j)961 4243 y Fj(\000)p Fl(1)1064 4277 y Fn(P)1129 4185 y Fg(n)1184 4277 y Fp(!)c Fm(2)d Fo(\012)37 b(:)g Fp(\025)1545 4289 y Fl(0)1597 4210 y Fg(\000)1635 4277 y Fp(H)1711 4243 y Fi(D)1704 4297 y Fl(\003)1771 4277 y Fo(\()p Fp(V)1851 4289 y Fi(!)1899 4277 y Fo(\))1931 4210 y Fg(\001)1993 4277 y Fp(<)22 b(E)2146 4185 y Fg(o)2225 4277 y Fm(\024)g Fp(N)9 b Fo(\()p Fp(E)c Fo(\))1272 4460 y Fm(\024)23 b(j)p Fo(\003)p Fm(j)1464 4425 y Fj(\000)p Fl(1)1553 4460 y Fp(N)f Fo(\()p Fp(E)5 b Fo(;)14 b Fp(H)1853 4420 y Fi(\037)1846 4484 y Fl(\003)1899 4460 y Fo(\(0\)\))60 b Fn(P)2162 4367 y Fg(n)2218 4460 y Fp(!)25 b Fm(2)f Fo(\012)37 b(:)f Fp(\025)2578 4472 y Fl(0)2630 4460 y Fo(\()p Fp(H)2738 4420 y Fi(\037)2731 4484 y Fl(\003)2784 4460 y Fo(\()p Fp(V)2864 4472 y Fi(!)2912 4460 y Fo(\)\))24 b Fp(<)f(E)3154 4367 y Fg(o)3306 4460 y Ft(\(23\))456 4613 y Ff(for)d(all)h(ener)m(gies)f Fp(E)28 b Fm(2)23 b Fn(R)p Ff(.)607 4749 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(F)o(or)16 b(the)h(lo)n(wer)f(bound)f(on)h Fp(N)9 b Ft(,)18 b(see)f([)p Fq(KM83a)n Ft(,)h(Eq.)e(\(4\))g(and)g(\(21\)])f (or)i([)p Fq(KS86)n Ft(,)h(Eq.)e(\(2\)].)456 4849 y(The)j(upper)g (bound)g(follo)n(ws)g(from)g([)p Fq(Mez87)o Ft(,)i(Eq.)e(\(29\)].)1260 b Fd(\003)605 4998 y Fq(Remark)22 b(3.3.)41 b Ft(Since)22 b(the)g(brack)o(eting)e([)p Fq(Mez87)o Ft(,)i(Prop.)f(1])h([)p Fq(CL90)o Ft(,)g(Probl.)f(I.7.19])f(applies)i(to)456 5098 y(Robin)i(boundary)e(conditions)h(with)i(more)f(general)f(real)i Fp(\037)31 b Fm(2)h Fo(L)2404 5068 y Fj(1)2474 5098 y Fo(\()p Fp(@)5 b Fo(\003\))25 b Ft(than)f(the)h(one)f(de\002ned)g(in) 456 5198 y(\(20\),)18 b(the)j(same)f(is)h(true)f(for)g(the)g(upper)f (bound)f(in)i(\(23\).)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 456 251 a Fr(10)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)605 450 y Fq(3.2.)40 b(Elementary)26 b(facts)f(about)h(mar)o(ginal)f(impurity)h(potentials.) 41 b Ft(K)n(e)o(y)25 b(quantities)h(in)g(our)456 550 y(proof)19 b(of)h(Theorem)f(2.8)i(are)f(the)h(mar)o(ginal)e(impurity)h (potentials)g Fp(f)2456 520 y Fl(\(1\))2569 550 y Fo(:)k Fn(R)2686 520 y Fi(d)2721 528 y Fv(1)2781 550 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ft(and)f Fp(f)3308 520 y Fl(\(2\))3421 550 y Fo(:)456 650 y Fn(R)526 620 y Fi(d)561 628 y Fv(2)624 650 y Fm(!)28 b Fo([0)p Fp(;)14 b Fm(1)p Fo([)23 b Ft(for)f(the)h Fp(x)1258 662 y Fl(1)1296 650 y Ft(-)g(and)f Fp(x)1537 662 y Fl(2)1575 650 y Ft(-direction,)g (respecti)n(v)o(ely)-5 b(.)30 b(F)o(or)23 b(the)g(gi)n(v)o(en)e Fp(f)37 b Fm(2)28 b Fo(L)3069 620 y Fl(1)3106 650 y Fo(\()p Fn(R)3208 620 y Fi(d)3247 650 y Fo(\))c Ft(the)o(y)456 749 y(are)c(de\002ned)f(as)i(follo)n(ws)1336 910 y Fp(f)1386 875 y Fl(\(1\))1474 910 y Fo(\()p Fp(x)1553 922 y Fl(1)1591 910 y Fo(\))84 b(:=)1817 843 y Fg(R)1857 939 y Fe(R)1911 920 y Fk(d)1942 932 y Fv(2)1996 910 y Fp(f)9 b Fo(\()p Fp(x)2125 922 y Fl(1)2163 910 y Fp(;)14 b(x)2247 922 y Fl(2)2284 910 y Fo(\))g Fp(dx)2420 922 y Fl(2)2458 910 y Fp(:)825 b Ft(\(24\))1336 1050 y Fp(f)1386 1016 y Fl(\(2\))1474 1050 y Fo(\()p Fp(x)1553 1062 y Fl(2)1591 1050 y Fo(\))95 b(:=)1829 983 y Fg(R)1868 1080 y Fe(R)1922 1061 y Fk(d)1953 1073 y Fv(1)2008 1050 y Fp(f)9 b Fo(\()p Fp(x)2137 1062 y Fl(1)2174 1050 y Fp(;)14 b(x)2258 1062 y Fl(2)2296 1050 y Fo(\))g Fp(dx)2432 1062 y Fl(1)3306 1050 y Ft(\(25\))456 1222 y(The)21 b(aim)g(of)g(this)h(Subsection)e(is) i(to)g(collect)f(properties)f(of)h Fp(f)2274 1192 y Fl(\(2\))2362 1222 y Ft(.)29 b(Since)21 b Fp(f)2668 1192 y Fl(\(1\))2779 1222 y Ft(results)h(from)e Fp(f)3251 1192 y Fl(\(2\))3361 1222 y Ft(by)456 1322 y(e)o(xchanging)d(the)j(role)g(of)g Fp(x)1267 1334 y Fl(1)1325 1322 y Ft(and)g Fp(x)1513 1334 y Fl(2)1551 1322 y Ft(,)g(analogous)f(properties)f(apply)i(to)g Fp(f)2643 1292 y Fl(\(1\))2732 1322 y Ft(.)605 1486 y Fq(Lemma)c(3.4.)33 b Ff(Assumption)14 b(2.4)h(with)i Fp(m)22 b Fo(=)h(2)16 b Ff(implies)g(that)f(ther)m(e)h(e)n(xist)h(two)f (constants)f Fo(0)22 b Fp(<)h(f)3387 1498 y Fl(1)3424 1486 y Ff(,)456 1586 y Fp(f)497 1598 y Fl(2)557 1586 y Fp(<)f Fm(1)f Ff(suc)o(h)f(that)948 1729 y Fp(f)989 1741 y Fl(1)p 779 1766 415 4 v 779 1843 a Fm(j)p Fp(x)849 1855 y Fl(2)887 1843 y Fm(j)910 1819 y Fi(\013)953 1827 y Fv(2)986 1819 y Fl(\(1)p Fj(\000)p Fi(\015)1132 1827 y Fv(1)1164 1819 y Fl(\))1227 1785 y Fm(\024)1315 1672 y Fg(Z)1361 1860 y Fj(j)p Fi(y)1415 1868 y Fv(2)1447 1860 y Fj(j)p Fi(<)1528 1838 y Fv(1)p 1528 1847 29 3 v 1528 1880 a(2)1492 1785 y Fp(f)1542 1750 y Fl(\(2\))1631 1785 y Fo(\()p Fp(y)1704 1797 y Fl(2)1759 1785 y Fm(\000)e Fp(x)1889 1797 y Fl(2)1927 1785 y Fo(\))c Fp(dy)2057 1797 y Fl(2)2094 1785 y Fp(;)180 b(f)2347 1750 y Fl(\(2\))2436 1785 y Fo(\()p Fp(x)2515 1797 y Fl(2)2553 1785 y Fo(\))23 b Fm(\024)2874 1729 y Fp(f)2915 1741 y Fl(2)p 2706 1766 415 4 v 2706 1843 a Fm(j)p Fp(x)2776 1855 y Fl(2)2814 1843 y Fm(j)2837 1819 y Fi(\013)2880 1827 y Fv(2)2912 1819 y Fl(\(1)p Fj(\000)p Fi(\015)3058 1827 y Fv(1)3091 1819 y Fl(\))3306 1785 y Ft(\(26\))456 2011 y Ff(for)d(lar)m(g)o(e)g (enough)f Fm(j)p Fp(x)1101 2023 y Fl(2)1138 2011 y Fm(j)k Fp(>)g Fo(0)p Ff(.)607 2176 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)21 b(lemma)g(follo)n(ws)g(by)g(elementary)e(inte)o (gration.)27 b(In)21 b(doing)f(so,)i(one)f(may)g(replace)456 2275 y(the)f(maximum)e(norm)h Fm(j)g(\001)f(j)j Ft(by)f(the)g(equi)n(v) n(alent)f(Euclidean)f Fo(2)p Ft(-norm)h(in)h(both)f(\(12\))g(and)h (\(26\).)201 b Fd(\003)605 2461 y Fq(Lemma)32 b(3.5.)48 b Ff(Assumption)30 b(2.4)h(with)h Fp(m)44 b Fo(=)g(2)32 b Ff(implies)g(that)f(ther)m(e)g(e)n(xists)i(some)f(constant)456 2561 y Fo(0)22 b Fp(<)h(f)649 2573 y Fl(3)709 2561 y Fp(<)f Fm(1)f Ff(suc)o(h)f(that)1363 2646 y Fg(Z)1409 2835 y Fj(j)p Fi(x)1467 2843 y Fv(2)1499 2835 y Fj(j)p Fi(>L)1542 2759 y Fp(f)1592 2725 y Fl(\(2\))1680 2759 y Fo(\()p Fp(x)1759 2771 y Fl(2)1797 2759 y Fo(\))14 b Fp(dx)1933 2771 y Fl(2)1994 2759 y Fm(\024)23 b Fp(f)2123 2771 y Fl(3)2174 2759 y Fp(L)2231 2725 y Fj(\000)p Fi(\013)2326 2733 y Fv(2)2358 2725 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3306 2759 y Ft(\(27\))456 2975 y Ff(for)d(suf)o(\002ciently)f (lar)m(g)o(e)h Fp(L)j(>)g Fo(0)p Ff(.)607 3139 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(By)15 b(Lemma)g(3.4)f(we)i(ha)n(v)o(e)1690 3072 y Fg(R)1730 3168 y Fj(j)p Fi(x)1788 3176 y Fv(2)1819 3168 y Fj(j)p Fi(>L)1955 3139 y Fp(f)2005 3109 y Fl(\(2\))2093 3139 y Fo(\()p Fp(x)2172 3151 y Fl(2)2210 3139 y Fo(\))e Fp(dx)2346 3151 y Fl(2)2407 3139 y Fm(\024)23 b Fp(f)2536 3151 y Fl(2)2587 3072 y Fg(R)2626 3168 y Fj(j)p Fi(x)2684 3176 y Fv(2)2716 3168 y Fj(j)p Fi(>L)2851 3139 y Fm(j)p Fp(x)2921 3151 y Fl(2)2959 3139 y Fm(j)2982 3109 y Fj(\000)p Fi(\013)3077 3117 y Fv(2)3109 3109 y Fl(\(1)p Fj(\000)p Fi(\015)3255 3117 y Fv(1)3287 3109 y Fl(\))3331 3139 y Fp(dx)3421 3151 y Fl(2)456 3248 y Ft(for)f(suf)n(\002ciently)g(lar)o (ge)f Fp(L)28 b(>)f Fo(0)p Ft(.)33 b(The)22 b(assertion)h(follo)n(ws)f (by)h(elementary)e(inte)o(gration)g(and)h(the)h(f)o(act)456 3348 y(that)d Fp(\013)654 3360 y Fl(2)691 3348 y Fo(\(1)e Fm(\000)h Fp(\015)910 3360 y Fl(1)947 3348 y Fo(\))f Fm(\000)g Fp(d)1123 3360 y Fl(2)1184 3348 y Fo(=)23 b Fp(\013)1325 3360 y Fl(2)1362 3348 y Fo(\(1)18 b Fm(\000)g Fp(\015)5 b Fo(\))p Ft(.)1742 b Fd(\003)605 3534 y Fq(Remark)31 b(3.6.)47 b Ft(One)30 b(consequence)f(of)h(Lemma)g(3.5,)i(which)e(will) i(be)e(useful)g(belo)n(w)-5 b(,)32 b(is)g(the)456 3633 y(follo)n(wing)18 b(inequality)1101 3833 y Fo(sup)1029 3907 y Fj(j)p Fi(y)1083 3915 y Fv(2)1114 3907 y Fj(j\024)p Fi(L=)p Fl(2)1313 3720 y Fg(Z)1359 3909 y Fj(j)p Fi(x)1417 3917 y Fv(2)1448 3909 y Fj(j)p Fi(>L)1566 3892 y Fk(\014)1530 3833 y Fp(f)1580 3799 y Fl(\(2\))1669 3833 y Fo(\()p Fp(x)1748 3845 y Fl(2)1805 3833 y Fm(\000)g Fp(y)1929 3845 y Fl(2)1966 3833 y Fo(\))c Fp(dx)2102 3845 y Fl(2)2163 3833 y Fm(\024)22 b Fp(f)2291 3845 y Fl(3)2356 3766 y Fg(\000)2394 3833 y Fo(2)p Fp(=L)2535 3799 y Fi(\014)2578 3766 y Fg(\001)2616 3779 y Fi(\013)2659 3787 y Fv(2)2692 3779 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3306 3833 y Ft(\(28\))456 4056 y(v)n(alid)16 b(for)h(all)h Fp(\014)27 b Fm(\025)c Fo(1)18 b Ft(and)e(suf)n(\002ciently)h(lar)o(ge)f Fp(L)23 b(>)f Fo(1)p Ft(.)i(It)18 b(is)g(obtained)e(by)h(observing)e (that)j(the)f(inte)o(gral)456 4156 y(in)j(\(28\))f(equals)1135 4208 y Fg(Z)1181 4396 y Fj(j)p Fi(x)1239 4404 y Fv(2)1271 4396 y Fl(+)p Fi(j)1349 4404 y Fv(2)1382 4396 y Fj(j)p Fi(>L)1500 4380 y Fk(\014)1464 4321 y Fp(f)1514 4286 y Fl(\(2\))1602 4321 y Fo(\()p Fp(x)1681 4333 y Fl(2)1719 4321 y Fo(\))14 b Fp(dx)1855 4333 y Fl(2)1917 4321 y Fm(\024)2004 4208 y Fg(Z)2050 4396 y Fj(j)p Fi(x)2108 4404 y Fv(2)2140 4396 y Fj(j\025)p Fi(L)2258 4380 y Fk(\014)2296 4396 y Fi(=)p Fl(2)2312 4321 y Fp(f)2362 4286 y Fl(\(2\))2451 4321 y Fo(\()p Fp(x)2530 4333 y Fl(2)2568 4321 y Fo(\))g Fp(dx)2704 4333 y Fl(2)2742 4321 y Fp(:)541 b Ft(\(29\))456 4525 y(Here)19 b(the)h(last)g(inequality)f(results)h(from)e(the)i (triangle)f(inequality)f Fm(j)p Fp(x)2483 4537 y Fl(2)2537 4525 y Fo(+)e Fp(y)2659 4537 y Fl(2)2696 4525 y Fm(j)24 b(\024)e(j)p Fp(x)2900 4537 y Fl(2)2938 4525 y Fm(j)16 b Fo(+)h Fm(j)p Fp(y)3123 4537 y Fl(2)3160 4525 y Fm(j)j Ft(and)f(the)456 4624 y(f)o(act)h(that)g Fm(j)p Fp(y)809 4636 y Fl(2)846 4624 y Fm(j)p Fp(L=)p Fo(2)i Fm(\024)h Fp(L)1177 4594 y Fi(\014)1221 4624 y Fp(=)p Fo(2)p Ft(.)1662 4867 y Fq(4.)40 b(Upper)21 b(bound)605 5016 y Ft(F)o(or)28 b(an)g(asymptotic)g(e)n(v)n(aluation)f(of)h(the)g(upper)f(bound)g(in)h (Proposition)f(3.2)h(for)g(small)h(ener)n(-)456 5116 y(gies)23 b Fp(E)29 b Ft(we)24 b(distinguish)f(the)g(three)g(re)o (gimes)g(de\002ned)f(belo)n(w)h(Theorem)f(2.8:)31 b(quantum,)22 b(quantum-)456 5216 y(classical)f(and)e(classical.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(11)605 450 y Fq(4.1.)40 b(Regularisation)19 b(of)g(random)h(Bor)o(el)g(measur)o(e.)40 b Ft(In)20 b(all)g(of)g(the)g(abo)o(v)o(e)e(mentioned)g(cases)456 550 y(it)e(will)g(be)f(necessary)g(to)g(re)o(gularise)f(the)i(gi)n(v)o (en)e(random)f(Borel)j(measure)e Fp(\026)i Ft(by)f(introducing)e(a)j (cut)f(of)n(f.)456 650 y(F)o(or)g(this)h(purpose)f(we)h(de\002ne)f(a)h (re)o(gularised)e(random)g(Borel)i(measure)f Fp(\026)2585 620 y Fl(\()p Fi(h)p Fl(\))2703 650 y Fo(:)23 b(\012)r Fm(\002)r(B)s Fo(\()p Fn(R)3038 620 y Fi(d)3075 650 y Fo(\))h Fm(!)f Fo([0)p Fp(;)14 b Fm(1)p Fo([)456 765 y Ft(with)20 b(parameter)f Fp(h)k(>)f Fo(0)e Ft(by)g Fp(\026)1351 722 y Fl(\()p Fi(h)p Fl(\))1351 775 y Fi(!)1446 765 y Fo(\(\003\))j(:=)1702 703 y Fg(P)1790 790 y Fi(j)s Fj(2)p Fe(Z)1911 774 y Fk(d)1964 765 y Fp(\026)2014 722 y Fl(\()p Fi(h)p Fl(\))2014 775 y Fi(!)2109 698 y Fg(\000)2147 765 y Fo(\003)c Fm(\\)f Fo(\003)2355 777 y Fi(j)2390 698 y Fg(\001)2449 765 y Ft(where)1018 1088 y Fp(\026)1068 1054 y Fl(\()p Fi(h)p Fl(\))1068 1109 y Fi(!)1163 1021 y Fg(\000)1201 1088 y Fo(\003)h Fm(\\)f Fo(\003)1409 1100 y Fi(j)1444 1021 y Fg(\001)1505 1088 y Fo(:=)1616 893 y Fg(8)1616 968 y(>)1616 993 y(<)1616 1142 y(>)1616 1167 y(:)1731 958 y Fp(\026)1781 970 y Fi(!)1829 890 y Fg(\000)1867 958 y Fo(\003)g Fm(\\)h Fo(\003)2075 970 y Fi(j)2110 890 y Fg(\001)2405 958 y Fp(\026)2455 970 y Fi(!)2503 890 y Fg(\000)2541 958 y Fo(\003)2599 970 y Fi(j)2633 890 y Fg(\001)2695 958 y Fm(\024)j Fp(h)1731 1157 y(h)1812 1095 y(\026)1862 1107 y Fi(!)1910 1028 y Fg(\000)1948 1095 y Fo(\003)c Fm(\\)h Fo(\003)2156 1107 y Fi(j)2191 1028 y Fg(\001)p 1812 1138 417 4 v 1887 1218 a Fp(\026)1937 1230 y Fi(!)1985 1151 y Fg(\000)2023 1218 y Fo(\003)2081 1230 y Fi(j)2116 1151 y Fg(\001)2405 1157 y Ft(otherwise)3306 1088 y(\(30\))456 1403 y(for)g(all)i Fo(\003)i Fm(2)g(B)s Fo(\()p Fn(R)996 1373 y Fi(d)1034 1403 y Fo(\))e Ft(and)e(all)i Fp(!)26 b Fm(2)d Fo(\012)p Ft(.)605 1593 y Fq(Remark)32 b(4.1.)49 b Ft(Since)32 b Fp(\026)1370 1550 y Fl(\()p Fi(h)p Fl(\))1370 1603 y Fi(!)1479 1593 y Fo(\()p Fm(;)p Fo(\))45 b(=)g(0)33 b Ft(and)e Fp(\026)2017 1550 y Fl(\()p Fi(h)p Fl(\))2017 1603 y Fi(!)2112 1526 y Fg(\000)2164 1531 y(S)2233 1618 y Fi(n)2292 1593 y Fo(\003)2350 1563 y Fl(\()p Fi(n)p Fl(\))2447 1526 y Fg(\001)2531 1593 y Fo(=)2641 1531 y Fg(P)2728 1618 y Fi(n)2787 1593 y Fp(\026)2837 1550 y Fl(\()p Fi(h)p Fl(\))2837 1603 y Fi(!)2932 1526 y Fg(\000)2970 1593 y Fo(\003)3028 1563 y Fl(\()p Fi(n)p Fl(\))3125 1526 y Fg(\001)3196 1593 y Ft(for)h(an)o(y)456 1717 y(collection)d(of)h (disjoint)h Fo(\003)1253 1687 y Fl(\()p Fi(n)p Fl(\))1392 1717 y Fm(2)42 b(B)s Fo(\()p Fn(R)1649 1687 y Fi(d)1687 1717 y Fo(\))p Ft(,)34 b(each)c(realization)f Fp(\026)2388 1674 y Fl(\()p Fi(h)p Fl(\))2388 1727 y Fi(!)2514 1717 y Ft(is)j(indeed)d(a)i(measure)f(on)g(the)456 1833 y(Borel)18 b(sets)i Fm(B)s Fo(\()p Fn(R)963 1803 y Fi(d)1001 1833 y Fo(\))p Ft(.)25 b(It)19 b(is)g(locally)f(\002nite)h(and)f(hence)g(a)h (Borel)f(measure,)g(because)g Fp(\026)2918 1790 y Fl(\()p Fi(h)p Fl(\))2918 1843 y Fi(!)3013 1833 y Fo(\(\003)3103 1845 y Fi(j)3138 1833 y Fo(\))23 b Fm(\024)g Fp(h)c Ft(for)456 1935 y(all)h Fp(j)28 b Fm(2)c Fn(Z)759 1905 y Fi(d)819 1935 y Ft(and)19 b(all)i Fp(!)26 b Fm(2)d Fo(\012)p Ft(.)605 2109 y(F)o(or)d(future)f(reference)f(we)j(collect)f(some)g(properties)f (of)h Fp(\026)2337 2079 y Fl(\()p Fi(h)p Fl(\))2432 2109 y Ft(.)605 2283 y Fq(Lemma)h(4.2.)40 b Ff(Let)21 b Fp(h)i(>)f Fo(0)p Ff(.)j(Then)20 b(the)g(following)g(thr)m(ee)g(assertions)h(hold) e(true:)543 2431 y(\(i\))83 b Fp(\026)755 2388 y Fl(\()p Fi(h)p Fl(\))755 2441 y Fi(!)850 2431 y Fo(\(\003\))30 b Fm(\024)f Fo(min)1249 2364 y Fg(\010)1297 2431 y Fp(\026)1347 2443 y Fi(!)1395 2431 y Fo(\(\003\))p Fp(;)14 b(h)30 b Fo(#)1715 2364 y Fg(\010)1764 2431 y Fp(j)e Fm(2)23 b Fn(Z)1963 2401 y Fi(d)2039 2431 y Fo(:)37 b(\003)18 b Fm(\\)h Fo(\003)2307 2443 y Fi(j)2364 2431 y Fm(6)p Fo(=)k Fm(;)2494 2364 y Fg(\011)14 b(\011)2659 2431 y Ff(for)24 b(all)g Fo(\003)30 b Fm(2)g(B)s Fo(\()p Fn(R)3225 2401 y Fi(d)3263 2431 y Fo(\))25 b Ff(and)705 2531 y(all)20 b Fp(!)26 b Fm(2)d Fo(\012)p Ff(.)520 2696 y(\(ii\))83 b(the)22 b(intensity)g(measur)m(e)p 1434 2650 51 4 v 22 w Fp(\026)1484 2664 y Fl(\()p Fi(h)p Fl(\))1606 2696 y Fo(:)27 b Fm(B)s Fo(\()p Fn(R)1816 2665 y Fi(d)1853 2696 y Fo(\))g Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)23 b Ff(given)e(by)p 2556 2650 V 22 w Fp(\026)2606 2664 y Fl(\()p Fi(h)p Fl(\))2701 2696 y Fo(\(\003\))27 b(:=)g Fn(E)3030 2628 y Fg(\002)3064 2696 y Fp(\026)3114 2665 y Fl(\()p Fi(h)p Fl(\))3209 2696 y Fo(\(\003\))3331 2628 y Fg(\003)3389 2696 y Ff(is)705 2808 y(a)20 b(Bor)m(el)g(measur)m(e)g (whic)o(h)g(is)h Fn(Z)1621 2778 y Fi(d)1660 2808 y Ff(-periodic)e(and)g (obe)n(ys)p 2338 2763 V 20 w Fp(\026)2388 2776 y Fl(\()p Fi(h)p Fl(\))2483 2808 y Fo(\(\003)2573 2820 y Fl(0)2610 2808 y Fo(\))k Fp(>)g Fo(0)p Ff(.)497 2976 y(\(iii\))83 b(the)20 b(r)o(andom)f(variables)1434 2909 y Fg(\000)1472 2976 y Fp(\026)1522 2946 y Fl(\()p Fi(h)p Fl(\))1617 2976 y Fo(\(\003)1707 2988 y Fi(j)1742 2976 y Fo(\))1774 2909 y Fg(\001)1813 3009 y Fi(j)s Fj(2)p Fe(Z)1934 2993 y Fk(d)1994 2976 y Ff(ar)m(e)h(independent)e(and)h(identically)g (distrib)n(uted.)607 3159 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)21 b(\002rst)h(part)f(of)g(the)g(\002rst)h(assertion)f (is)h(immediate.)28 b(The)21 b(other)f(part)h(follo)n(ws)g(from)456 3259 y(the)h(monotonicity)e Fp(\026)1096 3271 y Fi(!)1144 3259 y Fo(\(\003)g Fm(\\)g Fo(\003)1387 3271 y Fi(j)1421 3259 y Fo(\))28 b Fm(\024)e Fp(\026)1622 3271 y Fi(!)1670 3259 y Fo(\(\003)1760 3271 y Fi(j)1795 3259 y Fo(\))h Fm(\024)g Fp(h)22 b Ft(for)g(all)h Fo(\003)j Fm(2)i(B)s Fo(\()p Fn(R)2569 3229 y Fi(d)2606 3259 y Fo(\))p Ft(,)c Fp(j)31 b Fm(2)d Fn(Z)2890 3229 y Fi(d)2951 3259 y Ft(and)22 b(all)h Fp(!)29 b Fm(2)f Fo(\012)p Ft(.)456 3362 y(The)c(claimed)f Fn(Z)956 3331 y Fi(d)995 3362 y Ft(-periodicity)f(of)i(the)h(intensity) f(measure)f(is)i(traced)f(back)f(to)i(the)f Fn(Z)3004 3331 y Fi(d)3043 3362 y Ft(-stationarity)456 3461 y(of)31 b Fp(\026)p Ft(.)59 b(The)31 b(inequality)f(in)h(the)h(second)e (assertion)h(holds,)i(since)f Fp(\026)2519 3473 y Fi(!)2567 3461 y Fo(\(\003)2657 3473 y Fl(0)2694 3461 y Fo(\))g Ft(is)h(not)e(identical)g(zero)456 3561 y(for)c Fn(P)p Ft(-almost)h(all)h Fp(!)40 b Fm(2)e Fo(\012)29 b Ft(\(confer)d (Assumption)h(2.1\).)48 b(The)28 b(third)f(assertion)h(follo)n(ws)g (from)f(the)456 3660 y(corresponding)16 b(property)i(of)i Fp(\026)h Ft(\(confer)d(Assumption)i(2.1\).)1129 b Fd(\003)605 3870 y Fq(4.2.)40 b(Quantum)19 b(r)o(egime.)40 b Ft(Throughout)16 b(this)k(subsection)f(we)g(suppose)g(that)g Fc(\(qm\))g Ft(holds.)25 b(As-)456 3970 y(sumption)18 b(2.4)h(on)g(the)g(impurity)f (potential)h(requires)g(the)g(e)o(xistence)g(of)g(some)g(constant)g Fp(f)3109 3982 y Fi(u)3175 3970 y Fp(>)k Fo(0)d Ft(and)456 4069 y(some)g(Borel)g(set)h Fp(F)35 b Fm(2)23 b(B)s Fo(\()p Fn(R)1296 4039 y Fi(d)1334 4069 y Fo(\))e Ft(with)f Fm(j)p Fp(F)12 b Fm(j)23 b Fp(>)g Fo(0)d Ft(such)g(that)1763 4246 y Fp(f)31 b Fm(\025)23 b Fp(f)1964 4258 y Fi(u)2007 4246 y Fp(\037)2059 4258 y Fi(F)2114 4246 y Fp(:)1169 b Ft(\(31\))456 4423 y(W)m(ithout)25 b(loss)i(of)f(generality)-5 b(,)25 b(we)i(will)f(additionally)f(suppose)g(that)h Fp(F)46 b Fm(\032)33 b Fo(\003)2781 4435 y Fl(0)2818 4423 y Ft(.)43 b(W)-7 b(e)28 b(start)e(by)g(con-)456 4523 y(structing)18 b(a)j(lo)n(wer)e(bound)f(on)h(the)h(lo)n(west)g (Mezincescu)f(eigen)m(v)n(alue)e Fp(\025)2558 4535 y Fl(0)2610 4523 y Fo(\()p Fp(H)2718 4483 y Fi(\037)2711 4547 y Fl(\003)2763 4523 y Fo(\()p Fp(V)2843 4535 y Fi(!)2892 4523 y Fo(\)\))k Ft(sho)n(wing)d(up)i(in)456 4622 y(the)g(right-hand)d (side)k(of)f(\(23\))f(when)g(choosing)g(the)h(interior)f(of)h(the)h (closure)1670 4848 y Fo(\003)h(:=)p 1861 4751 275 4 v 1899 4769 a Fg([)1861 4951 y Fj(j)p Fi(j)s Fj(j)p Fi()g Fo(1)d Ft(from)e(the)i(origin.) 27 b(By)22 b(construction,)e(the)456 5216 y(cube)f Fo(\003)i Ft(is)g(open)e(and)g(compatible)g(with)h(the)h(lattice.)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 456 251 a Fr(12)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)605 450 y Ft(4.2.1.)39 b Ff(Lower)29 b(bound)d(on)h(the)g(lowest)i(Mezincescu)e(eig)o(en)m (value.)39 b Ft(From)27 b(Lemma)g(4.2\(i\))g(and)456 550 y(\(31\))19 b(we)h(conclude)f(that)h(the)g(potential)g Fp(V)1680 562 y Fi(!)r(;h)1810 550 y Fo(:)j Fn(R)1926 520 y Fi(d)1987 550 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ft(gi)n(v)o(en)d(by)960 733 y Fp(V)1008 745 y Fi(!)r(;h)1116 733 y Fo(\()p Fp(x)p Fo(\))24 b(:=)e Fp(f)1402 745 y Fi(u)1459 620 y Fg(Z)1505 809 y Fe(R)1559 792 y Fk(d)1612 733 y Fp(\037)1664 745 y Fi(F)1719 733 y Fo(\()p Fp(x)e Fm(\000)e Fp(y)s Fo(\))c Fp(\026)2041 699 y Fl(\()p Fi(h)p Fl(\))2041 754 y Fi(!)2135 733 y Fo(\()p Fp(dy)s Fo(\))24 b(=)f Fp(f)2439 745 y Fi(u)2505 733 y Fp(\026)2555 699 y Fl(\()p Fi(h)p Fl(\))2555 754 y Fi(!)2650 666 y Fg(\000)2688 733 y Fp(x)c Fm(\000)f Fp(F)2902 666 y Fg(\001)3306 733 y Ft(\(33\))456 942 y(in)26 b(terms)g(of)g(the)g(re)o(gularised)e(Borel)i(measure)g Fp(\026)1941 899 y Fl(\()p Fi(h)p Fl(\))1941 952 y Fi(!)2035 942 y Ft(,)i(pro)o(vides)d(a)h(lo)n(wer)g(bound)e(on)i Fp(V)3065 954 y Fi(!)3140 942 y Ft(for)f(e)n(v)o(ery)456 1042 y Fp(h)32 b(>)g Fo(0)26 b Ft(and)f Fp(!)35 b Fm(2)e Fo(\012)p Ft(.)41 b(The)25 b(f)o(act)h(that)f(the)h(pointwise)f(dif)n (ference)e Fp(x)g Fm(\000)f Fp(F)37 b Ft(is)27 b(contained)d(in)h(a)h (cube,)456 1142 y(which)18 b(consists)i(of)g(\(at)f(most\))g Fo(3)1409 1111 y Fi(d)1467 1142 y Ft(unit)h(cubes,)e(together)g(with)i (Lemma)f(4.2\(i\))f(implies)h(the)h(estimate)1655 1281 y Fp(V)1703 1293 y Fi(!)r(;h)1810 1281 y Fo(\()p Fp(x)p Fo(\))k Fm(\024)f Fo(3)2075 1247 y Fi(d)2113 1281 y Fp(f)2154 1293 y Fi(u)2197 1281 y Fp(h)1061 b Ft(\(34\))456 1422 y(for)16 b(all)i Fp(!)26 b Fm(2)d Fo(\012)18 b Ft(and)e(all)i Fp(x)24 b Fm(2)f Fn(R)1362 1392 y Fi(d)1400 1422 y Ft(.)i(T)-7 b(aking)16 b Fp(h)i Ft(small)f(enough)e(thus)i(ensures)g(that)g(the)h (maximum)d(of)i(the)456 1522 y(potential)23 b Fp(V)819 1534 y Fi(!)r(;h)951 1522 y Ft(is)j(smaller)e(than)g(the)g(ener)o(gy)f (dif)n(ference)f(of)i(the)h(lo)n(west)g(and)f(the)g(\002rst)h(eigen)m (v)n(alue)456 1621 y(of)18 b Fp(H)620 1581 y Fi(\037)613 1646 y Fl(\003)665 1621 y Fo(\(0\))p Ft(.)25 b(This)19 b(enables)f(one)f(to)i(mak)o(e)f(use)h(of)f(T)-6 b(emple')h(s)18 b(inequality)f(to)i(obtain)e(a)i(lo)n(wer)f(bound)e(on)456 1721 y(the)k(lo)n(west)g(Mezincescu)g(eigen)m(v)n(alue)e(in)i(the)g (quantum)e(re)o(gime.)605 1878 y Fq(Pr)o(oposition)j(4.3.)42 b Ff(Let)23 b Fo(\003)f Ff(denote)g(the)g(open)g(cube)f(\(32\).)31 b(Mor)m(eo)o(ver)-9 b(,)22 b(let)i Fp(h)j Fo(:=)f(\()q Fp(r)3063 1890 y Fl(0)3100 1878 y Fp(L)p Fo(\))3189 1836 y Fj(\000)p Fl(2)3301 1878 y Ff(with)456 1978 y Fp(r)493 1990 y Fl(0)553 1978 y Fp(>)d Fo(0)p Ff(.)i(Then)20 b(the)g(lowest)h (eig)o(en)m(value)d(of)j Fp(H)1809 1938 y Fi(\037)1802 2002 y Fl(\003)1854 1978 y Fo(\()p Fp(V)1934 2001 y Fi(!)r(;h)2041 1978 y Fo(\))g Ff(is)g(bounded)d(fr)l(om)j(below)f(accor)m(ding)e(to) 1196 2172 y Fp(\025)1244 2184 y Fl(0)1282 2105 y Fg(\000)1320 2172 y Fp(H)1396 2132 y Fi(\037)1389 2197 y Fl(\003)1441 2172 y Fo(\()p Fp(V)1521 2193 y Fi(!)r(;h)1629 2172 y Fo(\))1661 2105 y Fg(\001)1722 2172 y Fm(\025)1878 2116 y Fo(1)p 1820 2153 160 4 v 1820 2229 a(2)c Fm(j)p Fo(\003)p Fm(j)2003 2059 y Fg(Z)2049 2248 y Fl(\003)2112 2172 y Fp(V)2160 2193 y Fi(!)r(;h)2267 2172 y Fo(\()p Fp(x)p Fo(\))g Fp( )s Fo(\()p Fp(x)p Fo(\))2560 2138 y Fl(2)2613 2172 y Fp(dx)603 b Ft(\(35\))456 2354 y Ff(for)28 b(all)g Fp(!)41 b Fm(2)d Fo(\012)p Ff(,)30 b(all)f Fp(L)37 b(>)g Fo(1)28 b Ff(and)f(lar)m(g)o(e)h(enough)e Fp(r)2040 2366 y Fl(0)2116 2354 y Fp(>)37 b Fo(0)p Ff(.)49 b Ft([)p Ff(Recall)27 b(the)h(de\002nition)f(of)h Fp( )k Ff(at)c(the)456 2454 y(be)m(ginning)17 b(of)j(Subsection)f(3.1.)p Ft(])607 2605 y(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(By)26 b(construction)e Fp( )1521 2617 y Fi(L)1605 2605 y Fo(:=)34 b Fm(j)p Fo(\003)p Fm(j)1831 2575 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)2001 2605 y Fp( )j Fm(2)d Fp(L)2238 2575 y Fl(2)2275 2605 y Fo(\(\003\))27 b Ft(is)g(the)f(normalised)e(ground-state)456 2705 y(eigenfunction)k (of)i Fp(H)1120 2665 y Fi(\037)1113 2729 y Fl(\003)1166 2705 y Fo(\(0\))h Ft(which)f(satis\002es)j Fp(H)1908 2665 y Fi(\037)1901 2729 y Fl(\003)1953 2705 y Fo(\(0\))p Fp( )2113 2717 y Fi(L)2206 2705 y Fo(=)42 b(0)p Ft(.)57 b(Choosing)30 b(this)h(function)e(as)j(the)456 2805 y(v)n(ariational)18 b(function)h(in)h(T)-6 b(emple')h(s)20 b(inequality)f([)p Fq(RS78)o Ft(,)h(Thm.)g(XIII.5])e(yields)i(the)g(lo)n(wer)g(bound)845 3007 y Fp(\025)893 3019 y Fl(0)931 2940 y Fg(\000)969 3007 y Fp(H)1045 2967 y Fi(\037)1038 3032 y Fl(\003)1091 3007 y Fo(\()p Fp(V)1171 3028 y Fi(!)r(;h)1278 3007 y Fo(\))1310 2940 y Fg(\001)1371 3007 y Fm(\025)1459 2940 y Fg(\012)1498 3007 y Fp( )1552 3019 y Fi(L)1602 3007 y Fp(;)14 b(V)1687 3028 y Fi(!)r(;h)1808 3007 y Fp( )1862 3019 y Fi(L)1911 2940 y Fg(\013)1969 3007 y Fm(\000)2223 2873 y Fg(\012)2262 2940 y Fp(V)2310 2963 y Fi(!)r(;h)2431 2940 y Fp( )2485 2952 y Fi(L)2535 2940 y Fp(;)g(V)2620 2963 y Fi(!)r(;h)2741 2940 y Fp( )2795 2952 y Fi(L)2845 2873 y Fg(\013)p 2062 2988 983 4 v 2062 3069 a Fp(\025)2110 3081 y Fl(1)2148 3001 y Fg(\000)2186 3069 y Fp(H)2262 3029 y Fi(\037)2255 3093 y Fl(\003)2307 3069 y Fo(\(0\))2413 3001 y Fg(\001)2470 3069 y Fm(\000)2553 3001 y Fg(\012)2592 3069 y Fp( )2646 3081 y Fi(L)2696 3069 y Fp(;)g(V)2781 3094 y Fi(!)r(;h)2902 3069 y Fp( )2956 3081 y Fi(L)3005 3001 y Fg(\013)3306 3007 y Ft(\(36\))456 3209 y(pro)o(vided)21 b(the)j(denominator)e(in)i(\(36\))f(is)i(strictly)f(positi)n(v)o(e.)36 b(T)-7 b(o)24 b(check)f(this)i(we)f(note)g(that)g([)p Fq(Mez87)o Ft(,)456 3309 y(Prop.)19 b(4])h(implies)g(that)g(there)g(is) h(some)f(constant)g Fp(c)1947 3321 y Fl(0)2007 3309 y Fp(>)j Fo(0)d Ft(such)g(that)1075 3447 y Fp(\025)1123 3459 y Fl(1)1160 3379 y Fg(\000)1198 3447 y Fp(H)1274 3407 y Fi(\037)1267 3471 y Fl(\003)1320 3447 y Fo(\(0\))1426 3379 y Fg(\001)1487 3447 y Fo(=)i Fp(\025)1622 3459 y Fl(1)1660 3379 y Fg(\000)1698 3447 y Fp(H)1774 3407 y Fi(\037)1767 3471 y Fl(\003)1820 3447 y Fo(\(0\))1926 3379 y Fg(\001)1982 3447 y Fm(\000)c Fp(\025)2113 3459 y Fl(0)2151 3379 y Fg(\000)2189 3447 y Fp(H)2265 3407 y Fi(\037)2258 3471 y Fl(\003)2310 3447 y Fo(\(0\))2416 3379 y Fg(\001)2478 3447 y Fm(\025)k Fo(2)p Fp(c)2643 3459 y Fl(0)2680 3447 y Fp(L)2737 3412 y Fj(\000)p Fl(2)3306 3447 y Ft(\(37\))456 3593 y(for)32 b(all)i Fp(L)46 b(>)g Fo(1)p Ft(.)64 b(Moreo)o(v)o(er)m(,)33 b(we)g(estimate)1863 3526 y Fg(\012)1902 3593 y Fp( )1956 3605 y Fi(L)2006 3593 y Fp(;)14 b(V)2091 3616 y Fi(!)r(;h)2212 3593 y Fp( )2266 3605 y Fi(L)2316 3526 y Fg(\013)2402 3593 y Fm(\024)46 b Fo(3)2555 3563 y Fi(d)2593 3593 y Fp(f)2634 3605 y Fi(u)2691 3593 y Fp(h)h Fm(\024)f Fp(c)2933 3605 y Fl(0)2970 3593 y Fp(L)3027 3563 y Fj(\000)p Fl(2)3150 3593 y Ft(for)32 b(lar)o(ge)456 3693 y(enough)g Fp(r)772 3705 y Fl(0)858 3693 y Fp(>)49 b Fo(0)p Ft(.)67 b(T)-7 b(o)34 b(bound)e(the)i(numerator)e(in)i(\(50\))f(from)h(abo)o(v)o(e,)h (we)g(use)f(the)g(inequality)456 3725 y Fg(\012)495 3792 y Fp(V)543 3816 y Fi(!)r(;h)664 3792 y Fp( )718 3804 y Fi(L)768 3792 y Fp(;)14 b(V)853 3816 y Fi(!)r(;h)973 3792 y Fp( )1027 3804 y Fi(L)1077 3725 y Fg(\013)1154 3792 y Fm(\024)1256 3725 y Fg(\012)1295 3792 y Fp( )1349 3804 y Fi(L)1399 3792 y Fp(;)g(V)1484 3816 y Fi(!)r(;h)1605 3792 y Fp( )1659 3804 y Fi(L)1709 3725 y Fg(\013)1762 3792 y Fo(3)1804 3762 y Fi(d)1842 3792 y Fp(f)1883 3804 y Fi(u)1926 3792 y Fp(h)37 b Fm(\024)2113 3725 y Fg(\012)2153 3792 y Fp( )2207 3804 y Fi(L)2256 3792 y Fp(;)14 b(V)2341 3816 y Fi(!)r(;h)2462 3792 y Fp( )2516 3804 y Fi(L)2566 3725 y Fg(\013)2619 3792 y Fp(c)2655 3804 y Fl(0)2692 3792 y Fp(=)p Fo(\(2)p Fp(L)2865 3762 y Fl(2)2901 3792 y Fo(\))29 b Ft(v)n(alid)f(for)f(lar)o(ge)456 3892 y(enough)18 b Fp(r)758 3904 y Fl(0)819 3892 y Fp(>)k Fo(0)p Ft(.)2411 b Fd(\003)605 4047 y Ft(W)-7 b(e)29 b(proceed)d(by)h(constructing)f(a)i (lo)n(wer)g(bound)e(on)h(the)h(right-hand)d(side)j(of)f(\(36\).)47 b(F)o(or)27 b(this)456 4146 y(purpose)18 b(we)j(de\002ne)e(the)i(cube) 1680 4242 y Fg(e)1675 4263 y Fo(\003)h(:=)1946 4184 y Fg([)1866 4366 y Fj(j)p Fi(j)s Fj(j)p Fi()h Fo(1)d Ff(and)f(all)i Fp(h)i(>)f Fo(0)p Ff(.)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(13)607 450 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(Pulling)24 b(out)g(the)g(strictly)h(positi)n(v)o(e)f(in\002mum)f(of)i Fp( )2428 420 y Fl(2)2490 450 y Ft(and)f(using)g(its)i Fn(Z)3003 420 y Fi(d)3042 450 y Ft(-periodicity)-5 b(,)456 550 y(we)20 b(estimate)768 602 y Fg(Z)815 791 y Fl(\003)878 715 y Fp(V)926 736 y Fi(!)r(;h)1033 715 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1326 681 y Fl(2)1379 715 y Fp(dx)84 b Fm(\025)110 b Fo(inf)1700 769 y Fi(z)r Fj(2)p Fl(\003)1824 777 y Fv(0)1871 715 y Fp( )s Fo(\()p Fp(z)t Fo(\))2035 681 y Fl(2)2086 715 y Fp(f)2127 727 y Fi(u)2184 602 y Fg(Z)2230 791 y Fe(R)2284 774 y Fk(d)2337 715 y Fm(j)p Fo(\003)18 b Fm(\\)h Fo(\()p Fp(F)30 b Fo(+)18 b Fp(y)s Fo(\))p Fm(j)28 b Fp(\026)2885 681 y Fl(\()p Fi(h)p Fl(\))2885 736 y Fi(!)2980 715 y Fo(\()p Fp(dy)s Fo(\))1553 905 y Fm(\025)110 b Fo(inf)1700 959 y Fi(z)r Fj(2)p Fl(\003)1824 967 y Fv(0)1871 905 y Fp( )s Fo(\()p Fp(z)t Fo(\))2035 871 y Fl(2)2086 905 y Fp(f)2127 917 y Fi(u)2184 905 y Fm(j)p Fp(F)12 b Fm(j)23 b Fp(\026)2368 871 y Fl(\()p Fi(h)p Fl(\))2368 926 y Fi(!)2463 838 y Fg(\000)2507 884 y(e)2501 905 y Fo(\003)2558 838 y Fg(\001)3306 905 y Ft(\(40\))456 1083 y(by)d(omitting)f(positi)n(v)o(e)h(terms)h (and)f(using)g(Fubini')-5 b(s)20 b(theorem)f(together)g(with)i(the)f(f) o(act)h(that)g Fp(F)35 b Fm(\032)24 b Fo(\003)3387 1095 y Fl(0)3424 1083 y Ft(.)456 1183 y(The)19 b(proof)g(is)i(completed)e (with)h(the)g(help)g(of)g(the)g(inequality)625 1348 y Fp(\026)675 1314 y Fl(\()p Fi(h)p Fl(\))675 1368 y Fi(!)770 1281 y Fg(\000)814 1327 y(e)808 1348 y Fo(\003)866 1281 y Fg(\001)927 1348 y Fo(=)1078 1269 y Fg(X)1015 1464 y Fi(j)s Fj(2)p Fe(Z)1136 1447 y Fk(d)1171 1464 y Fj(\\)1220 1449 y Fb(e)1216 1464 y Fl(\003)1275 1348 y Fo(min)1428 1281 y Fg(\010)1476 1348 y Fp(h;)14 b(\026)1611 1360 y Fi(!)1659 1348 y Fo(\(\003)1749 1360 y Fi(j)1784 1348 y Fo(\))1816 1281 y Fg(\011)1887 1348 y Fm(\025)23 b Fp(h)g Fo(#)2115 1256 y Fg(n)2171 1348 y Fp(j)28 b Fm(2)23 b Fn(Z)2370 1314 y Fi(d)2427 1348 y Fm(\\)2507 1327 y Fg(e)2501 1348 y Fo(\003)37 b(:)g Fp(\026)2706 1360 y Fi(!)2754 1281 y Fg(\000)2792 1348 y Fo(\003)2850 1360 y Fi(j)2884 1281 y Fg(\001)2946 1348 y Fm(\025)22 b Fp(h)3081 1256 y Fg(o)3306 1348 y Ft(\(41\))456 1616 y(and)d Fm(j)p Fo(\003)p Fm(j)k(\024)g Fo(3)853 1586 y Fi(d)891 1616 y Fm(j)920 1595 y Fg(e)914 1616 y Fo(\003)p Fm(j)d Ft(v)n(alid)g(for)g (all)h Fp(L)h(>)h Fo(1)p Ft(.)1728 b Fd(\003)605 1777 y Ft(4.2.2.)39 b Ff(Pr)l(oof)18 b(of)f(Theor)m(em)g(2.8)f(\226)i(\002r) o(st)g(part:)23 b(quantum)16 b(r)m(e)m(gime.)40 b Ft(W)-7 b(e)18 b(\002x)g Fp(r)2810 1789 y Fl(0)2871 1777 y Fp(>)k Fo(0)c Ft(lar)o(ge)e(enough)456 1877 y(to)k(ensure)f(the)i(v)n(alidity) e(of)h(\(35\))f(in)h(Proposition)f(4.3.)24 b(F)o(or)c(a)h(gi)n(v)o(en)e (ener)o(gy)f Fp(E)28 b(>)23 b Fo(0)d Ft(we)g(then)g(pick)1639 2090 y Fp(L)j Fo(:=)1830 1973 y Fg(\022)1949 2034 y Fp(c)1985 2046 y Fl(1)p 1901 2071 185 4 v 1901 2147 a Fo(4)p Fp(r)1982 2118 y Fl(2)1980 2169 y(0)2019 2147 y Fp(E)2095 1973 y Fg(\023)2156 1990 y Fl(1)p Fi(=)p Fl(2)3306 2090 y Ft(\(42\))456 2277 y(where)g(the)h(constant)f Fp(c)1146 2289 y Fl(1)1208 2277 y Ft(has)h(been)f(\002x)o(ed)g(in)h(Lemma)f(4.4.) 36 b(Finally)-5 b(,)24 b(we)g(choose)f(the)h(cube)f Fo(\003)h Ft(from)456 2377 y(\(32\))19 b(and)g(set)i Fp(h)i Fo(:=)g(\()p Fp(r)1119 2389 y Fl(0)1157 2377 y Fp(L)p Fo(\))1246 2346 y Fj(\000)p Fl(2)1335 2377 y Ft(.)i(Proposition)19 b(4.3)g(and)h (\(39\))f(yield)h(the)g(estimate)927 2544 y Fn(P)992 2452 y Fg(n)1048 2544 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\025)1409 2556 y Fl(0)1460 2544 y Fo(\()q Fp(H)1569 2504 y Fi(\037)1562 2569 y Fl(\003)1614 2544 y Fo(\()p Fp(V)1694 2556 y Fi(!)1743 2544 y Fo(\)\))23 b Fp(<)g(E)1984 2452 y Fg(o)1093 2777 y Fm(\024)g Fn(P)1246 2635 y Fg(\()1313 2777 y Fp(!)j Fm(2)d Fo(\012)37 b(:)g(#)1695 2685 y Fg(n)1750 2777 y Fp(j)28 b Fm(2)c Fn(Z)1950 2742 y Fi(d)2007 2777 y Fm(\\)2087 2756 y Fg(e)2081 2777 y Fo(\003)37 b(:)g Fp(\026)2286 2789 y Fi(!)2334 2709 y Fg(\000)2372 2777 y Fo(\003)2430 2789 y Fi(j)2464 2709 y Fg(\001)2526 2777 y Fm(\025)22 b Fp(h)2661 2685 y Fg(o)2739 2777 y Fp(<)2844 2720 y Fo(2)p Fp(E)p 2837 2758 122 4 v 2837 2834 a(c)2873 2846 y Fl(1)2910 2834 y Fp(h)2968 2777 y Fm(j)2997 2756 y Fg(e)2991 2777 y Fo(\003)p Fm(j)3072 2635 y Fg(\))1093 3059 y Fo(=)h Fn(P)1246 2917 y Fg(\()1313 3059 y Fp(!)j Fm(2)d Fo(\012)37 b(:)g(#)1695 2967 y Fg(n)1750 3059 y Fp(j)28 b Fm(2)c Fn(Z)1950 3025 y Fi(d)2007 3059 y Fm(\\)2087 3038 y Fg(e)2081 3059 y Fo(\003)37 b(:)g Fp(\026)2286 3071 y Fi(!)2334 2992 y Fg(\000)2372 3059 y Fo(\003)2430 3071 y Fi(j)2464 2992 y Fg(\001)2526 3059 y Fp(<)22 b(h)2661 2967 y Fg(o)2739 3059 y Fp(>)2837 3003 y Fm(j)2866 2982 y Fg(e)2860 3003 y Fo(\003)p Fm(j)p 2837 3040 104 4 v 2868 3116 a Fo(2)2951 2917 y Fg(\))3018 3059 y Fp(:)265 b Ft(\(43\))456 3276 y(Here)24 b(the)h(last)g(equality)e(uses)j(the)e (f)o(act)h(that)f Fp(h)31 b Fo(=)g(4)p Fp(E)5 b(=c)2155 3288 y Fl(1)2191 3276 y Ft(.)39 b(In)24 b(case)p 2513 3231 51 4 v 25 w Fp(\026)p Fo(\(\003)2653 3288 y Fi(j)2688 3276 y Fo(\))31 b Fp(>)g(h)p Ft(,)26 b(that)e(is,)i(for)e(suf-)456 3376 y(\002ciently)e(small)g Fp(E)5 b Ft(,)23 b(the)g(right-hand)c (side)k(is)g(the)f(probability)e(of)i(a)h(lar)o(ge)e(de)n(viation)g(e)n (v)o(ent)g([)p Fq(DZ98)o Ft(].)456 3476 y(Consequently)c(\(confer)g([)p Fq(KS86)n Ft(,)i(Prop.)f(4]\),)g(there)h(e)o(xists)g(a)g(constant)f Fo(0)23 b Fp(<)f(c)2719 3488 y Fl(2)2780 3476 y Fp(<)g Fm(1)p Ft(,)e(such)e(that)h(\(43\))456 3575 y(is)i(estimated)f(from)f (abo)o(v)o(e)f(by)1051 3740 y Fo(exp)1191 3648 y Fg(h)1231 3740 y Fm(\000)p Fp(c)1332 3752 y Fl(2)1368 3740 y Fm(j)1397 3719 y Fg(e)1391 3740 y Fo(\003)p Fm(j)1472 3648 y Fg(i)1534 3740 y Fm(\024)23 b Fo(exp)1763 3673 y Fg(\002)1797 3740 y Fm(\000)p Fp(c)1898 3752 y Fl(2)1935 3740 y Fp(n)1985 3752 y Fi(u)2028 3740 y Fp(L)2085 3706 y Fi(d)2123 3673 y Fg(\003)2181 3740 y Fo(=)g(exp)2409 3648 y Fg(h)2449 3740 y Fm(\000)p Fp(c)2550 3752 y Fl(3)2586 3740 y Fp(E)2652 3706 y Fj(\000)p Fi(d=)p Fl(2)2810 3648 y Fg(i)3306 3740 y Ft(\(44\))456 3926 y(Here)f(the)g(inequality)f(follo)n(ws)h(from)f (the)i(estimate)f Fm(j)2026 3905 y Fg(e)2020 3926 y Fo(\003)p Fm(j)27 b(\025)g Fp(n)2270 3938 y Fi(u)2313 3926 y Fp(L)2370 3896 y Fi(d)2431 3926 y Ft(for)22 b(some)g(constant)f Fp(n)3098 3938 y Fi(u)3169 3926 y Fp(>)26 b Fo(0)d Ft(and)456 4026 y(all)k Fp(L)33 b(>)h Fo(2)p Ft(.)44 b(The)26 b(e)o(xistence)f(of) h(a)h(constant)f Fp(c)1852 4038 y Fl(3)1923 4026 y Fp(>)34 b Fo(0)27 b Ft(ensuring)d(the)j(v)n(alidity)e(of)h(the)h(last)g (equality)456 4126 y(follo)n(ws)20 b(from)g(\(42\).)27 b(Inserting)19 b(this)j(estimate)f(in)g(the)g(right-hand)e(side)i(of)g (\(23\))f(completes)g(the)h(\002rst)456 4225 y(part)h(of)g(the)h(proof) e(of)h(Theorem)f(2.8)h(for)g(the)h(quantum-classical)e(re)o(gime,)h (since)g(the)h(pre-f)o(actor)e(in)456 4325 y(the)f(upper)f(bound)f(in)i (Proposition)f(3.2)g(is)j(ne)o(gligible.)1320 b Fd(\003)605 4491 y Fq(4.3.)40 b(Quantum-classical)20 b(r)o(egime.)40 b Ft(W)m(ithout)19 b(loss)i(of)f(generality)f(we)i(suppose)e(that)i Fc(\(qm/cl\))456 4591 y Ft(holds)g(throughout)d(this)k(subsection,)f (that)h(is)g Fp(d)1847 4603 y Fl(1)1885 4591 y Fp(=)p Fo(2)i Fm(\025)i Fp(\015)2127 4603 y Fl(1)2164 4591 y Fp(=)p Fo(\(1)19 b Fm(\000)g Fp(\015)5 b Fo(\))22 b Ft(and)f Fp(d)2670 4603 y Fl(2)2707 4591 y Fp(=)p Fo(2)k Fp(<)g(\015)2949 4603 y Fl(2)2986 4591 y Fp(=)p Fo(\(1)19 b Fm(\000)g Fp(\015)5 b Fo(\))p Ft(.)30 b(W)-7 b(e)456 4690 y(start)29 b(by)g(constructing)e(a)i(lo)n(wer)f(bound)f(on)i(the)g(lo)n(west)g (Mezincescu)f(eigen)m(v)n(alue)f Fp(\025)3046 4702 y Fl(0)3097 4690 y Fo(\()q Fp(H)3206 4650 y Fi(\037)3199 4715 y Fl(\003)3251 4690 y Fo(\()p Fp(V)3331 4702 y Fi(!)3380 4690 y Fo(\)\))456 4790 y Ft(sho)n(wing)19 b(up)g(in)i(the)f (right-hand)e(side)i(of)g(\(23\))f(when)h(choosing)1588 4982 y Fo(\003)j(:=)p 1780 4884 437 4 v 1832 4903 a Fg([)1780 5085 y Fj(j)p Fi(j)1827 5093 y Fv(1)1859 5085 y Fj(j)p Fi()g Fo(1)p Ft(.)i(By)20 b(construction)e(it)j(is)g(open)f(and)f (compatible)g(with)h(the)g(lattice.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 456 251 a Fr(14)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)605 450 y Ft(4.3.1.)39 b Ff(Lower)31 b(bound)e(on)g(the)i(lowest)g(Mezincescu)f(eig)o(en)m (value.)39 b Ft(From)29 b(Lemma)h(4.2\(i\))f(we)456 550 y(conclude)18 b(that)i(for)g(e)n(v)o(ery)f Fp(R)k(>)g Fo(0)d Ft(and)g Fp(!)26 b Fm(2)d Fo(\012)e Ft(the)f(potential)f Fp(V)2336 573 y Fi(!)r(;R)2478 550 y Fo(:)k Fn(R)2594 520 y Fi(d)2656 550 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)20 b Ft(gi)n(v)o(en)f(by)1348 740 y Fp(V)1396 761 y Fi(!)r(;R)1515 740 y Fo(\()p Fp(x)p Fo(\))24 b(:=)1761 627 y Fg(Z)1807 816 y Fj(j)p Fi(y)1861 824 y Fv(2)1893 816 y Fj(j)p Fi(>R)1940 740 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(\026)2311 706 y Fl(\(1\))2311 761 y Fi(!)2400 740 y Fo(\()p Fp(dy)s Fo(\))755 b Ft(\(46\))456 961 y(in)22 b(terms)h(of)g(the)f(re)o(gularised)f(Borel)i(measure)f Fp(\026)1917 917 y Fl(\(1\))1917 970 y Fi(!)2006 961 y Ft(,)i(pro)o(vides)d(a)i(lo)n(wer)f(bound)f(on)i Fp(V)3015 973 y Fi(!)3063 961 y Ft(.)33 b(Therefore)456 1060 y Fp(\025)504 1072 y Fl(0)555 1060 y Fo(\()p Fp(H)663 1020 y Fi(\037)656 1085 y Fl(\003)709 1060 y Fo(\()p Fp(V)789 1072 y Fi(!)838 1060 y Fo(\)\))23 b Fm(\025)g Fp(\025)1061 1072 y Fl(0)1099 993 y Fg(\000)1137 1060 y Fp(H)1213 1020 y Fi(\037)1206 1085 y Fl(\003)1258 1060 y Fo(\()p Fp(V)1338 1083 y Fi(!)r(;R)1457 1060 y Fo(\))1489 993 y Fg(\001)1527 1060 y Ft(.)j(It)20 b(will)h(be)f(useful)g(to)g(collect) g(some)g(f)o(acts)h(related)f(to)g Fp(V)3198 1072 y Fi(!)r(;R)3316 1060 y Ft(.)605 1223 y Fq(Lemma)h(4.5.)40 b Ff(Let)21 b Fp(R)j(>)e Fo(1)e Ff(and)g(de\002ne)f Fp(V)1840 1246 y Fi(R)1918 1223 y Fo(:)k Fn(R)2034 1193 y Fi(d)2095 1223 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ff(by)1375 1377 y Fp(V)1423 1397 y Fi(R)1478 1377 y Fo(\()p Fp(x)p Fo(\))j(:=)1807 1298 y Fg(X)1758 1490 y Fi(j)1785 1498 y Fv(1)1818 1490 y Fj(2)p Fe(Z)1909 1465 y Fk(d)1940 1477 y Fv(1)1724 1553 y Fj(j)p Fi(j)1771 1561 y Fv(2)1803 1553 y Fj(j)p Fi(>R)p Fj(\000)p Fl(1)2040 1377 y Fo(sup)2024 1446 y Fi(y)r Fj(2)p Fl(\003)2150 1454 y Fk(j)2195 1377 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))p Fp(:)781 b Ft(\(47\))456 1676 y Ff(Then)19 b(the)i(following)e(thr)m (ee)h(assertions)h(hold)e(true:)543 1813 y(\(i\))83 b Fp(V)753 1835 y Fi(!)r(;R)894 1813 y Fm(\024)23 b Fp(V)1030 1835 y Fi(R)1106 1813 y Ff(for)d(e)o(very)g Fp(!)26 b Fm(2)e Fo(\012)p Ff(.)520 1980 y(\(ii\))83 b Fp(V)753 2003 y Fi(R)828 1980 y Ff(is)21 b Fn(Z)963 1950 y Fi(d)998 1958 y Fv(1)1035 1980 y Ff(-periodic)e(with)i(r)m(espect)f(to)h(tr)o (anslations)e(in)h(the)g Fp(x)2537 1992 y Fl(1)2575 1980 y Ff(-dir)m(ection.)497 2140 y(\(iii\))83 b(ther)m(e)16 b(e)n(xists)i(some)e(constant)f Fp(c)23 b(>)g Fo(0)16 b Ff(suc)o(h)g(that)g Fo(sup)2218 2161 y Fi(x)p Fj(2)p Fl(\003)2346 2169 y Fv(0)2396 2140 y Fp(V)2444 2163 y Fi(R)2499 2140 y Fo(\()p Fp(x)p Fo(\))24 b Fm(\024)f Fp(c)14 b(R)2836 2110 y Fj(\000)p Fi(\013)2931 2118 y Fv(2)2963 2110 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3159 2140 y Ff(for)j(lar)m(g)o(e)705 2240 y(enough)h Fp(R)24 b(>)e Fo(1)p Ff(.)607 2389 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)20 b(\002rst)g(assertion)g(follo)n(ws)g(from)f(the)h (inequalities)1229 2564 y Fp(V)1277 2584 y Fi(!)r(;R)1396 2564 y Fo(\()p Fp(x)p Fo(\))k Fm(\024)1702 2485 y Fg(X)1653 2677 y Fi(j)1680 2685 y Fv(1)1713 2677 y Fj(2)p Fe(Z)1804 2652 y Fk(d)1835 2664 y Fv(1)1619 2741 y Fj(j)p Fi(j)1666 2749 y Fv(2)1698 2741 y Fj(j)p Fi(>R)p Fj(\000)p Fl(1)1919 2451 y Fg(Z)1965 2640 y Fl(\003)2010 2648 y Fk(j)2059 2564 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(\026)2430 2530 y Fl(\(1\))2430 2584 y Fi(!)2519 2564 y Fo(\()p Fp(dy)s Fo(\))636 b Ft(\(48\))456 2894 y(and)21 b Fp(\026)648 2850 y Fl(\(1\))648 2903 y Fi(!)737 2894 y Fo(\(\003)827 2906 y Fi(j)862 2894 y Fo(\))27 b Fm(\024)f Fo(1)c Ft(v)n(alid)g(for)f(all)i Fp(!)29 b Fm(2)e Fo(\012)p Ft(.)k(The)21 b(second)g(assertion)h(holds)g(true)f(by)h(de\002nition.) 29 b(The)456 2993 y(third)19 b(assertion)h(deri)n(v)o(es)f(from)g (\(12\))g(and)h(is)h(the)f(\223summation\224)f(analogue)f(of)i(Lemma)g (3.5.)219 b Fd(\003)605 3143 y Ft(The)24 b(cut-of)n(f)f Fp(R)j Ft(guarantees)d(that)i(the)g(potential)f Fp(V)2118 3166 y Fi(!)r(;R)2261 3143 y Ft(does)h(not)f(e)o(xceed)f(a)j(certain)e (v)n(alue.)37 b(In)456 3243 y(particular)m(,)21 b(taking)h Fp(R)i Ft(lar)o(ge)d(enough)g(ensures)h(that)h(this)g(v)n(alue)f(is)h (smaller)g(than)f(the)h(ener)o(gy)d(dif)n(fer)n(-)456 3342 y(ence)28 b(of)g(the)h(lo)n(west)g(and)f(the)h(\002rst)g(eigen)m (v)n(alue)d(of)j Fp(H)2106 3302 y Fi(\037)2099 3367 y Fl(\003)2151 3342 y Fo(\(0\))p Ft(.)51 b(This)29 b(enables)f(one)g(to)h (mak)o(e)f(use)h(of)456 3442 y(T)-6 b(emple')h(s)24 b(inequality)g(to)i (obtain)e(a)h(lo)n(wer)g(bound)e(on)i(the)g(lo)n(west)h(Mezincescu)e (eigen)m(v)n(alue)f(in)i(the)456 3541 y(quantum-classical)18 b(re)o(gime.)605 3703 y Fq(Pr)o(oposition)23 b(4.6.)43 b Ff(Let)25 b Fo(\003)g Ff(denote)e(the)i(cuboid)e(\(45\).)37 b(Mor)m(eo)o(ver)-9 b(,)25 b(let)g Fp(R)32 b Fo(:=)e(\()q Fp(r)2996 3715 y Fl(0)3033 3703 y Fp(L)p Fo(\))3122 3661 y Fl(2)p Fi(=\013)3232 3669 y Fv(2)3265 3661 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))456 3803 y Ff(with)19 b Fp(r)655 3815 y Fl(0)716 3803 y Fp(>)j Fo(0)p Ff(.)j(Then)18 b(the)g(lowest)i (eig)o(en)m(value)c(of)j Fp(H)1962 3763 y Fi(\037)1955 3827 y Fl(\003)2008 3803 y Fo(\()p Fp(V)2088 3825 y Fi(!)r(;R)2206 3803 y Fo(\))h Ff(is)f(bounded)d(fr)l(om)k(below)e(accor)m(ding)f(to) 1185 3993 y Fp(\025)1233 4005 y Fl(0)1270 3926 y Fg(\000)1308 3993 y Fp(H)1384 3953 y Fi(\037)1377 4018 y Fl(\003)1430 3993 y Fo(\()p Fp(V)1510 4014 y Fi(!)r(;R)1629 3993 y Fo(\))1661 3926 y Fg(\001)1722 3993 y Fm(\025)1878 3937 y Fo(1)p 1820 3974 160 4 v 1820 4050 a(2)d Fm(j)p Fo(\003)p Fm(j)2003 3880 y Fg(Z)2049 4069 y Fl(\003)2112 3993 y Fp(V)2160 4014 y Fi(!)r(;R)2279 3993 y Fo(\()p Fp(x)p Fo(\))g Fp( )s Fo(\()p Fp(x)p Fo(\))2572 3959 y Fl(2)2625 3993 y Fp(dx)591 b Ft(\(49\))456 4172 y Ff(for)28 b(all)g Fp(!)41 b Fm(2)d Fo(\012)p Ff(,)30 b(all)f Fp(L)37 b(>)g Fo(1)28 b Ff(and)f(lar)m(g)o(e)h(enough)e Fp(r)2040 4184 y Fl(0)2116 4172 y Fp(>)37 b Fo(0)p Ff(.)49 b Ft([)p Ff(Recall)27 b(the)h(de\002nition)f(of)h Fp( )k Ff(at)c(the)456 4272 y(be)m(ginning)17 b(of)j(Subsection)f(3.1.)p Ft(])607 4421 y(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)35 b(proof)f(parallels)i (the)f(one)h(of)f(Proposition)f(4.6.)71 b(By)36 b(construction)e Fp( )3255 4433 y Fi(L)3357 4421 y Fo(:=)456 4521 y Fm(j)p Fo(\003)p Fm(j)560 4491 y Fj(\000)p Fl(1)p Fi(=)p Fl(2)729 4521 y Fp( )27 b Fm(2)c Fp(L)945 4491 y Fl(2)982 4521 y Fo(\(\003\))d Ft(is)g(the)g(normalised)e(ground-state)f (eigenfunction)g(of)i Fp(H)2787 4481 y Fi(\037)2780 4545 y Fl(\003)2832 4521 y Fo(\(0\))h Ft(which)f(satis\002es)456 4621 y Fp(H)532 4581 y Fi(\037)525 4645 y Fl(\003)577 4621 y Fo(\(0\))p Fp( )737 4633 y Fi(L)817 4621 y Fo(=)29 b(0)p Ft(.)36 b(Choosing)23 b(this)h(function)e(as)i(the)g(v)n (ariational)f(function)f(in)i(T)-6 b(emple')h(s)23 b(inequality)456 4720 y([)p Fq(RS78)n Ft(,)e(Thm.)e(XIII.5])g(yields)h(the)g(lo)n(wer)g (bound)828 4919 y Fp(\025)876 4931 y Fl(0)914 4852 y Fg(\000)952 4919 y Fp(H)1028 4879 y Fi(\037)1021 4943 y Fl(\003)1073 4919 y Fo(\()p Fp(V)1153 4939 y Fi(!)r(;R)1272 4919 y Fo(\))1304 4852 y Fg(\001)1366 4919 y Fm(\025)1453 4852 y Fg(\012)1492 4919 y Fp( )1546 4931 y Fi(L)1596 4919 y Fp(;)14 b(V)1681 4939 y Fi(!)r(;R)1813 4919 y Fp( )1867 4931 y Fi(L)1917 4852 y Fg(\013)1975 4919 y Fm(\000)2223 4785 y Fg(\012)2262 4852 y Fp(V)2310 4875 y Fi(!)r(;R)2443 4852 y Fp( )2497 4864 y Fi(L)2546 4852 y Fp(;)g(V)2631 4875 y Fi(!)r(;R)2764 4852 y Fp( )2818 4864 y Fi(L)2867 4785 y Fg(\013)p 2068 4900 994 4 v 2068 4980 a Fp(\025)2116 4992 y Fl(1)2154 4913 y Fg(\000)2192 4980 y Fp(H)2268 4940 y Fi(\037)2261 5005 y Fl(\003)2313 4980 y Fo(\(0\))2419 4913 y Fg(\001)2476 4980 y Fm(\000)2559 4913 y Fg(\012)2598 4980 y Fp( )2652 4992 y Fi(L)2702 4980 y Fp(;)g(V)2787 5005 y Fi(!)r(;R)2919 4980 y Fp( )2973 4992 y Fi(L)3023 4913 y Fg(\013)3306 4919 y Ft(\(50\))456 5116 y(pro)o(vided)21 b(the)j(denominator)e(in)i(\(50\))f(is)i (strictly)f(positi)n(v)o(e.)36 b(T)-7 b(o)24 b(check)g(this)g(we)h (note)e(that)h(a)h(simple)456 5216 y(e)o(xtension)e(of)i([)p Fq(Mez87)n Ft(,)i(Prop.)d(4])g(from)g(cubes)g(to)h(cuboids)f(implies)h (that)g(there)f(is)i(some)f(constant)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(15)456 450 y Fp(c)492 462 y Fl(0)569 450 y Fp(>)39 b Fo(0)29 b Ft(such)g(that)h Fp(\025)1129 462 y Fl(1)1166 383 y Fg(\000)1204 450 y Fp(H)1280 410 y Fi(\037)1273 475 y Fl(\003)1326 450 y Fo(\(0\))1432 383 y Fg(\001)1510 450 y Fo(=)39 b Fp(\025)1662 462 y Fl(1)1700 383 y Fg(\000)1738 450 y Fp(H)1814 410 y Fi(\037)1807 475 y Fl(\003)1859 450 y Fo(\(0\))1965 383 y Fg(\001)2029 450 y Fm(\000)24 b Fp(\025)2166 462 y Fl(0)2204 383 y Fg(\000)2242 450 y Fp(H)2318 410 y Fi(\037)2311 475 y Fl(\003)2363 450 y Fo(\(0\))2469 383 y Fg(\001)2547 450 y Fm(\025)40 b Fo(2)p Fp(c)2730 462 y Fl(0)2766 450 y Fp(L)2823 420 y Fj(\000)p Fl(2)2942 450 y Ft(for)29 b(all)g Fp(L)40 b(>)f Fo(1)p Ft(.)456 550 y(Moreo)o(v)o(er)m(,)17 b(using)i(Lemma)h(4.5)f(and)h(the)g(de\002nition)f(of)h Fp(R)h Ft(we)g(estimate)572 657 y Fg(\012)611 724 y Fp( )665 736 y Fi(L)715 724 y Fp(;)14 b(V)800 745 y Fi(!)r(;R)932 724 y Fp( )986 736 y Fi(L)1036 657 y Fg(\013)1098 724 y Fm(\024)1186 657 y Fg(\012)1225 724 y Fp( )1279 736 y Fi(L)1329 724 y Fp(;)g(V)1414 745 y Fi(R)1482 724 y Fp( )1536 736 y Fi(L)1586 657 y Fg(\013)1648 724 y Fo(=)1736 611 y Fg(Z)1782 800 y Fl(\003)1827 808 y Fv(0)1878 724 y Fp(V)1926 745 y Fi(R)1981 724 y Fo(\()p Fp(x)p Fo(\))g Fp( )s Fo(\()p Fp(x)p Fo(\))2274 690 y Fl(2)2313 724 y Fp(dx)24 b Fm(\024)e Fp(c)2564 657 y Fg(\000)2602 724 y Fp(r)2639 736 y Fl(0)2677 724 y Fp(L)2734 657 y Fg(\001)2771 674 y Fj(\000)p Fl(2)2884 724 y Fm(\024)g Fp(c)3007 736 y Fl(0)3044 724 y Fp(L)3101 690 y Fj(\000)p Fl(2)3306 724 y Ft(\(51\))456 908 y(for)29 b(lar)o(ge)g(enough)e Fp(r)1089 920 y Fl(0)1168 908 y Fp(>)40 b Fo(0)p Ft(.)54 b(T)-7 b(o)30 b(bound)e(the)i(numerator)d(in)j(\(50\))f(from)g(abo)o(v) o(e,)h(we)g(use)g(the)g(in-)456 1008 y(equality)749 940 y Fg(\012)788 1008 y Fp(V)836 1030 y Fi(!)r(;R)968 1008 y Fp( )1022 1020 y Fi(L)1072 1008 y Fp(;)14 b(V)1157 1030 y Fi(!)r(;R)1289 1008 y Fp( )1343 1020 y Fi(L)1393 940 y Fg(\013)1464 1008 y Fm(\024)1560 940 y Fg(\012)1599 1008 y Fp( )1653 1020 y Fi(L)1703 1008 y Fp(;)g(V)1788 1030 y Fi(!)r(;R)1920 1008 y Fp( )1974 1020 y Fi(L)2024 940 y Fg(\013)2091 1008 y Fo(sup)2216 1028 y Fi(x)p Fj(2)p Fl(\003)2362 1008 y Fp(V)2410 1030 y Fi(R)2465 1008 y Fo(\()p Fp(x)p Fo(\))p Ft(.)40 b(Lemma)24 b(4.5)g(ensures)h(that)456 1112 y Fo(sup)581 1133 y Fi(x)p Fj(2)p Fl(\003)726 1112 y Fp(V)774 1135 y Fi(R)829 1112 y Fo(\()p Fp(x)p Fo(\))f(=)f(sup)1177 1133 y Fi(x)p Fj(2)p Fl(\003)1305 1141 y Fv(0)1355 1112 y Fp(V)1403 1135 y Fi(R)1458 1112 y Fo(\()p Fp(x)p Fo(\))f Ft(and)e(thus)g(yields)g(the)g(bound)619 1209 y Fg(\012)658 1276 y Fp(V)706 1297 y Fi(!)r(;R)838 1276 y Fp( )892 1288 y Fi(L)942 1276 y Fp(;)14 b(V)1027 1297 y Fi(!)r(;R)1159 1276 y Fp( )1213 1288 y Fi(L)1263 1209 y Fg(\013)1325 1276 y Fm(\024)1413 1209 y Fg(\012)1452 1276 y Fp( )1506 1288 y Fi(L)1556 1276 y Fp(;)g(V)1641 1297 y Fi(!)r(;R)1773 1276 y Fp( )1827 1288 y Fi(L)1877 1209 y Fg(\013)1930 1276 y Fp(c)1980 1209 y Fg(\000)2018 1276 y Fp(r)2055 1288 y Fl(0)2093 1276 y Fp(L)2150 1209 y Fg(\001)2187 1226 y Fj(\000)p Fl(2)2299 1276 y Fm(\024)2387 1209 y Fg(\012)2426 1276 y Fp( )2480 1288 y Fi(L)2530 1276 y Fp(;)g(V)2615 1297 y Fi(!)r(;R)2747 1276 y Fp( )2801 1288 y Fi(L)2851 1209 y Fg(\013)2914 1220 y Fp(c)2950 1232 y Fl(0)p 2914 1257 74 4 v 2930 1333 a Fo(2)2997 1276 y Fp(L)3054 1242 y Fj(\000)p Fl(2)3306 1276 y Ft(\(52\))456 1433 y(for)19 b(lar)o(ge)g(enough)g Fp(r)1061 1445 y Fl(0)1121 1433 y Fp(>)k Fo(0)p Ft(.)2108 b Fd(\003)605 1583 y Ft(W)-7 b(e)29 b(proceed)d(by)h(constructing)f(a)i(lo)n(wer)g (bound)e(on)h(the)h(right-hand)d(side)j(of)f(\(49\).)47 b(F)o(or)27 b(this)456 1682 y(purpose)18 b(we)j(set)1638 1775 y Fg(e)1633 1796 y Fo(\003)h(:=)1946 1717 y Fg([)1861 1899 y Fj(j)p Fi(j)1908 1907 y Fv(1)1940 1899 y Fj(j\024)p Fi(L=)p Fl(8)1824 1965 y Fi(R<)p Fj(j)p Fi(j)1973 1973 y Fv(2)2006 1965 y Fj(j\024)p Fl(2)p Fi(R)2175 1796 y Fo(\003)2233 1808 y Fi(j)3306 1796 y Ft(\(53\))456 2071 y(a)e(union)f(of)h(disjoint)g(cuboids.)605 2220 y Fq(Lemma)h(4.7.)41 b Ff(Ther)m(e)20 b(e)n(xist)i(two)f(constants)f Fo(0)j Fp(<)g(c)2113 2232 y Fl(2)2150 2220 y Ff(,)e Fp(c)2228 2232 y Fl(3)2289 2220 y Fp(<)j Fm(1)d Ff(\(whic)o(h)f(ar)m(e)g (independent)e(of)j Fp(!)s Ff(,)456 2320 y Fp(L)f Ff(and)f Fp(R)q Ff(\))h(suc)o(h)g(that)872 2378 y Fg(Z)918 2567 y Fl(\003)981 2491 y Fp(V)1029 2512 y Fi(!)r(;R)1148 2491 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1441 2457 y Fl(2)1494 2491 y Fp(dx)24 b Fm(\025)1842 2435 y Fp(c)1878 2447 y Fl(2)p 1705 2472 348 4 v 1705 2550 a Fp(R)1769 2526 y Fi(\013)1812 2534 y Fv(2)1844 2526 y Fl(\(1)p Fj(\000)p Fi(\015)1990 2534 y Fv(1)2023 2526 y Fl(\))2086 2491 y Fp(\026)2136 2457 y Fl(\(1\))2136 2512 y Fi(!)2225 2424 y Fg(\000)2269 2470 y(e)2263 2491 y Fo(\003)2321 2424 y Fg(\001)2377 2491 y Fm(\000)18 b Fp(c)2496 2503 y Fl(3)2547 2491 y Fm(j)p Fo(\003)p Fm(j)c Fp(L)2722 2457 y Fj(\000)p Fi(\013)2817 2465 y Fv(1)2849 2457 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3306 2491 y Ft(\(54\))456 2667 y Ff(for)20 b(all)h Fp(!)k Fm(2)f Fo(\012)d Ff(and)e(lar)m(g)o(e)h(enough)e Fp(L)23 b(>)f Fo(1)f Ff(and)e Fp(R)24 b(>)e Fo(1)p Ff(.)605 2817 y Fq(Remark)e(4.8.)39 b Ft(An)19 b(important)f(consequence)f(of)j(this) g(lemma)f(reads)g(as)i(follo)n(ws.)j(There)19 b(e)o(xists)456 2924 y(some)h(constant)h Fp(n)1001 2936 y Fi(u)1069 2924 y Fp(>)j Fo(0)d Ft(such)f(that)i(the)f(number)e(of)h(lattice)i(points)f (in)2570 2903 y Fg(e)2565 2924 y Fo(\003)g Ft(is)h(estimated)f(from)f (belo)n(w)456 3031 y(by)27 b Fm(j)596 3010 y Fg(e)590 3031 y Fo(\003)p Fm(j)38 b(\025)g Fp(n)862 3043 y Fi(u)919 3031 y Fm(j)p Fo(\003)p Fm(j)p Fp(R)1087 3001 y Fi(d)1122 3009 y Fv(2)1186 3031 y Ft(for)27 b(all)i Fp(L)38 b(>)f Fo(1)28 b Ft(and)g Fp(R)39 b(>)e Fo(1)28 b Ft(and)g(some)g(constant)g Fp(n)2822 3043 y Fi(u)2903 3031 y Fp(>)37 b Fo(0)p Ft(.)49 b(Therefore)456 3141 y Fm(j)485 3120 y Fg(e)479 3141 y Fo(\003)o Fm(j)p Fp(=)p Fo(\()p Fm(j)p Fo(\003)p Fm(j)p Fp(R)801 3111 y Fi(\013)844 3119 y Fv(2)876 3111 y Fl(\(1)p Fj(\000)p Fi(\015)1022 3119 y Fv(1)1054 3111 y Fl(\))1084 3141 y Fo(\))38 b Fm(\025)e Fp(n)1305 3153 y Fi(u)1348 3141 y Fp(=R)1454 3111 y Fi(\013)1497 3119 y Fv(2)1529 3111 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))1708 3141 y Ft(.)48 b(Choosing)27 b Fp(R)37 b Fo(=)f(\()p Fp(r)2394 3153 y Fl(0)2432 3141 y Fp(L)p Fo(\))2521 3111 y Fl(2)p Fi(=\013)2631 3119 y Fv(2)2664 3111 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2871 3141 y Ft(as)29 b(in)f(Proposition)456 3241 y(4.6,)19 b(we)h(thus)h(arri)n(v)o(e)e(at)h(the)h(lo)n(wer)e (bound)674 3353 y Fo(1)p 643 3390 104 4 v 643 3466 a Fm(j)p Fo(\003)p Fm(j)771 3296 y Fg(Z)817 3484 y Fl(\003)880 3409 y Fp(V)928 3429 y Fi(!)r(;R)1047 3409 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1340 3374 y Fl(2)1393 3409 y Fp(dx)24 b Fm(\025)1631 3353 y Fp(c)1667 3365 y Fl(2)1718 3353 y Fp(n)1768 3365 y Fi(u)p 1604 3390 234 4 v 1604 3466 a Fo(\()p Fp(r)1673 3478 y Fl(0)1711 3466 y Fp(L)p Fo(\))1800 3442 y Fl(2)1911 3353 y Fo(1)p 1880 3390 104 4 v 1880 3481 a Fm(j)1909 3460 y Fg(e)1903 3481 y Fo(\003)p Fm(j)2071 3330 y Fg(X)2008 3525 y Fi(j)s Fj(2)p Fe(Z)2129 3508 y Fk(d)2165 3525 y Fj(\\)2214 3510 y Fb(e)2210 3525 y Fl(\003)2269 3409 y Fp(\026)2319 3374 y Fl(\(1\))2319 3429 y Fi(!)2408 3341 y Fg(\000)2446 3409 y Fo(\003)2504 3421 y Fi(j)2538 3341 y Fg(\001)2595 3409 y Fm(\000)18 b Fp(c)2714 3421 y Fl(3)2765 3409 y Fp(L)2822 3374 y Fj(\000)p Fi(\013)2917 3382 y Fv(1)2949 3374 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3306 3409 y Ft(\(55\))456 3645 y(v)n(alid)h(for)h(all)h Fp(r)899 3657 y Fl(0)959 3645 y Fp(>)i Fo(0)d Ft(and)g(lar)o(ge)f(enough)f Fp(L)23 b(>)g Fo(1)p Ft(.)607 3794 y(P)t FC(R)q(O)t(O)t(F)j(O)t(F)f Ft(L)t FC(E)t(M)t(M)t(A)g Ft(4)t(.)t(7)t(.)42 b(Pulling)21 b(out)g(the)h(strictly)g(positi)n(v)o(e)f(in\002mum)f(of)h Fp( )2962 3764 y Fl(2)3022 3794 y Ft(and)g(using)g(its)456 3894 y Fn(Z)515 3864 y Fi(d)554 3894 y Ft(-periodicity)-5 b(,)17 b(we)k(estimate)757 3955 y Fg(Z)803 4144 y Fl(\003)867 4068 y Fp(V)915 4088 y Fi(!)r(;R)1033 4068 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1326 4034 y Fl(2)1379 4068 y Fp(dx)84 b Fm(\025)110 b Fo(inf)1700 4122 y Fi(z)r Fj(2)p Fl(\003)1824 4130 y Fv(0)1871 4068 y Fp( )s Fo(\()p Fp(z)t Fo(\))2035 4034 y Fl(2)2086 3955 y Fg(Z)2132 4144 y Fl(\003)2195 4068 y Fp(V)2243 4088 y Fi(!)r(;R)2362 4068 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)1553 4297 y Fm(\025)110 b Fo(inf)1700 4351 y Fi(z)r Fj(2)p Fl(\003)1824 4359 y Fv(0)1871 4297 y Fp( )s Fo(\()p Fp(z)t Fo(\))2035 4263 y Fl(2)2086 4184 y Fg(Z)2136 4358 y Fb(e)2132 4373 y Fl(\003)2195 4180 y Fg(\022)2257 4184 y(Z)2303 4373 y Fl(\003)2366 4297 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx)2777 4180 y Fg(\023)2852 4297 y Fp(\026)2902 4263 y Fl(\(1\))2902 4318 y Fi(!)2991 4297 y Fo(\()p Fp(dy)s Fo(\))164 b Ft(\(56\))456 4476 y(by)23 b(omitting)g(positi)n(v)o(e)h(terms)g(and)f(using)h(Fubini')-5 b(s)24 b(theorem.)35 b(The)23 b(inner)h(inte)o(gral)f(in)h(the)g(last)h (line)456 4576 y(is)k(estimated)f(from)g(belo)n(w)f(with)i(the)g(help)e (of)i(Lemma)e(3.4)h(in)h(terms)f(of)g(the)h(mar)o(ginal)d(impurity)456 4676 y(potential)19 b Fp(f)817 4645 y Fl(\(2\))926 4676 y Ft(\(recall)h(de\002nition)f(\(25\)\))g(according)f(to)668 4737 y Fg(Z)714 4926 y Fl(\003)778 4850 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)h Fp(y)s Fo(\))14 b Fp(dx)83 b Fo(=)1420 4737 y Fg(Z)1466 4926 y Fj(j)p Fi(x)1524 4934 y Fv(2)1555 4926 y Fj(j)p Fi(<)1637 4903 y Fv(1)p 1637 4912 29 3 v 1637 4946 a(2)1601 4850 y Fp(f)1651 4816 y Fl(\(2\))1740 4850 y Fo(\()p Fp(x)1819 4862 y Fl(2)1875 4850 y Fm(\000)18 b Fp(y)1999 4862 y Fl(2)2036 4850 y Fo(\))c Fp(dx)2172 4862 y Fl(2)2229 4850 y Fm(\000)2354 4771 y Fg(X)2312 4953 y Fj(j)p Fi(k)2367 4961 y Fv(1)2399 4953 y Fj(j\025)p Fi(L)2530 4737 y Fg(Z)2576 4926 y Fl(\003)2621 4937 y Fv(\()p Fk(k)2674 4949 y Fv(1)2708 4937 y Fk(;)p Fv(0\))2708 4850 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)g Fp(y)s Fo(\))c Fp(dx)1272 5123 y Fm(\025)1689 5067 y Fp(f)1730 5079 y Fl(1)p 1430 5104 597 4 v 1430 5181 a Fo(\(2)p Fp(R)k Fo(+)h(1\))1743 5157 y Fi(\013)1786 5165 y Fv(2)1818 5157 y Fl(\(1)p Fj(\000)p Fi(\015)1964 5165 y Fv(1)1996 5157 y Fl(\))2055 5123 y Fm(\000)2180 5044 y Fg(X)2138 5226 y Fj(j)p Fi(k)2193 5234 y Fv(1)2225 5226 y Fj(j\025)p Fi(L)2356 5010 y Fg(Z)2402 5199 y Fl(\003)2447 5207 y Fv(0)2484 5123 y Fp(f)2534 5056 y Fg(\000)2572 5123 y Fp(x)g Fo(+)f(\()p Fp(k)2796 5135 y Fl(1)2833 5123 y Fp(;)c Fo(0\))19 b Fm(\000)f Fp(y)3090 5056 y Fg(\001)3141 5123 y Fp(dx)75 b Ft(\(57\))p eop end %%Page: 16 16 TeXDict begin 16 15 bop 456 251 a Fr(16)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)456 450 y Ft(for)19 b(all)h Fm(j)p Fp(y)740 462 y Fl(2)777 450 y Fm(j)j(\024)g Fo(2)p Fp(R)17 b Fo(+)g(1)j Ft(and)f(lar)o(ge)g(enough)f Fp(R)23 b(>)g Fo(0)p Ft(.)i(The)19 b(\002rst)i(term)e(on)h(the)g (right-hand)d(side)j(yields)456 550 y(the)15 b(\002rst)i(term)f(on)f (the)h(right-hand)d(side)j(of)g(\(54\).)22 b(T)-7 b(o)16 b(estimate)g(the)g(remainder)e(we)i(decompose)e(the)i Fp(y)s Ft(-)456 663 y(inte)o(gration)h(of)j(the)f(second)g(term)g(in)h (\(57\))f(with)h(respect)f(to)h Fp(\026)2270 620 y Fl(\(1\))2270 672 y Fi(!)2379 663 y Ft(and)f(use)h(the)g(f)o(act)g(that)f Fp(\026)3110 620 y Fl(\(1\))3110 672 y Fi(!)3200 663 y Fo(\(\003)3290 675 y Fi(j)3324 663 y Fo(\))24 b Fm(\024)456 762 y Fo(1)p Ft(.)h(This)20 b(yields)g(an)g(estimate)h(of)f(the)g(form) 677 850 y Fg(Z)727 1024 y Fb(e)723 1039 y Fl(\003)786 846 y Fg(\022)848 850 y(Z)894 1039 y Fl(\003)939 1047 y Fv(0)976 963 y Fp(g)s Fo(\()p Fp(x)e Fm(\000)g Fp(y)s Fo(\))c Fp(dx)1379 846 y Fg(\023)1455 963 y Fp(\026)1505 929 y Fl(\(1\))1505 984 y Fi(!)1594 963 y Fo(\()p Fp(dy)s Fo(\))84 b Fm(\024)2040 884 y Fg(X)1976 1079 y Fi(j)s Fj(2)p Fe(Z)2097 1063 y Fk(d)2133 1079 y Fj(\\)2182 1064 y Fb(e)2178 1079 y Fl(\003)2277 963 y Fo(sup)2260 1033 y Fi(y)r Fj(2)p Fl(\003)2386 1041 y Fv(0)2432 850 y Fg(Z)2478 1039 y Fl(\003)2523 1047 y Fv(0)2560 963 y Fp(g)s Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)j Fm(\000)d Fp(j)5 b Fo(\))14 b Fp(dx)1829 1245 y Fm(\024)82 b Fo(3)2018 1211 y Fi(d)2142 1166 y Fg(X)2070 1348 y Fj(j)p Fi(j)2117 1356 y Fv(1)2150 1348 y Fj(j\024)p Fi(L=)p Fl(2)2093 1425 y Fi(j)2120 1433 y Fv(2)2153 1425 y Fj(2)p Fe(Z)2244 1400 y Fk(d)2275 1412 y Fv(2)2348 1132 y Fg(Z)2394 1321 y Fl(\003)2439 1329 y Fv(0)2476 1245 y Fp(g)s Fo(\()p Fp(x)19 b Fm(\000)f Fp(j)5 b Fo(\))14 b Fp(dx)1829 1591 y Fo(=)82 b(3)2018 1556 y Fi(d)2142 1512 y Fg(X)2070 1694 y Fj(j)p Fi(j)2117 1702 y Fv(1)2150 1694 y Fj(j\024)p Fi(L=)p Fl(4)2348 1478 y Fg(Z)2394 1666 y Fj(j)p Fi(x)2452 1674 y Fv(1)2484 1666 y Fj(j)p Fi(<)p Fl(1)p Fi(=)p Fl(2)2627 1591 y Fp(g)2670 1556 y Fl(\(1\))2759 1591 y Fo(\()p Fp(x)2838 1603 y Fl(1)2895 1591 y Fm(\000)18 b Fp(j)3012 1603 y Fl(1)3049 1591 y Fo(\))c Fp(dx)3185 1603 y Fl(1)3306 1591 y Ft(\(58\))456 1850 y(v)n(alid)29 b(for)h(all)h Fp(g)44 b Fm(2)e Fo(L)1125 1819 y Fl(1)1162 1850 y Fo(\()p Fn(R)1264 1819 y Fi(d)1303 1850 y Fo(\))p Ft(.)56 b(Here)30 b(the)g(second)g(inequality)f(holds)g (for)h(e)n(v)o(ery)f Fp(L)41 b Fm(\025)g Fo(8)31 b Ft(\(so)f(that)456 1949 y Fp(L=)p Fo(4)23 b Fm(\000)h Fp(L=)p Fo(8)38 b Fm(\025)h Fo(1)p Ft(\))29 b(and)f(follo)n(ws)g(from)g(enlar)o(gening)e (the)j Fp(j)2292 1961 y Fl(2)2329 1949 y Ft(-summation)f(and)g(the)h(f) o(act)g(that)g(the)456 2049 y(pointwise)20 b(dif)n(ference)f Fo(\003)1214 2061 y Fl(0)1269 2049 y Fm(\000)g Fo(\003)1411 2061 y Fl(0)1469 2049 y Ft(is)j(contained)d(in)i(the)g(cube)f(centred)g (at)h(the)g(origin)e(and)i(consisting)456 2148 y(of)16 b Fo(3)584 2118 y Fi(d)639 2148 y Ft(unit)g(cubes.)24 b(The)16 b(last)h(equality)f(uses)h(the)f(de\002nition)f(\(24\))h(for)f (a)i(mar)o(ginal)e(impurity)g(potential.)456 2248 y(Substituting)21 b Fp(g)s Fo(\()p Fp(x)p Fo(\))27 b(=)f Fp(f)9 b Fo(\()p Fp(x)21 b Fo(+)e(\()p Fp(k)1461 2260 y Fl(1)1499 2248 y Fp(;)14 b Fo(0\)\))23 b Ft(in)f(the)g(abo)o(v)o(e)f(chain)g(of)h (inequalities,)g(performing)d(the)j Fp(k)3379 2260 y Fl(1)3417 2248 y Ft(-)456 2348 y(summation)c(and)i(enlar)o(gening)d (the)j Fp(x)1566 2360 y Fl(1)1604 2348 y Ft(-inte)o(gration)e(thus)i (yields)497 2541 y Fo(3)539 2507 y Fi(d)663 2462 y Fg(X)591 2644 y Fj(j)p Fi(j)638 2652 y Fv(1)671 2644 y Fj(j\024)p Fi(L=)p Fl(4)869 2428 y Fg(Z)915 2617 y Fj(j)p Fi(x)973 2625 y Fv(1)1005 2617 y Fj(j)p Fi(>L=)p Fl(2)1138 2541 y Fp(f)1188 2507 y Fl(\(1\))1277 2541 y Fo(\()p Fp(x)1356 2553 y Fl(1)1403 2541 y Fm(\000)10 b Fp(j)1512 2553 y Fl(1)1549 2541 y Fo(\))k Fp(dx)1685 2553 y Fl(1)1746 2541 y Fm(\024)23 b Fo(3)1876 2507 y Fi(d)1914 2541 y Fp(n)1964 2553 y Fl(0)2001 2541 y Fm(j)p Fo(\003)p Fm(j)97 b Fo(sup)2133 2615 y Fj(j)p Fi(j)2180 2623 y Fv(1)2212 2615 y Fj(j\024)p Fi(L=)p Fl(4)2410 2428 y Fg(Z)2456 2617 y Fj(j)p Fi(x)2514 2625 y Fv(1)2546 2617 y Fj(j)p Fi(>L=)p Fl(2)2679 2541 y Fp(f)2729 2507 y Fl(\(1\))2818 2541 y Fo(\()p Fp(x)2897 2553 y Fl(1)2945 2541 y Fm(\000)10 b Fp(j)3054 2553 y Fl(1)3091 2541 y Fo(\))k Fp(dx)3227 2553 y Fl(1)3306 2541 y Ft(\(59\))456 2786 y(as)24 b(an)g(upper)f (bound)f(for)i(the)g(remainder)e(for)h(all)i Fp(L)k Fm(\025)h Fo(8)p Ft(.)37 b(Here)24 b(the)g(inequality)e(follo)n(ws)i(from)f(the) 456 2886 y(estimate)16 b Fo(#)p Fm(fj)p Fp(j)917 2898 y Fl(1)955 2886 y Fm(j)23 b(\024)f Fp(L=)p Fo(2)p Fm(g)f(\024)i Fp(n)1430 2898 y Fl(0)1467 2886 y Fm(j)p Fo(\003)p Fm(j)17 b Ft(for)f(some)g Fp(n)1944 2898 y Fl(0)2004 2886 y Fp(<)23 b Fm(1)17 b Ft(and)f(all)h Fp(L)23 b(>)f Fo(1)p Ft(.)i(The)16 b(proof)f(is)j(completed)456 2988 y(by)h(emplo)o(ying)f(a)j(result)f (for)g Fp(f)1365 2958 y Fl(\(1\))1474 2988 y Ft(analogous)f(to)h (\(28\).)1302 b Fd(\003)605 3165 y Ft(4.3.2.)39 b Ff(Pr)l(oof)33 b(of)g(Theor)m(em)g(2.8)f(\226)h(\002r)o(st)h(part:)50 b(quantum-classical)30 b(r)m(e)m(gime.)40 b Ft(W)-7 b(e)34 b(\002x)g Fp(r)3296 3177 y Fl(0)3380 3165 y Fp(>)456 3266 y Fo(1)p Fp(=)p Fo(\(2)p 628 3220 51 4 v 14 w Fp(\026)676 3234 y Fl(\(1\))765 3266 y Fo(\(\003)855 3278 y Fl(0)893 3266 y Fo(\)\))23 b Ft(lar)o(ge)f(enough)e(to)j(ensure)f(the)g(v)n (alidity)g(of)g(\(49\))f(in)i(Proposition)e(4.6.)31 b(F)o(or)22 b(a)h(gi)n(v)o(en)456 3366 y(ener)o(gy)18 b Fp(E)28 b(>)23 b Fo(0)d Ft(we)g(then)g(pick)1639 3554 y Fp(L)j Fo(:=)1830 3437 y Fg(\022)1903 3498 y Fp(c)1939 3510 y Fl(2)1990 3498 y Fp(n)2040 3510 y Fi(u)p 1901 3535 185 4 v 1901 3611 a Fo(2)p Fp(r)1982 3583 y Fl(3)1980 3634 y(0)2019 3611 y Fp(E)2095 3437 y Fg(\023)2156 3455 y Fl(1)p Fi(=)p Fl(2)3306 3554 y Ft(\(60\))456 3735 y(where)j(the)h(constants)g Fp(c)1188 3747 y Fl(2)1253 3735 y Ft(and)f Fp(n)1450 3747 y Fi(u)1521 3735 y Ft(ha)n(v)o(e)h(been)f(\002x)o(ed)h(in)g(Lemma) f(4.7)h(and)f(Remark)h(4.8.)45 b(Finally)-5 b(,)456 3835 y(we)21 b(choose)e(the)i(cuboid)e Fo(\003)i Ft(from)f(\(45\))f(and)h (set)i Fp(R)i Fo(:=)g(\()p Fp(r)2139 3847 y Fl(0)2177 3835 y Fp(L)p Fo(\))2266 3805 y Fl(2)p Fi(=\013)2376 3813 y Fv(2)2408 3805 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2588 3835 y Ft(.)j(Proposition)19 b(4.6)h(and)g(\(55\))456 3935 y(then)f(yield)h(the)g(estimate)854 4110 y Fn(P)919 4018 y Fg(n)974 4110 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\025)1335 4122 y Fl(0)1387 4110 y Fo(\()p Fp(H)1495 4071 y Fi(\037)1488 4135 y Fl(\003)1540 4110 y Fo(\()p Fp(V)1620 4122 y Fi(!)1669 4110 y Fo(\)\))24 b Fp(<)e(E)1910 4018 y Fg(o)854 4343 y Fm(\024)g Fn(P)1006 4201 y Fg(\()1073 4343 y Fp(!)k Fm(2)d Fo(\012)37 b(:)1427 4287 y(1)p 1396 4324 104 4 v 1396 4415 a Fm(j)1425 4394 y Fg(e)1419 4415 y Fo(\003)p Fm(j)1587 4264 y Fg(X)1524 4459 y Fi(j)s Fj(2)p Fe(Z)1645 4442 y Fk(d)1680 4459 y Fj(\\)1729 4444 y Fb(e)1725 4459 y Fl(\003)1784 4343 y Fp(\026)1834 4309 y Fl(\(1\))1834 4363 y Fi(!)1924 4276 y Fg(\000)1962 4343 y Fo(\003)2020 4355 y Fi(j)2054 4276 y Fg(\001)2115 4343 y Fp(<)2213 4287 y Fo(\()p Fp(r)2282 4299 y Fl(0)2320 4287 y Fp(L)p Fo(\))2409 4257 y Fl(2)p 2213 4324 234 4 v 2246 4400 a Fp(c)2282 4412 y Fl(2)2319 4400 y Fp(n)2369 4412 y Fi(u)2470 4251 y Fg(\020)2519 4343 y Fo(2)p Fp(E)23 b Fo(+)18 b Fp(c)2764 4355 y Fl(2)2802 4343 y Fp(L)2859 4309 y Fj(\000)p Fi(\013)2954 4317 y Fv(1)2986 4309 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3165 4251 y Fg(\021)3228 4201 y(\))854 4657 y Fm(\024)k Fn(P)1006 4515 y Fg(\()1073 4657 y Fp(!)k Fm(2)d Fo(\012)37 b(:)1427 4601 y(1)p 1396 4638 104 4 v 1396 4729 a Fm(j)1425 4708 y Fg(e)1419 4729 y Fo(\003)p Fm(j)1587 4578 y Fg(X)1524 4773 y Fi(j)s Fj(2)p Fe(Z)1645 4757 y Fk(d)1680 4773 y Fj(\\)1729 4758 y Fb(e)1725 4773 y Fl(\003)1784 4657 y Fp(\026)1834 4623 y Fl(\(1\))1834 4678 y Fi(!)1924 4590 y Fg(\000)1962 4657 y Fo(\003)2020 4669 y Fi(j)2054 4590 y Fg(\001)2115 4657 y Fp(<)2230 4601 y Fo(2)p 2213 4638 75 4 v 2213 4714 a Fp(r)2250 4726 y Fl(0)2298 4515 y Fg(\))3306 4657 y Ft(\(61\))456 4912 y(pro)o(vided)20 b Fp(E)32 b(>)27 b Fo(0)22 b Ft(is)i(small)f(enough,)e(equi)n(v)n(alently)f Fp(L)i Ft(is)i(lar)o(ge)d(enough.)30 b(Here)22 b(the)h(last)g (inequality)456 5012 y(results)28 b(from)f(\(60\))f(and)i(from)f(the)g (\002rst)i(inequality)e(in)h Fc(\(qm/cl\))o Ft(,)i(which)e(implies)g (that)g Fp(c)3135 5024 y Fl(3)3172 5012 y Fp(r)3211 4981 y Fl(3)3209 5032 y(0)3249 5012 y Fp(L)3306 4981 y Fl(2)3380 5012 y Fm(\024)456 5116 y Fp(c)492 5128 y Fl(2)529 5116 y Fp(n)579 5128 y Fi(u)622 5116 y Fp(L)679 5086 y Fi(\013)722 5094 y Fv(1)754 5086 y Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))963 5116 y Ft(for)h(lar)o(ge)f(enough)f Fp(L)39 b(>)g Fo(0)p Ft(.)51 b(Since)29 b Fo(2)p Fp(=r)2207 5128 y Fl(0)2284 5116 y Fm(\024)p 2388 5070 51 4 v 39 w Fp(\026)2438 5084 y Fl(\(1\))2527 5116 y Fo(\(\003)2617 5128 y Fl(0)2654 5116 y Fo(\))h Ft(by)e(assumption)g(on)h Fp(r)3386 5128 y Fl(0)3424 5116 y Ft(,)456 5216 y(the)f(right-hand)e(side)j(of)f (\(61\))f(is)i(the)g(probability)d(of)i(a)h(lar)o(ge-de)n(viation)c(e)n (v)o(ent)i([)p Fq(Dur96)o(,)i(DZ98)o Ft(].)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(17)456 450 y Ft(Consequently)-5 b(,)26 b(there)h(e)o(xists)h (some)f(constant)g Fp(c)1911 462 y Fl(4)1985 450 y Fp(>)35 b Fo(0)28 b Ft(\(which)e(is)j(independent)c(of)i Fp(L)p Ft(\))g(such)g(that)456 550 y(\(61\))19 b(is)i(estimated)f(from)f(abo)o (v)o(e)f(by)1008 706 y Fo(exp)1148 614 y Fg(h)1206 706 y Fm(\000)g Fp(c)1325 718 y Fl(4)1376 706 y Fm(j)1405 685 y Fg(e)1399 706 y Fo(\003)p Fm(j)1480 614 y Fg(i)1602 706 y Fm(\024)83 b Fo(exp)1890 614 y Fg(h)1930 706 y Fm(\000)p Fp(c)2031 718 y Fl(4)2081 706 y Fp(n)2131 718 y Fi(u)2188 706 y Fp(L)2245 672 y Fi(d)2280 680 y Fv(1)2330 706 y Fo(\()p Fp(r)2399 718 y Fl(0)2437 706 y Fp(L)p Fo(\))2525 665 y Fl(2)p Fi(\015)2593 673 y Fv(2)2626 665 y Fi(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2839 614 y Fg(i)1602 889 y Fo(=)g(exp)1890 797 y Fg(h)1930 889 y Fm(\000)p Fp(c)2031 901 y Fl(5)2081 889 y Fp(E)2147 855 y Fj(\000)p Fi(d)2234 863 y Fv(1)2266 855 y Fi(=)p Fl(2)p Fj(\000)p Fi(\015)2420 863 y Fv(2)2452 855 y Fi(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2666 797 y Fg(i)2719 889 y Fp(:)564 b Ft(\(62\))456 1043 y(Here)24 b(the)h(e)o(xistence)f (of)g(a)i(constant)e Fp(c)1597 1055 y Fl(5)1665 1043 y Fp(>)31 b Fo(0)25 b Ft(ensuring)e(the)i(v)n(alidity)f(of)g(the)h (last)h(equality)d(follo)n(ws)456 1142 y(from)k(\(60\).)48 b(Inserting)28 b(this)g(estimate)h(in)g(the)f(right-hand)e(side)j(of)f (\(23\))f(completes)h(the)g(\002rst)h(part)456 1242 y(of)24 b(the)h(proof)e(of)h(Theorem)f(2.8)h(for)g(the)g(quantum-classical)f (re)o(gime,)h(since)h(the)g(pre-f)o(actor)d(in)j(the)456 1342 y(upper)18 b(bound)h(in)h(Proposition)f(3.2)g(is)i(ne)o(gligible.) 1443 b Fd(\003)605 1491 y Fq(4.4.)40 b(Classical)17 b(r)o(egime.)40 b Ft(Throughout)14 b(this)j(Subsection)f(we)h(suppose)f(that)h Fc(\(cl\))f Ft(holds.)24 b(F)o(or)16 b(an)456 1591 y(asymptotic)j(e)n (v)n(aluation)f(of)i(the)g(upper)f(bound)f(in)j(Proposition)d(3.2)i(in) g(the)h(present)e(case,)h(we)h(de\002ne)1152 1762 y Fp(\014)1199 1774 y Fi(k)1263 1762 y Fo(:=)1404 1705 y(2)p 1383 1742 85 4 v 1383 1818 a Fp(d)1426 1830 y Fi(k)1554 1705 y Fp(\015)1597 1717 y Fi(k)p 1501 1742 191 4 v 1501 1818 a Fo(1)d Fm(\000)g Fp(\015)1725 1762 y Fo(=)1983 1705 y(2)p 1822 1742 363 4 v 1822 1818 a Fp(\013)1875 1830 y Fi(k)1930 1818 y Fo(\(1)g Fm(\000)g Fp(\015)5 b Fo(\))2195 1762 y Fp(;)180 b(k)26 b Fm(2)d(f)p Fo(1)p Fp(;)14 b Fo(2)p Fm(g)556 b Ft(\(63\))456 1953 y(and)16 b(construct)f(a)i(lo)n (wer)g(bound)d(on)j(the)f(lo)n(west)h(Mezincescu)f(eigen)m(v)n(alue)e Fp(\025)2681 1965 y Fl(0)2719 1886 y Fg(\000)2757 1953 y Fp(H)2833 1913 y Fi(\037)2826 1986 y Fl(\003)2871 1966 y Fv(in)o(t)2871 2004 y(0)2949 1953 y Fo(\()p Fp(V)3029 1965 y Fi(!)3078 1953 y Fo(\))3110 1886 y Fg(\001)3165 1953 y Ft(sho)n(wing)456 2081 y(up)19 b(in)i(the)f(right-hand)e(side)i (of)g(\(23\))f(when)h(choosing)e Fo(\003)23 b(=)g(\003)2285 2051 y Fl(in)n(t)2285 2102 y(0)2389 2081 y Ft(the)d(open)f(unit)h(cube) f(there.)605 2181 y(4.4.1.)39 b Ff(Lower)23 b(bound)d(on)h(the)h (lowest)h(Mezincescu)e(eig)o(en)m(value.)39 b Ft(F)o(or)22 b(e)n(v)o(ery)e Fp(L)26 b(>)f Fo(1)d Ft(and)f Fp(!)29 b Fm(2)456 2281 y Fo(\012)20 b Ft(the)h(potential)e Fp(V)1018 2293 y Fi(!)r(;L)1155 2281 y Fo(:)k Fn(R)1271 2251 y Fi(d)1332 2281 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ft(gi)n(v)o(en)e(by)1279 2461 y Fp(V)1327 2473 y Fi(!)r(;L)1441 2461 y Fo(\()p Fp(x)p Fo(\))24 b(:=)1687 2348 y Fg(Z)1733 2503 y Fj(j)p Fi(y)1787 2511 y Fv(1)1819 2503 y Fj(j)p Fi(>L)1937 2478 y Fk(\014)1970 2490 y Fv(1)1733 2580 y Fj(j)p Fi(y)1787 2588 y Fv(2)1819 2580 y Fj(j)p Fi(>L)1937 2555 y Fk(\014)1970 2567 y Fv(2)2009 2461 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(\026)2380 2426 y Fl(\(1\))2380 2481 y Fi(!)2469 2461 y Fo(\()p Fp(dy)s Fo(\))686 b Ft(\(64\))456 2724 y(in)22 b(terms)h(of)g(the)f(re) o(gularised)f(Borel)i(measure)f Fp(\026)1917 2681 y Fl(\(1\))1917 2734 y Fi(!)2006 2724 y Ft(,)i(pro)o(vides)d(a)i(lo)n(wer)f(bound)f(on) i Fp(V)3015 2736 y Fi(!)3063 2724 y Ft(.)33 b(Therefore)456 2824 y Fp(\025)504 2836 y Fl(0)541 2757 y Fg(\000)579 2824 y Fp(H)655 2784 y Fi(\037)648 2857 y Fl(\003)693 2837 y Fv(in)o(t)693 2875 y(0)771 2824 y Fo(\()p Fp(V)851 2836 y Fi(!)900 2824 y Fo(\))932 2757 y Fg(\001)993 2824 y Fm(\025)23 b Fp(\025)1129 2836 y Fl(0)1166 2757 y Fg(\000)1205 2824 y Fp(H)1281 2784 y Fi(\037)1274 2857 y Fl(\003)1319 2837 y Fv(in)o(t)1319 2875 y(0)1396 2824 y Fo(\()p Fp(V)1476 2836 y Fi(!)r(;L)1590 2824 y Fo(\))1622 2757 y Fg(\001)1661 2824 y Ft(.)h(It)19 b(will)g(be)f(useful)f(to)h(collect)h(some)e(f)o (acts)i(related)f(to)g Fp(V)3310 2836 y Fi(!)r(;L)3424 2824 y Ft(.)605 3001 y Fq(Lemma)j(4.9.)40 b Ff(Let)21 b Fp(L)h(>)h Fo(1)d Ff(and)g(de\002ne)f Fp(V)1833 3013 y Fi(L)1906 3001 y Fo(:)k Fn(R)2022 2971 y Fi(d)2083 3001 y Fm(!)g Fo([0)p Fp(;)14 b Fm(1)p Fo([)21 b Ff(by)1345 3152 y Fp(V)1393 3164 y Fi(L)1443 3152 y Fo(\()p Fp(x)p Fo(\))j(:=)1805 3073 y Fg(X)1689 3265 y Fj(j)p Fi(j)1736 3273 y Fv(1)1769 3265 y Fj(j)p Fi(>L)1887 3240 y Fk(\014)1920 3252 y Fv(1)1955 3265 y Fj(\000)p Fl(1)1689 3342 y Fj(j)p Fi(j)1736 3350 y Fv(2)1769 3342 y Fj(j)p Fi(>L)1887 3317 y Fk(\014)1920 3329 y Fv(2)1955 3342 y Fj(\000)p Fl(1)2070 3152 y Fo(sup)2054 3222 y Fi(y)r Fj(2)p Fl(\003)2180 3230 y Fk(j)2225 3152 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))p Fp(:)751 b Ft(\(65\))456 3471 y Ff(Then)25 b(we)h(have)f Fp(V)997 3483 y Fi(!)r(;L)1143 3471 y Fm(\024)33 b Fp(V)1289 3483 y Fi(L)1365 3471 y Ff(for)26 b(e)o(very)g Fp(!)35 b Fm(2)e Fo(\012)p Ff(.)42 b(Mor)m(eo)o(ver)-9 b(,)26 b(the)g(supr)m(emum)f Fo(sup)2975 3491 y Fi(x)p Fj(2)p Fl(\003)3103 3499 y Fv(0)3153 3471 y Fp(V)3201 3483 y Fi(L)3251 3471 y Fo(\()p Fp(x)p Fo(\))i Ff(is)456 3570 y(arbitr)o(arily)20 b(small)g(for)h(lar)m(g)o(e)f(enough)e Fp(L)23 b(>)f Fo(1)p Ff(.)607 3717 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)28 b(\002rst)i(assertion)e(follo)n(ws)g(analogously)f(as) i(in)g(Lemma)f(4.5.)50 b(The)29 b(second)f(one)456 3817 y(deri)n(v)o(es)19 b(from)g(the)h(second)f(inequality)g(in)i(\(12\).) 1512 b Fd(\003)605 3966 y Fq(Remark)22 b(4.10.)41 b Ft(It)22 b(is)g(actually)g(not)f(dif)n(\002cult)g(to)h(pro)o(v)o(e)e(that)i (there)f(e)o(xists)i(some)e(constant)g Fo(0)26 b Fp(<)456 4066 y(C)31 b(<)26 b Fm(1)c Ft(\(which)e(is)j(independent)c(of)i Fp(L)p Ft(\))g(such)h(that)f Fo(sup)2143 4086 y Fi(x)p Fj(2)p Fl(\003)2271 4094 y Fv(0)2322 4066 y Fp(V)2370 4078 y Fi(L)2420 4066 y Fo(\()p Fp(x)p Fo(\))26 b Fm(\024)g Fp(C)20 b(L)2784 4036 y Fj(\000)p Fl(2)2894 4066 y Ft(for)h(lar)o(ge)g (enough)456 4166 y Fp(L)h(>)h Fo(0)p Ft(.)605 4313 y(The)g(ne)o(xt)h (proposition)d(contains)i(the)h(k)o(e)o(y)f(estimate)h(on)f(the)h(lo)n (west)g(Mezincescu)f(eigen)m(v)n(alue)456 4412 y(in)f(the)g(classical)h (re)o(gime.)29 b(In)22 b(contrast)f(to)h(the)g(quantum-classical)f(re)o (gime,)g(the)h(speci\002c)g(choice)f(of)456 4512 y(the)k(cut-of)n(f)g (made)g(in)h(\(64\))f(is)h(irrele)n(v)n(ant)f(as)h(f)o(ar)g(as)h(the)f (applicability)e(of)i(T)-6 b(emple')h(s)25 b(inequality)g(in)456 4611 y(the)18 b(subsequent)e(Proposition)h(is)i(concerned.)j(The)17 b(chosen)h(length)f(scales)i Fp(L)2723 4581 y Fi(\014)2761 4589 y Fv(1)2815 4611 y Ft(and)f Fp(L)3011 4581 y Fi(\014)3049 4589 y Fv(2)3103 4611 y Ft(will)h(rather)456 4711 y(become)g(important) f(later)i(on.)605 4858 y Fq(Pr)o(oposition)31 b(4.11.)47 b Ff(Let)33 b Fo(\003)1459 4828 y Fl(in)n(t)1459 4879 y(0)1575 4858 y Ff(be)f(the)g(open)f(unit)h(cube)o(.)60 b(Then)32 b(the)g(lowest)h(eig)o(en)m(value)d(of)456 4958 y Fp(H)532 4918 y Fi(\037)525 4991 y Fl(\003)570 4971 y Fv(in)o(t)570 5009 y(0)647 4958 y Fo(\()p Fp(V)727 4970 y Fi(!)r(;L)841 4958 y Fo(\))21 b Ff(is)g(bounded)d(fr)l(om)j (below)f(accor)m(ding)e(to)1197 5165 y Fp(\025)1245 5177 y Fl(0)1283 5098 y Fg(\000)1321 5165 y Fp(H)1397 5125 y Fi(\037)1390 5198 y Fl(\003)1435 5178 y Fv(in)o(t)1435 5216 y(0)1512 5165 y Fo(\()p Fp(V)1592 5177 y Fi(!)r(;L)1706 5165 y Fo(\))1738 5098 y Fg(\001)1800 5165 y Fm(\025)1897 5109 y Fo(1)p 1897 5146 42 4 v 1897 5222 a(2)1963 5052 y Fg(Z)2009 5240 y Fl(\003)2054 5248 y Fv(0)2105 5165 y Fp(V)2153 5177 y Fi(!)r(;L)2266 5165 y Fo(\()p Fp(x)p Fo(\))c Fp( )s Fo(\()p Fp(x)p Fo(\))2559 5131 y Fl(2)2612 5165 y Fp(dx)604 b Ft(\(66\))p eop end %%Page: 18 18 TeXDict begin 18 17 bop 456 251 a Fr(18)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)456 450 y Ff(for)27 b(all)g Fp(!)39 b Fm(2)d Fo(\012)27 b Ff(and)f(lar)m(g)o(e)h(enough)e Fp(L)35 b(>)h Fo(1)p Ff(.)45 b Ft([)p Ff(Recall)26 b(the)h (de\002nition)f(of)h Fp( )j Ff(at)d(the)g(be)m(ginning)e(of)456 550 y(Subsection)18 b(3.1.)p Ft(])607 709 y(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(The)16 b(proof)e(again)i(parallels)g(that)g(of)g(Proposition) f(4.3.)23 b(In)16 b(a)h(slight)f(ab)n(use)g(of)g(notation,)456 809 y(let)21 b Fp( )j Ft(denote)19 b(the)h(restriction)g(of)g Fp( )k Ft(to)d Fo(\003)1672 779 y Fl(in)n(t)1672 829 y(0)1776 809 y Ft(throughout)d(this)j(proof.)j(T)-6 b(emple')h(s)20 b(inequality)f([)p Fq(RS78)o Ft(,)456 908 y(Thm.)g(XIII.5])f(together)h (with)i(the)f(f)o(act)g(that)h Fp(H)1857 868 y Fi(\037)1850 941 y Fl(\003)1895 921 y Fv(in)o(t)1895 959 y(0)1972 908 y Fo(\(0\))p Fp( )26 b Fo(=)d(0)d Ft(yields)g(the)h(lo)n(wer)e (bound)872 1138 y Fp(\025)920 1150 y Fl(0)958 1071 y Fg(\000)996 1138 y Fp(H)1072 1098 y Fi(\037)1065 1171 y Fl(\003)1110 1151 y Fv(in)o(t)1110 1189 y(0)1188 1138 y Fo(\()p Fp(V)1268 1150 y Fi(!)r(;L)1382 1138 y Fo(\))1414 1071 y Fg(\001)1475 1138 y Fm(\025)k(h)p Fp( )s(;)14 b(V)1737 1150 y Fi(!)r(;L)1865 1138 y Fp( )s Fm(i)k(\000)2258 1082 y(h)p Fp(V)2338 1094 y Fi(!)r(;L)2466 1082 y Fp( )s(;)c(V)2608 1094 y Fi(!)r(;L)2736 1082 y Fp( )s Fm(i)p 2065 1119 953 4 v 2065 1199 a Fp(\025)2113 1211 y Fl(1)2151 1132 y Fg(\000)2189 1199 y Fp(H)2265 1160 y Fi(\037)2258 1232 y Fl(\003)2303 1212 y Fv(in)o(t)2303 1250 y(0)2381 1199 y Fo(\(0\))2487 1132 y Fg(\001)2543 1199 y Fm(\000)k(h)q Fp( )s(;)c(V)2801 1211 y Fi(!)r(;L)2928 1199 y Fp( )s Fm(i)3306 1138 y Ft(\(67\))456 1367 y(pro)o(vided)24 b(that)k(the)f(denominator)e(is)j(strictly)f(positi)n(v)o(e.)46 b(T)-7 b(o)27 b(check)g(this)g(we)h(emplo)o(y)e(Lemma)h(4.9)456 1467 y(and)34 b(tak)o(e)g Fp(L)50 b(>)f Fo(1)35 b Ft(lar)o(ge)f(enough) e(such)j(that)g Fm(h)p Fp( )s(;)14 b(V)2082 1479 y Fi(!)r(;L)2209 1467 y Fp( )s Fm(i)51 b(\024)e Fp(\025)2511 1479 y Fl(1)2549 1400 y Fg(\000)2587 1467 y Fp(H)2663 1427 y Fi(\037)2656 1500 y Fl(\003)2701 1480 y Fv(in)o(t)2701 1518 y(0)2778 1467 y Fo(\(0\))2884 1400 y Fg(\001)2923 1467 y Fp(=)p Fo(2)p Ft(.)68 b(\(Note)34 b(that)456 1597 y Fp(\025)504 1609 y Fl(1)541 1530 y Fg(\000)579 1597 y Fp(H)655 1557 y Fi(\037)648 1630 y Fl(\003)693 1610 y Fv(in)o(t)693 1648 y(0)771 1597 y Fo(\(0\))877 1530 y Fg(\001)945 1597 y Ft(is)c(independent)c(of)j Fp(L)p Ft(.\))52 b(T)-7 b(o)29 b(estimate)h(the)f(numerator)e(in)i(\(67\))f(from)g(abo)o(v)o (e,)i(we)456 1718 y(use)19 b(the)f(bound)f Fm(h)p Fp(V)1013 1730 y Fi(!)r(;L)1141 1718 y Fp( )s(;)d(V)1283 1730 y Fi(!)r(;L)1410 1718 y Fp( )s Fm(i)24 b(\024)e(h)q Fp( )s(;)14 b(V)1785 1730 y Fi(!)r(;L)1912 1718 y Fp( )s Fm(i)28 b Fo(sup)2154 1739 y Fi(x)p Fj(2)p Fl(\003)2282 1747 y Fv(0)2332 1718 y Fp(V)2380 1730 y Fi(L)2431 1718 y Fo(\()p Fp(x)p Fo(\))p Ft(.)e(T)-7 b(ogether)17 b(with)i(Lemma)f(4.9) 456 1826 y(this)i(yields)g Fm(h)q Fp(V)896 1838 y Fi(!)r(;L)1023 1826 y Fp( )s(;)14 b(V)1165 1838 y Fi(!)r(;L)1293 1826 y Fp( )s Fm(i)23 b(\024)g(h)p Fp( )s(;)14 b(V)1667 1838 y Fi(!)r(;L)1795 1826 y Fp( )s Fm(i)p Fp(\025)1932 1838 y Fl(1)1970 1759 y Fg(\000)2008 1826 y Fp(H)2084 1786 y Fi(\037)2077 1859 y Fl(\003)2122 1839 y Fv(in)o(t)2122 1877 y(0)2199 1826 y Fo(\(0\))2305 1759 y Fg(\001)2344 1826 y Fp(=)p Fo(4)19 b Ft(for)h(lar)o(ge)f(enough)f Fp(L)23 b(>)f Fo(1)p Ft(.)135 b Fd(\003)605 2030 y Fq(Remark)21 b(4.12.)40 b Ft(The)20 b(simple)g(lo)n(wer)h(bound)d Fp(\025)2002 2042 y Fl(0)2040 1963 y Fg(\000)2078 2030 y Fp(H)2154 1990 y Fi(\037)2147 2063 y Fl(\003)2192 2043 y Fv(in)o(t)2192 2081 y(0)2270 2030 y Fo(\()p Fp(V)2350 2042 y Fi(!)r(;L)2464 2030 y Fo(\))2496 1963 y Fg(\001)2558 2030 y Fm(\025)23 b Fo(inf)2747 2042 y Fi(x)p Fj(2)p Fl(\003)2875 2050 y Fv(0)2926 2030 y Fp(V)2974 2042 y Fi(!)r(;L)3087 2030 y Fo(\()p Fp(x)p Fo(\))p Ft(,)f(which)456 2147 y(w)o(as)28 b(emplo)o(yed)e(in)i([)p Fq(KS86)n Ft(],)i(w)o(ould)c (yield)i(a)g(result)f(similar)h(to)g(\(72\))e(belo)n(w)-5 b(,)29 b(b)n(ut)e(at)h(the)g(price)f(of)456 2246 y(assuming)19 b(that)h(the)h(lo)n(wer)e(bound)g(in)h(\(12\))f(holds)h(pointwise.)605 2406 y(W)-7 b(e)29 b(proceed)d(by)h(constructing)f(a)i(lo)n(wer)g (bound)e(on)h(the)h(right-hand)d(side)j(of)f(\(66\).)47 b(F)o(or)27 b(this)456 2505 y(purpose)18 b(we)j(set)1557 2615 y Fg(e)1552 2636 y Fo(\003)h(:=)1946 2557 y Fg([)1743 2749 y Fl(2)p Fi(L)1822 2724 y Fk(\014)1855 2736 y Fv(1)1891 2749 y Fi(<)p Fj(j)p Fi(j)1990 2757 y Fv(1)2022 2749 y Fj(j\024)p Fl(4)p Fi(L)2173 2724 y Fk(\014)2206 2736 y Fv(1)1743 2826 y Fl(2)p Fi(L)1822 2801 y Fk(\014)1855 2813 y Fv(2)1891 2826 y Fi(<)p Fj(j)p Fi(j)1990 2834 y Fv(2)2022 2826 y Fj(j\024)p Fl(4)p Fi(L)2173 2801 y Fk(\014)2206 2813 y Fv(2)2256 2636 y Fo(\003)2314 2648 y Fi(j)3306 2636 y Ft(\(68\))456 2949 y(an)e(annulus-shaped)d(re)o (gion.)605 3108 y Fq(Lemma)k(4.13.)40 b Ff(Ther)m(e)20 b(e)n(xists)i(a)f(constant)e Fp(c)1922 3120 y Fl(6)1982 3108 y Fp(>)k Fo(0)e Ff(\(whic)o(h)f(is)h(independent)d(of)i Fp(!)k Ff(and)19 b Fp(L)p Ff(\))h(suc)o(h)456 3207 y(that)1215 3234 y Fg(Z)1261 3422 y Fl(\003)1306 3430 y Fv(0)1357 3347 y Fp(V)1405 3359 y Fi(!)r(;L)1519 3347 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1812 3312 y Fl(2)1865 3347 y Fp(dx)23 b Fm(\025)2191 3291 y Fp(c)2227 3303 y Fl(6)p 2076 3328 304 4 v 2076 3405 a Fp(L)2133 3381 y Fl(2)p Fi(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2412 3347 y Fp(\026)2462 3312 y Fl(\(1\))2462 3367 y Fi(!)2551 3279 y Fg(\000)2595 3326 y(e)2589 3347 y Fo(\003)2647 3279 y Fg(\001)3306 3347 y Ft(\(69\))456 3530 y Ff(for)d(lar)m(g)o(e)g (enough)f Fp(L)j(>)h Fo(0)p Ff(.)607 3690 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(Pulling)25 b(out)g(the)h(strictly)g(positi)n(v)o(e)f (in\002mum)f(of)i Fp( )2436 3660 y Fl(2)2473 3690 y Ft(,)i(using)d (Fubini')-5 b(s)25 b(theorem)g(and)456 3789 y(omitting)19 b(a)h(positi)n(v)o(e)g(term,)f(we)i(estimate)775 3875 y Fg(Z)822 4064 y Fl(\003)867 4072 y Fv(0)917 3988 y Fp(V)965 4000 y Fi(!)r(;L)1079 3988 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1372 3954 y Fl(2)1425 3988 y Fp(dx)24 b Fm(\025)50 b Fo(inf)1627 4042 y Fi(z)r Fj(2)p Fl(\003)1751 4050 y Fv(0)1797 3988 y Fp( )s Fo(\()p Fp(z)t Fo(\))1961 3954 y Fl(2)2012 3875 y Fg(Z)2062 4049 y Fb(e)2058 4064 y Fl(\003)2122 3871 y Fg(\022)2183 3875 y(Z)2229 4064 y Fl(\003)2274 4072 y Fv(0)2325 3988 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)g Fp(y)s Fo(\))c Fp(dx)2735 3871 y Fg(\023)2811 3988 y Fp(\026)2861 3954 y Fl(\(1\))2861 4009 y Fi(!)2950 3988 y Fo(\()p Fp(dy)s Fo(\))p Fp(:)182 b Ft(\(70\))456 4203 y(Assumption)21 b(2.4)g(implies)i(that)f(the)g (estimate)1845 4136 y Fg(R)1884 4233 y Fl(\003)1929 4241 y Fv(0)1980 4203 y Fp(f)9 b Fo(\()p Fp(x)20 b Fm(\000)f Fp(y)s Fo(\))14 b Fp(dx)28 b Fm(\025)e Fp(f)2553 4215 y Fi(u)2596 4203 y Fp(=)2638 4136 y Fg(\002)2672 4203 y Fo(\(3)p Fp(L)2803 4173 y Fi(\014)2841 4181 y Fv(1)2877 4203 y Fo(\))2909 4173 y Fi(\013)2952 4181 y Fv(1)3009 4203 y Fo(+)19 b(\(3)p Fp(L)3224 4173 y Fi(\014)3262 4181 y Fv(2)3298 4203 y Fo(\))3330 4173 y Fi(\013)3373 4181 y Fv(2)3410 4136 y Fg(\003)456 4330 y Ft(holds)26 b(for)g(all)i Fp(y)38 b Fm(2)1074 4309 y Fg(e)1068 4330 y Fo(\003)28 b Ft(and)e(lar)o(ge)g(enough)f Fp(L)35 b(>)h Fo(1)p Ft(.)45 b(This)27 b(completes)f(the)h(proof,)g(since)g Fp(\013)3215 4342 y Fi(k)3256 4330 y Fp(\014)3303 4342 y Fi(k)3380 4330 y Fo(=)456 4430 y(2)p Fp(=)p Fo(\(1)17 b Fm(\000)h Fp(\015)5 b Fo(\))21 b Ft(for)e(both)g Fp(k)26 b Fm(2)e(f)p Fo(1)p Fp(;)14 b Fo(2)p Fm(g)p Ft(.)1906 b Fd(\003)605 4604 y Fq(Remark)29 b(4.14.)45 b Ft(There)28 b(e)o(xists)h(some)f(constant)g Fp(n)2138 4616 y Fi(u)2220 4604 y Fp(>)39 b Fo(0)28 b Ft(such)h(that)g(the)f(number)f(of)h (lattice)456 4705 y(points)18 b(in)767 4685 y Fg(e)761 4705 y Fo(\003)h Ft(can)f(be)h(bounded)d(from)h(belo)n(w)h(according)f (to)i Fm(j)2235 4685 y Fg(e)2229 4705 y Fo(\003)o Fm(j)k(\025)g Fp(n)2470 4717 y Fi(u)2513 4705 y Fp(L)2570 4675 y Fi(\014)2608 4683 y Fv(1)2640 4675 y Fi(d)2675 4683 y Fv(1)2707 4675 y Fl(+)p Fi(\014)2796 4683 y Fv(2)2828 4675 y Fi(d)2863 4683 y Fv(2)2922 4705 y Fo(=)g Fp(n)3060 4717 y Fi(u)3103 4705 y Fp(L)3160 4675 y Fl(2)p Fi(\015)t(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))456 4805 y Ft(for)c(all)i Fp(L)h(>)h Fo(1)p Ft(.)i(Lemma)20 b(4.13)f(thus)h(implies)g(the)g(inequality)1173 4884 y Fg(Z)1219 5072 y Fl(\003)1264 5080 y Fv(0)1315 4997 y Fp(V)1363 5009 y Fi(!)r(;L)1477 4997 y Fo(\()p Fp(x)p Fo(\))14 b Fp( )s Fo(\()p Fp(x)p Fo(\))1770 4962 y Fl(2)1823 4997 y Fp(dx)23 b Fm(\025)2034 4940 y Fp(c)2070 4952 y Fl(6)2121 4940 y Fp(n)2171 4952 y Fi(u)p 2034 4977 181 4 v 2077 5053 a Fp(L)2134 5029 y Fl(2)2247 4997 y Fm(j)2276 4976 y Fg(e)2270 4997 y Fo(\003)p Fm(j)2351 4962 y Fj(\000)p Fl(1)2454 4997 y Fp(\026)2504 4962 y Fl(\(1\))2504 5017 y Fi(!)2593 4929 y Fg(\000)2637 4976 y(e)2631 4997 y Fo(\003)2689 4929 y Fg(\001)3306 4997 y Ft(\(71\))456 5198 y(for)c(lar)o(ge)g(enough)g Fp(L)j(>)h Fo(1)p Ft(.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(19)605 450 y Ft(4.4.2.)39 b Ff(Pr)l(oof)22 b(of)g(Theor)m(em)f(2.8)g(\226)h(\002r)o(st)g(part:)28 b(classical)21 b(r)m(e)m(gime.)41 b Ft(F)o(or)21 b(a)h(gi)n(v)o(en)e (ener)o(gy)g Fp(E)31 b(>)25 b Fo(0)456 575 y Ft(we)k(let)g Fp(L)38 b Fo(:=)915 508 y Fg(\000)953 575 y Fp(c)989 587 y Fl(6)1026 575 y Fp(n)1076 587 y Fi(u)p 1133 529 51 4 v 1133 575 a Fp(\026)1183 543 y Fl(\(1\))1272 575 y Fo(\(\003)1362 587 y Fl(0)1399 575 y Fo(\))p Fp(=)p Fo(4)p Fp(E)1581 508 y Fg(\001)1619 520 y Fl(1)p Fi(=)p Fl(2)1723 575 y Ft(,)31 b(where)d(the)h(constant)e Fp(c)2479 587 y Fl(6)2546 575 y Ft(and)h Fp(n)2745 587 y Fi(u)2817 575 y Ft(ha)n(v)o(e)g(been)g(\002x)o(ed)g(in)456 674 y(Lemma)18 b(4.13)h(and)g(Remark)g(4.14.)24 b(Proposition)18 b(4.11)g(and)h(Equation)f(\(71\))h(then)g(yield)g(the)h(estimate)498 893 y Fn(P)563 801 y Fg(n)618 893 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\025)979 905 y Fl(0)1030 801 y Fg(\020)1080 893 y Fp(H)1156 853 y Fi(\037)1149 926 y Fl(\003)1194 906 y Fv(in)o(t)1194 944 y(0)1272 893 y Fo(\()p Fp(V)1352 905 y Fi(!)1400 893 y Fo(\))1432 801 y Fg(\021)1505 893 y Fp(<)23 b(E)1659 801 y Fg(o)1737 893 y Fm(\024)g Fn(P)1890 751 y Fg(\()1957 893 y Fp(!)j Fm(2)d Fo(\012)37 b(:)2311 837 y(1)p 2280 874 104 4 v 2280 965 a Fm(j)2309 944 y Fg(e)2303 965 y Fo(\003)p Fm(j)2485 814 y Fg(X)2421 1009 y Fi(j)s Fj(2)p Fe(Z)2542 993 y Fk(d)2578 1009 y Fj(\\)2627 994 y Fb(e)2623 1009 y Fl(\003)2682 893 y Fp(\026)2732 859 y Fl(\(1\))2732 914 y Fi(!)2821 826 y Fg(\000)2859 893 y Fo(\003)2917 905 y Fi(j)2952 826 y Fg(\001)3013 893 y Fp(<)3110 837 y Fo(2)p Fp(E)19 b(L)3289 807 y Fl(2)p 3110 874 216 4 v 3128 950 a Fp(c)3164 962 y Fl(6)3215 950 y Fp(n)3265 962 y Fi(u)3336 751 y Fg(\))3306 1094 y Ft(\(72\))456 1193 y(pro)o(vided)k Fp(E)40 b(>)33 b Fo(0)27 b Ft(is)g(small)g(enough,)e(equi)n(v)n(alently)f Fp(L)j Ft(is)g(lar)o(ge)e(enough.)41 b(Since)26 b Fo(2)p Fp(E)5 b(L)3101 1163 y Fl(2)3138 1193 y Fp(=c)3216 1205 y Fl(6)3252 1193 y Fp(n)3302 1205 y Fi(u)3380 1193 y Fo(=)456 1309 y Fn(E)521 1242 y Fg(\002)555 1309 y Fp(\026)605 1266 y Fl(\(1\))605 1319 y Fi(!)694 1309 y Fo(\(\003)784 1321 y Fl(0)822 1309 y Fo(\))854 1242 y Fg(\003)888 1309 y Fp(=)p Fo(2)24 b Ft(and)g(the)g(random)e(v)n(ariables)h(are)h (independent)e(and)h(identically)g(distrib)n(uted,)h(the)456 1409 y(last)17 b(probability)d(is)j(that)g(of)f(a)h(lar)o(ge)e(de)n (viation)g(e)n(v)o(ent)g([)p Fq(Dur96)o(,)i(DZ98)o Ft(].)24 b(Consequently)-5 b(,)15 b(there)h(e)o(xists)456 1509 y(some)k Fp(c)688 1521 y Fl(7)748 1509 y Fp(>)i Fo(0)f Ft(such)f(that)g(the)g(right-hand)e(side)i(of)g(\(72\))f(is)i(bounded)d (from)h(abo)o(v)o(e)g(by)821 1677 y Fo(exp)961 1585 y Fg(h)1019 1677 y Fm(\000)f Fp(c)1138 1689 y Fl(7)1189 1677 y Fm(j)1218 1656 y Fg(e)1212 1677 y Fo(\003)p Fm(j)1293 1585 y Fg(i)1415 1677 y Fm(\024)83 b Fo(exp)1703 1585 y Fg(h)1761 1677 y Fm(\000)18 b Fp(c)1880 1689 y Fl(7)1917 1677 y Fp(n)1967 1689 y Fi(u)2010 1677 y Fp(L)2067 1643 y Fl(2)p Fi(\015)t(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))2352 1585 y Fg(i)1415 1883 y Fo(=)83 b(exp)1703 1791 y Fg(h)1761 1883 y Fm(\000)18 b Fp(c)1880 1895 y Fl(7)1917 1883 y Fp(n)1967 1895 y Fi(u)2024 1791 y Fg(\020)2074 1883 y Fp(c)2110 1895 y Fl(6)2147 1883 y Fp(n)2197 1895 y Fi(u)p 2254 1837 51 4 v 2254 1883 a Fp(\026)2304 1849 y Fl(\(1\))2393 1883 y Fo(\(\003)2483 1895 y Fl(0)2520 1883 y Fo(\))p Fp(=)p Fo(4)p Fp(E)2702 1791 y Fg(\021)2751 1808 y Fi(\015)t(=)p Fl(\(1)p Fj(\000)p Fi(\015)t Fl(\))3017 1791 y Fg(i)3056 1883 y Fp(:)227 b Ft(\(73\))456 2049 y(Since)32 b(the)g(pre-f)o(actor)e (in)i(the)g(upper)f(bound)f(in)i(Proposition)e(3.2)i(is)h(ne)o (gligible,)g(inserting)e(\(72\))456 2149 y(together)c(with)h(\(73\))f (in)i(the)f(right-hand)e(side)i(of)g(\(23\))f(completes)h(the)g (\002rst)h(part)f(of)g(the)h(proof)d(of)456 2248 y(Theorem)18 b(2.8)i(for)f(the)h(classical)h(re)o(gime.)1686 b Fd(\003)1660 2453 y Fq(5.)41 b(Lo)o(wer)20 b(bound)605 2602 y Ft(T)-7 b(o)19 b(complete)f(the)g(proof)g(of)g(Theorem)f(2.8,)h(it)i(remains)e (to)h(asymptotically)e(e)n(v)n(aluate)h(the)g(lo)n(wer)456 2702 y(bound)24 b(in)i(Proposition)f(3.2)h(for)g(small)g(ener)o(gies.) 42 b(This)27 b(is)g(the)f(topic)g(of)g(the)h(present)e(Section.)43 b(In)456 2801 y(order)14 b(to)h(do)g(so,)h(we)g(\002rst)g(construct)e (an)h(upper)f(bound)f(on)i(the)h(lo)n(west)f(Dirichlet)g(eigen)m(v)n (alue)e(sho)n(wing)456 2901 y(up)19 b(in)i(the)f(left-hand)e(side)j(of) f(\(23\))f(when)g(choosing)1634 3097 y Fo(\003)k(:=)p 1825 3000 346 4 v 1899 3018 a Fg([)1825 3200 y Fj(j)p Fi(j)s Fj(j)t Fi()f Fo(0)e Ft(there.)25 b(By)c(construction)d Fo(\003)i Ft(is)h(open)e(and)h(compatible)f(with)h(the)g(lattice.)605 3509 y Fq(5.1.)40 b(Upper)25 b(bound)f(on)h(lo)o(west)f(Dirichlet)g (eigen)m(v)o(alue.)40 b Ft(The)23 b(follo)n(wing)g(lemma)g(basically) 456 3609 y(repeats)c([)p Fq(KS86)o Ft(,)h(Prop.)g(5])g(and)f(its)j (corollary)-5 b(.)605 3765 y Fq(Lemma)19 b(5.1.)36 b Ff(Let)19 b Fo(\003)f Ff(denote)f(the)h(open)f(cube)g(\(74\).)23 b(Ther)m(e)18 b(e)n(xist)h(two)g(constant)d Fo(0)23 b Fp(<)g(C)3186 3777 y Fl(1)3223 3765 y Fp(;)14 b(C)3319 3777 y Fl(2)3380 3765 y Fp(<)456 3865 y Fm(1)20 b Ff(\(whic)o(h)g(ar)m (e)g(independent)e(of)i Fp(!)k Ff(and)19 b Fp(L)p Ff(\))h(suc)o(h)f (that)1110 4050 y Fp(\025)1158 4062 y Fl(0)1209 3983 y Fg(\000)1247 4050 y Fp(H)1323 4016 y Fi(D)1316 4071 y Fl(\003)1383 4050 y Fo(\()p Fp(V)1463 4062 y Fi(!)1512 4050 y Fo(\))1544 3983 y Fg(\001)1605 4050 y Fm(\024)k Fp(C)1752 4062 y Fl(1)1803 4050 y Fm(j)p Fo(\003)p Fm(j)1907 4016 y Fj(\000)p Fl(1)2010 3937 y Fg(Z)2056 4126 y Fl(\003)2120 4050 y Fp(V)2168 4062 y Fi(!)2216 4050 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)20 b Fo(+)e Fp(C)2593 4062 y Fl(2)2645 4050 y Fp(L)2702 4016 y Fj(\000)p Fl(2)3306 4050 y Ft(\(75\))456 4238 y Ff(for)i(all)h Fp(!)k Fm(2)f Fo(\012)d Ff(and)e(all)h Fp(L)j(>)f Fo(1)p Ff(.)607 4395 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(W)-7 b(e)24 b(let)g Fp(\022)31 b Fm(2)d(C)1348 4364 y Fj(1)1343 4415 y Fi(c)1418 4395 y Fo(\(\003)1508 4407 y Fl(0)1546 4395 y Fo(\))c Ft(denote)e(a)h(smoothed)f(indicator)g (function)f(of)i(the)g(cube)g Fm(f)p Fp(x)28 b Fm(2)456 4497 y Fn(R)526 4467 y Fi(d)587 4497 y Fo(:)23 b Fm(j)p Fp(x)p Fm(j)h Fp(<)f Fo(1)p Fp(=)p Fo(4)p Fm(g)e(\032)h Fo(\003)1172 4509 y Fl(0)1230 4497 y Ft(and)e(set)h Fp(\022)1523 4509 y Fi(L)1572 4497 y Fo(\()p Fp(x)p Fo(\))j(:=)f Fp(\022)1859 4430 y Fg(\000)1898 4497 y Fp(x=)p Fm(j)p Fo(\003)p Fm(j)2091 4467 y Fl(1)p Fi(=d)2196 4430 y Fg(\001)2255 4497 y Ft(for)c(all)i Fp(x)j Fm(2)f Fo(\003)p Ft(.)i(Choosing)20 b(the)g(product)456 4597 y(of)g Fp(\022)585 4609 y Fi(L)659 4597 y Fm(2)k(C)787 4567 y Fj(1)782 4617 y Fi(c)857 4597 y Fo(\(\003\))e Ft(and)e(the)h(ground-state)d(function)h Fp( )25 b Ft(of)20 b Fp(H)7 b Fo(\(0\))21 b Ft(as)h(the)f(v)n(ariational)e(function)g(in)i (the)456 4697 y(Rayleigh-Ritz)e(principle)g(we)i(obtain)764 4844 y Fp(\025)812 4856 y Fl(0)863 4777 y Fg(\000)901 4844 y Fp(H)977 4810 y Fi(D)970 4865 y Fl(\003)1037 4844 y Fo(\()p Fp(V)1117 4856 y Fi(!)1166 4844 y Fo(\))1198 4777 y Fg(\001)28 b(\012)1303 4844 y Fp(\022)1342 4856 y Fi(L)1392 4844 y Fp( )s(;)14 b(\022)1525 4856 y Fi(L)1574 4844 y Fp( )1631 4777 y Fg(\013)1694 4844 y Fm(\024)1781 4777 y Fg(\012)1821 4844 y Fp(\022)1860 4856 y Fi(L)1909 4844 y Fp( )s(;)g(H)2079 4810 y Fi(D)2072 4865 y Fl(\003)2139 4844 y Fo(\()p Fp(V)2219 4856 y Fi(!)2268 4844 y Fo(\))p Fp(\022)2339 4856 y Fi(L)2389 4844 y Fp( )2446 4777 y Fg(\013)1225 4977 y Fo(=)1312 4910 y Fg(\012)1352 4977 y Fp(\022)1391 4989 y Fi(L)1440 4977 y Fp( )s(;)g(V)1582 4989 y Fi(!)1631 4977 y Fp(\022)1670 4989 y Fi(L)1719 4977 y Fp( )1776 4910 y Fg(\013)1834 4977 y Fo(+)1917 4910 y Fg(\012)1970 4977 y Fo(\()p Fm(r)p Fp(\022)2110 4989 y Fi(L)2160 4977 y Fo(\))g Fp( )s(;)g Fo(\()q Fm(r)p Fp(\022)2441 4989 y Fi(L)2491 4977 y Fo(\))g Fp( )2594 4910 y Fg(\013)1225 5159 y Fm(\024)39 b Fo(sup)1312 5229 y Fi(y)r Fj(2)p Fl(\003)1438 5237 y Fv(0)1485 5159 y Fp( )s Fo(\()p Fp(y)s Fo(\))1650 5125 y Fl(2)1724 5042 y Fg(\024)1768 5046 y(Z)1814 5235 y Fl(\003)1877 5159 y Fp(V)1925 5171 y Fi(!)1974 5159 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)20 b Fo(+)e Fm(j)p Fo(\003)p Fm(j)2396 5125 y Fl(1)p Fj(\000)p Fl(2)p Fi(=d)2600 5046 y Fg(Z)2646 5235 y Fl(\003)2691 5243 y Fv(0)2742 5159 y Fm(jr)p Fp(\022)r Fo(\()p Fp(x)p Fo(\))p Fm(j)3009 5125 y Fl(2)3048 5159 y Fp(dx)3138 5042 y Fg(\025)3196 5159 y Fp(:)87 b Ft(\(76\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 456 251 a Fr(20)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)456 450 y Ft(Here)15 b(the)g(equality)g(uses)h Fp(H)1269 410 y Fi(\037)1262 475 y Fl(\003)1314 450 y Fo(\(0\))p Fp( )26 b Fo(=)d(0)16 b Ft(and)f(inte)o(gration)e(by)i(parts.)23 b(Observing)14 b(that)i Fm(h)p Fp(\022)3035 462 y Fi(L)3085 450 y Fp( )s(;)e(\022)3218 462 y Fi(L)3267 450 y Fp( )s Fm(i)24 b(\025)456 557 y Fo(2)498 526 y Fj(\000)p Fi(d)588 557 y Fm(j)p Fo(\003)p Fm(j)j Fo(inf)820 569 y Fi(x)p Fj(2)p Fl(\003)948 577 y Fv(0)998 557 y Fp( )s Fo(\()p Fp(x)p Fo(\))1166 526 y Fl(2)1227 557 y Ft(and)20 b(that)i(the)f(is)h(some)f(constant)g Fp(C)31 b(>)25 b Fo(0)c Ft(such)g(that)h Fm(j)p Fo(\003)p Fm(j)2879 526 y Fl(1)p Fi(=d)3009 557 y Fm(\025)j Fp(C)6 b(L)22 b Ft(for)e(all)456 656 y Fp(L)i(>)h Fo(1)p Ft(,)d(completes)f (the)h(proof.)1992 b Fd(\003)605 849 y Ft(Our)24 b(ne)o(xt)f(task)i(is) g(to)f(bound)e(the)j(inte)o(gral)e(in)h(the)g(right-hand)e(side)i(of)g (\(75\))f(from)g(abo)o(v)o(e.)35 b(F)o(or)456 948 y(this)20 b(purpose)f(it)i(will)g(be)f(useful)g(to)g(introduce)e(the)i(cuboid) 1644 1106 y Fg(e)1638 1127 y Fo(\003)j(:=)1935 1048 y Fg([)1829 1240 y Fj(j)p Fi(j)1876 1248 y Fv(1)1909 1240 y Fj(j)t(\024)p Fl(2)p Fi(L)2064 1215 y Fk(\014)2097 1227 y Fv(1)1829 1317 y Fj(j)p Fi(j)1876 1325 y Fv(2)1909 1317 y Fj(j)t(\024)p Fl(2)p Fi(L)2064 1292 y Fk(\014)2097 1304 y Fv(2)2146 1127 y Fo(\003)2204 1139 y Fi(j)2239 1127 y Fp(;)1044 b Ft(\(77\))456 1470 y(which)19 b(contains)h(the)g (cube)f Fo(\003)i Ft(de\002ned)e(in)h(\(74\).)k(Here)c(and)g(in)g(the)g (follo)n(wing)f(we)h(use)h(the)f(abbre)n(via-)456 1570 y(tion)g Fp(\014)653 1582 y Fi(k)716 1570 y Fo(:=)j(max)13 b Fm(f)p Fo(1)p Fp(;)h Fo(2)p Fp(=\013)1253 1582 y Fi(k)1292 1570 y Fo(\(1)19 b Fm(\000)f Fp(\015)5 b Fo(\))p Fm(g)22 b Fo(=)h(2)p Fp(=d)1827 1582 y Fi(k)1894 1570 y Fo(max)14 b Fm(f)o Fp(d)2147 1582 y Fi(k)2188 1570 y Fp(=)p Fo(2)p Fp(;)g(\015)2352 1582 y Fi(k)2392 1570 y Fp(=)p Fo(\(1)k Fm(\000)g Fp(\015)5 b Fo(\))p Fm(g)o Ft(,)21 b(for)e Fp(k)26 b Fm(2)e(f)p Fo(1)p Fp(;)14 b Fo(2)p Fm(g)p Ft(.)605 1737 y Fq(Lemma)21 b(5.2.)40 b Ff(Let)21 b Fp(L)h(>)h Fo(0)d Ff(and)g(de\002ne)f(the)h(r)o(andom)f(variable)1075 1947 y Fp(W)1153 1959 y Fi(!)1201 1947 y Fo(\()p Fp(L)p Fo(\))24 b(:=)e Fm(j)p Fo(\003)p Fm(j)1560 1913 y Fj(\000)p Fl(1)1663 1834 y Fg(Z)1709 2023 y Fe(R)1763 2007 y Fk(d)1798 2023 y Fj(n)1836 2008 y Fb(e)1832 2023 y Fl(\003)1896 1830 y Fg(\022)1957 1834 y(Z)2003 2023 y Fl(\003)2066 1947 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx)2477 1830 y Fg(\023)2552 1947 y Fp(\026)2602 1959 y Fi(!)2651 1947 y Fo(\()p Fp(dy)s Fo(\))p Fp(:)481 b Ft(\(78\))456 2168 y Ff(Then)19 b(the)i(following)e(thr)m(ee)h (assertions)h(hold)e(true:)543 2311 y(\(i\))83 b Fm(j)p Fo(\003)p Fm(j)809 2281 y Fj(\000)p Fl(1)911 2244 y Fg(R)951 2340 y Fl(\003)1014 2311 y Fp(V)1062 2323 y Fi(!)1111 2311 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)24 b Fm(\024)f(k)o Fp(f)9 b Fm(k)1570 2336 y Fl(1)1635 2311 y Fp(\026)1685 2323 y Fi(!)1733 2244 y Fg(\000)1777 2290 y(e)1771 2311 y Fo(\003)1829 2244 y Fg(\001)1886 2311 y Fo(+)18 b Fp(W)2047 2323 y Fi(!)2095 2311 y Fo(\()p Fp(L)p Fo(\))j Ff(for)g(all)f Fp(!)26 b Fm(2)d Fo(\012)e Ff(and)e(all)i Fp(L)h(>)h Fo(0)p Ff(.)520 2473 y(\(ii\))83 b(ther)m(e)24 b(e)n(xists)i(some)e (constant)g Fo(0)30 b Fp(<)g(C)1833 2485 y Fl(3)1902 2473 y Fp(<)g Fm(1)25 b Ff(\(whic)o(h)f(is)h(independent)d(of)i Fp(!)k Ff(and)23 b Fp(L)p Ff(\))h(suc)o(h)705 2572 y(that)1319 2721 y Fn(P)1398 2654 y Fg(\010)1447 2721 y Fp(!)h Fm(2)f Fo(\012)37 b(:)g Fp(W)1838 2733 y Fi(!)1886 2721 y Fo(\()p Fp(L)p Fo(\))23 b Fm(\025)g Fp(C)2177 2733 y Fl(3)2214 2721 y Fp(L)2271 2687 y Fj(\000)p Fl(2)2360 2654 y Fg(\011)2432 2721 y Fm(\024)2529 2665 y Fo(1)p 2529 2702 42 4 v 2529 2778 a(2)3306 2721 y Ft(\(79\))705 2891 y Ff(for)d(lar)m(g)o(e)g (enough)f Fp(L)p Ff(.)497 3059 y(\(iii\))83 b(the)20 b(r)o(andom)f(variables)g Fp(\026)1484 2992 y Fg(\000)1528 3038 y(e)1523 3059 y Fo(\003)1580 2992 y Fg(\001)1639 3059 y Ff(and)g Fp(W)12 b Fo(\()p Fp(L)p Fo(\))21 b Ff(ar)m(e)f (independent)e(for)j(all)f Fp(L)j(>)f Fo(0)p Ff(.)607 3226 y Ft(P)t FC(R)q(O)t(O)t(F)n Ft(.)45 b(F)o(or)17 b(a)i(proof)e(of)h(the)g(\002rst)h(assertion)f(we)g(decompose)e(the)j (domain)d(of)i(inte)o(gration)f(and)456 3326 y(use)j(Fubini')-5 b(s)20 b(theorem)f(to)h(obtain)495 3408 y Fg(Z)541 3596 y Fl(\003)605 3521 y Fp(V)653 3533 y Fi(!)701 3521 y Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)85 b Fo(=)1148 3408 y Fg(Z)1198 3582 y Fb(e)1194 3597 y Fl(\003)1258 3404 y Fg(\022)1319 3408 y(Z)1365 3596 y Fl(\003)1429 3521 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)g Fp(y)s Fo(\))c Fp(dx)1839 3404 y Fg(\023)1915 3521 y Fp(\026)1965 3533 y Fi(!)2013 3521 y Fo(\()p Fp(dy)s Fo(\))19 b(+)2266 3408 y Fg(Z)2312 3596 y Fe(R)2366 3580 y Fk(d)2401 3596 y Fj(n)2439 3581 y Fb(e)2435 3596 y Fl(\003)2498 3404 y Fg(\022)2560 3408 y(Z)2606 3596 y Fl(\003)2669 3521 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx)3080 3404 y Fg(\023)3155 3521 y Fp(\026)3205 3533 y Fi(!)3253 3521 y Fo(\()p Fp(dy)s Fo(\))1001 3725 y Fm(\024)82 b(k)p Fp(f)9 b Fm(k)1281 3750 y Fl(1)1346 3725 y Fp(\026)1396 3737 y Fi(!)1444 3658 y Fg(\000)1488 3704 y(e)1482 3725 y Fo(\003)1540 3658 y Fg(\001)1596 3725 y Fo(+)18 b Fm(j)p Fo(\003)p Fm(j)c Fp(W)1875 3737 y Fi(!)1923 3725 y Fo(\()p Fp(L)p Fo(\))p Fp(:)1239 b Ft(\(80\))456 3894 y(Here)29 b(the)h(inequality)e(results)i(from)f(the)h(estimate)2019 3827 y Fg(R)2058 3924 y Fl(\003)2122 3894 y Fp(f)9 b Fo(\()p Fp(x)25 b Fm(\000)h Fp(y)s Fo(\))14 b Fp(dx)41 b Fm(\024)2693 3827 y Fg(R)2732 3924 y Fe(R)2786 3907 y Fk(d)2839 3894 y Fp(f)9 b Fo(\()p Fp(x)p Fo(\))14 b Fp(dx)42 b Fo(=:)f Fm(k)p Fp(f)9 b Fm(k)3409 3906 y Fl(1)456 4003 y Ft(v)n(alid)23 b(for)g(all)h Fp(y)32 b Fm(2)d Fn(R)1099 3972 y Fi(d)1138 4003 y Ft(.)35 b(This)24 b(yields)f(Lemma)g (5.2\(i\))f(since)i Fo(1)29 b Fm(\024)g(j)p Fo(\003)p Fm(j)p Ft(.)35 b(F)o(or)23 b(a)h(proof)e(of)h(the)h(second)456 4102 y(assertion,)19 b(we)i(emplo)o(y)e(Chebyche)n(v')-5 b(s)18 b(inequality)481 4339 y Fn(P)546 4247 y Fg(n)602 4339 y Fp(!)25 b Fm(2)f Fo(\012)37 b(:)g Fp(W)993 4351 y Fi(!)1041 4339 y Fo(\()p Fp(L)p Fo(\))23 b Fm(\025)g Fp(C)1332 4351 y Fl(3)1369 4339 y Fp(L)1426 4305 y Fj(\000)p Fl(2)1515 4247 y Fg(o)1653 4339 y Fm(\024)1812 4283 y Fp(L)1869 4253 y Fl(2)p 1811 4320 97 4 v 1811 4396 a Fp(C)1870 4408 y Fl(3)1931 4339 y Fm(j)p Fo(\003)p Fm(j)2035 4305 y Fj(\000)p Fl(1)2138 4339 y Fn(E)2217 4197 y Fg(")2265 4226 y(Z)2312 4415 y Fe(R)2366 4398 y Fk(d)2401 4415 y Fj(n)2439 4400 y Fb(e)2435 4415 y Fl(\003)2498 4222 y Fg(\022)2559 4226 y(Z)2605 4415 y Fl(\003)2669 4339 y Fp(f)9 b Fo(\()p Fp(x)18 b Fm(\000)h Fp(y)s Fo(\))14 b Fp(dx)3080 4222 y Fg(\023)3155 4339 y Fp(\026)p Fo(\()p Fp(dy)s Fo(\))3356 4197 y Fg(#)1653 4621 y Fo(=)1812 4565 y Fp(L)1869 4535 y Fl(2)p 1811 4602 V 1811 4678 a Fp(C)1870 4690 y Fl(3)1931 4621 y Fm(j)p Fo(\003)p Fm(j)2035 4587 y Fj(\000)p Fl(1)2138 4508 y Fg(Z)2184 4697 y Fl(\003)2248 4479 y Fg( )2313 4508 y(Z)2360 4697 y Fe(R)2414 4680 y Fk(d)2449 4697 y Fj(n)2487 4682 y Fb(e)2483 4697 y Fl(\003)2546 4621 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx)2957 4479 y Fg(!)p 3037 4575 51 4 v 3037 4621 a Fp(\026)p Fo(\()p Fp(dy)s Fo(\))1653 4882 y Fm(\024)1812 4826 y Fp(L)1869 4796 y Fl(2)p 1811 4863 97 4 v 1811 4939 a Fp(C)1870 4951 y Fl(3)p 1931 4836 51 4 v 1931 4882 a Fp(\026)p Fo(\(\003)2071 4894 y Fl(0)2109 4882 y Fo(\))28 b(sup)2169 4952 y Fi(y)r Fj(2)p Fl(\003)2322 4769 y Fg(Z)2368 4958 y Fe(R)2422 4941 y Fk(d)2457 4958 y Fj(n)2495 4943 y Fb(e)2491 4958 y Fl(\003)2555 4882 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx:)317 b Ft(\(81\))456 5116 y(Here)27 b(the)h(inequality)e(uses)i(the)g(f)o(act)f(that)h(the)g (intensity)f(measure)p 2489 5070 V 27 w Fp(\026)h Ft(is)g Fn(Z)2709 5086 y Fi(d)2748 5116 y Ft(-periodic.)46 b(The)27 b(inner)456 5216 y(inte)o(gral)g(is)i(in)g(turn)e(estimated)h(from)g (abo)o(v)o(e)e(in)j(term)f(of)g(tw)o(o)h(inte)o(grals)e(in)m(v)n (olving)f(the)j(mar)o(ginal)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(21)456 450 y Ft(impurity)18 b(potentials)i Fp(f)1156 420 y Fl(\(1\))1265 450 y Ft(and)g Fp(f)1456 420 y Fl(\(2\))1566 450 y Ft(\(recall)f(the)i(de\002nitions)e(\(24\))g (and)h(\(25\)\))605 532 y Fg(Z)651 720 y Fe(R)705 704 y Fk(d)740 720 y Fj(n)778 705 y Fb(e)774 720 y Fl(\003)837 645 y Fp(f)9 b Fo(\()p Fp(x)19 b Fm(\000)f Fp(y)s Fo(\))c Fp(dx)84 b Fm(\024)1479 532 y Fg(Z)1525 720 y Fj(j)p Fi(x)1583 728 y Fv(1)1615 720 y Fj(j)p Fi(>L)1733 702 y Fk(\014)1766 714 y Fv(1)1727 645 y Fp(f)1777 610 y Fl(\(1\))1866 645 y Fo(\()p Fp(x)1945 657 y Fl(1)2002 645 y Fm(\000)18 b Fp(y)2126 657 y Fl(1)2162 645 y Fo(\))c Fp(dx)2298 657 y Fl(1)2355 645 y Fo(+)2438 532 y Fg(Z)2484 720 y Fj(j)p Fi(x)2542 728 y Fv(2)2574 720 y Fj(j)p Fi(>L)2692 702 y Fk(\014)2725 714 y Fv(2)2686 645 y Fp(f)2736 610 y Fl(\(2\))2825 645 y Fo(\()p Fp(x)2904 657 y Fl(2)2960 645 y Fm(\000)k Fp(y)3084 657 y Fl(2)3121 645 y Fo(\))c Fp(dx)3257 657 y Fl(2)1332 840 y Fm(\024)82 b Fp(C)6 b(L)1601 805 y Fj(\000)p Fl(2)1690 840 y Fp(:)1593 b Ft(\(82\))456 995 y(Here)19 b(the)g(e)o(xistence)f(of)h(some)g Fo(0)j Fp(<)h(C)29 b(<)23 b Fm(1)d Ft(ensuring)d(the)j(last)g (inequality)d(for)i(all)h Fm(j)p Fp(y)s Fm(j)i(\024)h Fp(L=)p Fo(2)18 b Ft(\(that)456 1094 y(is)23 b(in)f(particular;)h(for)e (all)i Fp(y)30 b Fm(2)e Fo(\003)p Ft(\))22 b(and)g(suf)n(\002ciently)f (lar)o(ge)g Fp(L)27 b Fm(\025)g Fo(4)22 b Ft(follo)n(ws)g(from)f (\(28\))h(and)f(the)i(f)o(act)456 1194 y(that)h Fp(\014)652 1206 y Fi(k)693 1194 y Fp(\013)746 1206 y Fi(k)787 1194 y Fo(\(1)d Fm(\000)g Fp(\015)5 b Fo(\))30 b Fm(\024)g Fo(2)p Ft(.)37 b(T)-7 b(aking)23 b Fp(C)1585 1206 y Fl(3)1647 1194 y Ft(in)h(\(81\))f(lar)o(ge)g(enough)g(yields)h(the)g(second)f (assertion.)36 b(The)456 1293 y(third)19 b(assertion)h(is)h(a)g (consequence)d(of)i(Assumption)f(2.1\(ii\))n(.)1114 b Fd(\003)605 1472 y Fq(5.2.)40 b(Pr)o(oof)19 b(of)h(Theor)o(em)g(2.8)f (\226)h(\002nal)h(parts.)41 b Ft(F)o(or)20 b(a)g(gi)n(v)o(en)f(ener)o (gy)f Fp(E)28 b(>)23 b Fo(0)d Ft(we)h(choose)1452 1697 y Fp(L)h Fo(:=)1642 1580 y Fg(\022)1713 1641 y Fo(3)14 b(max)o Fm(f)p Fp(C)2024 1653 y Fl(2)2061 1641 y Fp(;)g(C)2157 1653 y Fl(3)2194 1641 y Fm(g)p 1713 1678 523 4 v 1941 1754 a Fp(E)2246 1580 y Fg(\023)2307 1597 y Fl(1)p Fi(=)p Fl(2)2425 1697 y Fp(;)858 b Ft(\(83\))456 1896 y(where)27 b(the)h(constants)g Fp(C)1214 1908 y Fl(2)1280 1896 y Ft(and)g Fp(C)1488 1908 y Fl(3)1554 1896 y Ft(were)g(\002x)o(ed)f(in)i (Lemma)e(5.1)g(and)h(Lemma)f(5.2,)i(respecti)n(v)o(ely)-5 b(.)456 2003 y(Moreo)o(v)o(er)m(,)13 b(we)i(pick)g(the)g(cube)g Fo(\003)g Ft(from)f(\(74\))g(and)h(the)g(cuboid)2281 1982 y Fg(e)2275 2003 y Fo(\003)g Ft(from)g(\(77\).)22 b(Emplo)o(ying)13 b(Lemma)h(5.1)456 2103 y(and)19 b(Lemma)h(5.2)f(we)i (estimate)f(the)g(probability)f(in)h(the)g(right-hand)e(side)i(of)g (\(23\))f(according)f(to)509 2258 y Fn(P)588 2191 y Fg(\010)636 2258 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\025)997 2270 y Fl(0)1048 2191 y Fg(\000)1086 2258 y Fp(H)1162 2224 y Fi(D)1155 2279 y Fl(\003)1222 2258 y Fo(\()p Fp(V)1302 2270 y Fi(!)1351 2258 y Fo(\))1383 2191 y Fg(\001)1445 2258 y Fp(<)22 b(E)1598 2191 y Fg(\011)532 2421 y Fm(\025)g Fn(P)684 2329 y Fg(\020)748 2354 y(\010)796 2421 y Fp(!)k Fm(2)d Fo(\012)37 b(:)g Fp(\025)1157 2433 y Fl(0)1209 2354 y Fg(\000)1247 2421 y Fp(H)1323 2387 y Fi(D)1316 2442 y Fl(\003)1383 2421 y Fo(\()p Fp(V)1463 2433 y Fi(!)1511 2421 y Fo(\))1543 2354 y Fg(\001)1605 2421 y Fp(<)22 b(E)1758 2354 y Fg(\011)1825 2421 y Fm(\\)1899 2354 y Fg(\010)1948 2421 y Fp(!)j Fm(2)f Fo(\012)37 b(:)g Fp(W)2339 2433 y Fi(!)2387 2421 y Fo(\()p Fp(L)p Fo(\))23 b Fp(<)g(C)2678 2433 y Fl(3)2715 2421 y Fp(L)2772 2387 y Fj(\000)p Fl(2)2861 2354 y Fg(\011)2923 2329 y(\021)532 2653 y Fm(\025)f Fn(P)684 2511 y Fg( \()817 2653 y Fp(!)k Fm(2)d Fo(\012)37 b(:)g Fp(\026)1180 2665 y Fi(!)1228 2586 y Fg(\000)1272 2632 y(e)1266 2653 y Fo(\003)1324 2586 y Fg(\001)1385 2653 y Fp(<)1482 2597 y Fo(max)p Fm(f)p Fp(C)1738 2609 y Fl(2)1775 2597 y Fp(;)14 b(C)1871 2609 y Fl(3)1908 2597 y Fm(g)g Fp(L)2021 2567 y Fj(\000)p Fl(2)p 1482 2634 628 4 v 1656 2710 a Fp(C)1715 2722 y Fl(1)1766 2710 y Fm(k)p Fp(f)9 b Fm(k)1900 2722 y Fl(1)2119 2511 y Fg(\))2205 2653 y Fm(\\)2279 2561 y Fg(n)2334 2653 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(W)2725 2665 y Fi(!)2773 2653 y Fo(\()p Fp(L)p Fo(\))23 b Fp(<)g(C)3064 2665 y Fl(3)3102 2653 y Fp(L)3159 2619 y Fj(\000)p Fl(2)3247 2561 y Fg(o)3303 2511 y(!)3368 2653 y Fp(:)3306 2823 y Ft(\(84\))456 3001 y(Since)h(the)g(random)f(v)n(ariables)g Fp(\026)p Fo(\()1480 2980 y Fg(e)1474 3001 y Fo(\003\))i Ft(and)f Fp(W)12 b Fo(\()p Fp(L)p Fo(\))25 b Ft(are)f(independent,)e(the)i(probability)f (in)h(\(84\))f(f)o(ac-)456 3101 y(torises.)58 b(Thanks)31 b(to)g(\(79\))f(the)i(probability)d(of)i(the)g(second)g(e)n(v)o(ent)f (is)j(bounded)28 b(from)j(belo)n(w)f(by)456 3201 y Fo(1)p Fp(=)p Fo(2)23 b Ft(pro)o(vided)e(that)k Fp(L)f Ft(is)h(lar)o(ge)e (enough,)g(equi)n(v)n(alently)-5 b(,)22 b(that)i Fp(E)35 b(>)30 b Fo(0)24 b Ft(is)h(small)g(enough.)35 b(Emplo)o(y-)456 3311 y(ing)24 b(the)h(decomposition)d(\(74\))h(of)1492 3290 y Fg(e)1486 3311 y Fo(\003)i Ft(into)f Fm(j)1752 3290 y Fg(e)1746 3311 y Fo(\003)p Fm(j)h Ft(unit)g(cubes)f(of)g(the)h (lattice)g Fn(Z)2729 3280 y Fi(d)2768 3311 y Ft(,)h(we)f(ha)n(v)o(e)g Fp(\026)3167 3323 y Fi(!)3215 3243 y Fg(\000)3258 3290 y(e)3253 3311 y Fo(\003)3310 3243 y Fg(\001)3380 3311 y Fo(=)456 3356 y Fg(P)543 3447 y Fi(j)s Fj(2)622 3432 y Fb(e)618 3447 y Fl(\003)q Fj(\\)p Fe(Z)755 3430 y Fk(d)808 3418 y Fp(\026)858 3430 y Fi(!)906 3351 y Fg(\000)944 3418 y Fo(\003)1002 3430 y Fi(j)1037 3351 y Fg(\001)1092 3418 y Ft(such)17 b(that)g(the)g(probability)e(of)h(the)h(\002rst)h(e)n (v)o(ent)e(in)h(\(84\))f(is)h(bounded)e(from)h(belo)n(w)456 3523 y(by)824 3715 y Fn(P)889 3573 y Fg(\()956 3715 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\026)1319 3727 y Fi(!)1367 3647 y Fg(\000)1405 3715 y Fo(\003)1463 3727 y Fi(j)1498 3647 y Fg(\001)1559 3715 y Fp(<)1656 3659 y Fo(max)p Fm(f)p Fp(C)1912 3671 y Fl(2)1949 3659 y Fp(;)14 b(C)2045 3671 y Fl(3)2082 3659 y Fm(g)g Fp(L)2195 3628 y Fj(\000)p Fl(2)p 1656 3696 V 1771 3786 a Fp(C)1830 3798 y Fl(1)1881 3786 y Fm(k)p Fp(f)9 b Fm(k)2015 3798 y Fl(1)2065 3786 y Fm(j)2094 3766 y Fg(e)2088 3786 y Fo(\003)p Fm(j)2376 3715 y Ft(for)20 b(all)h Fp(j)28 b Fm(2)2744 3694 y Fg(e)2738 3715 y Fo(\003)18 b Fm(\\)h Fn(Z)2947 3685 y Fi(d)2986 3573 y Fg(\))3053 3715 y Fp(:)230 b Ft(\(85\))456 3945 y(By)32 b(construction)e(of)1135 3924 y Fg(e)1129 3945 y Fo(\003)j Ft(there)e(is)i(some)e(constant)h Fp(n)2073 3957 y Fl(0)2154 3945 y Fp(>)45 b Fo(0)32 b Ft(such)f(that)h Fm(j)2708 3924 y Fg(e)2702 3945 y Fo(\003)p Fm(j)45 b(\024)f Fp(n)2987 3957 y Fl(0)3038 3945 y Fp(L)3095 3915 y Fi(\014)3133 3923 y Fv(1)3165 3915 y Fi(d)3200 3923 y Fv(1)3232 3915 y Fl(+)p Fi(\014)3321 3923 y Fv(2)3353 3915 y Fi(d)3388 3923 y Fv(2)3424 3945 y Ft(.)456 4045 y(Abbre)n(viating)15 b Fp(C)969 4057 y Fl(4)1030 4045 y Fo(:=)23 b(max)o Fm(f)p Fp(C)1396 4057 y Fl(2)1433 4045 y Fp(;)14 b(C)1529 4057 y Fl(3)1567 4045 y Fm(g)p Fp(=)p Fo(\()p Fp(C)1742 4057 y Fl(1)1779 4045 y Fm(k)p Fp(f)9 b Fm(k)1913 4057 y Fl(1)1962 4045 y Fp(n)2012 4057 y Fl(0)2049 4045 y Fo(\))19 b Ft(and)f Fp(#)23 b Fo(:=)g(2)11 b(+)g Fp(\014)2598 4057 y Fl(1)2634 4045 y Fp(d)2677 4057 y Fl(1)2726 4045 y Fo(+)g Fp(\014)2849 4057 y Fl(2)2885 4045 y Fp(d)2928 4057 y Fl(2)2966 4045 y Ft(,)19 b(and)e(using)h(the)456 4145 y(f)o(act)i(that)g(the)g(random) e(v)n(ariables)h Fp(\026)p Fo(\(\003)1599 4157 y Fi(j)1634 4145 y Fo(\))h Ft(are)g(independent)d(and)j(identically)f(distrib)n (uted)g(\(by)g(virtue)456 4244 y(of)h(Assumption)f(2.1\),)g(the)h(last) h(e)o(xpression)d(\(85\))h(may)h(be)g(bounded)e(from)h(belo)n(w)h(by) 639 4455 y Fn(P)704 4363 y Fg(n)759 4455 y Fp(!)26 b Fm(2)d Fo(\012)37 b(:)g Fp(\026)1122 4467 y Fi(!)1170 4455 y Fo(\(\003)1260 4467 y Fl(0)1297 4455 y Fo(\))23 b Fp(<)g(C)1499 4467 y Fl(4)1537 4455 y Fp(L)1594 4420 y Fj(\000)p Fi(#)1689 4363 y Fg(o)1745 4380 y Fi(n)1786 4388 y Fv(0)1829 4380 y Fi(L)1875 4355 y Fk(\014)1908 4367 y Fv(1)1940 4355 y Fk(d)1971 4367 y Fv(1)2003 4355 y(+)p Fk(\014)2079 4367 y Fv(2)2111 4355 y Fk(d)2142 4367 y Fv(2)2206 4455 y Fm(\025)2293 4387 y Fg(\000)2331 4455 y Fp(C)2390 4467 y Fl(4)2428 4455 y Fp(L)2485 4420 y Fj(\000)p Fi(#)2581 4387 y Fg(\001)2619 4401 y Fi(\024n)2699 4409 y Fv(0)2742 4401 y Fi(L)2788 4376 y Fk(\014)2821 4388 y Fv(1)2853 4376 y Fk(d)2884 4388 y Fv(1)2916 4376 y(+)p Fk(\014)2992 4388 y Fv(2)3024 4376 y Fk(d)3055 4388 y Fv(2)1345 4637 y Fo(=)f(exp)1573 4545 y Fg(h)1612 4637 y Fp(C)1671 4649 y Fl(5)1723 4545 y Fg(\020)1772 4637 y Fo(log)14 b Fp(E)1959 4603 y Fi(#=)p Fl(2)2089 4637 y Fo(+)k(log)c Fp(C)2352 4649 y Fl(6)2390 4545 y Fg(\021)2453 4637 y Fp(E)2519 4603 y Fj(\000)p Fl(\()p Fi(\014)2635 4611 y Fv(1)2667 4603 y Fi(d)2702 4611 y Fv(1)2734 4603 y Fl(+)p Fi(\014)2823 4611 y Fv(2)2855 4603 y Fi(d)2890 4611 y Fv(2)2922 4603 y Fl(\))p Fi(=)p Fl(2)3019 4545 y Fg(i)3072 4637 y Fp(:)211 b Ft(\(86\))456 4817 y(Here)17 b(the)h(\002rst)g(inequality)f(deri)n(v)o(es)f(from)h (Assumption)f(2.1)h(on)h(the)f(probability)f(measure)h(of)g Fp(\026)p Fo(\(\003)3354 4829 y Fl(0)3391 4817 y Fo(\))p Ft(.)456 4917 y(Moreo)o(v)o(er)m(,)24 b(the)i(e)o(xistence)f(of)h(tw)o (o)h(constants)e Fo(0)34 b Fp(<)f(C)2103 4929 y Fl(5)2141 4917 y Ft(,)28 b Fp(C)2249 4929 y Fl(6)2320 4917 y Fp(<)34 b Fm(1)26 b Ft(ensuring)f(the)h(v)n(alidity)f(of)h(the)456 5016 y(equality)h(follo)n(ws)h(from)g(\(83\).)48 b(Since)29 b(the)f(choice)g(\(83\))f(of)i(the)f(ener)o(gy-dependence)23 b(of)28 b Fp(L)h Ft(guar)n(-)456 5116 y(antees)23 b(that)g(the)h(pre-f) o(actor)d(in)j(the)f(lo)n(wer)g(bound)e(in)j(Proposition)e(3.2)g(is)j (ne)o(gligible,)d(the)h(proof)f(of)456 5216 y(Theorem)c(2.8)i(is)h (completed)d(by)i(inserting)f(\(86\))h(in)g(the)g(left-hand)f(side)h (of)g(\(23\).)529 b Fd(\003)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 456 251 a Fr(22)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)969 450 y Fq(A)n(ppendix)20 b(A.)42 b(Pr)o(oof)18 b(of)i(mixing)g(of)g(random)g(Bor)o(el)g(measur)o (e)605 600 y Ft(The)h(purpose)e(of)h(this)i(short)e(appendix)f(is)j(to) f(proof)e(Lemma)h(2.3.)26 b(W)-7 b(e)22 b(let)g Fo(\003)2869 569 y Fl(\()p Fi(n)p Fl(\))2990 600 y Fo(:=)3101 537 y Fg(S)3171 624 y Fj(j)p Fi(j)s Fj(j\024)p Fi(n)3352 600 y Fo(\003)3410 612 y Fi(j)456 721 y Ft(with)i Fp(n)31 b Fm(2)h Fn(N)p Ft(.)38 b(Moreo)o(v)o(er)m(,)22 b(let)j Fm(M)1502 654 y Fg(\000)1540 721 y Fo(\003)1598 691 y Fl(\()p Fi(n)p Fl(\))1695 654 y Fg(\001)1764 721 y Fm(\032)31 b(M)1960 654 y Fg(\000)1997 721 y Fn(R)2067 691 y Fi(d)2106 654 y Fg(\001)2169 721 y Ft(denote)23 b(the)i(set)g(of)f(Borel)h (measures)f(with)456 832 y(support)j(in)i Fo(\003)886 802 y Fl(\()p Fi(n)p Fl(\))1012 832 y Ft(and)f(let)i Fm(B)s Fo(\()p Fm(M)1464 844 y Fi(n)1508 832 y Fo(\))g Ft(be)f(the)f(smallest)i Fp(\033)s Ft(-algebra,)g(which)e(renders)g (the)h(mappings)456 935 y Fm(M)556 868 y Fg(\000)593 935 y Fo(\003)651 905 y Fl(\()p Fi(n)p Fl(\))748 868 y Fg(\001)829 935 y Fm(3)44 b Fp(\027)k Fm(7!)c Fp(\027)5 b Fo(\(\003\))32 b Ft(measurable)d(for)i(all)g(Borel)g(sets)i Fo(\003)42 b Fm(\032)h Fo(\003)2635 905 y Fl(\()p Fi(n)p Fl(\))2732 935 y Ft(.)58 b(Their)30 b(union)g Fm(R)44 b Fo(:=)456 972 y Fg(S)525 1059 y Fi(n)p Fj(2)p Fe(N)684 1034 y Fm(B)s Fo(\()p Fm(M)874 1046 y Fi(n)918 1034 y Fo(\))21 b Ft(satis\002es:)585 1172 y(\(i\))82 b Fm(R)21 b Ft(generates)f(the)g Fp(\033)s Ft(-algebra)f Fm(B)s Fo(\()p Fm(M)p Fo(\))p Ft(.)562 1309 y(\(ii\))82 b Fm(R)21 b Ft(is)g(a)g(semiring.)456 1446 y(The)27 b(\002rst)i(assertion)e (holds)h(by)f(de\002nition)g(of)h Fm(B)s Fo(\()p Fm(M)p Fo(\))p Ft(.)48 b(T)-7 b(o)28 b(check)f(the)h(second)f(one)g(we)h(note) g(that)456 1545 y Fm(;)22 b(2)i(R)p Ft(.)h(Moreo)o(v)o(er)m(,)17 b(for)j(e)n(v)o(ery)f Fp(M)9 b Ft(,)20 b Fp(M)1619 1515 y Fj(0)1665 1545 y Fm(2)j(R)e Ft(there)f(e)o(xists)h(some)f Fp(n)j Fm(2)g Fn(N)e Ft(such)f(that)1642 1697 y Fp(M)t(;)27 b(M)1867 1663 y Fj(0)1913 1697 y Fm(2)d(B)s Fo(\()p Fm(M)2182 1709 y Fi(n)2226 1697 y Fo(\))1048 b Ft(\(87\))456 1849 y(and)19 b(hence)h Fp(M)27 b Fm(\\)18 b Fp(M)1082 1819 y Fj(0)1128 1849 y Fm(2)24 b(B)s Fo(\()p Fm(M)1397 1861 y Fi(n)1441 1849 y Fo(\))f Fm(\032)g(R)e Ft(and)f Fp(M)9 b Fm(n)p Fp(M)2038 1819 y Fj(0)2083 1849 y Fm(2)23 b(B)s Fo(\()p Fm(M)2351 1861 y Fi(n)2395 1849 y Fo(\))h Fm(\032)e(R)p Ft(.)605 1948 y(Our)e(ne)o(xt)f(aim)h(is)g(to)g(pro)o(v)o(e)e(the)i (claimed)f(limit)h(relation)f(\(11\))g(for)g(all)i Fp(M)9 b Ft(,)20 b Fp(M)2886 1918 y Fj(0)2931 1948 y Fm(2)k(B)s Fo(\()p Fm(M)3200 1960 y Fi(n)3244 1948 y Fo(\))d Ft(with)456 2051 y Fp(n)33 b Fm(2)g Fn(N)26 b Ft(arbitrary)-5 b(.)40 b(Assumption)24 b(2.1\(ii\))h(ensures)g(that)g(the)h(e)n(v)o(ents)f Fp(T)2564 2063 y Fi(j)2599 2051 y Fp(M)41 b Fm(\032)33 b(M)2919 1984 y Fg(\000)2957 2051 y Fo(\003)3015 2021 y Fl(\()p Fi(n)p Fl(\))3134 2051 y Fo(+)22 b Fp(j)3260 1984 y Fg(\001)3325 2051 y Ft(and)456 2162 y Fp(M)546 2132 y Fj(0)592 2162 y Fm(\032)g(M)779 2095 y Fg(\000)817 2162 y Fo(\003)875 2132 y Fl(\()p Fi(n)p Fl(\))972 2095 y Fg(\001)1028 2162 y Ft(are)c(stochastically)g(independent)e(for)h (all)i Fp(j)28 b Fm(2)23 b Fn(Z)2460 2132 y Fi(d)2518 2162 y Ft(with)2684 2095 y Fg(\000)2722 2162 y Fo(\003)2780 2132 y Fl(\()p Fi(n)p Fl(\))2888 2162 y Fo(+)11 b Fp(j)3003 2095 y Fg(\001)3051 2162 y Fm(\\)g Fo(\003)3175 2132 y Fl(\()p Fi(n)p Fl(\))3294 2162 y Fo(=)23 b Fm(;)p Ft(,)456 2262 y(such)d(that)970 2379 y Fm(P)h(f)o Fp(T)1139 2391 y Fi(j)1174 2379 y Fp(M)27 b Fm(\\)18 b Fp(M)1445 2344 y Fj(0)1468 2379 y Fm(g)23 b Fo(=)f Fm(P)f(f)o Fp(T)1789 2391 y Fi(j)1824 2379 y Fp(M)9 b Fm(g)k(P)20 b(f)p Fp(M)2179 2344 y Fj(0)2202 2379 y Fm(g)i Fo(=)h Fm(P)d(f)p Fp(M)9 b Fm(g)k(P)20 b(f)p Fp(M)2829 2344 y Fj(0)2851 2379 y Fm(g)14 b Fp(:)376 b Ft(\(88\))456 2513 y(Here)20 b(the)g(last)h (equality)e(is)i(a)g(consequence)d(of)i(Assumption)f(2.1\(i\))n(.)605 2613 y(Thanks)14 b(to)i(\(87\))e(we)h(ha)n(v)o(e)g(thus)g(pro)o(v)o(en) e(the)i(v)n(alidity)g(of)f(\(88\))h(for)f(all)i Fp(M)9 b Ft(,)16 b Fp(M)2832 2583 y Fj(0)2878 2613 y Fm(2)23 b(R)p Ft(.)i(Lemma)14 b(2.3)456 2712 y(no)n(w)19 b(follo)n(ws)h(from)f ([)p Fq(D)m(VJ88)n Ft(,)h(Lemma)g(10.3.II],)d(which)j(is)h(a)g (monotone-class)d(ar)o(gument.)202 b Fd(\003)605 2872 y Fq(Remark)30 b(A.1.)47 b Ft(W)-7 b(e)32 b(pro)o(v)o(ed)27 b(abo)o(v)o(e)i(that)h(the)g(random)f(potential)g Fp(V)2683 2884 y Fi(!)2762 2872 y Ft(is)i(mixing)e(under)g(our)456 2971 y(assumptions.)39 b(Note,)27 b(that)f(mixing)e(is)i(actually)f(a)h (property)d(of)j(the)f(probability)f(measure)g Fm(P)33 b Ft(with)456 3071 y(respect)22 b(to)h(the)g(shifts)h Fm(f)o Fp(T)1221 3083 y Fi(j)1256 3071 y Fm(g)o Ft(.)34 b(Ho)n(we)n(v)o(er)m(,)22 b(the)h(potential)f Fp(V)2180 3083 y Fi(!)2252 3071 y Ft(will)i(not)e(satify)h(stronger)f(mixing)g (con-)456 3170 y(dition)d(such)h(as)h Fp(\036)p Fm(\000)p Ft(mixing.)k(In)20 b(f)o(act,)g(as)h(a)g(rule,)f(the)g(potential)g(may) g(e)n(v)o(en)f(be)h(deterministic)g(\(in)g(the)456 3270 y(technical)d(sense)i(of)f(this)g(notion,)f(see)i(e.g.)f([)p Fq(KKS85)n Ft(]\),)g(which)g(allo)n(ws)g(mixing,)f(b)n(ut)h(not)g Fp(\036)p Fm(\000)p Ft(mixing.)456 3370 y(F)o(or)h(further)g (references)g(to)h(this)h(see)g([)p Fq(Bil68)o(,)f(KM83a)o Ft(].)1757 3588 y Fq(Refer)o(ences)456 3721 y FC([Bil68])134 b(P)-7 b(.)16 b(Billingsle)o(y)l(.)j Fs(Con)m(ver)n(g)o(ence)i(of)c(pr) m(obability)j(measur)n(es)p FC(.)c(W)m(ile)o(y)j(1968.)456 3804 y([BHKL95])33 b(K.)13 b(Broderix,)j(D.)e(Hundertmark,)i(W)-6 b(.)12 b(Kirsch,)j(and)g(H.)e(Leschk)o(e.)i(The)f(f)o(ate)h(of)f (Lifshits)h(tails)g(in)g(magnetic)h(\002elds.)780 3887 y Fs(J)n(.)h(Stat.)h(Phys.)p FC(,)e(80:1\22622,)i(1995.)456 3970 y([CL90])129 b(R.)20 b(Carmona)i(and)f(J.)e(Lacroix.)i Fs(Spectr)o(al)h(theory)g(of)f(r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)23 b(oper)o(ator)o(s)p FC(.)e(Birkh)t(\250)-26 b(auser)m(,)24 b(Boston,)780 4053 y(1990.)456 4136 y([Dur96])111 b(R.)17 b(Durrett.)h Fs(Pr)m(obability:)23 b(theory)18 b(and)g(e)o(xamples)p FC(.)g(Duxb)o(ury)l(,)e(Belmont,)i(1996.)456 4219 y([D)m(V75])121 b(M.)23 b(D.)g(Donsk)o(er)i(and)g(S.)e(R.)g(S.)g (V)-7 b(aradhan.)25 b(Asymptotics)g(of)f(the)h(Wiener)g(sausage.)f Fs(Commun.)g(Pur)n(e)f(Appl.)780 4302 y(Math.)p FC(,)17 b(28:525\226565,)i(1975.)e(Errata:)22 b Fs(ibid)p FC(,)c(pp.)e(677.)456 4385 y([D)m(VJ88])95 b(D.)16 b(J.)f(Dale)o(y)j(and)f(D.)f(V)-7 b(ere-Jones.)17 b Fs(An)f(intr)m(oduction)j(to)e(the)g(theory)h(of)e (point)i(pr)m(ocesses)p FC(.)f(Springer)m(,)h(Ne)n(w)f(Y)-7 b(ork,)780 4468 y(1988.)456 4551 y([DZ98])125 b(A.)14 b(Dembo)h(and)g(O.)e(Zeitouni.)j Fs(Lar)n(g)o(e)f(de)o(viations)i(tec)o (hniques)g(and)e(applications)p FC(.)i(Springer)m(,)g(Ne)n(w)e(Y)-7 b(ork,)14 b(1998.)456 4634 y([Erd98])118 b(L.)21 b(Erd)6 b(\005)-28 b(os.)21 b(Lifschitz)j(tail)g(in)e(a)g(magnetic)i(\002eld:) 33 b(the)22 b(nonclassical)k(re)o(gime.)d Fs(Pr)m(obab)m(.)e(Theory)i (Relat.)g(F)m(ields)p FC(,)780 4717 y(112:321\226371,)c(1998.)456 4800 y([Erd01])118 b(L.)17 b(Erd)6 b(\005)-28 b(os.)18 b(Lifschitz)i(tail)f(in)g(a)f(magnetic)i(\002eld:)25 b(coe)o(xistence)d(of)c(the)h(classical)i(and)e(quantum)g(beha)o(vior)h (in)e(the)780 4883 y(borderline)i(case.)d Fs(Pr)m(obab)m(.)g(Theory)h (Relat.)f(F)m(ields)p FC(,)h(121:219\226236,)g(2001.)456 4967 y([HKW03])55 b(D.)17 b(Hundertmark,)i(W)-6 b(.)16 b(Kirsch,)i(and)g(S.)f(W)-5 b(arzel.)18 b(Lifshits)g(tails)h(in)f (three)g(space)h(dimensions:)24 b(impurity)19 b(poten-)780 5050 y(tials)g(with)e(slo)n(w)h(anisotropic)h(decay)l(.)f Fs(Mark)o(o)o(v)h(Pr)m(ocess.)e(Relat.)g(F)m(ields)p FC(,)g(9:651\226660,)i(2003.)456 5133 y([HL)-5 b(W99])67 b(T)-5 b(.)13 b(Hupfer)m(,)i(H.)e(Leschk)o(e,)i(and)f(S.)f(W)-5 b(arzel.)14 b(Poissonian)h(obstacles)h(with)e(Gaussian)h(w)o(alls)f (discriminate)j(between)780 5216 y(classical)j(and)d(quantum)h (Lifshits)g(tailing)h(in)f(magnetic)h(\002elds.)e Fs(J)n(.)g(Stat.)g (Phys.)p FC(,)f(97:725\226750,)j(1999.)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 1257 251 a Fr(LIFSHITS)20 b(T)-5 b(AILS)18 b(CA)m(USED)g(BY)g(ANISO)n(TR)n(OPIC)h(DECA)-6 b(Y)742 b(23)456 450 y FC([HL)-5 b(W00])67 b(T)-5 b(.)19 b(Hupfer)m(,)i(H.)e(Leschk)o(e,)j(and)e(S.)f(W)-5 b(arzel.)21 b(The)f(multiformity)i(of)e(Lifshits)g(tails)i(caused)f(by)f(random)g (Landau)780 533 y(Hamiltonians)29 b(with)e(repulsi)n(v)o(e)i(impurity)f (potentials)h(of)d(dif)n(ferent)j(decay)f(at)f(in\002nity)l(.)g Fs(AMS/IP)f(Stud.)g(Adv)-5 b(.)780 616 y(Math.)p FC(,)17 b(16:233\226247,)i(2000.)456 699 y([Kal83])119 b(O.)16 b(Kallenber)o(g.)k Fs(Random)c(measur)n(es)p FC(.)h(Akademie-V)-7 b(erlag,)20 b(Berlin,)e(1983.)456 782 y([Kir89])126 b(W)-6 b(.)35 b(Kirsch.)i(Random)g(Schr)6 b(\250)-28 b(odinger)39 b(operators:)62 b(a)36 b(course.)h(In)f(H.)g(Holden)h(and)g(A.)f (Jensen,)41 b(editors,)780 865 y Fs(Sc)o(hr)5 b(\250)-27 b(oding)o(er)20 b(Oper)o(ator)o(s)p FC(,)d(v)o(olume)h(345)f(of)g Fs(Lectur)n(e)h(notes)g(in)f(physics)p FC(,)h(pages)g(264\226370.)f (Springer)m(,)i(1989.)456 948 y([Klo99])115 b(F)-5 b(.)23 b(Klopp.)h(Internal)i(Lifshits)e(tails)i(for)d(random)i(perturbations)i (of)d(periodic)i(Schr)6 b(\250)-28 b(odinger)26 b(operators.)f Fs(Duk)o(e)780 1031 y(Math.)18 b(J)n(.)p FC(,)e(98:335\226369,)j(1999.) e(Erratum:)22 b(mp)p 1894 1031 20 4 v 23 w(arc)c(00-389.)456 1114 y([Klo02])115 b(F)-5 b(.)17 b(Klopp.)i(Une)g(remarque)k(\264)-26 b(a)19 b(propos)g(des)f(asymptotiques)j(de)e(Lifshitz)g(internes.)h Fs(C.)e(R.)f(Acad.)h(Sci.)h(P)-5 b(aris)18 b(Ser)-7 b(.)780 1197 y(I)p FC(,)17 b(335:87\22692,)h(2002.)456 1280 y([KKS85])81 b(W)-6 b(.)27 b(Kirsch,)k(S.)c(K)n(otani,)k(and)e(B.)e(Simon)h(Absence) h(of)f(absolutely)j(continuous)f(spectrum)f(for)f(some)f(one-)780 1363 y(dimensional)e(random)f(b)o(ut)f(deterministic)j(Schr)6 b(\250)-28 b(odinger)25 b(operators.)g Fs(Ann.)d(Inst.)g(H.)g(P)-5 b(oincar)n(e)24 b(Phys.)e(Theor)-7 b(.)p FC(,)780 1446 y(42:383)18 b(\226)f(406,)g(1985.)456 1529 y([KM82])107 b(W)-6 b(.)17 b(Kirsch)i(and)f(F)-5 b(.)17 b(Martinelli.)k(On)d(the)h (er)o(godic)g(properties)h(of)e(the)h(spectrum)g(of)f(general)i(random) f(operators.)780 1612 y Fs(J)n(.)e(Reine)h(Ang)o(e)o(w)-5 b(.)18 b(Math.)p FC(,)f(334:141\226156,)h(1982.)456 1695 y([KM83a])78 b(W)-6 b(.)23 b(Kirsch)h(and)h(F)-5 b(.)22 b(Martinelli.)27 b(Lar)o(ge)d(de)n(viations)j(and)d(Lifshitz)h (singularity)i(of)c(the)i(inte)o(grated)i(density)e(of)780 1778 y(states)18 b(of)f(random)h(Hamiltonians.)h Fs(Commun.)e(Math.)g (Phys.)p FC(,)f(89:27\22640,)j(1983.)456 1861 y([KM83b])74 b(W)-6 b(.)22 b(Kirsch)i(and)g(F)-5 b(.)22 b(Martinelli.)k(On)d(the)h (essential)h(self)f(adjointness)h(of)f(stochastic)h(Schr)6 b(\250)-28 b(odinger)26 b(operators.)780 1944 y Fs(Duk)o(e)18 b(Math.)g(J)n(.)p FC(,)e(50:1255\2261260,)j(1983.)456 2028 y([KS86])129 b(W)-6 b(.)13 b(Kirsch)i(and)g(B.)f(Simon.)g (Lifshits)h(tails)g(for)g(periodic)h(plus)f(random)g(potentials.)i Fs(J)n(.)d(Stat.)h(Phys.)p FC(,)e(42:799\226808,)780 2111 y(1986.)456 2194 y([KS87])129 b(W)-6 b(.)15 b(Kirsch)h(and)g(B.)f (Simon.)g(Comparison)i(theorems)f(for)g(the)g(gap)g(of)f(Schr)6 b(\250)-28 b(odinger)18 b(operators.)f Fs(J)n(.)e(Funct.)h(Anal.)p FC(,)780 2277 y(75:396\226410,)j(1987.)456 2360 y([KW02])103 b(F)-5 b(.)14 b(Klopp)i(and)f(T)-5 b(.)14 b(W)-5 b(olf)n(f.)15 b(Lifshitz)i(tails)f(for)f FA(2)p FC(-dimensional)j(random)d(Schr)6 b(\250)-28 b(odinger)18 b(operators.)e Fs(J)n(.)f(Anal.)g(Math.)p FC(,)780 2443 y(88:63\226147,)k(2002.)456 2526 y([Lan91])111 b(R.)17 b(Lang.)h Fs(Spectr)o(al)h(theory)g(of)f(r)o(andom)f(Sc)o(hr)5 b(\250)-27 b(oding)o(er)20 b(oper)o(ator)o(s)p FC(,)f(v)o(olume)f(1498) g(of)f Fs(Lectur)n(e)i(notes)f(in)g(mathe-)780 2609 y(matics)p FC(.)g(Springer)m(,)g(Berlin,)g(1991.)456 2692 y([Lif63])133 b(I.)19 b(M.)g(Lifshitz.)h(Structure)h(of)f(the)g(ener)o(gy)h(spectrum) f(of)g(the)g(impurity)h(bands)f(in)f(disordered)j(solid)e(solutions.) 780 2775 y Fs(So)o(v)-5 b(.)18 b(Phys.)e(JETP)p FC(,)g (17:1159\2261170,)j(1963.)e(Russian)h(original:)24 b Fs(Zh.)16 b(Eksp.)g(T)-6 b(er)f(.)16 b(F)m(iz.)p FC(,)g (44:1723\2261741,)j(1963.)456 2858 y([LMW03])51 b(H.)17 b(Leschk)o(e,)h(P)-7 b(.)16 b(M)6 b(\250)-28 b(uller)m(,)19 b(and)f(S.)e(W)-5 b(arzel.)18 b(A)f(surv)o(e)o(y)h(of)g(rigorous)g (results)g(on)g(random)g(Schr)6 b(\250)-28 b(odinger)19 b(operators)780 2941 y(for)e(amorphous)h(solids.)f Fs(Mark)o(o)o(v)i (Pr)m(ocess.)e(Relat.)g(F)m(ields)p FC(,)g(9:729\226760,)i(2003.)456 3024 y([L)-5 b(W04])115 b(H.)17 b(Leschk)o(e)i(and)f(S.)e(W)-5 b(arzel.)18 b(Quantum-classical)k(transitions)e(in)d(Lifshits)h(tails)h (with)f(magnetic)i(\002elds.)e Fs(Phys.)780 3107 y(Re)o(v)-5 b(.)17 b(Lett.)p FC(,)g(8:086402)i(\(1-4\),)e(2004.)456 3190 y([Mez86])97 b(G.)16 b(A.)f(Mezincescu.)k(Internal)f(Lifshitz)g (singularities)i(for)c(disordered)i(\002nite-dif)n(ference)j (operators.)d Fs(Commun.)780 3273 y(Math.)g(Phys.)p FC(,)e (103:167\226176,)i(1986.)456 3356 y([Mez87])97 b(G.)16 b(A.)f(Mezincescu.)k(Lifschitz)g(singularities)g(for)e(periodic)i (operators)f(plus)e(random)h(potential.)j Fs(J)n(.)c(Stat.)h(Phys.)p FC(,)780 3439 y(49:1181\2261190,)i(1987.)456 3522 y([Mez93])97 b(G.)21 b(A.)f(Mezincescu.)k(Internal)f(Lifshitz)g(singularities)h(for) d(one)h(dimensional)i(Schr)6 b(\250)-28 b(odinger)23 b(operators.)g Fs(Com-)780 3605 y(mun.)17 b(Math.)g(Phys.)p FC(,)f(158:315\226325,)j(1993.)456 3688 y([Min02])104 b(T)-5 b(.)24 b(Mine.)i(The)f(uniqueness)i(of)e(the)h(inte)o(grated)i (density)f(of)e(states)h(for)g(the)g(Schr)6 b(\250)-28 b(odinger)27 b(operators)g(for)e(the)780 3771 y(Robin)18 b(boundary)h(conditions.)g Fs(Publ.)d(RIMS,)h(K)m(yoto)h(Univ)-5 b(.)p FC(,)17 b(38:355\226385,)i(2002.)456 3854 y([Nak77])104 b(S.)19 b(Nakao.)i(On)e(the)i(spectral)h(distrib)o(ution)h(of)d(the)g (Schr)6 b(\250)-28 b(odinger)22 b(operator)g(with)f(random)f (potential.)j Fs(J)n(apan.)d(J)n(.)780 3937 y(Math.)p FC(,)d(3:111\226139,)i(1977.)456 4020 y([P)o(as77])123 b(L.)18 b(A)h(P)o(astur)l(.)g(Beha)o(vior)i(of)e(some)g(Wiener)h(inte)o (grals)i(as)d FB(t)24 b Fx(!)f(1)18 b FC(and)i(the)g(density)g(of)f (states)i(of)e(Schr)6 b(\250)-28 b(odinger)780 4103 y(equations)17 b(with)f(random)g(potential.)h Fs(Theor)-7 b(.)15 b(Math.)g(Phys.)p FC(,)g(32:615\226620,)i(1977.)e(Russian)h(original:)23 b Fs(T)-6 b(eor)f(.)14 b(Mat.)780 4186 y(F)m(iz.)p FC(,)i(6:88-95,)i (1977.)456 4269 y([PF92])140 b(L.)16 b(P)o(astur)i(and)f(A.)f(Figotin.) j Fs(Spectr)o(a)f(of)f(r)o(andom)h(and)f(almost-periodic)j(oper)o(ator) o(s)p FC(.)e(Springer)m(,)g(Berlin,)g(1992.)456 4352 y([RS78])133 b(M.)22 b(Reed)i(and)f(B.)f(Simon.)h Fs(Methods)h(of)f (modern)g(mathematical)j(physics)e(IV)l(:)e(analysis)i(of)f(oper)o (ator)o(s)p FC(.)g(Aca-)780 4435 y(demic,)18 b(Ne)n(w)f(Y)-7 b(ork,)17 b(1978.)456 4518 y([Sim82])107 b(B.)17 b(Simon.)g(Schr)6 b(\250)-28 b(odinger)20 b(semigroups.)e Fs(Bull.)g(Amer)-7 b(.)16 b(Math.)i(Soc.)f(\(N.)g(S.\))p FC(,)g(7:447\226526,)i(1982.)f (Erratum:)k(Bull.)780 4601 y(Amer)l(.)17 b(Math.)g(Soc.)g(\(N.)g(S.\),) e(1982,)i(7,)g(447\226526.)456 4684 y([Sim85])107 b(B.)17 b(Simon.)f(Lifshitz)j(tails)f(for)f(the)h(Anderson)g(model.)f Fs(J)n(.)g(Stat.)g(Phys.)p FC(,)f(38:65\22676,)j(1985.)456 4767 y([Sim87])107 b(B.)17 b(Simon.)f(Internal)k(Lifshitz)e(tails.)g Fs(J)n(.)f(Stat.)g(Phys.)p FC(,)f(46:911\226918,)j(1987.)456 4850 y([SKM87])70 b(D.)17 b(Sto)o(yan,)h(W)-6 b(.)16 b(S.)g(K)n(endal,)j(and)f(J.)e(Meck)o(e.)i Fs(Stoc)o(hastic)i(g)o (eometry)g(and)d(its)h(applications)p FC(.)i(W)m(ile)o(y)l(,)d (Chichester)m(,)780 4933 y(1987.)456 5016 y([Sto99])126 b(P)-7 b(.)15 b(Stollmann.)i(Lifshitz)g(asymptotics)g(via)g(linear)g (coupling)h(of)e(disorder)l(.)h Fs(Math.)f(Phys.)f(Anal.)g(Geom.)p FC(,)h(2:2679\226)780 5099 y(289,)h(1999.)456 5182 y([Sto01])126 b(P)-7 b(.)16 b(Stollmann.)i Fs(Caught)h(by)e(disor)n(der:)k(bound)d (states)g(in)f(r)o(andom)g(media)p FC(.)h(Birkh)t(\250)-26 b(auser)m(,)19 b(Boston,)e(2001.)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 456 251 a Fr(24)886 b(WERNER)13 b(KIRSCH)h(AND)h(SIMONE)f(W)-7 b(ARZEL)456 450 y FC([V)g(es03])118 b(I.)28 b(V)-7 b(eselic.)30 b(Inte)o(grated)h(density)f(of)f(states)h (and)f(W)-5 b(e)o(gner)29 b(estimates)h(for)f(random)g(Schr)6 b(\250)-28 b(odinger)31 b(operators.)780 533 y Fs(pr)n(eprint)18 b(math-ph/0307062)p FC(,)i(2003.)456 616 y([W)-5 b(ar01])105 b(S.)16 b(W)-5 b(arzel.)17 b Fs(On)f(Lifshits)i(tails)f(in)g(ma)o (gnetic)i(\002elds)p FC(.)e(Logos,)e(Berlin,)j(2001.)f(PhD)f(thesis,)h (Uni)n(v)o(ersity)i(Erlangen-)780 699 y(N)6 b(\250)-28 b(urnber)o(g)18 b(2001.)605 797 y Fs(E-mail)f(addr)n(ess)p FC(:)22 b Fa(werner.kirsch@mathphys.ruhr-uni-bochum.de)607 938 y FC(I)t FD(N)t(S)t(T)t(I)t(T)t(U)t(T)28 b(F)947 932 y FC(\250)939 938 y FD(U)t(R)i FC(M)t FD(A)n(T)t(H)t(E)t(M)t(A)n(T) t(I)t(K)t FC(,)g(R)r FD(U)t(H)t(R)t FC(-)t(U)t FD(N)t(I)t(V)t(E)t(R)t (S)s(I)s(T)2019 932 y FC(\250)2011 938 y FD(A)t(T)c FC(B)t FD(O)t(C)t(H)t(U)t(M)k(U)t(N)t(D)h FC(S)t(F)t(B)f(T)t(R)h(1)t(2)t(,)i (4)t(4)t(7)t(8)t(0)d(B)t FD(O)t(C)t(H)t(U)t(M)t FC(,)458 1021 y(G)t FD(E)t(R)t(M)t(A)t(N)t(Y)605 1162 y Fs(E-mail)17 b(addr)n(ess)p FC(:)22 b Fa(swarzel@princeton.edu)605 1303 y Fs(Pr)n(esent)d(addr)n(ess:)26 b FC(P)t FD(R)t(I)t(N)t(C)t(E)t (T)s(O)t(N)19 b FC(U)t FD(N)t(I)t(V)t(E)t(R)t(S)t(I)t(T)t(Y)l FC(,)d(D)t FD(E)t(P)t(T)o(A)t(R)q(T)t(M)t(E)t(N)t(T)i(O)t(F)j FC(P)t FD(H)t(Y)t(S)t(I)t(C)t(S)t FC(,)16 b(J)p FD(A)t(D)r(W)t(I)t(N)k FC(H)t FD(A)t(L)t(L)t FC(,)g(P)t FD(R)t(I)t(N)t(C)t(E)t(T)s(O)t(N)t FC(,)458 1386 y(N)t(J)f(0)t(8)t(5)t(4)t(4)t(,)e(U)t(S)t(A)605 1527 y Fs(On)c(leave)i(fr)m(om:)r FC(,)i(I)t FD(N)t(S)t(T)t(I)t(T)t(U)t (T)12 b(F)1356 1521 y FC(\250)1348 1527 y FD(U)t(R)j FC(T)t FD(H)t(E)t(O)t(R)t(E)t(T)t(I)t(S)t(C)t(H)t(E)10 b FC(P)t FD(H)t(Y)t(S)t(I)t(K)t FC(,)j(U)t FD(N)t(I)t(V)t(E)t(R)t(S)t (I)t(T)2481 1521 y FC(\250)2473 1527 y FD(A)s(T)e FC(E)t FD(R)t(L)t(A)t(N)t(G)t(E)t(N)t FC(-)t(N)2967 1521 y(\250)2959 1527 y FD(U)t(R)t(N)t(B)t(E)t(R)s(G)t FC(,)g(S)t FD(T)o(A)q(U)t(D)t(T)o FC(-)458 1610 y FD(S)t(T)t(R)t FC(.)17 b(7)t(,)i(9)t(1)t(0)t(5)t(8)g(E) t FD(R)t(L)t(A)t(N)t(G)t(E)t(N)t FC(,)d(G)t FD(E)t(R)t(M)t(A)t(N)t(Y)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0402271119941--