Content-Type: multipart/mixed; boundary="-------------0309231123226" This is a multi-part message in MIME format. ---------------0309231123226 Content-Type: text/plain; name="03-439.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-439.keywords" Bose Einstein Condensation, Magnetic Field, Periodic Potentials ---------------0309231123226 Content-Type: application/postscript; name="brcoza_final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="brcoza_final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: brcoza_final.dvi %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR8 CMR12 CMMI12 CMSY10 CMR7 CMR10 MSBM10 CMMI8 %%+ CMMIB10 CMSY8 CMEX10 CMMI6 CMR6 CMTI12 MSBM7 CMSY6 CMBX8 MSAM10 %%+ CMMI10 CMSY7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /pack/texlive/bin/sparc-solaris2.7/old_dvips %+ brcoza_final -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.21:1436 %%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: texps.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 dup 102 /f put dup 107 /k put dup 120 /x put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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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 3 /asteriskmath put dup 48 /prime put dup 49 /infinity put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF1B263DB260571DCE3DFEC496E93 CB244233DDAA5033CD8D8B204D87ABD75B439379225D1A63D5AB6E69E23855AE F38E2BB7A6C2D5552261CE6C0009A08AD3A04A01BA0C1111B17CABAC3BDCD3DD 4E110C9CD1CD07B43C53B8EB3F5538C1E52D4C9EFBAE77F451041F09E3DD1AFD 873978F2F286853A80C6B732B7F13CB6E8C329BBB2050D2F3BCFA0DCD73B27B5 19FC79CBF299D917171E983307B2A1FF73B066E40FD464B939570E518939446D 90CC173201C3B5D6DDDD7260B3498B701ABCFAD82CF8868B4886A86CDBABDAC7 BEFB98292A8B053A11ADA107F07683E456C9254A2279A83E3EEC6D589CFDC9F4 A6D9FA509A6E54E6261AE3ED3B064A81D81C993D6AB43D2E0C37C3C03DF746C7 CB60B7949BC9A40192D7FBDB2F4578FEC7FA61C27BDDCAF30CC4813AFD8549A9 3E4CDF913C46969D166702783D4B2D64C440B0EED7211B7FF8E28485E65D77B5 6A56131B6E6E4D4C4D3D19B70A1E165DB49576B88D6BE0086ACB30610899EA31 3F61B6EB5BF71B60E4AF01107133BCCDB30076E3B5680D4BEB2FA9D43CCA0877 597F02A07258BAEB53240B707D08378088E17E2FFD04F44327FB614CA5C108FF CE79111F2CF01A6F97BBE274A02515C36D9984443CB4B6C9ECB48B4D6E806D1C 1B9A4A5863EFF8EC1FD52002D4F05B108C40FF2F4F0A9D63F4184736466B431D AFD89A2CE9232C8EC40B613FA5C2FF36B7AFCF08374A10E2229E5002BAB528B6 514532235C05E64F20DC039973B8EE6E12E6EBEF576D3F62D16CABFAA6654EEA 15FCCB126360F94A983D5BB26B49679E568BC1FF07957C96C1AC499D1F2BB5D3 2D892627F98156A382F63C1064CAADC7D010A52BF6DBE38A0914D804B581351E 9B078B7F3AA44ABE441FB33BF4B395697B327EC69E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % 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 (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 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 12 /fi put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 54 /six put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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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 21 /lambda put dup 59 /comma put dup 66 /B put dup 69 /E put dup 70 /F put dup 76 /L put dup 77 /M put dup 78 /N put dup 100 /d put dup 106 /j put dup 109 /m put dup 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 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 12 /vextendsingle 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 40 /braceleftBigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile 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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 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 8 /circleplus put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 37 /arrownortheast put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 70 /F put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % 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 (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 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 13 /gamma put dup 24 /xi put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 84 /T put dup 86 /V put dup 87 /W put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 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 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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%!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 49 /one put dup 50 /two put dup 51 /three 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 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 10 /circlemultiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 38 /arrowsoutheast put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 79 /O put dup 83 /S put dup 85 /U put dup 86 /V put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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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 (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 28 /oslash put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /Lambda put dup 4 /Xi put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 105 /i put dup 109 /m put dup 110 /n put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (brcoza_final.dvi) @start /Fa 206[66 49[{}1 58.1154 /CMSY7 rf /Fb 151[48 104[{}1 83.022 /CMMI10 rf /Fc 153[58 102[{}1 119.552 /CMMI12 rf /Fd 252[77 3[{}1 99.6264 /MSAM10 rf /Fe 133[60 71 71 97 71 75 52 53 55 1[75 67 75 112 37 71 1[37 75 67 41 61 75 60 75 65 13[75 5[128 81 2[50 3[88 1[97 96 102 6[37 4[67 1[67 67 67 2[37 45 32[75 12[{}40 119.552 /CMBX12 rf /Ff 135[43 12[43 4[25 37 101[{}4 66.4176 /CMBX8 rf /Fg 206[61 18 44[32 3[{}3 49.8132 /CMSY6 rf /Fh 165[44 7[48 82[{}2 66.4176 /MSBM7 rf /Fi 128[50 5[47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 3[30 1[30 60 2[97 72 72 70 55 71 75 66 75 72 87 61 1[51 38 72 75 64 66 74 70 69 72 5[30 30 4[50 50 50 50 50 50 1[30 35 30 2[40 40 30 24[86 1[55 12[{}67 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b(oundation;)40 b(e-mail:)324 4681 y(cornean@math.auc.dk)436 4750 y Fr(3)473 4781 y Fq(Univ)n(ersit)n(\023)-39 b(e)25 b(de)h(la)f(M)n(\023)-39 b(editerran)n(\023)g(ee,)24 b(and)h(Cen)n(tre)h(de)g(Ph)n(ysique)f(Th)n (\023)-39 b(eorique-CNRS,)24 b(Campus)i(de)324 4880 y(Lumin)n(y)-7 b(,)27 b(Case)g(907)f(13288)g(Marseille)g(cedex)i(9,)f(F)-7 b(rance;)27 b(e-mail:)36 b(zagrebno)n(v@cpt.univ-mrs.fr)1918 5251 y Fu(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 324 548 a Fu(these)42 b(exp)s(erimen)m(ts)i(w)m (as)f(reduced)f(to)f(preparation)h(and)f(trapping)h(neutral)g(b)s(oson) 324 668 y(atoms)33 b(for)f(further)g(co)s(oling.)470 789 y(The)46 b(question)h(in)f(the)g(title)f(and)h(the)f(presen)m(t)i (pap)s(er)f(are)f(motiv)-5 b(ated)46 b(b)m(y)g(an-)324 909 y(other)36 b(problem.)56 b(Namely)-8 b(,)39 b(ho)m(w)e(thermo)s (dynamic)h(prop)s(erties)f(of)f(\(c)m(harged\))h(quan-)324 1029 y(tum)24 b(gases)g(will)g(c)m(hange)g(in)g(the)g(presence)i(of)d (a)g(magnetic)h(\014eld)g(in)g(the)g(appro)m(ximation)324 1150 y(when)35 b(w)m(e)g(neglect)g(particle)g(in)m(teraction?)49 b(The)35 b(\014rst)f(rigorous)g(result)h(in)f(this)h(direc-)324 1270 y(tion)d(w)m(as)g(due)h(to)f([A-C],)g(who)g(studied)h(b)s(oth)f (Bose)h(and)f(F)-8 b(ermi)32 b(p)s(erfect)g(gases.)44 b(One)324 1391 y(of)31 b(their)i(conclusion)h(is)e(a)g(sort)h(of)e (\\no-go")g(theorem)i(forbidding)f(BEC)h(of)f(the)h(PBG)324 1511 y(in)j(dimension)i Ft(d)33 b Fu(=)h(3.)54 b(Since)37 b(at)f(the)g(same)h(time)g(their)f(result)h(allo)m(ws)g(the)g(BEC)g(at) 324 1631 y(higher)j(dimensions)h(\()p Ft(d)d(>)h Fu(4\),)i(it)e(is)h (clear)g(that)f(the)g(impact)h(of)f(the)g(homogeneous)324 1752 y(magnetic)33 b(\014eld)g(reduces)h(to)e(a)g(mo)s(di\014cation)h (of)f(the)h(one-particle)f(densit)m(y)j(of)d(states)324 1872 y(at)g(the)h(b)s(ottom)f(of)g(the)h(sp)s(ectrum.)470 1993 y(The)g(aim)g(of)e(the)i(presen)m(t)h(pap)s(er)e(is)h(to)f(\014nd) h(an)f(external)h(\(\\electric"\))g(p)s(oten)m(tial)324 2113 y(whic)m(h)28 b(is)g(able)f(to)f(restore)i(BEC)g(in)f Ft(d)g Fu(=)h(3.)41 b(Motiv)-5 b(ated)27 b(b)m(y)h(recen)m(t)g(exp)s (erimen)m(ts)i(with)324 2233 y(optical)f(lattices)h(\(see)h(e.g.)42 b([BeS-M])30 b(and)f(references)j(therein\))e(w)m(e)g(construct)h(a)d (class)324 2354 y(of)k(p)s(erio)s(dic)h(p)s(oten)m(tials)g(with)g(this) h(prop)s(ert)m(y)-8 b(.)470 2474 y(No)m(w)35 b(w)m(e)h(come)f(to)g(our) f(mathematical)i(mo)s(del.)50 b(Denote)34 b(b)m(y)i(\003)2879 2489 y Fv(1)2949 2474 y Fs(2)c Fp(R)3119 2438 y Fo(d)3194 2474 y Fu(an)i(op)s(en,)324 2594 y(con)m(v)m(ex)28 b(and)e(simply)i (connected)g(domain)e(with)h(smo)s(oth)f(b)s(oundary)g Ft(@)5 b Fu(\003)3025 2609 y Fv(1)3066 2594 y Fu(,)27 b(con)m(taining)324 2715 y(the)36 b(origin)f(of)g(co)s(ordinates;)j (here)e(1)d Fs(\024)g Ft(d)f Fs(\024)i Fu(3.)52 b(The)36 b(b)s(o)m(x)g(whic)m(h)h(traps)f(our)g(system)324 2835 y(is)d(giv)m(en)h(b)m(y)f(\()p Ft(L)28 b(>)g Fu(1\))1356 2956 y(\003)1424 2971 y Fo(L)1504 2956 y Fu(:=)f Fs(f)p Fw(x)h Fs(2)g Fp(R)1937 2914 y Fo(d)1978 2956 y Ft(;)44 b Fw(x)p Ft(=L)28 b Fs(2)g Fu(\003)2413 2971 y Fv(1)2452 2956 y Fs(g)p Ft(:)832 b Fu(\(1.1\))324 3130 y(In)31 b(this)g(pap)s(er)g(w)m(e)h(consider)g(con)m(tin)m(uous)g Fp(Z)1976 3094 y Fo(d)2017 3130 y Fs(\000)p Fu(p)s(erio)s(dic)g (external)f(p)s(oten)m(tials)h Ft(V)52 b Fu(\(i.e.)324 3250 y Fn(\015)34 b Fs(2)28 b Fp(Z)577 3214 y Fo(d)618 3250 y Ft(;)17 b(V)k Fu(\()p Fw(x)g Fu(+)h Fn(\015)6 b Fu(\))28 b(=)f Ft(V)22 b Fu(\()p Fw(x)p Fu(\))p Ft(;)17 b Fw(x)27 b Fs(2)h Fp(R)1700 3214 y Fo(d)1741 3250 y Fu(\),)k Ft(V)1895 3265 y Fo(L)1979 3250 y Fu(denotes)i(the)f (restriction)g(of)f Ft(V)53 b Fu(to)32 b(\003)3374 3265 y Fo(L)3426 3250 y Fu(.)44 b(If)324 3371 y Ft(d)27 b Fu(=)h(3)k(w)m(e)i(also)e(consider)i(a)e(magnetic)i(v)m(ector)f(p)s (oten)m(tial)g(of)f(the)h(form:)1442 3591 y Fw(a)p Fu(\()p Fw(x)p Fu(\))28 b(=)g Ft(!)t Fw(a)1882 3606 y Fv(0)1921 3591 y Fu(\()p Fw(x)p 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y Fu(+)g Ft(V)1918 5251 y Fu(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 324 548 a Fu(de\014ned)35 b(on)f Ft(C)875 512 y Fm(1)868 573 y Fv(0)950 548 y Fu(\()p Fp(R)1060 512 y Fo(d)1100 548 y Fu(\))g([R-SI)s(I].)g(Because)h(of)f (the)g(magnetic)h(\014eld,)g(the)f(nature)h(of)e(the)324 668 y(sp)s(ectrum)f(of)d Ft(h)912 683 y Fm(1)1018 668 y Fu(is)i(not)f(kno)m(wn)h(in)g(general;)h(but)e(since)i(b)m(y)f(our)g (assumptions)h Ft(h)3391 683 y Fm(1)3496 668 y Fu(is)324 789 y(b)s(ounded)39 b(from)f(b)s(elo)m(w)h(and)f(comm)m(utes)i(with)f (the)g(\(magnetic\))g(translations,)h(then)324 909 y Ft(h)380 924 y Fm(1)489 909 y Fu(has)35 b(no)f(discrete)i(sp)s(ectrum.) 50 b(Let)34 b(us)h(denote)g(b)m(y)g Ft(E)2459 924 y Fv(0)2529 909 y Fu(:=)c(inf)24 b Ft(\033)t Fu(\()p Ft(h)2951 924 y Fm(1)3025 909 y Fu(\).)49 b(Moreo)m(v)m(er,)324 1029 y(due)29 b(to)f(standard)h(argumen)m(ts)h(in)m(v)m(olving)g(the)f (min-max)g(principle,)h(for)e(all)h Ft(L)f(>)f Fu(1)i(w)m(e)324 1150 y(ha)m(v)m(e)34 b Ft(E)621 1165 y Fv(0)688 1150 y Fs(\024)28 b Ft(\025)850 1165 y Fv(1)922 1150 y Fu([R-SIV].)470 1270 y(W)-8 b(e)39 b(\014rst)f(consider)h(a)f(p)s(erfect)h(Bose)f(gas)g (\(PBG\))g(con\014ned)i(in)e(the)g(v)m(olume)i(\003)3483 1285 y Fo(L)3535 1270 y Fu(,)324 1391 y(eac)m(h)35 b(particle)h(of)e (the)h(gas)g(in)m(teracts)h(with)f(the)g(bac)m(kground)h(p)s(oten)m (tial)f Ft(V)3148 1406 y Fo(L)3234 1391 y Fu(and)g(the)324 1511 y(external)45 b(magnetic)g(\014eld.)79 b(The)46 b(case)f(of)f(an)g(imp)s(erfect)h(gas)f(with)h(a)f(Mean-Field)324 1631 y(\(MF\))28 b(t)m(yp)s(e)h(of)f(particle)h(p)s(oten)m(tial)f(will) h(b)s(e)g(considered)h(in)e(Section)h(5.)42 b(In)29 b(the)f(grand-)324 1752 y(canonical)23 b(ensem)m(ble,)28 b(the)23 b(pressure)i(of)d(a)g(p) s(erfect)i(gas)f(at)f(in)m(v)m(erse)j(temp)s(erature)f Ft(\014)33 b(>)28 b Fu(0)324 1872 y(and)f(c)m(hemical)j(p)s(oten)m (tial)d Ft(\026)h(<)f(E)1570 1887 y Fv(0)1637 1872 y Fu(is)h(giv)m(en)h(b)m(y)f(the)g(w)m(ell)h(kno)m(wn)g(expression,)i 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Ft(z)t(e)2151 4683 y Fo(\014)s(\025)p 1971 4764 354 4 v 1971 4855 a Fu(1)22 b Fs(\000)h Ft(z)t(e)2236 4826 y Fo(\014)s(\025)2335 4646 y Fl(\025)2414 4719 y Ft(N)2492 4734 y Fo(L)2545 4719 y Fu(\()p Ft(\025)p Fu(\))p 2414 4764 264 4 v 2458 4855 a Fs(j)p Fu(\003)2554 4870 y Fo(L)2606 4855 y Fs(j)2687 4787 y Ft(d\025:)539 b Fu(\(1.6\))1918 5251 y(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 324 548 a Fu(Recall)47 b(that)f(the)g(in)m (tegrated)i(densit)m(y)g(of)e(states)h(for)f Ft(h)2501 563 y Fm(1)2576 548 y Fu(,)k(denoted)d(b)m(y)g Ft(n)3242 563 y Fm(1)3317 548 y Fu(\()p Ft(\025)p Fu(\))f(is)324 668 y(de\014ned)34 b(as)f(a)f(w)m(eak)i(limit:)1500 910 y Ft(n)1558 925 y Fm(1)1633 910 y Fu(\()p Ft(\025)p Fu(\))27 b(=)55 b(lim)1897 972 y Fo(L)p Fm(!1)2113 842 y Ft(N)2191 857 y Fo(L)2243 842 y Fu(\()p Ft(\025)p Fu(\))p 2113 887 264 4 v 2156 978 a Fs(j)p Fu(\003)2252 993 y Fo(L)2304 978 y Fs(j)3361 910 y Fu(\(1.7\))324 1177 y(on)32 b(the)h(space)h(of)e (con)m(tin)m(uous)i(functions)g 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5 5 TeXDict begin 5 4 bop 470 548 a Fu(On)34 b(the)g(other)f(hand,)h(w)m(e) h(kno)m(w)g(from)e([A-C])h(that)f(for)g(a)g(p)s(erfect)h(Bose)g(gas,)g (the)324 668 y(BEC)46 b(do)s(es)h(not)e(exist)i(\(i.e.)84 b Ft(\032)1535 683 y Fo(c)1570 668 y Fu(\()p Ft(\014)6 b Fu(\))50 b(=)g Fs(1)p Fu(\))45 b(in)h(the)h(presence)h(of)d(a)g (homogeneous)324 789 y(magnetic)38 b(\014eld)h(\()p Ft(!)h Fs(6)p Fu(=)d(0,)i Ft(V)58 b Fu(=)37 b(0\).)59 b(W)-8 b(e)39 b(will)f(see)h(that)f(this)h(is)f(related)h(to)e(the)i(fact)324 909 y(that)789 1029 y Ft(n)847 1044 y Fm(1)922 1029 y Fu(\()p Ft(\025)p Fu(\))27 b(=)h Ft(B)1260 1044 y Fo(!)r(;d)1389 1029 y Fs(\001)21 b Fu(\()p Ft(\025)h Fs(\000)h Ft(E)1727 1044 y Fv(0)1767 1029 y Fu(\()p Ft(!)t Fu(\)\))1946 988 y Fo(d=)p Fv(2)p Fm(\000)p Fv(1)2168 1029 y Fu(+)f Ft(o)p Fu(\(\()p Ft(\025)g Fs(\000)h Ft(E)2640 1044 y Fv(0)2679 1029 y Fu(\()p Ft(!)t Fu(\)\))2858 988 y Fo(d=)p Fv(2)p Fm(\000)p Fv(1)3058 1029 y Fu(\))324 1204 y(for)32 b Ft(\025)27 b Fs(&)h Ft(E)757 1219 y Fv(0)796 1204 y Fu(\()p Ft(!)t Fu(\).)43 b(Hence,)34 b(in)m(tegral)f(\(1.9\))f(div)m(erges)i (for)f Ft(z)f Fu(=)27 b Ft(z)2661 1219 y Fo(c)2697 1204 y Fu(.)470 1324 y(In)34 b(what)h(follo)m(ws,)g(w)m(e)g(will)g(sho)m(w)g (that)f(b)m(y)h(adding)f(a)g(certain)g(external)h(p)s(erio)s(dic)324 1445 y(p)s(oten)m(tial,)47 b(w)m(e)e(can)f(restore)g(the)h(BEC)f(in)h (our)e(system)j(with)f(magnetic)f(\014eld.)78 b(In)324 1565 y(particular,)33 b(w)m(e)g(pro)m(v)m(e)h(the)f(follo)m(wing)g (main)g(theorem:)324 1768 y Fw(Theorem)47 b(1.2.)f Fi(Consider)41 b(a)h(thr)-5 b(e)g(e)42 b(dimensional)e(p)-5 b(erfe)g(ct)42 b(Bose)f(gas)h(in)g(a)g(homo-)324 1889 y(gene)-5 b(ous)28 b(magnetic)f(\014eld,)i(wher)-5 b(e)28 b(the)g(one-p)-5 b(article)28 b(Hamiltonian)f(is)i(given)e(by)i Ft(h)3351 1904 y Fv(0)p Fo(;L)3486 1889 y Fu(=)324 2009 y(\()p Fs(\000)p Ft(i)p Fs(r\000)p Fw(a)p Fu(\))724 1973 y Fv(2)790 2009 y Fi(on)24 b Ft(L)985 1973 y Fv(2)1025 2009 y Fu(\(\003)1131 2024 y Fo(L)1183 2009 y Fu(\))h 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y(This)h(statement)g(r)-5 b(emains)36 b(true)h(by)f(switching)g(on)g(a)g(me)-5 b(an-\014eld)35 b(p)-5 b(article)36 b(inter-)324 2731 y(action.)470 2935 y Fu(The)e(pro)s(of)e(of)g(the)i(Theorem)g(1.2)f(is) g(actually)h(based)f(on)g(the)h(follo)m(wing)f(remark.)324 3055 y(Since)38 b(the)f(diamagnetic)h(inequalit)m(y)h(\(see)g([S)e (2]\))g(yields)h Ft(h)2568 3070 y Fm(1)2679 3055 y Fs(\025)d(\000)p Fu(\001)26 b(+)f(min)q(\()p Ft(V)d Fu(\),)38 b(b)m(y)324 3176 y(the)k(W)-8 b(eyl)42 b(estimate)h(w)m(e)f(get)g(a)f(p)s (olynomial)h(upp)s(er)g(b)s(ound)f Fs(\030)j Ft(\025)2858 3139 y Fo(d=)p Fv(2)3010 3176 y Fu(for)d Ft(n)3226 3191 y Fm(1)3301 3176 y Fu(\()p Ft(\025)p Fu(\))g(at)324 3296 y(in\014nit)m(y)-8 b(.)74 b(Therefore,)46 b(the)d(only)g(factor)f(whic) m(h)h(can)g(decide)h(whether)g(the)f(limit)f(in)324 3416 y(\(1.10\))e(is)h(\014nite)g(or)g(not)f(is)i(the)f(b)s(eha)m(vior)g(of) g Ft(n)2153 3431 y Fm(1)2228 3416 y Fu(\()p Ft(\025)p Fu(\))f(near)h(the)g(b)s(ottom)g Ft(E)3227 3431 y Fv(0)3307 3416 y Fu(of)f(the)324 3537 y(sp)s(ectrum)h Ft(\033)t Fu(\()p Ft(h)910 3552 y Fm(1)985 3537 y Fu(\).)66 b(Indeed,)44 b(one)c(can)h(easily)g(see)g(that)f(a)g(su\016cien)m(t)i(condition)f (for)324 3657 y(ha)m(ving)33 b(a)f(\014nite)i(critical)f(densit)m(y)h (is)g(the)f(estimate:)929 3898 y Ft(n)987 3913 y Fm(1)1062 3898 y Fu(\()p Ft(\025)p Fu(\))28 b Fs(\024)g Ft(const)22 b Fs(\001)g Fu(\()p Ft(\025)g Fs(\000)h Ft(E)1917 3913 y Fv(0)1956 3898 y Fu(\))1994 3857 y Fv(1+)p Fo(\013)2134 3898 y Ft(;)114 b(\025)28 b Fs(2)g Fu(\()p Ft(E)2564 3913 y Fv(0)2603 3898 y Ft(;)17 b(E)2719 3913 y Fv(0)2781 3898 y Fu(+)22 b Ft(\017)p Fu(\))356 b(\(1.11\))324 4072 y(for)35 b(some)h Ft(\013)e(>)f Fu(0)i(and)h(\014nite)g Ft(\017)e(>)e Fu(0.)53 b(On)36 b(the)g(con)m(trary)-8 b(,)37 b(a)e(su\016cien)m(t)j(condition)e(for)324 4193 y(an)c(in\014nite)i(critical)f(densit)m(y)i(\(or)d(zero)h(critical)g (temp)s(erature\))g(is)g(the)g(estimate)999 4433 y Ft(n)1057 4448 y Fm(1)1132 4433 y Fu(\()p Ft(\025)p Fu(\))27 b Fs(\025)i Ft(const)22 b Fs(\001)g Fu(\()p Ft(\025)g Fs(\000)g Ft(E)1986 4448 y Fv(0)2026 4433 y Fu(\))p Ft(;)114 b(\025)28 b Fs(2)g Fu(\()p Ft(E)2494 4448 y Fv(0)2534 4433 y Ft(;)17 b(E)2650 4448 y Fv(0)2711 4433 y Fu(+)22 b Ft(\017)p Fu(\))426 b(\(1.12\))324 4608 y(for)32 b(some)h(\014nite)h Ft(\017)28 b(>)f Fu(0.)324 4861 y Fw(Remark)h(1.3.)35 b Fi(Mor)-5 b(e)27 b(gener)-5 b(al)5 b(ly,)27 b(a)g(ne)-5 b(c)g(essary)26 b(and)g(su\016cient)h(c)-5 b(ondition)26 b(for)h(having)324 4981 y(a)34 b(\014nite)h(critic)-5 b(al)35 b(density)f(for)h(every)g Ft(\014)e(>)28 b Fu(0)34 b Fi(is)h(the)g(fol)5 b(lowing)33 b(estimate:)1918 5251 y Fu(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 1399 575 a Fl(Z)1498 602 y Fo(E)1550 611 y Fj(0)1585 602 y Fv(+1)1454 801 y Fo(E)1506 810 y Fj(0)1776 644 y Ft(n)1834 659 y Fm(1)1908 644 y Fu(\()p Ft(\025)p Fu(\))p 1706 688 406 4 v 1706 779 a(\()p Ft(\025)22 b Fs(\000)g Ft(E)1994 794 y Fv(0)2034 779 y Fu(\))2072 751 y Fv(2)2121 711 y Ft(d\025)27 b(<)h Fs(1)p Ft(:)825 b Fu(\(1.13\))324 946 y Fi(This)28 b(c)-5 b(ondition)28 b(also)g(implies)f(that)i Ft(\032)1725 961 y Fo(c)1788 946 y Fu(:)f(\(0)p Ft(;)17 b Fs(1)p Fu(\))27 b Fs(!)g Fu(\(0)p Ft(;)17 b Fs(1)p Fu(\))28 b Fi(is)g(smo)-5 b(oth)28 b(and)g(invertible)324 1067 y(\(one)34 b(shows)g(that)h Ft(\032)1076 1082 y Fo(c)1146 1067 y Fi(is)g(de)-5 b(cr)g(e)g(asing)33 b(and)h(onto\).)470 1187 y(By)47 b(the)g(virtue)g(of)g(De\014nition)f (1.1,)j(an)e(e)-5 b(quivalent)46 b(way)h(of)g(de\014ning)f(BEC)g(is)324 1308 y(imp)-5 b(osing)31 b(that)h(the)h(critic)-5 b(al)32 b(temp)-5 b(er)g(atur)g(e)32 b Ft(\014)1996 1323 y Fo(c)2031 1308 y Fu(\()p Ft(\032)p Fu(\))2157 1271 y Fm(\000)p Fv(1)2284 1308 y Fi(exists)g(and)g(is)g(p)-5 b(ositive)32 b(for)g(every)324 1428 y(density.)44 b(If)35 b(the)g(critic)-5 b(al)34 b(density)h(is)f(in\014nite,)h(we)f(set)h Ft(\014)2445 1443 y Fo(c)2480 1428 y Fu(\()p Ft(\032)p Fu(\))28 b(=)f Fs(1)p Fi(.)470 1679 y Fu(W)-8 b(e)27 b(close)g(this)g(section)h(b)m(y) 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Fk(L)1749 4305 y Fu(])28 b(=)1953 4237 y(1)p 1918 4282 121 4 v 1918 4373 a Fs(j)p Fu(\004)p Fs(j)2064 4169 y Fl(Z)2120 4395 y Fv(\004)2188 4305 y Ft(f)2236 4320 y Fo(h)2277 4328 y Fg(1)2346 4305 y Fu(\(\()p 2422 4229 49 4 v(0)o Ft(;)17 b(x)p Fu(\);)g(\()p 2689 4229 V(0)p Ft(;)g(x)p Fu(\)\))p Ft(dx:)266 b Fu(\(1.17\))324 4592 y(where)40 b(no)m(w)g(\004)f Fs(\032)g Fp(R)1114 4555 y Fo(d)p Fm(\000)p Fo(d)1241 4532 y Fg(0)1307 4592 y Fu(is)h(the)f(elemen)m(tary)j(cell)e(in)f(the)h(subspace)h (orthogonal)d(to)324 4712 y Ft(R)399 4676 y Fo(d)435 4652 y Fg(0)462 4712 y Fu(.)470 4882 y(Our)29 b(pap)s(er)h(is)f (organized)h(as)f(follo)m(ws:)43 b(Section)30 b(2)f(is)h(dev)m(oted)g (to)f(a)g(discussion)j(on)324 5002 y(v)-5 b(arious)32 b(results)h(concerning)h(the)e(BEC)h(for)e(a)h(p)s(erfect)h(Bose)f(gas) g(in)g(the)h(presence)h(of)1918 5251 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 324 548 a Fu(p)s(erio)s(dic)28 b(external)h(p)s(oten)m(tial)g(without)f(magnetic)h(\014eld.)42 b(Most)29 b(of)f(the)g(facts)g(giv)m(en)h(in)324 668 y(this)34 b(section)h(are)f(kno)m(wn)h(but)f(they)h(are)f(instructiv)m (e)i(for)d(the)h(rest)h(of)e(the)h(pap)s(er.)47 b(In)324 789 y(Section)29 b(3)e(w)m(e)i(discuss)i(the)d(stabilit)m(y)h(of)f(the) g(BEC)h(after)f(an)g(external)h(magnetic)g(\014eld)324 909 y(is)37 b(switc)m(hed)j(on;)f(w)m(e)g(will)f(also)f(pro)m(v)m(e)h (there)g(the)g(\014rst)g(part)e(of)h(Theorem)i(1.2.)57 b(The)324 1029 y(results)41 b(of)e(Section)h(3)f(are)h(applied)g(in)g (Section)g(4)g(where)g(w)m(e)h(study)g(the)f(imp)s(erfect)324 1150 y(Bose)34 b(gas,)g(in)g(the)g(case)g(of)f(a)h(mean-\014eld)g(t)m (yp)s(e)h(in)m(teraction.)47 b(W)-8 b(e)34 b(collect)h(in)f(Section)324 1270 y(5)e(tec)m(hnical)i(App)s(endices)h(indinsp)s(ensable)g(for)d (pro)s(ofs)g(placed)i(in)f(Section)g(3.)324 1602 y Fx(2)213 b(BEC)52 b(for)g(a)h(Bose)e(gas)h(in)g(a)h(p)t(erio)t(dic)f(external) 566 1784 y(p)t(oten)l(tial)324 2003 y Fu(In)38 b(this)g(section)h(w)m (e)g(assume)g(that)f(eac)m(h)h(particle)f(of)f(the)i(Bose)f(gas)g(in)m (teracts)h(with)324 2124 y(a)44 b(con)m(tin)m(uous,)49 b Fp(Z)1013 2087 y Fo(d)1054 2124 y Fu(-)44 b(p)s(erio)s(dic)h(p)s (oten)m(tial)g Ft(V)22 b Fu(;)50 b(without)45 b(loss)g(of)g(generalit)m (y)g(w)m(e)h(will)324 2244 y(c)m(ho)s(ose)37 b(min\()p Ft(V)22 b Fu(\))34 b(=)g(0.)54 b(The)37 b(one)f(particle)h(Hamiltonian) g Ft(h)2591 2259 y Fo(L)2677 2244 y Fu(=)d Fs(\000)p Fu(\001)25 b(+)g Ft(V)57 b Fu(is)37 b(then)g(a)324 2364 y(self-adjoin)m(t)g(op)s(erator)f(on)g Ft(L)1432 2328 y Fv(2)1472 2364 y Fu(\(\003)1578 2379 y Fo(L)1630 2364 y Fu(\))g(\(with)h(DBC)g(on)f Ft(@)5 b Fu(\003)2482 2379 y Fo(L)2535 2364 y Fu(\))37 b(as)f(w)m(ell)i(as)f(the)g(in\014nite-)324 2485 y(v)m(olume)d(Hamiltonian)f Ft(h)1277 2500 y Fm(1)1379 2485 y Fu(=)28 b Fs(\000)p Fu(\001)23 b(+)f Ft(V)54 b Fu(on)32 b Ft(L)2074 2449 y Fv(2)2114 2485 y Fu(\()p Fp(R)2224 2449 y Fo(d)2264 2485 y Fu(\).)470 2605 y(W)-8 b(e)27 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Fw(k)p Ft(;)17 b(x)p 1970 3249 56 4 v 1 w Fu(\))27 b(:=)2246 3138 y Fl(X)2222 3358 y Fn(\015)6 b Fm(2)p Fh(Z)2378 3339 y Fk(d)2564 3165 y Fu(1)p 2441 3210 295 4 v 2441 3301 a(\(2)p Ft(\031)t Fu(\))2625 3272 y Fo(d=)p Fv(2)2746 3233 y Ft(e)2791 3192 y Fm(\000)p Fo(i)p Ff(k)p Fm(\001)p Fv(\()p Fo(x)p 2960 3204 40 3 v Fv(+)p Fn(\015)g Fv(\))3151 3233 y Ft(f)11 b Fu(\()p Ft(x)p 3248 3249 56 4 v 23 w Fu(+)22 b Fn(\015)6 b Fu(\))p Ft(;)324 3554 y Fu(where)43 b Fw(k)h Fs(2)h Fu(\012)899 3518 y Fm(\003)981 3554 y Fu(and)d Ft(x)p 1180 3570 V 45 w Fs(2)i Fu(\012.)73 b(Then)44 b(the)e(unitary)h (transformation)f(of)g Ft(h)3207 3569 y Fm(1)3324 3554 y Fu(is)h(de-)324 3674 y(comp)s(osable)e(in)m(to)f(the)g(direct)h(in)m (tegral:)59 b Ft(U)10 b(h)2059 3689 y Fm(1)2135 3674 y Ft(U)2211 3638 y Fm(\003)2291 3674 y Fu(=)2407 3594 y Fl(R)2454 3709 y Fv(\012)2505 3690 y Fg(\003)2562 3674 y Ft(h)p Fu(\()p Fw(k)p Fu(\))p Ft(d)p Fw(k)p Fu(.)66 b(Here)41 b(the)f(\014b)s(er)324 3795 y(Hamiltonians:)1230 4036 y Ft(h)p Fu(\()p Fw(k)p Fu(\))28 b(=)f(\()p Fs(\000)p Ft(i)p Fs(r)c Fu(+)f Fw(k)p Fu(\))2001 3994 y Fv(2)2063 4036 y Fu(+)g Ft(V)5 b(;)115 b Fw(k)27 b Fs(2)h Fu(\012)2615 3994 y Fm(\003)3361 4036 y Fu(\(2.1\))324 4205 y(liv)m(e)35 b(in)f Ft(L)685 4169 y Fv(2)725 4205 y Fu(\(\012\))g(with)h(p)s(erio)s (dic)g(b)s(oundary)f(conditions.)49 b(They)35 b(ha)m(v)m(e)h(purely)f (discrete)324 4326 y(sp)s(ectrum)j(whic)m(h)h(accum)m(ulates)h(at)c (in\014nit)m(y;)42 b(for)36 b(a)h(giv)m(en)i Fw(k)c Fs(2)h Fu(\012)2853 4290 y Fm(\003)2893 4326 y Fu(,)j(w)m(e)f(denote)g(the)324 4446 y(set)33 b(of)f(corresp)s(onding)i(eigen)m(v)-5 b(alues)34 b(b)m(y)g Fs(f)p Ft(\025)1961 4461 y Fo(j)1997 4446 y Fu(\()p Fw(k)p Fu(\))p Fs(g)2182 4461 y Fo(j)t Fm(\025)p Fv(1)2308 4446 y Fu(.)470 4567 y(An)d(imp)s(ortan)m(t)f (ingredien)m(t)i(for)e(us)h(is)h(a)e(result)h(due)g(to)g(Kirsc)m(h)g (and)g(Simon)g([K-S])324 4687 y(ab)s(out)h(the)h(band)g(structure)h(of) e Ft(\033)t Fu(\()p Ft(h)1697 4702 y Fm(1)1772 4687 y Fu(\):)324 4882 y Fw(Prop)s(osition)49 b(2.1.)e Fi(Assume)d(that)g(the) g(p)-5 b(otential)44 b Ft(V)65 b Fi(is)44 b Fp(Z)2624 4846 y Fo(d)2665 4882 y Fi(-p)-5 b(erio)g(dic,)45 b(c)-5 b(ontinuous)324 5002 y(and)39 b Fu(min)q(\()p Ft(V)21 b Fu(\))37 b(=)h(0)p Fi(.)59 b(Consider)39 b(the)h(op)-5 b(er)g(ator)40 b Ft(h)2156 5017 y Fm(1)2268 5002 y Fu(=)d Fs(\000)p Fu(\001)27 b(+)f Ft(V)61 b Fi(on)40 b Ft(L)2997 4966 y Fv(2)3037 5002 y Fu(\()p Fp(R)3147 4966 y Fo(d)3187 5002 y Fu(\))g Fi(and)g(let)1918 5251 y Fu(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 324 548 a Fs(f)p Ft(\025)431 563 y Fo(j)467 548 y Fu(\()p Fw(k)p Fu(\))p Fs(g)652 563 y Fo(j)t Fm(\025)p Fv(1)811 548 y Fi(b)-5 b(e)32 b(the)g(eigenvalues)f (of)h(the)g(\014b)-5 b(er)32 b(Hamiltonians)f Ft(h)p Fu(\()p Fw(k)p Fu(\))p Ft(;)17 b Fw(k)28 b Fs(2)g Fu(\012)3165 512 y Fm(\003)3205 548 y Fi(,)33 b(de\014ne)-5 b(d)324 668 y(in)34 b(\(2.1\).)44 b(Denoting)34 b(by)h Ft(E)1346 683 y Fv(0)1413 668 y Fu(=)28 b(inf)c Ft(\033)t Fu(\()p 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y(and)c(deca)m(y)j(p)s(olynomialy)e(with)g(resp)s(ect)h(to)e(the)h(v)-5 b(ariable)43 b Ft(x)p 2629 3199 V 30 w Fs(\000)29 b Ft(x)p 2820 3199 V 2876 3147 a Fm(0)2941 3183 y Fu(\(see)44 b(App)s(endix)324 3303 y(1\),)i(the)d(op)s(erator)g Ft(f)11 b Fu(\()p Ft(h)p Fu(\()p Fw(k)p Fu(\)\))43 b(admits)h(an)f(in)m(tegral) h(k)m(ernel)h(whic)m(h)g(due)f(to)f(the)g(\014b)s(er)324 3424 y(decomp)s(osition)34 b(has)e(for)h Ft(x)p 1290 3440 V 32 w Fu(and)g Ft(x)p 1567 3440 V 1623 3387 a Fm(0)1674 3424 y Fs(2)28 b Fu(\012)33 b(the)g(represen)m(tation:)963 3664 y Ft(f)1011 3680 y Fo(h)p Fv(\()p Ff(k)p Fv(\))1154 3664 y Fu(\()p Ft(x)p 1192 3680 V(;)17 b(x)p 1291 3680 V -41 x Fm(0)1370 3664 y Fu(\))27 b(=)1563 3570 y Fl(X)1539 3789 y Fn(\015)6 b Fm(2)p Fh(Z)1695 3770 y Fk(d)1748 3664 y Ft(f)1796 3679 y Fo(h)1837 3687 y Fg(1)1906 3664 y Fu(\()p Ft(x)p 1944 3680 V 23 w Fu(+)22 b Fn(\015)6 b Ft(;)17 b(x)p 2229 3680 V -41 x Fm(0)2307 3664 y Fu(\))28 b Ft(e)2418 3623 y Fm(\000)p Fo(i)p Ff(k)p Fm(\001)p 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b(r)p Ft(f)11 b Fu(\()p Ft(h)p Fu(\()p Fw(k)p Fu(\)\))27 b(=)881 4600 y Fl(X)892 4810 y Fo(j)t Fm(\025)p Fv(1)1042 4695 y Ft(f)11 b Fu(\()p Ft(\025)1196 4710 y Fo(j)1232 4695 y Fu(\()p Fw(k)p Fu(\)\))27 b(=)1536 4559 y Fl(Z)1591 4784 y Fv(\012)1663 4695 y Ft(f)1711 4710 y Fo(h)p Fv(\()p Ff(k)p Fv(\))1854 4695 y Fu(\()p Ft(x)p 1892 4710 V(;)17 b(x)p 1991 4710 V Fu(\))p Ft(dx)p 2135 4710 V 28 w Fu(=)2346 4600 y Fl(X)2321 4819 y Fn(\015)7 b Fm(2)p Fh(Z)2478 4800 y Fk(d)2531 4559 y Fl(Z)2586 4784 y Fv(\012)2658 4695 y Ft(f)2706 4710 y Fo(h)2747 4718 y Fg(1)2816 4695 y Fu(\()p Ft(x)p 2854 4710 V Fu(+)p Fn(\015)f Ft(;)17 b(x)p 3094 4710 V Fu(\))28 b Ft(e)3260 4653 y Fm(\000)p Fo(i)p Ff(k)p Fm(\001)p Fn(\015)3471 4695 y Ft(dx)p 3522 4710 V(:)3361 4932 y Fu(\(2.5\))1918 5251 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 324 548 a Fu(Then)33 b(in)m(tegrating)g(\(2.5\))f (with)i(resp)s(ect)f(to)g(the)g Fw(k)f Fu(v)-5 b(ariable)33 b(w)m(e)h(ha)m(v)m(e:)488 722 y Fl(X)498 932 y Fo(j)t Fm(\025)p 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Fu(,)j(and)f Ft(\025)e Fs(\024)h Fu(max)2592 1273 y Ff(k)p Fm(2)p Fv(\012)2733 1254 y Fg(\003)2789 1258 y Ft(\025)2846 1273 y Fv(1)2886 1258 y Fu(\()p Fw(k)p Fu(\).)55 b(Then)37 b(\(2.6\))324 1378 y(and)32 b(Prop)s(osition)h(2.1)f(imply:)1139 1643 y Ft(n)1197 1658 y Fm(1)1272 1643 y Fu(\()p Ft(\025)p Fu(\))c(=)1634 1575 y(1)p 1546 1620 225 4 v 1546 1711 a(\(2)p Ft(\031)t Fu(\))1730 1682 y Fo(d)1797 1507 y Fl(Z)1852 1733 y Fv(\012)1903 1714 y Fg(\003)1960 1643 y Ft(\037)2021 1658 y Fv([)p Fo(E)2093 1667 y Fj(0)2127 1658 y Fm(\000)p Fv(0)p Fo(;\025)p Fv(])2302 1643 y Fu(\()p Ft(\025)2397 1658 y Fv(1)2436 1643 y Fu(\()p Fw(k)p Fu(\)\))p Ft(d)p Fw(k)p Ft(;)615 b Fu(\(2.7\))324 1873 y(where)35 b Ft(\037)668 1888 y Fo(I)742 1873 y Fu(denotes)g(the)f(indicator)g(of) f(the)i(set)f Ft(I)k Fs(\032)30 b Fp(R)p Fu(.)47 b(No)m(w,)35 b(Prop)s(osition)f(\(2.1\))g(\(i\))324 1993 y(and)e(a)h(c)m(hange)g(of) f(v)-5 b(ariables)34 b(in)e(\(2.7\))g(giv)m(e:)1075 2213 y Ft(n)1133 2228 y Fm(1)1208 2213 y Fu(\()p Ft(\025)p Fu(\))c(=)f Ft(A)1545 2228 y Fo(d)1586 2213 y Fu(\()p Ft(\025)22 b Fs(\000)g Ft(E)1874 2228 y Fv(0)1914 2213 y Fu(\))1952 2172 y Fo(d=)p Fv(2)2085 2213 y Fu(+)g Ft(o)p Fu(\(\()p Ft(\025)g Fs(\000)h Ft(E)2557 2228 y Fv(0)2596 2213 y Fu(\))2634 2172 y Fo(d=)p Fv(2)2745 2213 y Fu(\))p Ft(;)551 b Fu(\(2.8\))324 2433 y(for)37 b Ft(\025)e Fs(&)h Ft(E)778 2448 y Fv(0)818 2433 y Fu(.)57 b(Notice)38 b(that)g(this)g(is) g(exactly)g(the)g(same)g(b)s(eha)m(vior)h(as)e(in)h(the)g(\\free")324 2554 y(case)33 b(\(i.e.)44 b Ft(V)49 b Fu(=)28 b(0\),)k(and)h(the)g (prop)s(osition)g(is)g(pro)m(v)m(en.)p 3491 2554 4 66 v 3495 2491 59 4 v 3495 2554 V 3553 2554 4 66 v 324 2887 a Fx(3)220 b(BEC)58 b(for)g(a)i(Bose)d(gas)i(in)f(presence)f(of)h(a)i (con-)566 3069 y(stan)l(t)53 b(magnetic)g(\014eld)324 3288 y Fu(In)38 b(this)h(section)g(w)m(e)h(pro)m(v)m(e)f(the)f(\014rst) h(part)f(of)g(our)g(main)g(Theorem)i(1.2.)59 b(As)39 b(it)f(has)324 3409 y(b)s(een)30 b(men)m(tioned)i(b)s(efore,)e(w)m(e)h (are)f(motiv)-5 b(ated)30 b(b)m(y)g(the)h(w)m(ork)f(of)f (Angelescu-Corco)m(v)m(ei)324 3529 y([A-C])41 b(who)h(sho)m(w)m(ed)i (that)d(for)g(a)g(free,)j(three-dimensional)f(Bose)f(gas,)i(the)e (critical)324 3649 y(densit)m(y)47 b(is)f(in\014nite)h(in)f(the)g (presence)h(of)e(a)h(constan)m(t)g(magnetic)g(\014eld,)k(i.e.)83 b(BEC)324 3770 y(disapp)s(ears.)69 b(The)41 b(mec)m(hanism)i(of)e(that) f(is)h(describ)s(ed)i(in)e(Section)g(1:)60 b(b)m(y)41 b(creating)324 3890 y(the)d(Landau)h(lev)m(els,)j(the)c(magnetic)h (\014elds)h(leads)f(to)f(increasing)h(of)f(the)h(in)m(tegrated)324 4011 y(densit)m(y)47 b(of)e(states)h(at)e(the)i(b)s(ottom)e(of)h(the)h (sp)s(ectrum)g Ft(\033)t Fu(\()p Ft(h)2651 4026 y Fv(0)2691 4011 y Fu(\()p Ft(!)52 b Fs(6)p Fu(=)d(0\)\).)81 b(W)-8 b(e)45 b(only)324 4131 y(consider)36 b(the)f(case)h(of)e(dimension)j Ft(d)31 b Fu(=)g(3,)k(for)g(whic)m(h)h(the)f(BEC)h(in)f(the)g(PBG)g (holds)324 4251 y(when)h Fw(a)31 b Fu(=)g(0)k(ev)m(en)h(for)e(a)h(p)s 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Fu(-p)s(erio)s(dic.)54 b(Throughout)324 1123 y(this)37 b(section,)j(w)m(e)e(use)g(the)f (Landau)g(gauge)g Fw(a)2063 1138 y Fv(1)2103 1123 y Fu(\()p Fw(x)p Fu(\))e(=)g(\(0)p Ft(;)17 b(x)2570 1138 y Fv(1)2610 1123 y Ft(;)g Fu(0\))34 b Fs(2)i Fp(R)2949 1087 y Fv(3)2988 1123 y Fu(.)57 b(Notice)38 b(that)324 1243 y(the)43 b(c)m(hoice)h(of)e (a)h(particular)g(gauge)f(is)h(irrelev)-5 b(an)m(t)44 b(since)g(the)f(densit)m(y)i(of)d(states)i(is)324 1364 y(gauge)32 b(in)m(v)-5 b(arian)m(t.)44 b(Under)34 b(these)f (conditions,)h(the)f(\\bulk")g(Hamiltonian)g(is:)473 1557 y Ft(h)529 1572 y Fm(1)632 1557 y Fu(=)27 b(\()p Fs(\000)p Ft(i)p Fs(r)c(\000)g Ft(!)t Fw(a)1208 1572 y Fv(1)1247 1557 y Fu(\))1285 1516 y Fv(2)1347 1557 y Fu(+)f Ft(v)31 b Fu(=)d Fs(\000)p Ft(@)1760 1516 y Fv(2)1755 1582 y Fo(x)1795 1591 y Fj(1)1856 1557 y Fu(+)22 b Ft(v)t Fu(\()p Ft(x)2098 1572 y Fv(1)2138 1557 y Fu(\))g(+)g(\()p Fs(\000)p Ft(i@)2495 1572 y Fo(x)2535 1581 y Fj(2)2597 1557 y Fs(\000)g Ft(!)t(x)2816 1572 y Fv(1)2855 1557 y Fu(\))2893 1516 y Fv(2)2955 1557 y Fs(\000)h Ft(@)3111 1516 y Fv(2)3106 1582 y Fo(x)3146 1591 y Fj(3)3185 1557 y Ft(;)149 b Fu(\(3.1\))324 1751 y(acting)32 b(on)h Ft(L)818 1715 y Fv(2)858 1751 y Fu(\()p Fp(R)968 1715 y Fv(3)1007 1751 y Fu(\),)g(where)g Ft(!)e Fs(\025)e Fu(0.)324 1932 y Fw(Prop)s(osition)37 b(3.1.)42 b Fi(Consider)33 b(a)h(p)-5 b(erfe)g(ct)35 b(Bose)e(gas)h(describ)-5 b(e)g(d)34 b(by)g(the)h(one)f (p)-5 b(article)324 2052 y(Hamiltonian)26 b Ft(h)935 2067 y Fo(L)1015 2052 y Fi(de\014ne)-5 b(d)27 b(as)g(the)g(r)-5 b(estriction)28 b(of)f(the)g(op)-5 b(er)g(ator)27 b(\(3.1\))g(to)h Ft(L)3122 2016 y Fv(2)3162 2052 y Fu(\(\003)3268 2067 y Fo(L)3319 2052 y Fu(\))g Fi(with)324 2172 y(DBC.)36 b(Then)g(for)g(every)h Ft(!)e(>)d Fu(0)p Fi(,)37 b(the)g(critic)-5 b(al)37 b(density)f(is)h(in\014nite,)g(i.e.)51 b(the)37 b(BEC)f(is)324 2293 y(destr)-5 b(oye)g(d.)324 2473 y Fu(Before)25 b(coming)g(to)f(the)h(pro)s(of)f(of)g(Prop)s(osition)g (3.1,)i(w)m(e)g(need)g(some)f(tec)m(hnical)h(results.)324 2594 y(W)-8 b(e)37 b(will)g(often)f(write)i(a)e(v)m(ector)i Fw(u)c Fs(2)h Fp(R)1839 2558 y Fv(3)1915 2594 y Fu(as)i Fw(u)d Fu(=)h(\()p Ft(u)2340 2609 y Fv(1)2378 2594 y Ft(;)23 b Fu(~)-55 b Ft(u)p Fu(\))36 b(with)43 b(~)-55 b Ft(u)34 b Fu(=)g(\()p Ft(u)3072 2609 y Fv(2)3111 2594 y Ft(;)17 b(u)3211 2609 y Fv(3)3250 2594 y Fu(\))34 b Fs(2)h Fp(R)3495 2558 y Fv(2)3535 2594 y Fu(.)324 2714 y(Decomp)s(ose)24 b Ft(L)892 2678 y Fv(2)932 2714 y Fu(\()p Fp(R)1042 2678 y Fv(3)1081 2714 y Fu(\))f(with)h(the)g(help)g(of)e(the) i(partial)f(F)-8 b(ourier)23 b(transform)g(with)h(resp)s(ect)324 2835 y(to)32 b Ft(x)498 2850 y Fv(2)570 2835 y Fu(and)h Ft(x)815 2850 y Fv(3)855 2835 y Fu(:)324 3067 y Ft(U)38 b Fu(:)28 b Ft(L)549 3025 y Fv(2)589 3067 y Fu(\()p Fp(R)699 3025 y Fv(3)738 3067 y Fu(\))g Fs(7!)931 2931 y Fl(Z)986 3157 y Fh(R)1034 3138 y Fj(2)1090 3067 y Ft(L)1156 3025 y Fv(2)1195 3067 y Fu(\()p Fp(R)p Fu(\))p Ft(d)1397 3040 y Fu(~)1394 3067 y Ft(k)s(;)44 b(U)39 b Fu(=)1727 2931 y Fl(Z)1783 3157 y Fh(R)1831 3138 y Fj(2)1886 3067 y Ft(U)1954 3077 y Fv(~)1952 3094 y Fo(k)1995 3067 y Ft(d)2049 3040 y Fu(~)2046 3067 y Ft(k)s(;)44 b Fu(\()p Ft(U)2277 3077 y Fv(~)2275 3094 y Fo(k)2318 3067 y Ft(f)11 b Fu(\)\()p Ft(t)p Fu(\))27 b(=)2696 2999 y(1)p 2667 3044 108 4 v 2667 3135 a(2)p Ft(\031)2801 2931 y Fl(Z)2856 3157 y Fh(R)2904 3138 y Fj(2)2960 3067 y Ft(e)3005 3025 y Fm(\000)p Fo(i)3086 3008 y Fv(~)3084 3025 y Fo(k)5 b Fv(~)-39 b Fo(x)3166 3067 y Ft(f)11 b Fu(\()p Ft(t;)23 b Fu(~)-55 b Ft(x)p Fu(\))p Ft(d)6 b Fu(~)-55 b Ft(x:)3361 3240 y Fu(\(3.2\))324 3360 y(Then)515 3588 y Ft(U)10 b(h)647 3603 y Fm(1)722 3588 y Ft(U)798 3547 y Fm(\003)866 3588 y Fu(=)970 3452 y Fl(Z)1025 3678 y Fh(R)1073 3659 y Fj(2)1128 3588 y Ft(h)p Fu(\()1225 3562 y(~)1222 3588 y Ft(k)s Fu(\))p Ft(d)1368 3562 y Fu(~)1365 3588 y Ft(k)s(;)114 b(h)p Fu(\()1657 3562 y(~)1654 3588 y Ft(k)t Fu(\))27 b(=)h Fs(\000)1983 3520 y Ft(d)2034 3484 y Fv(2)p 1965 3565 126 4 v 1965 3656 a Ft(dt)2051 3627 y Fv(2)2123 3588 y Fu(+)22 b(\()p Ft(!)t(t)f Fs(\000)i Ft(k)2531 3603 y Fv(2)2570 3588 y Fu(\))2608 3547 y Fv(2)2670 3588 y Fu(+)f Ft(v)t Fu(\()p Ft(t)p Fu(\))g(+)g Ft(k)3104 3547 y Fv(2)3101 3612 y(3)3143 3588 y Ft(:)191 b Fu(\(3.3\))324 3852 y(If)42 b Ft(z)48 b Fs(2)c Fp(C)28 b Fs(n)h Fp(R)42 b Fu(denote)g(b)m(y)h(\()p Ft(h)1489 3867 y Fm(1)1592 3852 y Fs(\000)29 b Ft(z)t Fu(\))1785 3815 y Fm(\000)p Fv(1)1880 3852 y Fu(\()p Ft(x)1973 3867 y Fv(1)2013 3852 y Ft(;)23 b Fu(~)-55 b Ft(x)p Fu(;)17 b Ft(x)2211 3815 y Fm(0)2211 3876 y Fv(1)2251 3852 y Ft(;)23 b Fu(~)-55 b Ft(x)2350 3815 y Fm(0)2373 3852 y Fu(\))42 b(and)g([)p Ft(h)p Fu(\()2776 3825 y(~)2773 3852 y Ft(k)s Fu(\))29 b Fs(\000)g Ft(z)t Fu(])3076 3815 y Fm(\000)p Fv(1)3171 3852 y Fu(\()p Ft(t;)17 b(t)3323 3815 y Fm(0)3346 3852 y Fu(\))42 b(the)324 3972 y(in)m(tegral)g(k)m(ernels)i(of)d(the)i (corresp)s(onding)f(op)s(erators.)72 b(W)-8 b(e)42 b(are)g(in)m (terested)i(here)e(in)324 4092 y(the)33 b(analog)f(of)g(\(2.4\).)324 4273 y Fw(Lemma)50 b(3.2.)d Fi(If)d Ft(f)55 b Fs(2)45 b Ft(C)1356 4237 y Fm(1)1349 4298 y Fv(0)1431 4273 y Fu(\()p Fp(R)p Fu(\))p Fi(,)h(then)e Ft(f)11 b Fu(\()p Ft(h)2034 4288 y Fm(1)2108 4273 y Fu(\))44 b Fi(admits)g(a)f(smo)-5 b(oth)44 b(inte)-5 b(gr)g(al)43 b(kernel)324 4394 y Ft(f)372 4409 y Fo(h)413 4417 y Fg(1)482 4394 y Fu(\()p Ft(t;)23 b Fu(~)-55 b Ft(y)s Fu(;)17 b Ft(t)729 4357 y Fm(0)752 4394 y Ft(;)23 b Fu(~)-55 b Ft(y)848 4357 y Fm(0)870 4394 y Fu(\))35 b Fi(and)f(we)h(have)f(the)h(r)-5 b(epr)g(esentation) 1149 4636 y Ft(f)1197 4663 y Fo(h)p Fv(\()1267 4646 y(~)1265 4663 y Fo(k)q Fv(\))1335 4636 y Fu(\()p Ft(t;)17 b(t)1487 4594 y Fm(0)1510 4636 y Fu(\))28 b(=)1679 4500 y Fl(Z)1735 4725 y Fh(R)1783 4707 y Fj(2)1838 4636 y Ft(e)1883 4594 y Fm(\000)p Fo(i)1964 4577 y Fv(~)1962 4594 y Fo(k)7 b Fv(~)-40 b Fo(y)2042 4636 y Ft(f)2090 4651 y Fo(h)2131 4659 y Fg(1)2200 4636 y Fu(\()p Ft(t;)24 b Fu(~)-56 b Ft(y)s Fu(;)17 b Ft(t)2447 4594 y Fm(0)2470 4636 y Ft(;)g Fw(0)p Fu(\))p Ft(d)7 b Fu(~)-56 b Ft(y)s(;)624 b Fu(\(3.4\))324 4882 y Fi(for)31 b(the)h(kernel)g(of)f(the)h(op)-5 b(er)g(ator)32 b(in)f(\(3.3\))g(wher)-5 b(e)32 b(the)g(inte)-5 b(gr)g(al)31 b(is)h(absolutely)g(c)-5 b(onver-)324 5002 y(gent.)1894 5251 y Fu(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 324 548 a Fi(Pr)-5 b(o)g(of)p Fu(.)48 b(Let)34 b Ft(f)41 b Fs(2)31 b Ft(C)1065 512 y Fm(1)1058 573 y Fv(0)1140 548 y Fu(\()p Fp(R)p Fu(\).)48 b(Then)35 b Ft(f)11 b Fu(\()p Ft(h)1772 563 y Fm(1)1847 548 y Fu(\))34 b(admits)h(a)e(smo)s(oth)i(in)m(tegral)f(k)m(ernel)i (whic)m(h)324 668 y(deca)m(ys)41 b(faster)e(than)h(an)m(y)g(p)s (olynomial)f(in)p 1972 613 52 4 v 40 w Ft(y)j Fu(for)d Ft(t)g Fu(and)h Ft(t)2524 632 y Fm(0)2586 668 y Fu(\014xed)h([S)e(1],)i ([G-Kl])e(\(see)324 789 y(also)g(App)s(endix)i(1\).)63 b(Moreo)m(v)m(er,)43 b(b)m(y)d(standard)g(argumen)m(ts)h([S)e(1],)i(it) e(is)h(enough)g(to)324 909 y(pro)m(v)m(e)34 b(\(3.4\))e(for)g(the)h (resolv)m(en)m(t)h(op)s(erator.)43 b(Let)33 b Ft(g)e Fs(2)d Ft(C)2401 873 y Fm(1)2394 934 y Fv(0)2476 909 y Fu(\()p Fp(R)2586 873 y Fv(3)2625 909 y Fu(\).)43 b(Then)34 b(w)m(e)g(ha)m(v)m(e:)661 1121 y([)p Ft(U)756 1131 y Fv(~)754 1149 y Fo(k)797 1121 y Fu(\()p Ft(h)891 1136 y Fm(1)988 1121 y Fs(\000)22 b Ft(z)t Fu(\))1174 1080 y Fm(\000)p Fv(1)1269 1121 y Ft(U)1345 1080 y Fm(\003)1386 1121 y Ft(g)t Fu(]\()p Ft(t)p Fu(\))1786 b(\(3.5\))502 1348 y(=)661 1212 y Fl(Z)716 1438 y Fh(R)785 1348 y Ft(dt)871 1307 y Fm(0)911 1212 y Fl(Z)966 1438 y Fh(R)1014 1419 y Fj(2)1069 1348 y Ft(d)6 b Fu(~)-55 b Ft(x)1175 1307 y Fm(0)1215 1212 y Fl(Z)1271 1438 y Fh(R)1319 1419 y Fj(2)1374 1348 y Ft(d)6 b Fu(~)-55 b Ft(x)1490 1280 y(e)1535 1244 y Fm(\000)p Fo(i)1616 1227 y Fv(~)1614 1244 y Fo(k)6 b Fv(~)-39 b Fo(x)p 1490 1325 207 4 v 1540 1416 a Fu(2)p Ft(\031)1707 1348 y Fu(\()p Ft(h)1801 1363 y Fm(1)1898 1348 y Fs(\000)22 b Ft(z)t Fu(\))2084 1307 y Fm(\000)p Fv(1)2179 1348 y Fu(\()p Ft(t;)h Fu(~)-55 b 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Fv(~)1121 2642 y Fo(k)1164 2614 y Fu(\()p Ft(h)1258 2629 y Fm(1)1355 2614 y Fs(\000)23 b Ft(z)t Fu(\))1542 2573 y Fm(\000)p Fv(1)1637 2614 y Ft(U)1713 2573 y Fm(\003)1753 2614 y Ft(g)t Fu(]\()p Ft(t)p Fu(\))870 2819 y(=)1028 2684 y Fl(Z)1084 2909 y Fh(R)1152 2819 y Ft(dt)1238 2778 y Fm(0)1278 2679 y Fl(\032)1353 2684 y(Z)1408 2909 y Fh(R)1456 2890 y Fj(2)1512 2819 y Ft(e)1557 2778 y Fm(\000)p Fo(i)1638 2761 y Fv(~)1636 2778 y Fo(k)6 b Fv(~)-40 b Fo(y)1716 2819 y Fu(\()p Ft(h)1810 2834 y Fm(1)1907 2819 y Fs(\000)22 b Ft(z)t Fu(\))2093 2778 y Fm(\000)p Fv(1)2188 2819 y Fu(\()p Ft(t;)i Fu(~)-56 b Ft(y)s Fu(;)17 b Ft(t)2435 2778 y Fm(0)2459 2819 y Ft(;)g Fw(0)p Fu(\))p Ft(d)7 b Fu(~)-56 b Ft(y)2698 2679 y Fl(\033)2789 2819 y Ft(g)t Fu(\()p Ft(t)2913 2778 y Fm(0)2936 2819 y Ft(;)2982 2793 y Fu(~)2980 2819 y Ft(k)s Fu(\))p Ft(:)262 b Fu(\(3.6\))324 3089 y(By)33 b(virtue)g(of)f(\(3.2\)) h(and)f(\(3.3\),)g(this)i(yields)g(the)f(equalit)m(y)831 3353 y([)p Ft(h)p Fu(\()955 3327 y(~)952 3353 y Ft(k)t Fu(\))22 b Fs(\000)g Ft(z)t Fu(])1242 3312 y Fm(\000)p Fv(1)1337 3353 y Fu(\()p Ft(t;)17 b(t)1489 3312 y Fm(0)1513 3353 y Fu(\))27 b(=)1682 3217 y Fl(Z)1737 3443 y Fh(R)1785 3424 y Fj(2)1841 3353 y Ft(e)1886 3312 y Fm(\000)p Fo(i)1967 3294 y Fv(~)1965 3312 y Fo(k)6 b Fv(~)-40 b Fo(y)2045 3353 y Fu(\()p Ft(h)2139 3368 y Fm(1)2236 3353 y Fs(\000)22 b Ft(z)t Fu(\))2422 3312 y Fm(\000)p Fv(1)2517 3353 y Fu(\()p Ft(t;)i Fu(~)-56 b Ft(y)s Fu(;)17 b Ft(t)2764 3312 y Fm(0)2788 3353 y Ft(;)g Fw(0)p Fu(\))p Ft(d)7 b Fu(~)-56 b Ft(y)r(;)307 b Fu(\(3.7\))324 3571 y(whic)m(h)42 b(has)g(to)f(b)s(e)g(understo)s(o)s(d)h(as)f(equalit)m(y)i(b)s(et)m(w)m (een)g(smo)s(oth)e(functions)i(outside)324 3691 y(the)33 b(diagonal)f Ft(t)c Fu(=)f Ft(t)1083 3655 y Fm(0)1107 3691 y Fu(.)p 3491 3691 4 66 v 3495 3629 59 4 v 3495 3691 V 3553 3691 4 66 v 324 3859 a Fw(Pro)s(of)49 b(of)h(Prop)s (osition)g(3.1)p Fu(.)77 b(By)43 b(the)h(conditions)h(on)e(the)h (external)g(p)s(oten)m(tial)324 3979 y Ft(v)t Fu(\()p Ft(t)p Fu(\),)32 b(the)h(\014b)s(er)g(op)s(erator)f Ft(h)p Fu(\()1428 3953 y(~)1425 3979 y Ft(k)s Fu(\))h(has)g(purely)g(discrete) i(sp)s(ectrum)f(for)e(an)m(y)3145 3953 y(~)3142 3979 y Ft(k)f Fs(2)d Fp(R)3390 3943 y Fv(2)3429 3979 y Fu(.)470 4100 y(W)-8 b(e)31 b(denote)f(b)m(y)h Fs(f)p Ft(\025)1187 4115 y Fo(n)1234 4100 y Fu(\()p Ft(k)1323 4115 y Fv(2)1362 4100 y Fu(\))p Fs(g)1450 4115 y Fo(n)p Fm(\025)p Fv(1)1617 4100 y Fu(the)g(nondegenerate)g(eigen)m(v)-5 b(alues)32 b(of)e(the)g(op)s(erator)324 4220 y Ft(h)p Fu(\()p Ft(k)469 4235 y Fv(2)508 4220 y Ft(;)17 b Fu(0\))27 b(=)h Fs(\000)p Ft(d)898 4184 y Fv(2)937 4220 y Ft(=t)1021 4184 y Fv(2)1072 4220 y Fu(+)11 b(\()p Ft(!)t(t)g Fs(\000)g Ft(k)1447 4235 y Fv(2)1486 4220 y Fu(\))1524 4184 y Fv(2)1575 4220 y Fu(+)g Ft(v)t Fu(\()p Ft(t)p Fu(\),)28 b(and)f(b)m(y)h Fs(f)p Ft( )2306 4235 y Fo(n)2354 4220 y Fu(\()p Fs(\001)p Ft(;)17 b(k)2515 4235 y Fv(2)2553 4220 y Fu(\))p Fs(g)2641 4235 y Fo(n)p Fm(\025)p Fv(1)2805 4220 y Fu(the)28 b(corresp)s(onding) 324 4341 y(eigenfunctions.)45 b(Let)33 b Ft(f)38 b Fs(2)28 b Ft(C)1431 4304 y Fm(1)1424 4365 y Fv(0)1506 4341 y Fu(\()p Fp(R)p Fu(\).)43 b(Then)34 b(w)m(e)g(ha)m(v)m(e)906 4570 y Ft(f)954 4598 y Fo(h)p Fv(\()1024 4580 y(~)1022 4598 y Fo(k)q Fv(\))1092 4570 y Fu(\()p Ft(t;)17 b(t)1244 4529 y Fm(0)1267 4570 y Fu(\))28 b(=)1436 4475 y Fl(X)1442 4686 y Fo(n)p Fm(\025)p Fv(1)1597 4570 y Ft(f)11 b Fu(\()p Ft(\025)1751 4585 y Fo(n)1797 4570 y Fu(\()p Ft(k)1886 4585 y Fv(2)1926 4570 y Fu(\))22 b(+)g Ft(k)2138 4529 y Fv(2)2135 4595 y(3)2177 4570 y Fu(\))p Ft( )2278 4585 y Fo(n)2326 4570 y Fu(\()p Ft(t;)17 b(k)2494 4585 y Fv(2)2533 4570 y Fu(\))p 2571 4489 67 4 v Ft( )2637 4594 y Fo(n)2684 4570 y Fu(\()p Ft(t)2757 4529 y Fm(0)2781 4570 y Ft(;)g(k)2876 4585 y Fv(2)2915 4570 y Fu(\))p Ft(:)381 b Fu(\(3.8\))324 4882 y(Here)22 b(the)g(sum)h(o)m(v)m(er)g Ft(n)f Fu(is)g(\014nite,)j (since)e Ft(f)32 b Fu(has)22 b(compact)h(supp)s(ort,)h(but)e(lim)3063 4897 y Fo(n)p Fm(!1)3267 4882 y Ft(\025)3324 4897 y Fo(n)3371 4882 y Fu(\()p Ft(k)3460 4897 y Fv(2)3499 4882 y Fu(\))28 b(=)324 5002 y Fs(1)36 b Fu(uniformly)h(in)f Ft(k)1076 5017 y Fv(2)1149 5002 y Fs(2)e Fp(R)p Fu(.)54 b(Hence,)39 b Ft(f)1772 5030 y Fo(h)p Fv(\()1842 5012 y(~)1840 5030 y Fo(k)r Fv(\))1946 5002 y Fu(is)e(a)f(\014nite-rank)h(op)s(erator.)54 b(This)37 b(can)f(b)s(e)1894 5251 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 324 548 a Fu(explicitely)30 b(seen)g(from)e (the)g(fact)g(that)g(the)h(\014b)s(er)f(op)s(erator)g Ft(h)p Fu(\()2647 522 y(~)2644 548 y Ft(k)s Fu(\))g(in)h(\(3.3\))e(is)i (unitarily)324 668 y(equiv)-5 b(alen)m(t)34 b(to)e(the)h(op)s(erator:) 1259 920 y Fs(\000)1364 853 y Ft(d)1415 817 y Fv(2)p 1346 898 126 4 v 1346 989 a Ft(dt)1432 960 y Fv(2)1504 920 y Fu(+)22 b Ft(!)1667 879 y Fv(2)1706 920 y Ft(t)1741 879 y Fv(2)1802 920 y Fu(+)g Ft(v)t Fu(\()p Ft(t)g Fu(+)g Ft(k)2195 935 y Fv(2)2235 920 y Ft(=!)t Fu(\))f(+)h Ft(k)2560 879 y Fv(2)2557 945 y(3)2599 920 y Ft(;)324 1144 y Fu(whic)m(h)38 b(is)f(a)g(harmonic)g(oscilator)g(plus)h(a)f(b)s(ounded)g(p)s (erturbation.)56 b(Moreo)m(v)m(er,)40 b(this)324 1265 y(represen)m(tation)28 b(mak)m(es)f(eviden)m(t)h(that)e Fp(Z)p Fs(\000)p Fu(p)s(erio)s(dicit)m(y)i(of)e Ft(v)j Fu(implies)f Fp(Z)2978 1280 y Fo(!)3029 1265 y Fs(\000)p Fu(p)s(erio)s(dicit)m(y)324 1385 y(of)k Ft(\025)492 1400 y Fo(n)539 1385 y Fu(\()p Ft(k)628 1400 y Fv(2)667 1385 y Fu(\))g(for)g(all)h Ft(n)28 b Fs(\025)g Fu(1.)470 1505 y(Notice)38 b(that)f(w)m(e)i(are)e(only)h(in)m(terested)i(in)d(what)h (happ)s(ens)h(near)e(the)h(b)s(ottom)f(of)324 1626 y(the)31 b(sp)s(ectrum,)h Ft(E)1013 1641 y Fv(0)1080 1626 y Fu(=)c(inf)23 b Ft(\033)t Fu(\()p Ft(h)1471 1641 y Fm(1)1546 1626 y Fu(\))28 b(=)f(inf)1833 1641 y Fo(k)1870 1650 y Fj(2)1926 1626 y Ft(\025)1983 1641 y Fv(1)2022 1626 y Fu(\()p Ft(k)2111 1641 y Fv(2)2150 1626 y Fu(\).)43 b(Because)32 b(of)d(the)i (non-degeneracy)324 1746 y(of)h(the)g(eigen)m(v)-5 b(alues)35 b Fs(f)p Ft(\025)1214 1761 y Fo(n)1260 1746 y Fu(\()p Ft(k)1349 1761 y Fv(2)1388 1746 y Fu(\))p Fs(g)1476 1761 y Fo(n)p Fm(\025)p Fv(1)1613 1746 y Fu(,)e Ft(E)1745 1761 y Fv(0)1817 1746 y Fu(is)g(isolated)f(from)h(the)f(other)h(bands,) g(i.e)g(from)324 1867 y(Ran\()p Ft(\025)594 1882 y Fo(n)640 1867 y Fu(\))39 b(with)h Ft(n)e Fs(\025)h Fu(2.)63 b(Applying)40 b(usual)f(argumen)m(ts)i(\(see)f([R-S)e(IV]\),)h Ft( )3197 1882 y Fv(1)3276 1867 y Fu(can)g(b)s(e)324 1987 y(c)m(hosen)k(p)s (ositiv)m(e.)70 b(Hence,)45 b(if)c Ft(f)52 b Fu(is)41 b(supp)s(orted)h(close)h(enough)e(to)g Ft(E)2955 2002 y Fv(0)2995 1987 y Fu(,)i(b)m(y)f(\(3.8\))f(w)m(e)324 2107 y(obtain:)1181 2228 y Ft(f)1229 2255 y Fo(h)p Fv(\()1299 2238 y(~)1297 2255 y Fo(k)q Fv(\))1367 2228 y Fu(\()p Ft(t;)17 b(t)p Fu(\))28 b(=)f Ft(f)11 b Fu(\()p Ft(\025)1842 2243 y Fv(1)1881 2228 y Fu(\()p Ft(k)1970 2243 y Fv(2)2009 2228 y Fu(\))22 b(+)g Ft(k)2221 2187 y Fv(2)2218 2252 y(3)2261 2228 y Fu(\))27 b Ft( )2393 2187 y Fv(2)2389 2252 y(1)2433 2228 y Fu(\()p Ft(t;)17 b(k)2601 2243 y Fv(2)2640 2228 y Fu(\))p Ft(:)324 2393 y Fu(Then)33 b(taking)g(the)g(F) -8 b(ourier)33 b(transform)f(in)h(\(3.4\))f(w)m(e)i(get)453 2572 y(1)p 404 2617 148 4 v 404 2708 a(4)p Ft(\031)512 2679 y Fv(2)578 2504 y Fl(Z)633 2730 y Fh(R)681 2711 y Fj(2)736 2640 y Ft(f)784 2667 y Fo(h)p Fv(\()854 2650 y(~)852 2667 y Fo(k)r Fv(\))923 2640 y Fu(\()p Ft(t;)17 b(t)p Fu(\))p Ft(d)1167 2613 y Fu(~)1164 2640 y Ft(k)30 b Fu(=)1408 2572 y(1)p 1359 2617 V 1359 2708 a(4)p Ft(\031)1467 2679 y Fv(2)1532 2504 y Fl(Z)1588 2730 y Fh(R)1636 2711 y Fj(2)1691 2640 y Ft(f)11 b Fu(\()p Ft(\025)1845 2655 y Fv(1)1884 2640 y Fu(\()p Ft(k)1973 2655 y Fv(2)2012 2640 y Fu(\))22 b(+)g Ft(k)2224 2599 y Fv(2)2221 2664 y(3)2264 2640 y Fu(\))27 b Ft( )2396 2599 y Fv(2)2392 2664 y(1)2436 2640 y Fu(\()p Ft(t;)17 b(k)2604 2655 y Fv(2)2643 2640 y Fu(\))p Ft(d)2735 2613 y Fu(~)2732 2640 y Ft(k)30 b Fu(=)e Ft(f)2965 2655 y Fo(h)3006 2663 y Fg(1)3075 2640 y Fu(\()p Ft(t;)17 b Fw(0)p Fu(;)g Ft(t;)g Fw(0)p Fu(\))p Ft(:)3361 2813 y Fu(\(3.9\))324 2933 y(Note)32 b(that)h(the)g(second)h(in)m(tegral)f(in)f(\(3.9\))g(con)m(v)m(erges,)j (since)f(b)m(y)g(standard)f(metho)s(ds)324 3053 y(w)m(e)27 b(can)f(pro)m(v)m(e)h(that)f Ft( )1157 3068 y Fv(1)1222 3053 y Fu(is)h(sharply)g(lo)s(calized)f(near)g Ft(k)2304 3068 y Fv(2)2344 3053 y Ft(=!)j Fu(and)c(that)h(it)g(has)g(a)g (gaussian)324 3174 y(deca)m(y)36 b(of)e(the)h(form)g Ft(const)c(e)1417 3138 y Fm(\000)p Fo(\013)p Fv(\()p Fo(t)p Fm(\000)p Fo(k)1661 3147 y Fj(2)1697 3138 y Fo(=!)r Fv(\))1805 3114 y Fj(2)1879 3174 y Fu(\(see)36 b([Ag]\).)50 b(No)m(w)35 b(using)h(\(1.16\),)e(\(1.17\))g(for)324 3294 y Ft(\025)c Fu(close)h(to)f Ft(E)831 3309 y Fv(0)871 3294 y Fu(,)h(w)m(e)g(ev)m(en)m(tually)i(get)e(\(in)f(a)g(w)m(a)m(y)i (similar)f(to)f(\(2.7\)\))g(that)g(\(3.9\))g(implies)628 3637 y Ft(n)686 3652 y Fm(1)761 3637 y Fu(\()p Ft(\025)p Fu(\))d(=)1084 3569 y(1)p 1035 3614 V 1035 3705 a(4)p Ft(\031)1143 3676 y Fv(2)1208 3501 y Fl(Z)1264 3726 y Fv(\004)1332 3501 y Fl(Z)1387 3726 y Fh(R)1435 3708 y Fj(2)1490 3637 y Ft(\037)1551 3652 y Fv([)p Fo(E)1623 3661 y Fj(0)1657 3652 y Fm(\000)p Fv(0)p Fo(;\025)p Fv(])1832 3637 y Fu(\()p Ft(\025)1927 3652 y Fv(1)1966 3637 y Fu(\()p Ft(k)2055 3652 y Fv(2)2094 3637 y Fu(\))22 b(+)g Ft(k)2306 3595 y Fv(2)2303 3661 y(3)2346 3637 y Fu(\))28 b Ft( )2479 3595 y Fv(2)2475 3661 y(1)2518 3637 y Fu(\()p Ft(t;)17 b(k)2686 3652 y Fv(2)2725 3637 y Fu(\))p Ft(dt)27 b(d)2930 3610 y Fu(~)2927 3637 y Ft(k)s(;)304 b Fu(\(3.10\))324 3888 y(where)34 b(\004)27 b(=)h([)p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2].)470 4008 y(In)43 b(the)g(particular)g(case)h(when)g Ft(v)49 b Fu(=)c(0,)g(w)m(e)f(ha)m(v)m(e)g Ft(E)2521 4023 y Fv(0)2605 4008 y Fu(=)h Ft(!)k Fu(=)44 b Ft(\025)3013 4023 y Fv(1)3053 4008 y Fu(\()p Ft(k)3142 4023 y Fv(2)3181 4008 y Fu(\).)74 b(Fix)43 b Ft(\025)324 4128 y Fu(b)s(et)m(w)m(een)36 b(the)g(\014rst)f(t)m(w)m(o)g(Landau)g(lev)m(els:)49 b Ft(\025)31 b Fs(2)h Fu(\()p Ft(!)t(;)17 b Fu(3)p Ft(!)t Fu(\).)48 b(Then)35 b(b)m(y)h(in)m(tegrating)f(with)324 4249 y(resp)s(ect)f(to)e Ft(k)828 4264 y Fv(2)900 4249 y Fu(w)m(e)h(obtain)1523 4276 y Fl(Z)1579 4502 y Fh(R)1647 4412 y Ft( )1714 4371 y Fv(2)1710 4437 y(1)1754 4412 y Fu(\()p Ft(t;)17 b(k)1922 4427 y Fv(2)1961 4412 y Fu(\))p Ft(dk)2101 4427 y Fv(2)2167 4412 y Fu(=)28 b Ft(!)t(;)324 4641 y Fu(th)m(us)40 b Ft(n)603 4656 y Fm(1)678 4641 y Fu(\()p Ft(\025)p Fu(\))e(=)1007 4602 y Fo(!)p 974 4618 113 4 v 974 4675 a Fv(2)p Fo(\031)1052 4656 y Fj(2)1097 4641 y Fu(\()p Ft(\025)26 b Fs(\000)h Ft(!)t Fu(\))1425 4605 y Fv(1)p Fo(=)p Fv(2)1573 4641 y Fu(for)39 b(the)g(pure)h (magnetic)g(case.)64 b(Notice)39 b(the)h(\\bad")324 4761 y(exp)s(onen)m(t)34 b(1)p Ft(=)p Fu(2)e(whic)m(h)i(mak)m(es)g(the)f (critical)g(densit)m(y)i(to)d(div)m(erge,)i(see)g(\(1.12\).)470 4882 y(No)m(w)43 b(w)m(e)h(sho)m(w)g(that)e(ev)m(en)i(if)f Ft(v)48 b Fs(6)p Fu(=)d(0,)f(the)f(in)m(tegrated)h(densit)m(y)g(of)e (states)i(still)324 5002 y(b)s(eha)m(v)m(es)36 b(lik)m(e)f(in)f (\(1.12\).)46 b(First,)34 b(the)g(general)g(theory)h(insures)g(that)f Ft(\025)2984 5017 y Fv(1)3023 5002 y Fu(\()p Ft(k)3112 5017 y Fv(2)3151 5002 y Fu(\))g(is)g(a)f(real)1894 5251 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 324 548 a Fu(analytic)28 b(function)g(of)g Ft(k)1225 563 y Fv(2)1292 548 y Fu([R-S)f(IV].)h(If)g(it)f(is)i (constan)m(t,)g(then)g(w)m(e)f(are)g(essen)m(tially)j(bac)m(k)324 668 y(to)h(the)h(case)g Ft(v)f Fu(=)27 b(0,)33 b(since)h(one)f(has)g(a) f(lo)m(w)m(er)i(b)s(ound)e(for)g Ft(n)2543 683 y Fm(1)2651 668 y Fu(of)g(the)h(form:)397 903 y Ft(n)455 918 y Fm(1)530 903 y Fu(\()p Ft(\025)p Fu(\))27 b Fs(\025)854 836 y Fu(1)p 805 880 148 4 v 805 971 a(4)p Ft(\031)913 943 y Fv(2)979 763 y Fl(\022)1052 767 y(Z)1108 993 y Fh(R)1176 903 y Ft(\037)1237 918 y Fv([)p Fo(E)1309 927 y Fj(0)1343 918 y Fm(\000)p Fv(0)p Fo(;\025)p Fv(])1518 903 y Fu(\()p Ft(E)1628 918 y Fv(0)1690 903 y Fu(+)22 b Ft(k)1842 862 y Fv(2)1839 928 y(3)1881 903 y Fu(\))p Ft(dk)2021 918 y Fv(3)2060 763 y Fl(\023)2320 903 y Fu(inf)2178 969 y Fm(\000)p Fv(1)p Fo(=)p Fv(2)p Fm(\024)p Fo(t)p Fm(\024)p Fv(1)p Fo(=)p Fv(2)2596 767 y Fl(Z)2651 993 y Fh(R)2720 903 y Ft( )2787 862 y Fv(2)2783 928 y(1)2827 903 y Fu(\()p Ft(t;)17 b(k)2995 918 y Fv(2)3034 903 y Fu(\))p Ft(dk)3174 918 y Fv(2)3212 903 y Ft(;)73 b Fu(\(3.11\))324 1147 y(for)25 b(ev)m(ery)i Ft(\025)h Fs(2)g Fu(\()p Ft(E)1005 1162 y Fv(0)1044 1147 y Ft(;)17 b(E)1160 1162 y Fv(0)1208 1147 y Fu(+)8 b Ft(\017)p Fu(\).)41 b(Since)26 b Ft( )1751 1111 y Fv(2)1747 1172 y(1)1791 1147 y Fu(\()p Ft(t;)17 b(k)1959 1162 y Fv(2)1998 1147 y Fu(\))25 b(is)h(join)m(tly)h(smo)s (oth)e(in)h(b)s(oth)f(argumen)m(ts)324 1268 y(and)32 b(p)s(ositiv)m(e,)i(the)g(mapping)1148 1494 y([)p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2])26 b Fs(3)i Ft(t)g Fs(!)1928 1358 y Fl(Z)1983 1584 y Fh(R)2052 1494 y Ft( )2119 1452 y Fv(2)2115 1518 y(1)2158 1494 y Fu(\()p Ft(t;)17 b(k)2326 1509 y Fv(2)2365 1494 y Fu(\))p Ft(dk)2505 1509 y Fv(2)2572 1494 y Fs(2)28 b Fp(R)324 1740 y Fu(has)f(a)f(p)s(ositiv)m(e)i(minim)m(um,)i(so)d Ft(n)1553 1755 y Fm(1)1628 1740 y Fu(\()p Ft(\025)p Fu(\))g Fs(\025)i Ft(a)p Fu(\()p Ft(\025)10 b Fs(\000)g Ft(E)2209 1755 y Fv(0)2249 1740 y Fu(\))2287 1704 y Fv(1)p Fo(=)p Fv(2)2407 1740 y Fu(+)g Ft(o)p Fu(\(\()p Ft(\025)g Fs(\000)g Ft(E)2842 1755 y Fv(0)2882 1740 y Fu(\))2920 1704 y Fv(1)p Fo(=)p Fv(2)3030 1740 y Fu(\))27 b(for)f Ft(\025)i Fs(&)f Ft(E)3522 1755 y Fv(0)324 1861 y Fu(and)32 b(for)h(some)g Ft(a)28 b(>)f Fu(0,)33 b(i.e.)44 b(w)m(e)33 b(get)g(bac)m(k)h(to)e (case)h(\(1.12\).)470 1981 y(Let)38 b Ft(\025)707 1996 y Fv(1)746 1981 y Fu(\()p Ft(k)835 1996 y Fv(2)874 1981 y Fu(\))f(b)s(e)h(not)f(constan)m(t.)58 b(Since)39 b(it)e(is)h(a)f (real)g(analytic)h(function)g(and)f Fp(Z)3478 1996 y Fo(!)3529 1981 y Fu(-)324 2101 y(p)s(erio)s(dic,)27 b(there)g(exists)h (a)d(\014nite)i(set)f(of)g(p)s(oin)m(ts)g Fs(f)p Ft(\030)2171 2116 y Fv(1)2210 2101 y Ft(;)17 b(:::;)g(\030)2422 2116 y Fo(N)2488 2101 y Fs(g)28 b(\032)g Fu([0)p Ft(;)17 b(!)t Fu(\),)26 b(where)h Ft(\025)3279 2116 y Fv(1)3344 2101 y Fu(tak)m(es)324 2222 y(its)35 b(minimal)g(v)-5 b(alue)35 b Ft(E)1169 2237 y Fv(0)1209 2222 y Fu(.)50 b(Let)35 b Ft(\025)1520 2237 y Fv(1)1559 2222 y Fu(\()p Ft(\030)1640 2237 y Fo(j)1676 2222 y Fu(\))c(=)g Ft(E)1924 2237 y Fv(0)1964 2222 y Fu(,)k Ft(j)i Fs(2)32 b(f)p Fu(1)p Ft(;)17 b(:)g(:)g(:)e(;)i(N)10 b Fs(g)p Fu(.)50 b(Then)36 b(there)f(exists)i(a) 324 2342 y(p)s(ositiv)m(e)d(in)m(teger)f Ft(n)1065 2357 y Fo(j)1130 2342 y Fs(\025)28 b Fu(1)k(and)h(a)f(constan)m(t)h Ft(C)2050 2357 y Fo(j)2119 2342 y Fu(so)g(that)f(for)g Ft(k)2650 2357 y Fv(2)2722 2342 y Fu(close)i(to)e Ft(\030)3118 2357 y Fo(j)1322 2525 y Ft(\025)1379 2540 y Fv(1)1418 2525 y Fu(\()p Ft(k)1507 2540 y Fv(1)1546 2525 y Fu(\))c Fs(\030)g Ft(E)1789 2540 y Fv(0)1851 2525 y Fu(+)22 b Ft(C)2019 2540 y Fo(j)2055 2525 y Fu(\()p Ft(k)2144 2540 y Fv(2)2205 2525 y Fs(\000)h Ft(\030)2348 2540 y Fo(j)2384 2525 y Fu(\))2422 2484 y Fv(2)p Fo(n)2500 2494 y Fk(j)2537 2525 y Ft(:)324 2708 y Fu(T)-8 b(o)29 b(get)g(a)g(lo)m(w)m(er)i(b)s (ound)e(for)g Ft(n)1454 2723 y Fm(1)1529 2708 y Fu(\()p Ft(\025)p Fu(\))g(w)m(e)h(ma)m(y)g(tak)m(e)g(the)f(in)m(tegral)h (\(3.10\))e(with)i(resp)s(ect)324 2828 y(to)454 2802 y(~)451 2828 y Ft(k)44 b Fu(o)m(v)m(er)e(compact)f(domains)g(around)g (the)g(minima)h(of)e(the)h(function)g Ft(\025)3172 2843 y Fv(1)3211 2828 y Fu(\()p Ft(k)3300 2843 y Fv(2)3340 2828 y Fu(\).)67 b(In)324 2949 y(fact,)34 b(for)g Ft(\025)g Fu(close)h(to)f Ft(E)1215 2964 y Fv(0)1289 2949 y Fu(w)m(e)h(can)g(b)s (ound)f Ft(n)1975 2964 y Fm(1)2085 2949 y Fu(from)g(b)s(elo)m(w)h(b)m (y)g(taking)f(in)m(to)h(accoun)m(t)324 3069 y(just)e(one)g(of)f(those)h (minima:)324 3334 y Ft(n)382 3349 y Fm(1)457 3334 y Fu(\()p Ft(\025)p Fu(\))27 b Fs(\025)h Ft(const)967 3163 y Fl( )1046 3198 y(Z)1146 3224 y Fv(1)p Fo(=)p Fv(2)1101 3424 y Fm(\000)p Fv(1)p Fo(=)p Fv(2)1283 3334 y Ft( )1350 3293 y Fv(2)1346 3358 y(1)1389 3334 y Fu(\()p Ft(t;)17 b(\030)1549 3349 y Fo(j)1585 3334 y Fu(\))p Ft(dt)1709 3163 y Fl(!)1807 3334 y Fs(\001)1854 3198 y Fl(Z)1909 3424 y Fh(R)1957 3405 y Fj(2)2013 3334 y Ft(\037)2074 3349 y Fv([)p Fo(E)2146 3358 y Fj(0)2180 3349 y Fo(;\025)p Fv(])2264 3334 y Fu(\()p Ft(E)2374 3349 y Fv(0)2433 3334 y Fu(+)i Ft(\016)t(C)2645 3349 y Fo(j)2681 3334 y Fu(\()p Ft(k)2770 3349 y Fv(2)2828 3334 y Fs(\000)h Ft(\030)2968 3349 y Fo(j)3004 3334 y Fu(\))3042 3293 y Fv(2)p Fo(n)3120 3303 y Fk(j)3176 3334 y Fu(+)f Ft(k)3325 3293 y Fv(2)3322 3358 y(3)3364 3334 y Fu(\))28 b Ft(d)3484 3307 y Fu(~)3481 3334 y Ft(k)s(;)3312 3541 y Fu(\(3.12\))324 3662 y(for)k(some)h Ft(\016)f(>)27 b Fu(1.)44 b(This)33 b(leads)h(to)1317 3857 y Ft(n)1375 3872 y Fm(1)1450 3857 y Fu(\()p Ft(\025)p Fu(\))28 b Fs(\025)g Ft(const)p Fu(\()p Ft(\025)22 b Fs(\000)h Ft(E)2233 3872 y Fv(0)2272 3857 y Fu(\))2320 3767 y Fj(1)p 2320 3779 31 3 v 2320 3820 a(2)2361 3794 y Fv(+)2461 3767 y Fj(1)p 2426 3779 102 3 v 2426 3820 a(2)p Fk(n)2494 3836 y(j)2541 3857 y Ft(;)324 4040 y Fu(whic)m(h)35 b(clearly)g (implies)g(\(1.12\))f(for)f Ft(\025)23 b Fs(\000)g Ft(E)1935 4055 y Fv(0)2009 4040 y Fu(small)34 b(enough.)48 b(Therefore,)35 b(the)g(prop)s(o-)324 4160 y(sition)e(is)g(pro)m(v)m(en.)p 3491 4160 4 66 v 3495 4097 59 4 v 3495 4160 V 3553 4160 4 66 v 324 4450 a Fw(Remark)41 b(3.3.)i Fi(Pr)-5 b(op)g(osition)37 b(3.1)g(c)-5 b(an)37 b(b)-5 b(e)38 b(e)-5 b(asily)37 b(extende)-5 b(d)36 b(to)i Fp(Z)2815 4414 y Fv(2)2855 4450 y Fs(\000)p Fi(p)-5 b(erio)g(dic)37 b(p)-5 b(oten-)324 4570 y(tials)29 b Ft(v)i Fu(=)d Ft(v)t Fu(\()p Ft(x)856 4585 y Fv(1)896 4570 y Ft(;)17 b(x)995 4585 y Fv(3)1034 4570 y Fu(\))p Fi(.)43 b(In)29 b(this)g(c)-5 b(ase,)30 b(a)f(similar)g(analysis)f(shows)h(that)h(the)f(c)-5 b(orr)g(esp)g(ond-)324 4690 y(ing)36 b(Hamiltonian)h Ft(h)1112 4705 y Fm(1)1224 4690 y Fi(is)g(unitarily)g(e)-5 b(quivalent)37 b(to)g(the)g(op)-5 b(er)g(ator)2855 4610 y Fl(R)2902 4725 y Fh(R)p Fm(\002)p Fv(\()p Fm(\000)p Fo(\031)r(;\031)r Fv(\))3241 4690 y Ft(h)p Fu(\()3338 4664 y(~)3335 4690 y Ft(k)s Fu(\))p Ft(d)3481 4664 y Fu(~)3478 4690 y Ft(k)s Fi(,)324 4819 y(wher)g(e)34 b(now)965 5002 y Ft(h)p Fu(\()1062 4976 y(~)1059 5002 y Ft(k)s Fu(\))28 b(=)g Fs(\000)p Ft(@)1416 4961 y Fv(2)1411 5027 y Fo(t)1479 5002 y Fu(+)22 b(\()p Ft(!)t(t)f Fs(\000)i Ft(k)1887 5017 y Fv(2)1926 5002 y Fu(\))1964 4961 y Fv(2)2025 5002 y Fu(+)g(\()p Fs(\000)p Ft(i@)2323 5017 y Fo(s)2383 5002 y Fu(+)f Ft(k)2532 5017 y Fv(3)2571 5002 y Fu(\))2609 4961 y Fv(2)2670 5002 y Fu(+)g Ft(v)t Fu(\()p Ft(t;)17 b(s)p Fu(\))292 b(\(3.13\))1894 5251 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 324 548 a Fi(on)39 b Ft(L)534 512 y Fv(2)574 548 y Fu(\()p Fp(R)26 b Fs(\002)g Fu(\()p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2\)\))39 b Fi(with)g(p)-5 b(erio)g(dic)39 b(b)-5 b(oundary)40 b(c)-5 b(onditions)39 b(on)g Fp(R)26 b Fs(\002)g(f\006)p Fu(1)p Ft(=)p Fu(2)p Fs(g)p Fi(.)324 668 y(Notic)-5 b(e)43 b(that)h(for)f(every)1277 642 y Fu(~)1275 668 y Ft(k)j Fs(2)e Fp(R)28 b Fs(\002)h Fu(\()p Fs(\000)p Ft(\031)t(;)17 b(\031)t Fu(\))p Fi(,)46 b(the)d(\014b)-5 b(er)43 b(Hamiltonnian)f Ft(h)p Fu(\()3196 642 y(~)3193 668 y Ft(k)t Fu(\))h Fi(has)g(a)324 789 y(c)-5 b(omp)g(act)28 b(r)-5 b(esolvent)29 b(which)f(is)h(p)-5 b(ositivity)30 b(impr)-5 b(oving)28 b(\(se)-5 b(e)29 b([R-S)f(IV]\).)h(L)-5 b(et)30 b Fs(f)p Ft(\025)p Fu(\()3286 762 y(~)3283 789 y Ft(k)s Fu(\))p Fs(g)3425 804 y Fo(n)p Fm(\025)p Fv(1)324 909 y Fi(denote)36 b(the)g(set)h(of)f(eigenvalues)f (of)h Ft(h)p Fu(\()1795 883 y(~)1792 909 y Ft(k)s Fu(\))h Fi(and)f Fs(f)p Ft( )2225 924 y Fo(n)2272 909 y Fu(\()p Fs(\001)p Ft(;)2384 883 y Fu(~)2382 909 y Ft(k)r Fu(\))p Fs(g)2523 924 y Fo(n)p Fm(\025)p Fv(1)2697 909 y Fi(b)-5 b(e)36 b(the)g(c)-5 b(orr)g(esp)g(onding)324 1029 y(eigenve)g(ctors.)42 b(Then)30 b Ft(\025)1216 1044 y Fv(1)1256 1029 y Fu(\()1297 1003 y(~)1294 1029 y Ft(k)s Fu(\))h Fi(is)g(c)-5 b(ontinuous)31 b(and)f(nonde)-5 b(gener)g(ate)30 b(for)h(any)3161 1003 y Fu(~)3158 1029 y Ft(k)s Fi(.)44 b(L)-5 b(et)31 b Ft(E)3522 1044 y Fv(0)324 1150 y Fi(b)-5 b(e)30 b(the)g(minimal)f(value)h(of)g Ft(\025)1389 1165 y Fv(1)1429 1150 y Fu(\()1470 1124 y(~)1467 1150 y Ft(k)s Fu(\))p Fi(.)43 b(Then)30 b(ther)-5 b(e)30 b(exists)g(a)g(p)-5 b(oint)30 b Fu(\()p Ft(\030)5 b(;)17 b(\020)8 b Fu(\))26 b Fs(2)i Fp(R)12 b Fs(\002)g Fu(\()p Fs(\000)p Ft(\031)t(;)17 b(\031)t Fu(\))p Fi(,)324 1270 y Fu(\()p Ft(n;)g(m)p Fu(\))42 b Fs(2)g Fp(N)809 1234 y Fv(2)891 1270 y Fi(and)g Fu(\()p Ft(C)r(;)17 b(D)s Fu(\))41 b Fs(2)h Fp(R)1585 1234 y Fv(2)1667 1270 y Fi(two)h(non-ne)-5 b(gative)41 b(c)-5 b(onstants)42 b(such)g(that)h(in)g(the)324 1391 y(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fu(\()p Ft(\030)5 b(;)17 b(\020)8 b Fu(\))33 b Fi(we)i(have)f(the)h(exp)-5 b(ansion)374 1594 y Ft(\025)431 1609 y Fv(1)470 1594 y Fu(\()511 1567 y(~)508 1594 y Ft(k)s Fu(\))28 b(=)f Ft(E)803 1609 y Fv(0)860 1594 y Fu(+)16 b Ft(C)7 b Fu(\()p Ft(k)1118 1609 y Fv(2)1174 1594 y Fs(\000)17 b Ft(\030)5 b Fu(\))1354 1552 y Fv(2)p Fo(n)1453 1594 y Fu(+)16 b Ft(D)s Fu(\()p Ft(k)1718 1609 y Fv(3)1774 1594 y Fs(\000)h Ft(\020)8 b Fu(\))1957 1552 y Fv(2)p Fo(m)2075 1594 y Fu(+)16 b Ft(o)p Fu(\(\()p Ft(k)2341 1609 y Fv(2)2397 1594 y Fs(\000)h Ft(\030)5 b Fu(\))2577 1552 y Fv(2)p Fo(n)2676 1594 y Fu(+)16 b(\()p Ft(k)2857 1609 y Fv(3)2913 1594 y Fs(\000)i Ft(\020)8 b Fu(\))3097 1552 y Fv(2)p Fo(m)3198 1594 y Fu(\))p Ft(:)49 b Fu(\(3.14\))324 1797 y Fi(Notic)-5 b(e)38 b(that)h(by)f(a)g(standar)-5 b(d)38 b(Thomas')f(ar)-5 b(gument)38 b(\(se)-5 b(e)37 b([R-S)h(IV]\))g(c)-5 b(onc)g(erning)37 b(the)324 1917 y Ft(k)375 1932 y Fv(3)459 1917 y Fi(variable)44 b(one)h(c)-5 b(oncludes)44 b(that)h Ft(\025)1741 1932 y Fv(1)1826 1917 y Fi(c)-5 b(annot)44 b(b)-5 b(e)45 b(c)-5 b(onstant)45 b(in)g Ft(k)2862 1932 y Fv(3)2901 1917 y Fi(,)j(which)c(implies)324 2037 y Ft(D)30 b(>)e Fu(0)34 b Fi(and)h Ft(m)28 b Fs(\025)g Fu(1)p Fi(.)470 2158 y(If)34 b(we)h(cho)-5 b(ose)34 b Ft(\025)g Fi(close)h(to)g Ft(E)1546 2173 y Fv(0)1585 2158 y Fi(,)g(formula)f(\(3.10\))g(now)h(takes)f(the)h(form)679 2409 y Ft(n)737 2424 y Fm(1)812 2409 y Fu(\()p Ft(\025)p Fu(\))27 b(=)1115 2342 y(1)p 1085 2386 108 4 v 1085 2478 a(2)p Ft(\031)1220 2274 y Fl(Z)1275 2499 y Fv(\004)1322 2480 y Fj(2)1377 2409 y Ft(dsdt)1577 2274 y Fl(Z)1632 2499 y Fh(R)p Fm(\002)p Fv(\()p Fm(\000)p Fo(\031)r(;\031)r Fv(\))1971 2409 y Ft(d)2025 2383 y Fu(~)2022 2409 y Ft(k)j(\037)2164 2425 y Fv([)p Fo(E)2236 2434 y Fj(0)2270 2425 y Fm(\000)p Fv(0)p Fo(;\025)p Fv(])2445 2409 y Fu(\()p Ft(\025)2540 2424 y Fv(1)2579 2409 y Fu(\()2620 2383 y(~)2617 2409 y Ft(k)s Fu(\)\))e Ft( )2842 2368 y Fv(2)2838 2434 y(1)2881 2409 y Fu(\()p Ft(s;)17 b(t;)3090 2383 y Fu(~)3088 2409 y Ft(k)s Fu(\))p Ft(:)324 2676 y Fi(Then)32 b(the)h(r)-5 b(est)32 b(of)h(the)g(r)-5 b(e)g(asoning)31 b(fol)5 b(lows)32 b(the)h(same)f(lines)g(as)g(ab)-5 b(ove.)43 b(F)-7 b(or)32 b Ft(C)j Fu(=)27 b(0)33 b Fi(in)324 2796 y(\(3.14\),)k(we)f(use)i(the)f (ar)-5 b(gument)37 b(as)g(the)g(one)g(for)g(\(3.11\),)g(while)f(for)h Ft(C)i(>)32 b Fu(0)37 b Fi(we)g(take)324 2917 y(the)e(estimate)f (\(3.12\).)44 b(This)34 b(gives)1306 3133 y Ft(n)1364 3148 y Fm(1)1439 3133 y Fu(\()p Ft(\025)p Fu(\))28 b Fs(\025)g Ft(const)p Fu(\()p Ft(\025)22 b Fs(\000)h Ft(E)2222 3148 y Fv(0)2261 3133 y Fu(\))2329 3065 y Fj(1)p 2309 3077 69 3 v 2309 3118 a(2)p Fk(n)2388 3092 y Fv(+)2480 3065 y Fj(1)p 2453 3077 85 3 v 2453 3118 a(2)p Fk(m)2552 3133 y Ft(;)324 3336 y Fi(for)42 b Ft(\025)f Fs(&)f Ft(E)797 3351 y Fv(0)837 3336 y Fi(,)k(which)d(implies)g(\(1.12\),)i(even)e(for) h(non-de)-5 b(gener)g(ate)40 b(minimum)h Ft(n)h Fu(=)324 3456 y Ft(m)28 b Fu(=)f(1)35 b Fi(in)g(\(3.14\).)324 3743 y Fe(3.2)135 b(An)52 b(example)i(of)g(\014nite)f(critical)h (densit)l(y)g(for)f(non-zero)630 3892 y(uniform)45 b(magnetic)h (\014eld)324 4077 y Fu(The)35 b(previous)g(subsection)h(sho)m(w)m(ed)g (that)e(the)g(Bose)h(condensate)g(can)f(b)s(e)g(destro)m(y)m(ed)324 4197 y(b)m(y)f(turning)g(on)f(a)g(no)g(matter)g(ho)m(w)h(w)m(eak)h (constan)m(t)f(magnetic)g(\014eld.)44 b(Here)33 b(w)m(e)h(w)m(an)m(t) 324 4318 y(to)39 b(sho)m(w)j(that)d(this)i(is)g(not)f(alw)m(a)m(ys)h (true.)66 b(Let)41 b(dimension)g Ft(d)f Fu(=)g(3.)66 b(W)-8 b(e)40 b(c)m(ho)s(ose)h(in)324 4438 y(this)e(subsection)i(the)f (gauge)f Fw(a)1511 4453 y Fv(0)1551 4438 y Fu(\()p Ft(x)1644 4453 y Fv(1)1683 4438 y Ft(;)17 b(x)1782 4453 y Fv(2)1822 4438 y Fu(\))38 b(=)h(1)p Ft(=)p Fu(2\()p Fs(\000)p Ft(x)2330 4453 y Fv(2)2370 4438 y Ft(;)17 b(x)2469 4453 y Fv(1)2508 4438 y Ft(;)g Fu(0\))38 b(and)i(w)m(e)g(construct)g(an)324 4558 y(external)g(p)s(erio)s(dic)g(p)s(oten)m(tial,)h(whic)m(h)g(dep)s (ends)g(on)e(all)g(three)h(v)-5 b(ariables)40 b(suc)m(h)h(that)324 4679 y(the)33 b(critical)g(densit)m(y)i(b)s(ecomes)f(\014nite.)470 4799 y(W)-8 b(e)33 b(assume)h(that)f(the)g(external)g(p)s(oten)m(tial)g (has)g(the)g(follo)m(wing)g(form:)1162 5002 y Ft(V)1219 5017 y Fo(\017)1252 5002 y Fu(\()p Fw(x)p Fu(\))28 b(=)f Ft(\017)c Fs(\001)e Fu([)p Ft(v)1703 5017 y Fv(1)1743 5002 y Fu(\()p Ft(x)1836 5017 y Fv(1)1876 5002 y Fu(\))h(+)g Ft(v)2081 5017 y Fv(2)2121 5002 y Fu(\()p Ft(x)2214 5017 y Fv(2)2253 5002 y Fu(\)])h(+)f Ft(v)2486 5017 y Fv(3)2525 5002 y Fu(\()p Ft(x)2618 5017 y Fv(3)2658 5002 y Fu(\))p Ft(;)589 b Fu(\(3.15\))1894 5251 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 324 548 a Fu(where)36 b Ft(\017)d(>)g Fu(0)i(and)h(small,)h(eac)m(h)f(of)f(the)h(functions)g Fs(f)p Ft(v)2379 563 y Fo(j)2416 548 y Fs(g)2466 512 y Fv(3)2466 573 y Fo(j)t Fv(=1)2628 548 y Fu(is)g(smo)s(oth)f Fp(Z)p Fs(\000)p Fu(p)s(erio)s(dic)324 668 y(p)s(oten)m(tial,)e(and)f (w)m(e)i(also)f(supp)s(ose)h(that)e(neither)i(one)e(of)g Ft(v)2537 683 y Fv(1)2610 668 y Fu(and)g Ft(v)2846 683 y Fv(2)2918 668 y Fu(is)h(constan)m(t.)470 789 y(T)-8 b(ak)m(e)39 b(the)f(magnetic)g(\014eld)g(in)m(tensit)m(y)i Ft(!)f Fu(=)d(2)p Ft(\031)t Fu(,)i(then)g(the)g(\\bulk")g(Hamiltonian) 324 909 y(can)33 b(b)s(e)f(written)i(as)678 1107 y Ft(h)734 1122 y Fm(1)837 1107 y Fu(=)28 b(\()p Fs(\000)p Ft(i)p Fs(r)1172 1122 y Ff(x)1242 1107 y Fs(\000)22 b Fu(2)p Ft(\031)t Fw(a)1503 1122 y Fv(0)1543 1107 y Fu(\()p Ft(x)1636 1122 y Fv(1)1676 1107 y Ft(;)17 b(x)1775 1122 y Fv(2)1814 1107 y Fu(\)\))1890 1066 y Fv(2)1952 1107 y Fu(+)22 b Ft(V)2107 1122 y Fo(\017)2167 1107 y Fu(=)27 b Ft(h)2326 1122 y Fo(\017)2381 1107 y Fs(\012)c Fw(1)f Fu(+)g Fw(1)g Fs(\012)h Ft(h)2891 1122 y Fv(3)2931 1107 y Ft(;)354 b Fu(\(3.16\))324 1305 y(where)30 b(the)g(op)s(erator)f Ft(h)1213 1320 y Fo(\017)1273 1305 y Fu(=)f(\()p Fs(\000)p Ft(i)p Fs(r)16 b(\000)g Fu(2)p Ft(\031)t Fw(a)1879 1320 y Fv(0)1919 1305 y Fu(\))1957 1268 y Fv(2)2012 1305 y Fu(+)g Ft(\017)p Fu(\()p Ft(v)2228 1320 y Fv(1)2282 1305 y Fu(+)g Ft(v)2421 1320 y Fv(2)2460 1305 y Fu(\))30 b(liv)m(es)h(in)e Ft(L)2918 1268 y Fv(2)2958 1305 y Fu(\()p Fp(R)3068 1268 y Fv(2)3107 1305 y Fu(\))h(while)g(the)324 1425 y(op)s(erator)i Ft(h)773 1440 y Fv(3)840 1425 y Fu(=)c Fs(\000)p Ft(d)1072 1389 y Fv(2)1111 1425 y Ft(=dx)1266 1389 y Fv(2)1266 1450 y(3)1328 1425 y Fu(+)22 b Ft(v)1473 1440 y Fv(3)1545 1425 y Fu(liv)m(es)34 b(in)f Ft(L)1942 1389 y Fv(2)1982 1425 y Fu(\()p Fp(R)p Fu(\).)470 1545 y(First,)j(let)g(us)g(in)m(tro)s (duce)g(some)g(notation.)52 b(W)-8 b(e)36 b(write)g(an)f(arbitrary)g(v) m(ector)i Fw(x)32 b Fs(2)324 1666 y Fp(R)396 1630 y Fv(3)482 1666 y Fu(as)47 b Fw(x)53 b Fu(=)f(\()p 894 1611 56 4 v Ft(x;)17 b(x)1048 1681 y Fv(3)1088 1666 y Fu(\))47 b(where)p 1469 1611 V 48 w Ft(x)53 b Fu(:=)f(\()p Ft(x)1825 1681 y Fv(1)1865 1666 y Ft(;)17 b(x)1964 1681 y Fv(2)2003 1666 y Fu(\).)87 b(W)-8 b(e)47 b(often)g(use)h(the)g(notation)e(\004)53 b(=)324 1786 y(\()p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2\).)54 b(The)38 b(elemen)m(tary)g(cell)g(is)f(\012) e(=)f(\004)2163 1750 y Fv(3)2239 1786 y Fu(and)j(the)g(one)g(of)f(the)h (dual)g(lattice)324 1906 y(\012)394 1870 y Fm(\003)466 1906 y Fu(is)c(giv)m(en)h(b)m(y)1282 2027 y(\012)1352 1986 y Fm(\003)1420 2027 y Fu(:=)28 b Fs(f)p Fu(2)p Ft(\031)t Fn(\030)r Ft(;)114 b Fn(\030)30 b Fs(2)e Fu(\004)2143 1986 y Fv(3)2183 2027 y Fs(g)f Fu(=)h(2)p Ft(\031)t Fu(\004)2537 1986 y Fv(3)2576 2027 y Ft(:)470 2192 y Fu(According)51 b(to)f(App)s(endix)i(2,)j(the)c(op)s(erator)e Ft(h)2336 2207 y Fm(1)2462 2192 y Fu(is)i(unitarily)g(equiv)-5 b(alen)m(t)52 b(to)324 2232 y Fl(R)371 2347 y Fv(\004)418 2328 y Fj(3)473 2312 y Ft(h)p Fu(\()p Fn(\030)s Fu(\))p Ft(d)p Fn(\030)r Fu(,)33 b(where)h(the)f(\014b)s(er)g(op)s(erator)e (can)i(b)s(e)g(further)g(written)g(as)1300 2521 y Ft(h)p Fu(\()p Fn(\030)s Fu(\))27 b(=)h Ft(h)1673 2536 y Fo(\017)1706 2521 y Fu(\()p 1744 2440 48 4 v Ft(\030)t Fu(\))22 b Fs(\012)h Fw(1)f Fu(+)g Fw(1)g Fs(\012)h Ft(h)2361 2536 y Fv(3)2400 2521 y Fu(\()p Ft(\030)2481 2536 y Fv(3)2520 2521 y Fu(\))p Ft(:)324 2719 y Fu(Here)40 b(the)f(op)s(erators)g Ft(h)1229 2734 y Fo(\017)1262 2719 y Fu(\()p 1300 2638 V Ft(\030)t Fu(\))g(=)f Ft(h)1594 2734 y Fv(0)1634 2719 y Fu(\()p 1672 2638 V Ft(\030)t Fu(\))27 b(+)f Ft(\017V)c Fu(\()p 2042 2664 56 4 v Ft(x)q Fu(\),)40 b Ft(h)2259 2734 y Fv(0)2299 2719 y Fu(\()p 2337 2638 48 4 v Ft(\030)t Fu(\))f(:=)g([)p Fs(\000)p Ft(i)p Fs(r)p 2823 2697 44 3 v 15 x Fo(x)2894 2719 y Fs(\000)27 b Fw(a)p Fu(\()p 3090 2664 56 4 v Ft(x)q Fu(\))f(+)h Fw(k)p Fu(\()p 3410 2638 48 4 v Ft(\030)t Fu(\)])3522 2683 y Fv(2)324 2840 y Fu(liv)m(e)34 b(in)e Ft(L)682 2803 y Fv(2)722 2840 y Fu(\(\004)825 2803 y Fv(2)865 2840 y Fu(\))g(with)h(\\magnetic")g(p)s (erio)s(dic)g(b)s(oundary)g(conditions)g(\(see)h(App)s(endix)324 2960 y(2)e(for)g(de\014nition\),)i(and)e Ft(h)1298 2975 y Fv(3)1338 2960 y Fu(\()p Ft(\030)1419 2975 y Fv(3)1458 2960 y Fu(\))c(=)f(\()p Fs(\000)p Ft(id=dx)1981 2975 y Fv(3)2043 2960 y Fu(+)22 b Ft(\030)2184 2975 y Fv(3)2223 2960 y Fu(\))2261 2924 y Fv(2)2323 2960 y Fu(+)g Ft(v)2468 2975 y Fv(3)2507 2960 y Fu(\()p Ft(x)2600 2975 y Fv(3)2640 2960 y Fu(\))32 b(in)h Ft(L)2890 2924 y Fv(2)2930 2960 y Fu(\(\004\).)470 3080 y(Recall)f(that)e Fw(a)p Fu(\()p 1064 3025 56 4 v Ft(x)q Fu(\))e(=)f(2)p Ft(\031)t Fw(a)1451 3095 y Fv(0)1491 3080 y Fu(\()p 1529 3025 V Ft(x)p Fu(\))k(and)g Fw(k)p Fu(\()p 1938 2999 48 4 v Ft(\030)5 b Fu(\))28 b(=)f(2)p Ft(\031)t Fu(\()p Fw(e)2352 3095 y Fv(1)2391 3080 y Ft(\030)2434 3095 y Fv(1)2492 3080 y Fu(+)19 b Fw(e)2638 3095 y Fv(2)2678 3080 y Ft(\030)2721 3095 y Fv(2)2760 3080 y Fu(\).)43 b(If)31 b Ft(f)41 b Fu(is)32 b(a)f Ft(C)3307 3044 y Fm(1)3300 3105 y Fv(0)3381 3080 y Fu(\()p Fp(R)p Fu(\)-)324 3201 y(function)i(then)g(the)g(in)m(tegral) g(k)m(ernel)h(of)e Ft(f)11 b Fu(\()p Ft(h)p Fu(\()p Fn(\030)s Fu(\)\))32 b(is)h(giv)m(en)h(b)m(y)423 3416 y Ft(f)471 3451 y Fo(h)p Fv(\()p Fn(\030)s Fv(\))624 3416 y Fu(\()p 662 3361 56 4 v Ft(x)q(;)17 b(x)817 3431 y Fv(3)856 3416 y Fu(;)p 900 3361 V 17 w Ft(x)955 3375 y Fm(0)979 3416 y Ft(;)g(x)1078 3375 y Fm(0)1078 3440 y Fv(3)1117 3416 y Fu(\))84 b(=)1410 3321 y Fl(X)p 1399 3501 41 3 v 1399 3538 a Fo(\015)t Fm(2)p Fh(Z)1530 3519 y Fj(2)1591 3321 y Fl(X)1581 3533 y Fo(\015)1617 3542 y Fj(3)1652 3533 y Fm(2)p Fh(Z)1760 3416 y Ft(e)1805 3375 y Fm(\000)p Fo(i\036)p Fv(\()p 1953 3337 40 3 v Fo(x)q(;)p 2014 3337 41 3 v(\015)s Fv(\))p Fm(\000)p Fo(ib)p Fv(\()p 2216 3337 V Fo(\015)6 b Fv(\))p Fm(\000)p Fv(2)p Fo(\031)r(i)p 2442 3320 34 3 v(\030)s Fm(\001)p Fv(\()p 2523 3337 40 3 v Fo(x)p Fv(+)p 2618 3337 41 3 v Fo(\015)t Fm(\000)p 2713 3337 40 3 v Fo(x)2752 3351 y Fg(0)2775 3375 y Fv(\))2806 3416 y Ft(e)2851 3375 y Fm(\000)p Fv(2)p Fo(\031)r(i\030)3039 3384 y Fj(3)3074 3375 y Fv(\()p Fo(x)3141 3384 y Fj(3)3176 3375 y Fv(+)p Fo(\015)3267 3384 y Fj(3)3302 3375 y Fm(\000)p Fo(x)3397 3351 y Fg(0)3397 3395 y Fj(3)3431 3375 y Fv(\))1238 3672 y Fs(\002)84 b Ft(f)1447 3687 y Fo(h)1488 3695 y Fg(1)1557 3672 y Fu(\()p 1595 3618 56 4 v Ft(x)22 b Fu(+)p 1770 3618 57 4 v 22 w Ft(\015)5 b(;)17 b(x)1925 3687 y Fv(3)1987 3672 y Fu(+)22 b Ft(\015)2136 3687 y Fv(3)2175 3672 y Fu(;)p 2219 3618 56 4 v 17 w Ft(x)2274 3631 y Fm(0)2297 3672 y Ft(;)17 b(x)2396 3631 y Fm(0)2396 3697 y Fv(3)2436 3672 y Fu(\))p Ft(;)811 b Fu(\(3.17\))324 3870 y(for)44 b(ev)m(ery)i Fw(x)p Ft(;)17 b Fw(x)916 3834 y Fm(0)987 3870 y Fs(2)49 b Fu(\012,)f(where)d Ft(\036)p Fu(\()p 1636 3815 V Ft(x)q(;)p 1736 3815 57 4 v 17 w(\015)t Fu(\))j(=)g Ft(\031)t Fu(\()p Ft(x)2153 3885 y Fv(2)2193 3870 y Ft(n)30 b Fs(\000)h Ft(x)2444 3885 y Fv(1)2484 3870 y Ft(m)p Fu(\))44 b(and)h Ft(b)p Fu(\()p 2932 3815 V Ft(\015)5 b Fu(\))48 b(=)g Ft(\031)t(mn)d Fu(for)p 324 3936 V 324 3990 a Ft(\015)50 b Fu(=)45 b Ft(m)p Fw(e)682 4005 y Fv(1)751 3990 y Fu(+)29 b Ft(n)p Fw(e)965 4005 y Fv(2)1005 3990 y Fu(,)45 b(see)f(App)s(endix)g(2.)74 b(Notice)44 b(that)e(the)h(third)h(co)s(ordinate)e(is)i(not)324 4111 y(in\015uenced)25 b(b)m(y)e(the)h(magnetic)f(\014eld,)j(while)e (the)f(\014rst)g(t)m(w)m(o)h(co)s(ordinates)f(are)g(essen)m(tially)324 4231 y(treated)35 b(in)h(the)f(App)s(endix)i(2;)f(see)g(for)f(instance) h(\(5.47\).)51 b(Then)36 b(b)m(y)g(in)m(tegrating)f(the)324 4352 y(trace)e(of)f Ft(f)727 4387 y Fo(h)p Fv(\()p Fn(\030)s Fv(\))913 4352 y Fu(with)h(resp)s(ect)h(to)e Fn(\030)j Fu(w)m(e)f(obtain)592 4492 y Fl(Z)647 4717 y Fv(\004)694 4698 y Fj(3)750 4627 y Fu(T)-8 b(r)28 b Ft(f)926 4662 y Fo(h)p Fv(\()p Fn(\030)s Fv(\))1079 4627 y Ft(d)p Fn(\030)i Fu(=)1315 4492 y Fl(Z)1370 4717 y Fv(\004)1417 4698 y Fj(3)1472 4492 y Fl(Z)1528 4717 y Fv(\012)1600 4627 y Ft(f)1648 4662 y Fo(h)p Fv(\()p Fn(\030)r Fv(\))1801 4627 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)p Fu(\))p Ft(d)p Fw(x)p Ft(d)p Fn(\030)30 b Fu(=)2384 4492 y Fl(Z)2440 4717 y Fv(\012)2511 4627 y Ft(f)2559 4642 y Fo(h)2600 4650 y Fg(1)2669 4627 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)p Fu(\))p Ft(d)p Fw(x)p Ft(:)268 b Fu(\(3.18\))324 4882 y(No)m(w,)33 b(since)i(w)m(e)f(put)f(the)g(magnetic)g(\015ux)h(through) f(\004)2348 4846 y Fv(2)2420 4882 y Fu(to)g(b)s(e)g(exactly)h(2)p Ft(\031)t Fu(,)f(all)f(eigen-)324 5002 y(v)-5 b(alues)26 b(of)f Ft(h)767 5017 y Fo(\017)800 5002 y Fu(\()p 838 4921 48 4 v Ft(\030)t Fu(\))h(are)f(simple)i(and)f(b)s(elong)f(to)g(an) h(in)m(terv)-5 b(al)26 b(of)f(width)h(of)f(order)h Ft(\017)g Fu(around)1894 5251 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 324 548 a Fu(the)39 b(old)f(Landau)h(lev)m(els) i(due)e(to)f(the)h(analysis)h(of)e(App)s(endix)i(2.)62 b(Denote)38 b(them)i(b)m(y)324 668 y(\(see)33 b(also)g(\(5.54\)\))1155 876 y Fs(f)p Fu(2)p Ft(\031)t Fu(\(2)p Ft(n)22 b Fu(+)g(1\))g(+)g Ft(\017a)1875 891 y Fo(n)1922 876 y Fu(\()p Ft(\017;)p 2043 795 48 4 v 17 w(\030)5 b Fu(\))p Ft(;)p 2270 795 V 114 w(\030)32 b Fs(2)c Fu(\004)2504 835 y Fv(2)2543 876 y Fs(g)2593 891 y Fo(n)p Fm(\025)p Fv(0)3312 876 y Fu(\(3.19\))324 1083 y(and)33 b(b)m(y)g Ft(l)678 1098 y Fo(m)745 1083 y Fu(\()p Ft(\030)826 1098 y Fv(3)865 1083 y Fu(\),)g Ft(m)28 b Fs(\025)h Fu(1)j(the)h(eigen)m(v)-5 b(alues)35 b(of)d Ft(h)2103 1098 y Fv(3)2143 1083 y Fu(\()p Ft(\030)2224 1098 y Fv(3)2263 1083 y Fu(\).)44 b(Then)34 b(the)f(sp)s(ectrum)h(of)e Ft(h)3388 1098 y Fm(1)3496 1083 y Fu(is)324 1203 y(giv)m(en)h(b)m(y)h(the)f(closure)h(of)e(the)h (range)f(of)g(the)h(function)534 1411 y(2)p Ft(\031)t Fu(\(2)p Ft(n)22 b Fu(+)g(1\))g(+)g Ft(\017a)1204 1426 y Fo(n)1251 1411 y Fu(\()p Ft(\017;)p 1372 1330 V 17 w(\030)5 b Fu(\))22 b(+)g Ft(l)1607 1426 y Fo(m)1673 1411 y Fu(\()p Ft(\030)1754 1426 y Fv(3)1793 1411 y Fu(\))p Ft(;)114 b Fn(\030)31 b Fu(=)c(\()p 2195 1330 V Ft(\030)5 b(;)17 b(\030)2330 1426 y Fv(3)2368 1411 y Fu(\))28 b Fs(2)g Fu(\004)2593 1370 y Fv(3)2633 1411 y Ft(;)114 b(n)28 b Fs(\025)g Fu(0)p Ft(;)17 b(m)27 b Fs(\025)i Fu(1)p Ft(:)470 1618 y Fu(Notice)43 b(that)e(b)m(y)i(Prop)s(osition)g (2.1)e(w)m(e)i(kno)m(w)g(that)f(the)g(branc)m(h)h Ft(l)3028 1633 y Fv(1)3109 1618 y Fu(reac)m(hes)h(its)324 1738 y(minim)m(um)e(at)e(zero)h(and)g Ft(l)1336 1753 y Fv(1)1376 1738 y Fu(\()p Ft(\030)1457 1753 y Fv(3)1496 1738 y Fu(\))g Fs(\030)h Ft(l)1723 1753 y Fv(1)1763 1738 y Fu(\(0\))27 b(+)g Ft(C)7 b(\030)2143 1702 y Fv(2)2138 1763 y(3)2223 1738 y Fu(in)41 b(its)g(neigh)m(b)s(orho)s(o)s(d.)67 b(Then)42 b(the)324 1859 y(b)s(ottom)32 b(of)g(the)h(sp)s(ectrum)h Ft(\033)t Fu(\()p Ft(h)1526 1874 y Fm(1)1601 1859 y Fu(\))e(is)h(equal) h(to)1271 2066 y Ft(E)1343 2081 y Fv(0)1410 2066 y Fu(=)28 b(2)p Ft(\031)e Fu(+)c Ft(\017)17 b Fu(min)p 1798 2089 34 3 v 1798 2144 a Fo(\030)s Fm(2)p Fv(\004)1926 2125 y Fj(2)1977 2066 y Ft(a)2028 2081 y Fv(0)2068 2066 y Fu(\()p Ft(\017;)p 2189 1985 48 4 v 17 w(\030)t Fu(\))22 b(+)g Ft(l)2423 2081 y Fv(1)2463 2066 y Fu(\(0\))p Ft(:)324 2331 y Fu(Moreo)m(v)m(er,)41 b Ft(E)858 2346 y Fv(0)936 2331 y Fu(is)d(isolated)h(from)f(the)h(other)f(bands)h(if)f Ft(\017)g Fu(is)h(small)g(enough,)h(as)e(it)g(is)324 2451 y(in)33 b(the)g(nonmagnetic)g(case.)45 b(Th)m(us,)34 b(w)m(e)g(can)f(rep)s(eat)g(our)f(argumen)m(ts)i(leading)f(to)f(the)324 2572 y(estimate)41 b(of)e(the)h(in)m(tegral)g(in)g(\(2.7\))f(for)h Ft(\025)f Fs(&)h Ft(E)2203 2587 y Fv(0)2282 2572 y Fu(and)g(w)m(e)h (get)e(for)h(the)g(in)m(tegrated)324 2692 y(densit)m(y)31 b(of)d(states)i(in)f(this)g(limit)h(exp)s(onen)m(t)g(3)p Ft(=)p Fu(2,)f(pro)m(vided)h(the)f(minim)m(um)i(of)d Ft(a)3335 2707 y Fv(0)3375 2692 y Fu(\()p Ft(\017;)17 b Fs(\001)p Fu(\))324 2813 y(is)48 b(nondegenerate.)91 b(\(See)49 b(Lemma)g(3.4)e(for)h(the)g(de\014nition)h(of)f(a)f (nondegenerate)324 2933 y(minim)m(um)34 b(for)e Ft(\017)c Fu(=)g(0\).)470 3053 y(This)j(w)m(ould)h(pro)m(v)m(e)f(the)g(\014rst)g (part)f(of)f(Theorem)j(1.2.)42 b(Th)m(us)32 b(w)m(e)f(con)m(tin)m(ue)h (b)m(y)f(the)324 3174 y(follo)m(wing)i(statemen)m(t.)324 3366 y Fw(Lemma)40 b(3.4.)i Fi(Assume)35 b Ft(a)1353 3381 y Fv(0)1393 3366 y Fu(\(0)p Ft(;)17 b Fs(\001)p Fu(\))34 b Fi(has)h(a)g(nonde)-5 b(gener)g(ate)34 b(\(lo)-5 b(c)g(al\))35 b(minimum)f(at)p 3380 3285 V 36 w Ft(\030)3427 3390 y Fv(0)3495 3366 y Fs(2)324 3487 y Fu(\004)389 3451 y Fv(2)428 3487 y Fi(,)k(i.e.)52 b(ther)-5 b(e)37 b(exists)g(a)g (symmetric)g(and)f(p)-5 b(ositive)37 b(matrix)g Ft(Q)32 b Fs(2)h(M)2949 3502 y Fv(2)2988 3487 y Fu(\()p Fp(R)p Fu(\))k Fi(such)g(that)324 3607 y(for)d(smal)5 b(l)34 b Fs(j)p 766 3526 V Ft(\030)27 b Fs(\000)p 935 3526 V 22 w Ft(\030)983 3631 y Fv(0)1022 3607 y Fs(j)35 b Fi(we)f(have)645 3814 y Ft(a)696 3829 y Fv(0)736 3814 y Fu(\(0)p Ft(;)p 867 3733 V 17 w(\030)t Fu(\))27 b(=)h Ft(a)1134 3829 y Fv(0)1173 3814 y Fu(\(0)p Ft(;)p 1304 3733 V 17 w(\030)1351 3838 y Fv(0)1391 3814 y Fu(\))22 b(+)g Fs(h)p 1588 3733 V Ft(\030)k Fs(\000)p 1757 3733 V 23 w Ft(\030)1804 3838 y Fv(0)1843 3814 y Ft(;)17 b(Q)p Fu(\()p 2002 3733 V Ft(\030)27 b Fs(\000)p 2172 3733 V 23 w Ft(\030)2219 3838 y Fv(0)2258 3814 y Fu(\))p Fs(i)22 b Fu(+)g Fs(O)s Fu(\()p Fs(j)p 2603 3733 V Ft(\030)k Fs(\000)p 2772 3733 V 23 w Ft(\030)2820 3838 y Fv(0)2859 3814 y Fs(j)2887 3773 y Fv(3)2926 3814 y Fu(\))p Ft(:)321 b Fu(\(3.20\))324 4022 y Fi(Her)-5 b(e)35 b Fs(h\001)p Ft(;)17 b Fs(\001i)33 b Fi(is)i(the)g(sc)-5 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Fu(giv)m(en)g(b)m(y)f([)p Ft(D)1593 4590 y Fo(\030)1631 4575 y Fu(\()p Fs(r)1752 4590 y Fo(\030)1790 4575 y Ft(a)1841 4590 y Fv(0)1881 4575 y Fu(\)]\(0)p Ft(;)p 2077 4494 V 17 w(\030)2123 4599 y Fv(0)2163 4575 y Fu(\))h(=)g Ft(Q)e Fu(is)i(in)m(v)m(ertible.)72 b(Due)42 b(to)f(the)324 4696 y(smo)s(othness)j(prop)s(erties)g(of)f Ft(a)1495 4711 y Fv(0)1534 4696 y Fu(\()p Ft(\017;)p 1655 4615 V 17 w(\030)5 b Fu(\))42 b(with)i(resp)s(ect)g(to)f(all)f (its)i(v)-5 b(ariables,)46 b(one)d(can)324 4816 y(easily)29 b(v)m(erify)g(the)f(h)m(yp)s(otheses)i(of)d(the)h(Implicit)h(F)-8 b(unction)28 b(Theorem,)i(whic)m(h)g(\014nishes)324 4937 y(the)j(pro)s(of.)p 3491 4937 4 66 v 3495 4874 59 4 v 3495 4937 V 3553 4937 4 66 v 1894 5251 a(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 324 548 a Fw(Remark)44 b(3.5.)h Fi(.)59 b(Assume)40 b(that)g Ft(a)1687 563 y Fv(0)1727 548 y Fu(\(0)p Ft(;)17 b Fs(\001)p Fu(\))39 b Fi(has)g(a)h(\014nite)f (numb)-5 b(er)40 b(of)f(nonde)-5 b(gener)g(ate)324 668 y(lo)g(c)g(al)41 b(minima,)h(only.)64 b(Then)41 b(due)h(to)f(the)h (joint)f(analyticity)h(of)g Ft(a)2856 683 y Fv(0)2937 668 y Fi(with)f(r)-5 b(esp)g(e)g(ct)41 b(to)324 789 y(\()p Ft(\017;)p 447 708 48 4 v 17 w(\030)t Fi(\),)i(ther)-5 b(e)41 b(exists)g Ft(\017)1168 804 y Fv(0)1248 789 y Ft(>)e Fu(0)i Fi(smal)5 b(l)41 b(enough)g(such)g(that)h(for)f Fs(j)p Ft(\017)p Fs(j)e Ft(<)h(\017)2942 804 y Fv(0)2982 789 y Fi(,)j(the)e(absolute)324 909 y(minimum)29 b(of)i Ft(a)922 924 y Fv(0)961 909 y Fu(\()p Ft(\017;)17 b Fs(\001)p Fu(\))30 b Fi(wil)5 b(l)30 b(also)g(b)-5 b(e)31 b(nonde)-5 b(gener)g(ate)29 b(and)h(ther)-5 b(e)30 b(ar)-5 b(e)30 b(no)h(other)f(critic)-5 b(al)324 1029 y(p)g(oints)34 b(than)h(the)g(ones)f(given)g(by)h(L)-5 b(emma)34 b(3.4.)470 1337 y Fu(Therefore,)h(the)e(only)h(thing)f(whic)m(h)h(remains)h(to)d (b)s(e)i(studied)g(is)g(the)f(b)s(eha)m(vior)h(of)324 1457 y Ft(a)375 1472 y Fv(0)414 1457 y Fu(\(0)p Ft(;)17 b Fs(\001)p Fu(\))35 b(near)h(its)h(absolute)f(minim)m(um.)55 b(But)36 b(\(see)h(App)s(endix)h(2\),)e(w)m(e)h(ha)m(v)m(e)g(already) 324 1577 y(got)k(a)h(fairly)f(explicit)j(expression)g(for)d(it)h(in)g (\(5.55\),)h(where)g(w)m(e)g(ha)m(v)m(e)g(no)m(w)g(to)e(put)324 1698 y Ft(V)21 b Fu(\()p Ft(x)495 1713 y Fv(1)535 1698 y Ft(;)c(x)634 1713 y Fv(1)674 1698 y Fu(\))27 b(=)h Ft(v)890 1713 y Fv(1)929 1698 y Fu(\()p Ft(x)1022 1713 y Fv(1)1062 1698 y Fu(\))22 b(+)g Ft(v)1267 1713 y Fv(2)1307 1698 y Fu(\()p Ft(x)1400 1713 y Fv(1)1440 1698 y Fu(\),)32 b Ft(!)f Fu(=)d(2)p Ft(\031)36 b Fu(and)d Ft(n)28 b Fu(=)f(0.)43 b(W)-8 b(e)33 b(use)h(the)f(notation)1180 1973 y(\()t(^)-53 b Ft(v)s Fu(\))1306 1988 y Fo(\015)1378 1973 y Fu(=)1482 1837 y Fl(Z)1581 1864 y Fv(1)p Fo(=)p Fv(2)1537 2063 y Fm(\000)p Fv(1)p Fo(=)p Fv(2)1719 1973 y Ft(e)1764 1932 y Fm(\000)p Fv(2)p Fo(\031)r(i\015)t(x)2005 1973 y Ft(v)t Fu(\()p Ft(x)p Fu(\))p Ft(dx;)115 b(\015)32 b Fs(2)c Fp(Z)p Ft(;)324 2245 y Fu(for)k(the)h(discrete)h(F)-8 b(ourier)32 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y Ft(;)918 2892 y(b)959 2907 y Fv(0)p Fo(;)p Fv(2)1054 2892 y Fu(\()p 1092 2811 V Ft(\030)t Fu(\))28 b(=)f Ft(b)1349 2907 y Fv(0)p Fo(;)p Fv(2)1444 2892 y Fu(\()p Ft(\030)1525 2907 y Fv(1)1564 2892 y Fu(\))83 b(=)1853 2797 y Fl(X)1844 3009 y Fo(\015)1880 3018 y Fj(1)1915 3009 y Fm(2)p Fh(Z)2023 2892 y Ft(e)2068 2851 y Fv(2)p Fo(\031)r(i\030)2201 2860 y Fj(1)2236 2851 y Fo(\015)2272 2860 y Fj(1)2339 2892 y Ft(e)2384 2851 y Fm(\000)p Fo(\031)r(\015)2522 2827 y Fj(2)2518 2872 y(1)2557 2851 y Fo(=)p Fv(2)2631 2892 y Fu(\()t(^)-53 b Ft(v)2716 2907 y Fv(2)2756 2892 y Fu(\))2794 2907 y Fo(\015)2830 2916 y Fj(1)2869 2892 y Ft(:)324 3192 y Fu(It)42 b(is)h(easy)h(to)e(see)i(that)e(b)m(y)h(a)g(judicios)g (c)m(hoice)h(of)e Ft(v)2360 3207 y Fv(1)2442 3192 y Fu(and)g Ft(v)2688 3207 y Fv(2)2770 3192 y Fu(w)m(e)i(can)f(create)g(an)m(y)324 3313 y(pro\014le)d(w)m(e)h(w)m(an)m(t)g(for)e(the)i(functions)g Ft(b)1827 3328 y Fv(0)p Fo(;)p Fv(1)1961 3313 y Fu(and)f Ft(b)2199 3328 y Fv(0)p Fo(;)p Fv(2)2294 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Fu(\))2142 4932 y Fo(k)2185 4917 y Ft(;)44 b(x)28 b Fs(2)g Fu(\()p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2\))p Ft(:)1894 5251 y Fu(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 324 548 a Fu(Then)39 b(b)m(y)g(\(3.21\))f(it)g (follo)m(ws)h(that)e Ft(b)1699 563 y Fv(0)p Fo(;)p Fv(1)1794 548 y Fu(\()p Ft(\030)1875 563 y Fv(2)1914 548 y Fu(\))g(=)46 b(~)-58 b Ft(p)2151 563 y Fo(M)2230 548 y Fu(\()p Fs(\000)p Ft(\030)2388 563 y Fv(2)2428 548 y Fu(\).)60 b(Th)m(us,)41 b Ft(b)2875 563 y Fv(0)p Fo(;)p Fv(1)3008 548 y Fu(has)e(a)e(nonde-)324 668 y(generate)30 b(absolute)f(minim)m(um)i(at)e Fs(\000)p Ft(\030)1761 683 y Fo(M)s(;p)1891 668 y Fu(.)43 b(Similar)29 b(line)h(of)f(reasoning)g(in)m(v)m(olving)i(the)324 789 y(function)g Ft(q)j Fu(implies)d(the)g(same)g(conclusion)h(for)d Ft(b)2174 804 y Fv(0)p Fo(;)p Fv(2)2269 789 y Fu(.)43 b(This)31 b(\014nishes)h(the)f(pro)s(of)e(of)h(the)324 909 y(\014rst)j(part)f(of)g(Theorem)i(1.2.)324 1104 y Fw(Remark)j(3.6.)k Fi(By)34 b(insp)-5 b(e)g(ction)33 b(of)h(this)g(line)g(of)f(r)-5 b(e)g(asoning,)33 b(one)h(\014nds)f (that)i(ther)-5 b(e)34 b(is)324 1224 y(no)h(ne)-5 b(e)g(d)36 b(to)g(r)-5 b(estrict)36 b(the)g(p)-5 b(otential)36 b(p)-5 b(erio)g(dic)35 b(in)g(dir)-5 b(e)g(ctions)36 b Fu(\()p Ft(x)2726 1239 y Fv(1)2765 1224 y Ft(;)17 b(x)2864 1239 y Fv(2)2904 1224 y Fu(\))36 b Fi(to)g(a)g(sep)-5 b(ar)g(able)324 1344 y(form,)43 b(se)-5 b(e)42 b(\(3.15\).)67 b(In)41 b(fact.)67 b(our)43 b(pr)-5 b(o)g(of)42 b(go)-5 b(es)42 b(thr)-5 b(ough)42 b(verb)-5 b(atim)42 b(for)g Fp(Z)3157 1308 y Fv(2)3197 1344 y Fi(-p)-5 b(erio)g(dic)324 1465 y(p)g(otential)1243 1585 y Ft(\017)23 b Fs(\001)e Fu([)p Ft(v)1428 1600 y Fv(1)1468 1585 y Fu(\()p Ft(x)1561 1600 y Fv(1)1601 1585 y Fu(\))h(+)g Ft(v)1806 1600 y Fv(2)1846 1585 y Fu(\()p Ft(x)1939 1600 y Fv(2)1978 1585 y Fu(\))g(+)g Ft(\016)k Fs(\001)c Ft(v)t Fu(\()p Ft(x)2399 1600 y Fv(1)2439 1585 y Ft(;)17 b(x)2538 1600 y Fv(2)2577 1585 y Fu(\)])324 1755 y Fi(for)34 b Ft(\016)39 b Fi(smal)5 b(l)34 b(enough.)324 2086 y Fx(4)161 b(Imp)t(erfect)50 b(Bose)f(gas:)70 b(mean-\014eld)49 b(in)l(teraction)324 2305 y Fu(T)-8 b(o)31 b(discuss)j(whether)f(the)f (BEC)g(found)g(in)g(the)g(t)m(w)m(o)g(previous)h(sections)g(surviv)m (es)h(the)324 2426 y(switc)m(hing)j(on)d(of)h(a)g(particle)g(in)m (teraction,)i(w)m(e)f(consider)g(here)g(the)f(simplest)i(v)m(ersion)324 2546 y(of)32 b(it,)h(kno)m(wn)g(as)g(the)g(Mean-Field)g(\(MF\))g(in)m (teraction,)g(see)h(e.g.)44 b([Z-B].)470 2666 y(T)-8 b(o)44 b(this)h(end)g(w)m(e)g(need)g(the)g(second)g(quan)m(tized)h (form)e(of)g(the)h(in)m(teracting)g(gas)324 2787 y(Hamiltonian)33 b(in)f(the)h(b)s(oson)g(F)-8 b(o)s(c)m(k)33 b(space)g Fs(F)2020 2751 y Fo(B)2081 2787 y Fu(\()p Ft(L)2185 2751 y Fv(2)2241 2787 y Fu(\(\003)2347 2802 y Fo(L)2399 2787 y Fu(\)\):)958 2996 y Ft(H)1039 3011 y Fv(\003)1088 3022 y Fk(L)1138 2996 y Fu(\()p Ft(\026)p Fu(\))27 b(:=)h Ft(H)1512 3011 y Fv(\003)1561 3022 y Fk(L)1633 2996 y Fs(\000)23 b Ft(\026N)1870 3011 y Fv(\003)1919 3022 y Fk(L)1997 2996 y Fu(=)k Ft(T)2157 3011 y Fv(\003)2206 3022 y Fk(L)2279 2996 y Fs(\000)c Ft(\026N)2516 3011 y Fv(\003)2565 3022 y Fk(L)2637 2996 y Fu(+)f Ft(U)2801 3011 y Fv(\003)2850 3022 y Fk(L)2901 2996 y Ft(:)433 b Fu(\(4.1\))324 3206 y(Here)1391 3350 y Ft(T)1448 3365 y Fv(\003)1497 3376 y Fk(L)1575 3350 y Fu(:=)1706 3214 y Fl(Z)1761 3440 y Fv(\003)1810 3451 y Fk(L)1861 3350 y Fu(d)p Fw(x)p Ft(a)2025 3309 y Fm(\003)2065 3350 y Fu(\()p Fw(x)p Fu(\))p Ft(h)2256 3365 y Fo(L)2308 3350 y Ft(a)p Fu(\()p Fw(x)p Fu(\))867 b(\(4.2\))324 3584 y(is)31 b(the)h(kinetic-energy)h(part)e(with)g(one-particle)h(op)s (erator)e Ft(h)2621 3599 y Fo(L)2705 3584 y Fu(de\014ned)i(b)m(y)g (\(1.3\))e(and)874 3842 y Ft(U)940 3857 y Fv(\003)989 3868 y Fk(L)1068 3842 y Fu(:=)1208 3774 y(1)p 1208 3819 49 4 v 1208 3910 a(2)1284 3706 y Fl(Z)1339 3932 y Fv(\()q(\003)1416 3943 y Fk(L)1462 3932 y Fv(\))1489 3902 y Fj(2)1528 3842 y Fu(d)p Fw(x)p Fu(d)p Fw(y)35 b Ft(a)1840 3801 y Fm(\003)1879 3842 y Fu(\()p Fw(x)p Fu(\))p Ft(a)2065 3801 y Fm(\003)2105 3842 y Fu(\()p Fw(y)r Fu(\))p Ft(v)t Fu(\()p Fw(x)22 b Fs(\000)g Fw(y)r Fu(\))p Ft(a)p Fu(\()p Fw(y)r Fu(\))p Ft(a)p Fu(\()p Fw(x)p Fu(\))p Ft(;)350 b Fu(\(4.3\))324 4121 y(is)22 b(the)g(in)m(teraction)h(de\014ned)g(b)m(y)g(the)f(t)m(w)m (o-b)s(o)s(dy)g(p)s(oten)m(tial)g Ft(v)t Fu(\()p Fw(x)p Fs(\000)p Fw(y)r Fu(\),)i(where)f Ft(a)3173 4084 y Fm(\003)3229 4121 y Fu(\()p Fw(x)p Fu(\))17 b Ft(;)g(a)p Fu(\()p Fw(x)p Fu(\))324 4241 y(are)32 b(usual)i Fi(b)-5 b(oson-\014eld)41 b Fu(op)s(erators,)1427 4493 y Ft(N)1505 4508 y Fv(\003)1554 4519 y Fk(L)1632 4493 y Fu(:=)1762 4358 y Fl(Z)1818 4583 y Fv(\003)1867 4594 y Fk(L)1917 4493 y Fu(d)p Fw(x)17 b Ft(a)2098 4452 y Fm(\003)2138 4493 y Fu(\()p Fw(x)p Fu(\))p Ft(a)p Fu(\()p Fw(x)p Fu(\))902 b(\(4.4\))324 4761 y(is)33 b(the)g(particle-n)m(um)m(b)s(er)h(op)s(erator,)e(and)h Ft(\026)f Fu(is)h(the)g(c)m(hemical)h(p)s(oten)m(tial.)470 4882 y(T)-8 b(o)34 b(ensure)h(the)f(existence)j(of)c(the)h(thermo)s (dynamics)i(of)d(the)i(Bose)f(gas)g(\(4.1\))f(for)324 5002 y(all)24 b(parameters)i(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))24 b(of)g(the)h(grand-canonical)g(ensem)m(ble,)k(it)24 b(used)i(to)f(supp)s(ose)h(that)1894 5251 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 324 548 a Fu(the)32 b(in)m(teraction)h Ft(v)t Fu(\()p Fw(x)21 b Fs(\000)h Fw(y)r Fu(\))32 b(is)g(sup)s (erstable)i([Ru].)43 b(F)-8 b(or)32 b(example,)h(let)g(the)f(pair)g(in) m(ter-)324 668 y(action)39 b(p)s(oten)m(tial)h Ft(v)t Fu(\()p Fw(x)p Fu(\))e(=)h Ft(v)t Fu(\()p Fs(\000)p Fw(x)p Fu(\))h(b)s(e)f(a)g(real,)i(non-negativ)m(e)f(con)m(tin)m(uous)h (function)324 789 y(from)32 b Ft(L)620 753 y Fv(1)660 789 y Fu(\()p Fp(R)770 753 y Fo(d)810 789 y Fu(\).)44 b(Since)33 b Ft(v)f Fs(2)c Ft(L)1412 753 y Fv(1)1452 789 y Fu(\()p Fp(R)1562 753 y Fo(d)1602 789 y Fu(\),)33 b(the)g(F)-8 b(ourier)32 b(transform)k(^)-52 b Ft(v)s Fu(\()p Fw(q)p Fu(\))33 b(exists,)h(and)788 1025 y(^)-53 b Ft(v)t Fu(\()p Fw(0)p Fu(\))28 b(=)1098 889 y Fl(Z)1154 1115 y Fh(R)1202 1096 y Fk(d)1242 1025 y Fu(d)p Fw(x)33 b Ft(v)t Fu(\()p Fw(x)p Fu(\))27 b Ft(>)h Fu(0)97 b(with)102 b(^)-53 b Ft(v)t Fu(\()p Fw(0)p Fu(\))28 b Fs(\025)j Fu(^)-52 b Ft(v)s Fu(\()p Fw(q)p Fu(\))p Ft(;)115 b Fw(q)28 b Fs(2)g Fp(R)3034 984 y Fo(d)3074 1025 y Ft(:)260 b Fu(\(4.5\))324 1265 y(It)38 b(is)h(kno)m(wn)g([Ru])g(that)f(the)g (corresp)s(onding)h(in)m(teraction)g(is)g(sup)s(erstable,)i(i.e.)61 b(the)324 1385 y Ft(n)p Fu(-b)s(o)s(dy)32 b(p)s(oten)m(tial)h (satis\014es)h(the)f(inequalit)m(y)1280 1521 y Fl(X)1202 1732 y Fv(1)p Fm(\024)p Fo(i)f Fu(0,)i Ft(B)j Fs(\025)c Fu(0,)g(for)f(all)g Ft(n)g Fs(2)h Fp(N)p Fu(,)h Fw(x)2578 1914 y Fo(i)2606 1899 y Ft(;)17 b Fw(x)2709 1914 y Fo(j)2786 1899 y Fs(2)41 b Fu(\003)2961 1914 y Fo(L)3052 1899 y Fu(and)f Ft(L)h Fu(large)324 2020 y(enough)32 b(whic)m(h)h(implies)g(that)e(the)h (thermo)s(dynamic)h(p)s(oten)m(tials)f(exist)h(for)e(all)h(v)-5 b(alues)324 2140 y(of)32 b(the)h(c)m(hemical)h(p)s(oten)m(tial)f Ft(\026)p Fu(.)470 2261 y(T)-8 b(o)33 b(in)m(tro)s(duce)g(the)g(MF)g (in)m(teraction)g(consider)h(the)f(scaled)h(p)s(oten)m(tial)f(:)1429 2448 y Ft(v)1476 2463 y Fo(\025)1521 2448 y Fu(\()p Ft(x)p Fu(\))28 b(:=)g Ft(\025)1868 2407 y Fo(d)1908 2448 y Ft(v)t Fu(\()p Ft(\025x)p Fu(\))p Ft(;)17 b(\025)28 b Fs(\025)g Fu(0)p Ft(:)904 b Fu(\(4.7\))324 2636 y(Denote)32 b(b)m(y)780 2863 y Ft(p)829 2878 y Fv(\003)878 2889 y Fk(L)945 2782 y Fl(\002)986 2863 y Ft(H)1075 2821 y Fo(\025)1067 2887 y Fv(\003)1116 2898 y Fk(L)1167 2782 y Fl(\003)1225 2863 y Fu(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))26 b(:=)1725 2795 y(1)p 1632 2840 236 4 v 1632 2931 a Ft(\014)6 b Fs(j)p Fu(\003)1789 2946 y Fo(L)1840 2931 y Fs(j)1894 2863 y Fu(ln)17 b Ft(T)d(r)2107 2881 y Fm(F)2164 2862 y Fk(B)2217 2881 y Fv(\()p Fo(L)2292 2862 y Fj(2)2327 2881 y Fv(\(\003)2403 2892 y Fk(L)2449 2881 y Fv(\))q(\))2508 2863 y Ft(e)2553 2812 y Fm(\000)p Fo(\014)s Fv(\()p Fo(H)2741 2789 y Fk(\025)2736 2835 y Fj(\003)2777 2852 y Fk(L)2829 2812 y Fm(\000)p Fo(\026N)2982 2823 y Fj(\003)3023 2840 y Fk(L)3074 2812 y Fv(\))3361 2863 y Fu(\(4.8\))324 3114 y(the)33 b(grand-canonical)f(pressure)i(de\014ned)g(b)m(y)f(the)g (Hamiltonian)f Ft(H)2864 3078 y Fo(\025)2856 3139 y Fv(\003)2905 3150 y Fk(L)2988 3114 y Fu(with)h(the)f(t)m(w)m(o-)324 3234 y(b)s(o)s(dy)g(in)m(teraction)i Ft(v)1108 3249 y Fo(\025)1153 3234 y Fu(\()p Ft(x)p Fu(\).)44 b(Then)34 b(the)f(limit)1122 3422 y(lim)1116 3484 y Fo(\025)p Fm(!)p Fv(0)1307 3422 y Fu(lim)1280 3484 y Fo(L)p Fm(!1)1486 3422 y Ft(p)1535 3437 y Fv(\003)1584 3448 y Fk(L)1651 3341 y Fl(\002)1692 3422 y Ft(H)1781 3381 y Fo(\025)1773 3446 y Fv(\003)1822 3457 y Fk(L)1872 3341 y Fl(\003)1931 3422 y Fu(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))26 b(=)i Ft(p)2350 3381 y Fo(v)r(dW)2503 3422 y Fu(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))p Ft(;)591 b Fu(\(4.9\))324 3645 y(exists)39 b(and)e(it)h(is)g(kno)m(wn)g(as)g(the)f Fi(van)i(der)g (Waals)g(limit)47 b Fu([deSm-Z].)37 b(If)h(one)f(c)m(ho)s(oses)324 3765 y(the)c(scaled)h(t)m(w)m(o-b)s(o)s(dy)e(p)s(oten)m(tial)h(\(4.7\)) f(in)h(the)g(form)1573 3953 y Ft(v)1620 3968 y Fo(L)1672 3953 y Fu(\()p Fw(x)p Fu(\))28 b(:=)g Ft(g)t Fs(j)p Fu(\003)2113 3968 y Fo(L)2163 3953 y Fs(j)2191 3912 y Fm(\000)p Fv(1)2285 3953 y Ft(;)1000 b Fu(\(4.10\))324 4141 y(Then)33 b(the)h(limit)1230 4261 y(lim)1203 4323 y Fo(L)p Fm(!1)1408 4261 y Ft(p)1457 4276 y Fv(\003)1506 4287 y Fk(L)1573 4150 y Fl(h)1620 4261 y Ft(H)1709 4215 y Fo(\025)1750 4226 y Fk(L)1701 4288 y Fv(\003)1750 4299 y Fk(L)1801 4150 y Fl(i)1864 4261 y Fu(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))27 b(=)g Ft(p)2283 4220 y Fo(M)7 b(F)2417 4261 y Fu(\()p Ft(\014)f(;)17 b(\026)p Fu(\))p Ft(;)628 b Fu(\(4.11\))324 4456 y(exists)26 b(and)e(it)g(is)g(kno)m(wn)h(as)g(the)f Fi(Me)-5 b(an-Field)26 b(limit)33 b Fu([deSm-Z].)24 b(Notice)h(that)f(b)m(y)h(virtue)324 4576 y(of)32 b(\(4.4\))g(the)h(in)m(teraction)g(\(4.3\))f(in)h(this)g (case)h(tak)m(es)g(the)f(form)f(:)342 4813 y Ft(U)418 4771 y Fo(M)7 b(F)408 4837 y Fv(\003)457 4848 y Fk(L)580 4813 y Fu(=)694 4745 y(1)p 694 4790 49 4 v 694 4881 a(2)769 4677 y Fl(Z)824 4902 y Fv(\()q(\003)901 4913 y Fk(L)947 4902 y Fv(\))974 4873 y Fj(2)1013 4813 y Fu(d)p Fw(x)p Fu(d)p Fw(y)35 b Ft(a)1325 4771 y Fm(\003)1364 4813 y Fu(\()p Fw(x)p Fu(\))p Ft(a)1550 4771 y Fm(\003)1590 4813 y Fu(\()p Fw(y)r Fu(\))p Ft(v)1774 4828 y Fo(L)1826 4813 y Fu(\()p Fw(x)22 b Fs(\000)h Fw(y)r Fu(\))p Ft(a)p Fu(\()p Fw(y)r Fu(\))p Ft(a)p Fu(\()p Fw(x)p Fu(\))k(=)2659 4745 y(1)p 2659 4790 V 2659 4881 a(2)2790 4745 y Ft(g)p 2727 4790 176 4 v 2727 4881 a Fs(j)p Fu(\003)2823 4896 y Fo(L)2875 4881 y Fs(j)2913 4813 y Ft(N)2991 4828 y Fv(\003)3040 4839 y Fk(L)3090 4813 y Fu(\()p Ft(N)3206 4828 y Fv(\003)3255 4839 y Fk(L)3328 4813 y Fs(\000)22 b Ft(I)8 b Fu(\))p Ft(;)3312 5002 y Fu(\(4.12\))1894 5251 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 324 548 a Fu(i.e.,)29 b(the)g(corresp)s(onding) f(Hamiltonian)h(in)f(the)g(F)-8 b(o)s(c)m(k)28 b(space)h Fs(F)2683 512 y Fo(B)2743 548 y Fu(\()p Ft(L)2847 512 y Fv(2)2903 548 y Fu(\(\003)3009 563 y Fo(L)3061 548 y Fu(\)\))f(is)g(de\014ned)324 668 y(b)m(y)33 b Ft(H)548 632 y Fo(M)7 b(F)540 693 y Fv(\003)589 704 y Fk(L)709 668 y Fu(:=)28 b Ft(T)897 683 y Fv(\003)946 694 y Fk(L)1018 668 y Fu(+)22 b Ft(U)1192 632 y Fo(M)7 b(F)1182 693 y Fv(\003)1231 704 y Fk(L)1326 668 y Fu(.)470 789 y(Since)25 b(the)g(sp)s(ectrum)g(of)f(the)g(one-particle)g(kinetic-energy)j(op)s (erator)c(is)h(suc)m(h)i(that)324 909 y(inf)d Ft(\033)t Fu(\()p Ft(h)611 924 y Fm(1)686 909 y Fu(\))39 b(=)f Ft(E)949 924 y Fv(0)988 909 y Fu(,)j(see)f(Section)g(1,)g(thermo)s (dynamic)g(b)s(eha)m(viour)g(of)e(the)i(b)s(oson)f(gas)324 1029 y(\(4.1\))32 b(is)h Ft(E)727 1044 y Fv(0)767 1029 y Fu(-dep)s(enden)m(t.)324 1228 y Fw(Lemma)c(4.1)e(\(Thermo)s(dynamic)i (F)-9 b(unctions\).)35 b Fi(The)26 b(gr)-5 b(and-c)g(anonic)g(al)25 b(pr)-5 b(essur)g(e)324 1349 y Ft(p)373 1312 y Fo(M)7 b(F)r(;E)567 1321 y Fj(0)604 1349 y Fu(\()p Ft(\014)f(;)17 b(\026)p Fu(\))40 b Fi(\(4.11\))g(of)g(the)h(M-F)g(b)-5 b(oson)40 b(gas)g(\(4.1\))g(exists)h(for)g(al)5 b(l)40 b Ft(\014)k Fs(\025)c Fu(0)p Ft(;)17 b(\026)38 b Fs(2)h Fp(R)324 1469 y Fi(and)34 b(is)h(given)f(by)h(the)g(L)-5 b(e)g(gendr)g(e)34 b(tr)-5 b(ansformation:)1092 1684 y Ft(p)1141 1642 y Fo(M)7 b(F)r(;E)1335 1651 y Fj(0)1373 1684 y Fu(\()p Ft(\014)f(;)17 b(\026)p Fu(\))26 b(=)i(sup)1753 1764 y Fo(\032)p Fm(\025)p Fv(0)1906 1603 y Fl(\000)1952 1684 y Ft(\026\032)22 b Fs(\000)h Ft(f)2242 1642 y Fo(M)7 b(F)r(;E)2436 1651 y Fj(0)2473 1684 y Fu(\()p Ft(\014)f(;)17 b(\032)p Fu(\))2704 1603 y Fl(\001)2766 1684 y Ft(;)519 b Fu(\(4.13\))324 1971 y Fi(wher)-5 b(e)39 b(the)h(c)-5 b(anonic)g(al)38 b(fr)-5 b(e)g(e-ener)g(gy)39 b(density)h Ft(f)2100 1935 y Fo(M)7 b(F)r(;E)2294 1944 y Fj(0)2331 1971 y Fu(\()p Ft(\014)f(;)17 b(\032)p Fu(\))40 b Fi(at)g(inverse)f (temp)-5 b(er)g(atur)g(e)324 2091 y Ft(\014)40 b Fi(and)35 b(density)f Ft(\032)h Fi(is)g(given)f(by)1180 2323 y Ft(f)1239 2282 y Fo(M)7 b(F)r(;E)1433 2291 y Fj(0)1471 2323 y Fu(\()p Ft(\014)f(;)17 b(\032)p Fu(\))27 b(=)g Ft(f)1891 2282 y Fo(P)10 b(B)s(G;E)2129 2291 y Fj(0)2167 2323 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\))22 b(+)2528 2256 y Ft(g)p 2528 2300 51 4 v 2529 2392 a Fu(2)2588 2323 y Ft(\032)2638 2282 y Fv(2)2678 2323 y Ft(:)607 b Fu(\(4.14\))324 2576 y Fi(Her)-5 b(e)41 b Ft(f)622 2539 y Fo(P)10 b(B)s(G;E)860 2548 y Fj(0)898 2576 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\))41 b Fi(is)f(the)i(fr)-5 b(e)g(e-ener)g(gy)40 b(density)h(of)g(the)g(PBG,)g(c)-5 b(orr)g(esp)g(onding)39 b(to)324 2696 y(\(4.2\).)324 2895 y(Pr)-5 b(o)g(of)p Fu(.)79 b(The)46 b(grand-canonical)f(thermo)s (dynamic)h(pressure)h(of)d(the)h(PBG)g(\(4.2\))f(is)324 3015 y(giv)m(en)33 b(b)m(y)h(the)f(limit)352 3370 y Ft(p)401 3328 y Fo(P)10 b(B)s(G;E)639 3337 y Fj(0)677 3370 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\))27 b(=)54 b(lim)1047 3432 y Fo(L)p Fm(!1)1253 3370 y Ft(p)1302 3322 y Fo(P)10 b(B)s(G;E)1540 3331 y Fj(0)1302 3397 y Fv(\003)1351 3408 y Fk(L)1578 3370 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\))27 b(=)54 b(lim)1948 3432 y Fo(L)p Fm(!1)2257 3302 y Fu(1)p 2164 3347 236 4 v 2164 3438 a Ft(\014)6 b Fs(j)p Fu(\003)2321 3453 y Fo(L)2372 3438 y Fs(j)2410 3370 y Ft(T)14 b(r)2525 3388 y Fm(F)2582 3369 y Fk(B)2634 3388 y Fv(\()p Fo(L)2709 3369 y Fj(2)2744 3388 y Fv(\()q(\003)2821 3399 y Fk(L)2867 3388 y Fv(\)\))2926 3370 y Ft(e)2971 3328 y Fm(\000)p Fo(\014)s Fv(\()p Fo(T)3137 3339 y Fj(\003)3178 3356 y Fk(L)3229 3328 y Fm(\000)p Fo(\026N)3382 3339 y Fj(\003)3423 3356 y Fk(L)3475 3328 y Fv(\))3506 3370 y Ft(;)3312 3546 y Fu(\(4.15\))324 3666 y(whic)m(h)30 b(implies)h(that)e(in)h(order)f (to)g(b)s(e)g(w)m(ell)h(de\014ned,)i(the)d(c)m(hemical)i(p)s(oten)m (tial)f Ft(\026)e Fu(m)m(ust)324 3787 y(b)s(e)33 b(b)s(ounded)g(from)f (ab)s(o)m(v)m(e:)45 b Ft(\026)27 b(<)g(E)1661 3802 y Fv(0)1701 3787 y Fu(,)33 b(see)g(Section)h(1.)43 b(On)33 b(the)g(other)f(hand,)h(one)g(has)389 4072 y Ft(p)438 4024 y Fo(M)7 b(F)r(;E)632 4033 y Fj(0)438 4099 y Fv(\003)487 4110 y Fk(L)670 4072 y Fu(\()p Ft(\014)f(;)17 b(\026)p Fu(\))26 b(=)1143 4004 y(1)p 1050 4049 V 1050 4140 a Ft(\014)6 b Fs(j)p Fu(\003)1207 4155 y Fo(L)1258 4140 y Fs(j)1312 4072 y Fu(ln)1451 3947 y Fm(1)1415 3977 y Fl(X)1410 4188 y Fo(N)h Fv(=0)1580 4072 y Ft(e)1625 4030 y Fo(\014)s Fv(\()p Fo(\026N)g Fm(\000)p Fo(g)r(N)g Fv(\()p Fo(N)g Fm(\000)p Fv(1\))p Fo(=)p Fv(2)p Fm(j)p Fv(\003)2300 4041 y Fk(L)2349 4030 y Fm(j)p Fv(\))2400 4072 y Ft(T)14 b(r)2515 4094 y Fm(F)2572 4071 y Fk(B)2565 4117 y(N)2624 4094 y Fv(\()p Fo(L)2699 4075 y Fj(2)2734 4094 y Fv(\()q(\003)2811 4105 y Fk(L)2857 4094 y Fv(\)\))2916 4072 y Ft(e)2961 4021 y Fm(\000)p Fo(\014)s(T)3110 3985 y Fj(\()p Fk(N)5 b Fj(\))3100 4044 y(\003)3141 4061 y Fk(L)3220 4072 y Ft(;)65 b Fu(\(4.16\))324 4401 y(where)44 b Ft(T)687 4351 y Fv(\()p Fo(N)7 b Fv(\))673 4428 y(\003)722 4439 y Fk(L)852 4401 y Fu(is)44 b(a)e(restriction)j(of)d(the)i (kinetic-energy)h(op)s(erator)d(\(4.3\))h(on)g(the)h Ft(N)10 b Fu(-)324 4522 y(particle)38 b(sector)h Fs(F)1051 4486 y Fo(B)1041 4547 y(N)1111 4522 y Fu(\()p Ft(L)1215 4486 y Fv(2)1272 4522 y Fu(\()o(\003)1377 4537 y Fo(L)1429 4522 y Fu(\)\))f(of)f(the)i(F)-8 b(o)s(c)m(k)38 b(space)h Fs(F)2418 4486 y Fo(B)2478 4522 y Fu(\()p Ft(L)2582 4486 y Fv(2)2638 4522 y Fu(\(\003)2744 4537 y Fo(L)2796 4522 y Fu(\)\),)g Ft(N)47 b Fu(=)37 b(0)p Ft(;)17 b Fu(1)p Ft(;)g Fu(2)p Ft(;)g(:::)p Fu(.)324 4642 y(Put)1199 4857 y Ft(f)1258 4816 y Fo(P)10 b(B)s(G;E)1500 4825 y Fj(0)1534 4816 y Fv(=0)1628 4857 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\))27 b(:=)h Ft(f)2076 4816 y Fo(P)10 b(B)s(G)2246 4857 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\))p Ft(;)1219 5002 y(p)1268 4961 y Fo(P)10 b(B)s(G;E)1510 4970 y Fj(0)1543 4961 y Fv(=0)1638 5002 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\))26 b(:=)i Ft(p)2084 4961 y Fo(P)10 b(B)s(G)2254 5002 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\))p Ft(:)791 b Fu(\(4.17\))1894 5251 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 324 548 a Fu(Since)33 b(the)f(canonical)g (free-energy)h(densit)m(y)g Ft(f)2064 512 y Fo(P)10 b(B)s(G;E)2302 521 y Fj(0)2340 548 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\),)32 b(is)g(the)g(Legendre)h(trans-)324 668 y(formation)f(of)g Ft(p)931 632 y Fo(P)10 b(B)s(G;E)1169 641 y Fj(0)1207 668 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\),)32 b(b)m(y)h (de\014nitions)h(\(4.17\))e(one)h(gets:)1016 1009 y Ft(f)1075 968 y Fo(P)10 b(B)s(G;E)1317 977 y Fj(0)1354 1009 y Fu(\()p Ft(\014)c(;)17 b(\032)p Fu(\))27 b(=)46 b(sup)1716 1090 y Fo(\026)p Fm(\024)p Fo(E)1865 1099 y Fj(0)1916 928 y Fl(\000)1962 1009 y Ft(\032\026)22 b Fs(\000)g Ft(p)2241 968 y Fo(P)10 b(B)s(G;E)2483 977 y Fj(0)2521 1009 y Fu(\()p Ft(\014)c(;)17 b(\026)p Fu(\))2761 928 y Fl(\001)897 1233 y Fu(=)46 b(sup)1001 1314 y Fo(\026)p Fm(\024)p Fo(E)1150 1323 y Fj(0)1201 1152 y Fl(\000)1247 1233 y Ft(\032)p Fu(\()p Ft(\026)22 b Fs(\000)g Ft(E)1587 1248 y Fv(0)1627 1233 y Fu(\))g Fs(\000)h Ft(p)1836 1192 y Fo(P)10 b(B)s(G)2006 1233 y Fu(\()p 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y(prop)s(er)33 b(space)h(of)f(the)h(particle-n)m(um)m(b)s(er)h(op)s(erator)d Ft(N)2357 2615 y Fv(\003)2406 2626 y Fk(L)2490 2600 y Fu(with)i(the)f(prop)s(er)g(v)-5 b(alue)34 b Ft(N)10 b Fu(,)324 2721 y(the)32 b(mean-\014eld)h(in)m(teraction)g(term)g(on)e (this)i(space)g(is)g(constan)m(t.)44 b(Th)m(us,)34 b(w)m(e)f(immedi-) 324 2841 y(ately)g(\014nd)g(\(4.14\))f(in)h(the)g(thermo)s(dynamic)h (limit:)923 3082 y(lim)896 3144 y Fo(L)p Fm(!1)1102 3082 y Ft(f)1150 3097 y Fv(\003)1199 3108 y Fk(L)1250 3082 y Fu([)p Ft(H)1366 3041 y Fo(M)7 b(F)1358 3106 y Fv(\003)1407 3117 y Fk(L)1499 3082 y Fu(]\()p Ft(\014)f(;)17 b(\032)p Fu(\))27 b(=)54 b(lim)1887 3144 y Fo(L)p Fm(!1)2093 3082 y Ft(f)2141 3097 y Fv(\003)2190 3108 y Fk(L)2241 3082 y Fu([)p Ft(T)2325 3097 y Fv(\003)2374 3108 y Fk(L)2424 3082 y Fu(]\()p Ft(\014)6 b(;)17 b(\032)p Fu(\))22 b(+)2812 3014 y Ft(g)p 2812 3059 51 4 v 2813 3150 a Fu(2)2872 3082 y Ft(\032)2922 3041 y Fv(2)2962 3082 y Ft(:)323 b Fu(\(4.20\))324 3291 y(By)29 b(\(4.16\))f(the)h(pressure)h(of)f(the)g (mean-\014eld)g(gas)g(is)g(w)m(ell-de\014ned)i(for)d(all)h Ft(\026)e Fs(2)h Fp(R)p Fu(,)h(and)324 3411 y(it)j(is)h(again)g(the)g (Legendre)g(transform)g(of)f Ft(f)1984 3375 y Fo(M)7 b(F)2117 3411 y Fu(\()p Ft(\014)f(;)17 b(\032)p Fu(\),)32 b(yielding)i(form)m(ula)f(\(4.13\).)76 b Fd(\003)324 3614 y Fw(Corollary)39 b(4.2.)j(\(Pressure)37 b(of)h(the)f(M-F)i(Bose)e (Gas\))324 3735 y Fi(The)d(gr)-5 b(and-c)g(anonic)g(al)33 b(pr)-5 b(essur)g(e)34 b(of)h(a)g(me)-5 b(an-\014eld)33 b(Bose)h(Gas)h(\(4.13\))f(is)g(given)g(by)426 4042 y Ft(p)475 4001 y Fo(M)7 b(F)r(;E)669 4010 y Fj(0)706 4042 y Fu(\()p Ft(\014)f(;)17 b(\026)p Fu(\))27 b(=)1076 3872 y Fl(\()1198 3957 y Ft(\026)p 1257 3902 V(\032)p Fu(\()p Ft(\014)6 b(;)17 b(\026)p Fu(\))k Fs(\000)i Ft(f)1727 3921 y Fo(M)7 b(F)r(;E)1921 3930 y Fj(0)1958 3957 y Fu(\()p Ft(\014)f(;)p 2101 3902 V 17 w(\032)p Fu(\()p Ft(\014)g(;)17 b(\026)p Fu(\)\))p Ft(;)81 b Fi(for)28 b Ft(\026)f Fs(\024)h Ft(E)2949 3972 y 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y(\(4.23\))32 b(of)g(our)g(theorem)h (follo)m(ws)h(directly)g(from)e(\(4.21\))g(and)h(\(4.22\).)470 2181 y(The)48 b(part)f(\(4.24\))f(is)h(a)g(more)g(delicate)h(matter.)86 b(It)47 b(is)h(based)f(on)g(the)h(strong)324 2301 y(equiv)-5 b(alence)41 b(of)e(ensem)m(bles)j(for)d(the)h(mean-\014eld)g(Bose)g (gas)f(and)g(the)h(fact)f(that)g(ex-)324 2421 y(p)s(ectations)34 b(in)g(the)h(canonical)f(ensem)m(ble)i(coincide)f(with)g(those)f(for)f (the)i(PBG.)e(This)324 2542 y(implies)40 b([vdB-L-deSm],[P-Z])f(that)g (the)g(particle)g(densit)m(y)i(in)e(the)g(states)h(with)f(the)324 2662 y(energies)34 b(higher)f(than)g(some)g Ft(\016)f(>)27 b Fu(0)32 b(is)h(equal)h(to)389 2888 y(lim)363 2950 y Fo(L)p Fm(!1)641 2820 y Fu(1)p 578 2865 176 4 v 578 2956 a Fs(j)p Fu(\003)674 2971 y Fo(L)726 2956 y Fs(j)927 2793 y Fl(X)780 3009 y Fm(f)p Fo(j)t Fv(:)11 b Fo(\025)920 3019 y Fk(j)953 3009 y Fo(>E)1060 3018 y Fj(0)1094 3009 y Fv(+)p Fo(\016)r Fm(g)1234 2888 y Fs(h)p Ft(a)1324 2847 y Fm(\003)1364 2888 y Fu(\()p 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b(k)l(ernel)g(of)f Fc(f)1513 963 y Fb(h)1561 975 y Fa(1)324 1130 y Fu(F)-8 b(or)37 b(simplicit)m(y)j(w)m(e)e(consider)h(her)g(only) f(the)g(dimension)h Ft(d)d Fu(=)h(3,)h(and)g(the)g(op)s(erator)324 1251 y Ft(h)380 1266 y Fm(1)482 1251 y Fu(=)28 b(\()p Fs(\000)p Ft(i)p Fs(r)23 b(\000)f Fw(a)p Fu(\))1031 1214 y Fv(2)1093 1251 y Fu(+)g Ft(V)54 b Fu(as)33 b(in)g(In)m(tro)s (duction.)44 b(Let)33 b Ft(f)38 b Fs(2)28 b Ft(C)2572 1214 y Fm(1)2565 1275 y Fv(0)2647 1251 y Fu(\()p Fp(R)p Fu(\).)43 b(W)-8 b(e)33 b(write)607 1441 y Ft(f)11 b Fu(\()p Ft(h)760 1456 y Fm(1)835 1441 y Fu(\))27 b(=)h(exp)q(\()p Fs(\000)p Ft(h)1324 1456 y Fm(1)1399 1441 y Fu(\))22 b Fs(\001)1530 1414 y Fu(~)1509 1441 y Ft(f)11 b Fu(\()p Ft(h)1662 1456 y Fm(1)1737 1441 y Fu(\))22 b Fs(\001)f Fu(exp)q(\()p Fs(\000)p Ft(h)2166 1456 y Fm(1)2242 1441 y Fu(\))p Ft(;)2442 1414 y Fu(~)2421 1441 y Ft(f)11 b Fu(\()p Ft(t)p Fu(\))27 b(=)h Ft(e)2767 1400 y Fv(2)p Fo(t)2832 1441 y Ft(f)11 b Fu(\()p Ft(t)p Fu(\))p Ft(:)283 b Fu(\(5.29\))324 1631 y(It)33 b(is)h(w)m(ell-kno)m(wn)i(\(see)e(e.g.) 46 b(the)34 b(argumen)m(ts)h(via)e(F)-8 b(eynman-Kac-Ito)33 b(form)m(ula)h(in)f([S)324 1751 y(2]\))g(that)g(exp)q(\()p Fs(\000)p Ft(h)1003 1766 y Fm(1)1078 1751 y Fu(\))h(admits)g(an)f(in)m (tegral)h(k)m(ernel,)h(denoted)f(b)m(y)g Ft(e)2828 1715 y Fm(\000)p Fo(h)2924 1723 y Fg(1)2993 1751 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)3193 1715 y Fm(0)3217 1751 y Fu(\),)33 b(whic)m(h)324 1872 y(ob)s(eys)g(the)g(so-called)g(diamagnetic)h (inequalit)m(y)1023 2047 y Fl(\014)1023 2107 y(\014)1056 2132 y Ft(e)1101 2091 y Fm(\000)p Fo(h)1197 2099 y Fg(1)1266 2132 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1466 2091 y Fm(0)1489 2132 y Fu(\))1527 2047 y Fl(\014)1527 2107 y(\014)1588 2132 y Fs(\024)28 b Ft(e)1738 2091 y Fm(\000)12 b Fv(min\()p Fo(V)k Fv(\))2061 2132 y Fs(\001)2243 2065 y Fu(1)p 2120 2109 294 4 v 2120 2200 a(\(4)p Ft(\031)t Fu(\))2304 2171 y Fv(3)p Fo(=)p Fv(2)2424 2132 y Ft(e)2469 2091 y Fm(\000j)p Ff(x)p Fm(\000)p Ff(x)2685 2067 y Fg(0)2707 2091 y Fm(j)2727 2067 y Fj(2)2761 2091 y Fo(=)p Fv(4)2836 2132 y Ft(:)324 2354 y Fu(Moreo)m(v)m(er,)30 b(let)f Ft(K)34 b Fs(2)29 b Fp(R)1196 2317 y Fv(3)1263 2354 y Fu(b)s(e)f(a)g(compact)g(set,)i (and)e Ft(\013)2276 2369 y Fv(1)2316 2354 y Ft(;)17 b(\013)2422 2369 y Fv(2)2488 2354 y Fs(2)29 b Fp(N)2655 2317 y Fv(3)2694 2354 y Fu(.)42 b(Under)29 b(the)f(smo)s(oth-)324 2474 y(ness)33 b(conditions)h(w)m(e)f(assumed)h(for)d Fw(a)i Fu(and)f Ft(V)22 b Fu(,)32 b(the)h(semigroup)g(k)m(ernel)h(ob)s(eys)f (\(see)g([S)324 2594 y(2]\))530 2700 y Fl(\014)530 2760 y(\014)563 2784 y Ft(@)619 2743 y Fo(\013)664 2752 y Fj(1)614 2809 y Ff(x)704 2784 y Ft(@)760 2741 y Fo(\013)805 2750 y Fj(2)755 2813 y Ff(x)798 2794 y Fg(0)845 2784 y Ft(e)890 2743 y Fm(\000)p Fo(h)986 2751 y Fg(1)1055 2784 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1255 2743 y Fm(0)1278 2784 y Fu(\))1316 2700 y Fl(\014)1316 2760 y(\014)1377 2784 y Fs(\024)28 b Ft(C)1552 2799 y Fv(1)1613 2784 y Fs(\001)22 b Ft(e)1708 2743 y Fm(\000)p Fo(C)1813 2752 y Fj(2)1848 2743 y Fm(j)p Ff(x)p Fm(\000)p Ff(x)2009 2720 y Fg(0)2031 2743 y Fm(j)2082 2784 y Ft(<)28 b Fs(1)p Ft(;)114 b Fw(x)2486 2743 y Fm(0)2537 2784 y Fs(2)28 b Fp(R)2703 2743 y Fv(3)2742 2784 y Ft(;)45 b Fw(x)28 b Fs(2)g Ft(K)r(;)205 b Fu(\(5.30\))324 2975 y(where)34 b Ft(C)676 2990 y Fv(1)747 2975 y Fu(and)f Ft(C)1007 2990 y Fv(2)1079 2975 y Fu(are)f(constan)m(ts)i(whic)m(h)g(ma)m(y)g (dep)s(end)f(on)g Ft(\013)q Fu('s)g(and)f Ft(K)7 b Fu(.)470 3095 y(By)32 b(virtue)g(of)f(the)h(previous)h(estimate,)f(w)m(e)h(can)e (then)h(write)g(do)m(wn)g(the)g(k)m(ernel)h(of)324 3215 y Ft(f)11 b Fu(\()p Ft(h)477 3230 y Fm(1)551 3215 y Fu(\))33 b(as)685 3454 y Ft(f)733 3469 y Fo(h)774 3477 y Fg(1)843 3454 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1043 3413 y Fm(0)1066 3454 y Fu(\))28 b(=)1235 3318 y Fl(Z)1291 3544 y Fh(R)1339 3525 y Fj(3)1394 3454 y Ft(e)1439 3413 y Fm(\000)p Fo(h)1535 3421 y Fg(1)1604 3454 y Fu(\()p Fw(x)p Ft(;)17 b Fw(y)r Fu(\))1861 3343 y Fl(h)1928 3428 y Fu(~)1907 3454 y Ft(f)11 b Fu(\()p Ft(h)2060 3469 y Fm(1)2135 3454 y Fu(\))p Ft(e)2218 3413 y Fm(\000)p Fo(h)2314 3421 y Fg(1)2383 3454 y Fu(\()p Fs(\001)p Ft(;)17 b Fw(x)2552 3413 y Fm(0)2574 3454 y Fu(\))2612 3343 y Fl(i)2676 3454 y Fu(\()p Fw(y)r Fu(\))p Ft(d)p Fw(y)r Ft(:)360 b Fu(\(5.31\))324 3697 y(By)37 b(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)j(inequalit)m(y)f(with)e(resp)s (ect)h(to)f(the)g Fw(y)h Fu(v)-5 b(ariable)37 b(in)g(\(5.31\),)324 3817 y(w)m(e)32 b(get)f(that)g(the)g(ab)s(o)m(v)m(e)h(k)m(ernel)h(b)s (elongs)f(to)e Ft(L)2096 3781 y Fm(1)2171 3817 y Fu(\()p Fp(R)2281 3781 y Fv(3)2340 3817 y Fs(\002)19 b Fp(R)2508 3781 y Fv(3)2548 3817 y Fu(\).)43 b(Since)32 b(this)f(is)h(also)f(true) 324 3937 y(for)494 3911 y(~)473 3937 y Ft(f)11 b Fu(,)32 b(w)m(e)i(can)f(then)g(rewrite)g(\(5.31\))f(as)696 4176 y Ft(f)744 4191 y Fo(h)785 4199 y Fg(1)854 4176 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1054 4135 y Fm(0)1077 4176 y Fu(\))28 b(=)1246 4040 y Fl(Z)1302 4266 y Fh(R)1350 4247 y Fj(6)1405 4176 y Ft(e)1450 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i(adapted)f(to)f(our)g(setting)h(assures)h(suc)m(h)g(that)324 4784 y(for)32 b(ev)m(ery)i Ft(N)k Fs(\025)29 b Fu(1,)j(there)h(exists)i (a)d(p)s(ositiv)m(e)i(constan)m(t)f Ft(C)2484 4799 y Fo(N)s(;f)2641 4784 y Fu(so)g(that)870 4974 y Fs(j)p Ft(f)946 4989 y Fo(h)987 4997 y Fg(1)1056 4974 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1256 4933 y Fm(0)1279 4974 y Fu(\))p Fs(j)27 b(\024)h Ft(C)1547 4989 y Fo(N)s(;f)1693 4974 y Fs(\001)22 b Fu(\(1)g(+)g Fs(j)p Fw(x)g Fs(\000)h Fw(x)2218 4933 y Fm(0)2241 4974 y Fs(j)p Fu(\))2307 4933 y Fm(\000)p Fo(N)2429 4974 y Ft(;)114 b Fw(x)p Ft(;)17 b Fw(x)2732 4933 y Fm(0)2783 4974 y Fs(2)28 b Fp(R)2949 4933 y Fv(3)2988 4974 y Ft(:)1894 5251 y Fu(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 324 548 a Fe(5.2)135 b(App)t(endix)36 b(2:)56 b(Magnetic)38 b(\014eld)f(in)g(t)l(w)l(o)h(dimensions)g(and)630 697 y(banding)45 b(of)g(Landau)g(lev)l(els)324 932 y Fu(Consider)27 b(a)f(2-dimensional)h(particle)g(sub)5 b(jected)29 b(to)d(a)f(constan)m(t)j(magnetic)e(\014eld)h Fw(B)h Fu(=)324 1052 y(\(0)p Ft(;)17 b Fu(0)p Ft(;)g(!)t Fu(\),)26 b(whic)m(h)k(is)e(orthogonal)e(to)h(the)h(plane)g Fp(R)2158 1016 y Fv(2)2198 1052 y Fu(,)g(where)h(the)f(particle)g(is)g (allo)m(w)m(ed)h(to)324 1173 y(mo)m(v)m(e.)52 b(W)-8 b(e)35 b(use)h(here)g(the)f(transv)m(erse)i(\(symmetric\))g(gauge)e (i.e.)51 b Fw(a)p Fu(\()p Fw(x)p Fu(\))32 b(=)3156 1133 y Fv(1)p 3156 1150 36 4 v 3156 1207 a(2)3201 1173 y Fw(B)24 b Fs(^)g Fw(x)32 b Fu(=)324 1293 y Ft(!)t Fw(a)443 1308 y Fv(0)482 1293 y Fu(\()p Fw(x)p Fu(\))d(=)h Ft(!)t(=)p Fu(2\()p Fs(\000)p Ft(x)1085 1308 y Fv(2)1124 1293 y Ft(;)17 b(x)1223 1308 y Fv(1)1262 1293 y Fu(\).)47 b(Therefore,)34 b(it)g(is)g(a)f(t)m(w)m(o)i(dimensional)g(restriction)f(of)f(the)324 1413 y(mo)s(del)g(w)m(e)g(consider)h(in)f(Section)g(3.2.)470 1534 y(T)-8 b(ak)m(e)34 b Fw(e)757 1549 y Fv(1)829 1534 y Fu(and)f Fw(e)1070 1549 y Fv(2)1143 1534 y Fu(as)f(elemen)m(ts)k(of)c (the)h(standard)g(orthonormal)g(basis)g(in)g Fp(R)3332 1498 y Fv(2)3404 1534 y Fu(and)324 1654 y(consider)j(the)f(lattice)h Fp(Z)1243 1618 y Fv(2)1282 1654 y Fu(.)51 b(Let)35 b(\004)c(=)g(\()p Fs(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)17 b Fu(1)p Ft(=)p Fu(2\),)35 b(the)g(w)m(e)h(denote)f(the)h(elemen)m(tary)324 1775 y(cell)d(in)g Fp(R)683 1738 y Fv(2)755 1775 y Fu(b)m(y)g(\004)23 b Fs(\002)f Fu(\004.)470 1895 y(W)-8 b(e)34 b(denote)h(the)f(dual)g (lattice)g(b)m(y)h(\()p Fp(Z)1882 1859 y Fv(2)1922 1895 y Fu(\))1960 1859 y Fm(\003)1999 1895 y Fu(,)f(and)g(de\014ne)h(the)f (dual)g(elemen)m(tary)i(cell)324 2015 y(b)m(y)728 2136 y(\(\004)831 2095 y Fv(2)870 2136 y Fu(\))908 2095 y Fm(\003)975 2136 y Fu(:=)28 b Fs(f)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))27 b(:=)g Ft(\030)1539 2151 y Fv(1)1578 2136 y Fw(k)1637 2151 y Fv(1)1699 2136 y Fu(+)22 b Ft(\030)1840 2151 y Fv(2)1879 2136 y Fw(k)1938 2151 y Fv(2)1978 2136 y Ft(;)44 b(\030)32 b Fu(=)c(\()p Ft(\030)2309 2151 y Fv(1)2348 2136 y Ft(;)17 b(\030)2435 2151 y Fv(2)2473 2136 y Fu(\))28 b Fs(2)g Fu(\004)2698 2095 y Fv(2)2738 2136 y Fs(g)f Fu(=)h(2)p Ft(\031)t Fu(\004)3092 2095 y Fv(2)3131 2136 y Ft(;)324 2310 y Fu(where)34 b Fw(k)665 2325 y Fv(1)p Fo(;)p Fv(2)787 2310 y Fu(=)27 b(2)p Ft(\031)t Fw(e)1049 2325 y Fv(1)p Fo(;)p Fv(2)1143 2310 y Fu(.)470 2430 y(W)-8 b(e)44 b(supp)s(ose)g(here)g(that)f(the)g(magnetic)h (\014eld)g(satis\014es)h(the)f(\\rationalit)m(y)f(con-)324 2551 y(dition",)h(i.e.)73 b(there)42 b(exists)i Ft(N)54 b Fs(2)45 b Fp(N)1742 2515 y Fm(\003)1823 2551 y Fu(suc)m(h)e(that)f (the)h(magnetic)f(\015ux)h(through)f(\004)3522 2515 y Fv(2)324 2671 y Fu(is:)1221 2792 y Fw(B)21 b Fs(\001)h Fu(\()p Fw(e)1461 2807 y Fv(1)1523 2792 y Fs(^)g Fw(e)1662 2807 y Fv(2)1702 2792 y Fu(\))28 b(=)f Ft(!)t Fs(j)p Fw(e)2015 2807 y Fv(1)2076 2792 y Fs(^)22 b Fw(e)2215 2807 y Fv(2)2255 2792 y Fs(j)27 b Fu(=)h(2)p Ft(\031)t(N)38 b(:)647 b Fu(\(5.32\))470 2966 y(It)37 b(is)h(w)m(ell-kno)m(wn)h(\(see) f(e.g.)57 b([B-E-S],)37 b([J-P])g(and)g(references)i(therein\))f(that)f (for)324 3086 y(the)49 b(particle)h(restricted)g(to)f(the)g(plane)g Fp(R)1995 3050 y Fv(2)2035 3086 y Fu(,)k(the)c(\\free")g(magnetic)g (Hamiltonian)324 3207 y Ft(h)380 3222 y Fv(0)464 3207 y Fu(=)c(\()p Fs(\000)p Ft(i)p Fs(r)29 b(\000)h Fw(a)p Fu(\))1044 3171 y Fv(2)1126 3207 y Fu(has)43 b(only)g(pure)g(p)s(oin)m (t)g(sp)s(ectrum)h(\(Landau)f(lev)m(els\),)k(whic)m(h)d(is)324 3327 y(giv)m(en)32 b(b)m(y)g(the)f(set)h Ft(\033)t Fu(\()p Ft(h)1181 3342 y Fv(0)1220 3327 y Fu(\))c(=)f Fs(f)p Fu(\(2)p Ft(n)19 b Fu(+)f(1\))p Ft(!)59 b Fu(:)27 b Ft(n)h Fs(2)g(f)p Fu(0)p Ft(;)17 b Fu(1)p Ft(;)g(:)g(:)g(:)e Fs(gg)p Fu(.)43 b(Let)31 b Ft(V)49 b Fs(2)28 b Ft(C)3124 3291 y Fv(0)3164 3327 y Fu(\()p Fp(R)3274 3291 y Fv(2)3313 3327 y Fu(\))j(b)s(e)g(a)324 3447 y Fp(Z)390 3411 y Fv(2)430 3447 y Fu(-p)s(erio)s(dic)h(external)i(p)s(oten)m(tial.)44 b(F)-8 b(or)31 b Ft(\017)d Fs(\025)h Fu(0)j(the)h(p)s(erturb)s(ed)g (Hamiltonian)1666 3667 y Ft(h)1722 3682 y Fo(\017)1782 3667 y Fu(=)28 b Ft(h)1942 3682 y Fv(0)2004 3667 y Fu(+)22 b Ft(\017V)324 3888 y Fu(acts)33 b(on)h Ft(L)728 3851 y Fv(2)767 3888 y Fu(\()p Fp(R)877 3851 y Fv(2)917 3888 y Fu(\).)45 b(W)-8 b(e)33 b(no)m(w)h(are)f(in)m(terested)j(in)d(t)m(w)m (o)h(questions:)47 b(\014rst,)33 b(to)g(justify)h(the)324 4008 y(represen)m(tation)j(\(3.17\))e(and)h(second,)i(to)e(in)m(v)m (estigate)h(the)g(nature)f(of)f(the)h(sp)s(ectrum)324 4128 y(of)c Ft(h)491 4143 y Fo(\017)524 4128 y Fu(.)470 4249 y(F)-8 b(or)32 b Fw(x)p Ft(;)44 b Fw(y)30 b Fs(2)e Fp(R)1030 4212 y Fv(2)1069 4249 y Fu(,)33 b(de\014ne)h(the)f (\\magnetic)g(phase")1385 4508 y Ft(\036)p Fu(\()p Fw(x)p Ft(;)17 b Fw(y)r Fu(\))27 b(:=)h Fs(\000)1928 4440 y Fu(1)p 1928 4485 49 4 v 1928 4576 a(2)2015 4508 y Fw(B)21 b Fs(\001)h Fu(\()p Fw(x)g Fs(^)h Fw(y)r Fu(\))839 b(\(5.33\))324 4753 y(and)32 b(recall)i(that)e(its)h(main)g(prop)s(ert)m(y)g(is:)446 4973 y(exp)18 b([)p Fs(\000)p Ft(i\036)p Fu(\()p Fw(x)p Ft(;)f Fw(y)r Fu(\)]\()p Fs(\000)p Ft(i)p Fs(r)1305 4988 y Ff(x)1375 4973 y Fs(\000)23 b Fw(a)p Fu(\()p Fw(x)p Fu(\)\))17 b(exp)g([)p Ft(i\036)p Fu(\()p Fw(x)p Ft(;)g Fw(y)r Fu(\)])28 b(=)f Fs(\000)p Ft(i)p Fs(r)2593 4988 y Ff(x)2663 4973 y Fs(\000)c Fw(a)p Fu(\()p Fw(x)g Fs(\000)f Fw(y)r Fu(\))28 b Ft(:)122 b Fu(\(5.34\))1894 5251 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 324 548 a Fu(F)-8 b(or)33 b(all)h Fw(e)c Fu(=)g Ft(m)p Fw(e)960 563 y Fv(1)1023 548 y Fu(+)23 b Ft(n)p Fw(e)1231 563 y Fv(2)1301 548 y Fs(2)31 b Fp(Z)1464 512 y Fv(2)1504 548 y Fu(,)j(de\014ne)h Ft(b)p Fu(\()p Fw(e)p Fu(\))c(:=)f Ft(\031)t(N)10 b(mn)p Fu(.)49 b(Let)34 b Fw(f)41 b Fu(=)30 b Ft(p)p Fw(e)3003 563 y Fv(1)3066 548 y Fu(+)23 b Ft(q)t Fw(e)3263 563 y Fv(2)3332 548 y Fs(2)31 b Fp(Z)3495 512 y Fv(2)3535 548 y Fu(.)324 668 y(Then)i(b)m(y)h(virtue)g(of)e(\(5.32\))1052 933 y Ft(\036)p Fu(\()p Fw(e)p Ft(;)17 b Fw(f)10 b Fu(\))27 b(=)h Fs(\000)1543 865 y Fu(1)p 1543 910 49 4 v 1543 1001 a(2)1630 933 y Fw(B)21 b Fs(\001)h Fu(\()p Fw(e)g Fs(^)h Fw(f)10 b Fu(\))28 b(=)g Ft(\031)t(N)10 b Fu(\()p Ft(pn)22 b Fs(\000)h Ft(q)t(m)p Fu(\))28 b Ft(:)478 b Fu(\(5.35\))324 1184 y(These)29 b(imply)f(that)g Ft(b)p Fu(\()p Fw(e)p Fu(\))12 b(+)g Ft(b)p Fu(\()p Fw(f)e Fu(\))i Fs(\000)g Ft(b)p Fu(\()p Fw(e)g Fu(+)g Fw(f)e Fu(\))i Fs(\000)g Ft(\036)p Fu(\()p Fw(e)p Ft(;)17 b Fw(f)10 b Fu(\))27 b Fs(2)h Fu(2)p Ft(\031)t Fp(Z)p Fu(,)g(and)g(that)f(the)h (mo)s(di\014ed)324 1304 y(magnetic)33 b(translations)419 1524 y(\()p Ft(T)514 1539 y Ff(e)555 1524 y Ft( )t Fu(\)\()p Fw(x)p Fu(\))28 b(:=)g(exp)17 b([)p Ft(i\036)p Fu(\()p Fw(x)p Ft(;)g Fw(e)p Fu(\))23 b(+)f Ft(ib)p Fu(\()p Fw(e)p Fu(\)])p Ft( )t Fu(\()p Fw(x)h Fs(\000)f Fw(e)p Fu(\))p Ft(;)115 b Fw(e)27 b Fs(2)h Fu(\000)p Ft(;)115 b( )31 b Fs(2)d Ft(L)2962 1483 y Fv(2)3002 1524 y Fu(\()p Fp(R)3112 1483 y Fv(2)3152 1524 y Fu(\))122 b(\(5.36\))324 1744 y(form)26 b(an)g(ab)s(elian)g(group,)h(i.e.)42 b Ft(T)1530 1759 y Ff(e)1571 1744 y Ft(T)1628 1759 y Ff(f)1693 1744 y Fu(=)28 b Ft(T)1854 1759 y Ff(e)p Fv(+)p Ff(f)1983 1744 y Fu(,)f(whic)m(h)h(comm)m(utes)g(with)e(the)h(p)s(erturb)s(ed)324 1865 y(Hamiltonian)36 b Ft(h)944 1880 y Fo(\017)977 1865 y Fu(,)h Ft(\017)d Fs(\025)h Fu(0.)54 b(This)37 b(means)g(that)f(in)g (the)h(case)g(of)f(\\rational")f(magnetic)324 1985 y(\014elds)f(some)f (sort)g(of)f(the)h(Flo)s(quet)g(banding)f(of)h(Landau)f(lev)m(els)j (should)e(exist.)470 2105 y(No)m(w)c(w)m(e)h(explicitely)h(decomp)s (ose)f(the)f(op)s(erator)f Ft(h)2386 2120 y Fo(\017)2448 2105 y Fu(in)m(to)g(a)h(direct)g(\014b)s(er)g(in)m(tegral.)324 2226 y(This)f(will)g(b)s(e)g(done)g(\014rst)g(for)e Ft(h)1471 2241 y Fv(0)1538 2226 y Fu(\(in)i(fact)f(for)g(its)h(resolv)m(en)m(t\)) h(and)f(then)g(for)e Ft(h)3190 2241 y Fo(\017)3223 2226 y Fu(.)42 b(De\014ne)324 2346 y(a)25 b(direct)h(\014b)s(er)g(in)m (tegral)f(of)g Ft(L)1402 2310 y Fv(2)1442 2346 y Fu(\(\004)1545 2310 y Fv(2)1584 2346 y Fu(\)-spaces,)j Fs(H)h Fu(:=)2219 2266 y Fl(R)2285 2292 y Fm(\010)2266 2381 y Fv(\004)2313 2362 y Fj(2)2368 2346 y Ft(L)2434 2310 y Fv(2)2474 2346 y Fu(\(\004)2577 2310 y Fv(2)2616 2346 y Fu(\))f Ft(d\030)5 b Fu(,)26 b(together)f(with)h(the)324 2466 y(unitary)39 b(op)s(erator)f Ft(U)49 b Fu(:)38 b Ft(L)1319 2430 y Fv(2)1358 2466 y Fu(\()p Fp(R)1468 2430 y Fv(2)1508 2466 y Fu(\))g Fs(7!)f(H)j Fu(whose)g(action)e(on)h(smo)s(oth)g(and)f (compactly)324 2587 y(supp)s(orted)33 b(functions)h(is:)544 2819 y(\()p Ft(U)10 b( )t Fu(\)\()p Ft(\030)5 b(;)17 b(x)p 893 2835 56 4 v Fu(\))28 b(=)1127 2724 y Fl(X)1117 2941 y Ff(e)p Fm(2)p Fh(Z)1245 2922 y Fj(2)1297 2819 y Fu(exp)18 b([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Fw(e)g Fs(\000)h Ft(i\036)p Fu(\()p Ft(x)p 2157 2835 V(;)17 b Fw(e)p Fu(\))22 b Fs(\000)h Ft(ib)p Fu(\()p Fw(e)p Fu(\)])q Ft( )t Fu(\()p Ft(x)p 2801 2835 V 22 w Fu(+)f Fw(e)p Fu(\))p Ft(;)220 b Fu(\(5.37\))324 3142 y(here)41 b Ft(x)p 543 3158 V 41 w Fu(denotes)g(the)g(p)s(osition) g(v)-5 b(ariable)41 b(\()p Ft(x)2022 3157 y Fv(1)2061 3142 y Ft(;)17 b(x)2160 3157 y Fv(2)2200 3142 y Fu(\))41 b Fs(2)h Fu(\004)2452 3106 y Fv(2)2491 3142 y Fu(.)67 b(F)-8 b(orm)m(ula)40 b(\(5.37\))g(is)h(then)324 3263 y(extended)34 b(b)m(y)g(con)m(tin)m(uit)m(y)h(on)d Ft(L)1536 3226 y Fv(2)1576 3263 y Fu(\()p Fp(R)1686 3226 y Fv(2)1725 3263 y Fu(\).)44 b(Its)33 b(adjoin)m(t)f(reads)h(as)g(\()p Fw(e)28 b Fs(2)g Fp(Z)2960 3226 y Fv(2)3000 3263 y Fu(\):)483 3531 y(\()p Ft(U)597 3490 y Fm(\003)637 3531 y Ft( )t Fu(\)\()p Ft(x)p 780 3547 V 23 w Fu(+)22 b Fw(e)p Fu(\))28 b(=)1176 3395 y Fl(Z)1231 3621 y Fv(\004)1278 3602 y Fj(2)1334 3531 y Ft(d\030)1433 3490 y Fm(0)1472 3531 y Fu(exp)17 b([)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)1842 3490 y Fm(0)1866 3531 y Fu(\))22 b Fs(\001)g Fw(e)g Fu(+)g Ft(i\036)p Fu(\()p Ft(x)p 2276 3547 V(;)17 b Fw(e)p Fu(\))22 b(+)g Ft(ib)p Fu(\()p Fw(e)p Fu(\)])q Ft( )t Fu(\()p Ft(\030)2966 3490 y Fm(0)2989 3531 y Ft(;)17 b(x)p 3033 3547 V Fu(\))p Ft(:)159 b Fu(\(5.38\))470 3804 y(It)36 b(is)h(kno)m(wn)g(\(see)g([J-P],)f([C-N])g(and)g(references)i (therein\))f(that)f(for)f Ft(z)40 b Fu(from)c(the)324 3924 y(resolv)m(en)m(t)46 b(set)f Ft(\032)p Fu(\()p Ft(h)1053 3939 y Fv(0)1093 3924 y Fu(\),)i(the)e(resolv)m(en)m(t)h(\()p Ft(h)1900 3939 y Fv(0)1970 3924 y Fs(\000)31 b Ft(z)t Fu(\))2165 3888 y Fm(\000)p Fv(1)2304 3924 y Fu(admits)45 b(the)g(follo)m(wing)f(in)m(tegral)324 4044 y(k)m(ernel)34 b Ft(K)694 4059 y Fv(0)733 4044 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)933 4008 y Fm(0)957 4044 y Fu(;)g Ft(z)t Fu(\):)770 4264 y Ft(K)853 4279 y Fv(0)892 4264 y Fu(\()p Fw(x)p Ft(;)g Fw(x)1092 4223 y Fm(0)1115 4264 y Fu(;)g Ft(z)t Fu(\))85 b(=)e Ft(e)1535 4223 y Fo(i\036)p Fv(\()p Ff(x)p Fo(;)p Ff(x)1734 4200 y Fg(0)1757 4223 y Fv(\))1810 4264 y Ft(G)1887 4279 y Fv(0)1927 4264 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)2127 4223 y Fm(0)2150 4264 y Fu(;)g Ft(z)t Fu(\))1330 4465 y Fs(\021)1500 4397 y Ft(\015)5 b Fu(\()p Ft(\013)q Fu(\))p 1500 4442 195 4 v 1533 4533 a(4)21 b Ft(\031)1705 4465 y(e)1750 4423 y Fo(i\036)p Fv(\()p Ff(x)p Fo(;)p Ff(x)1949 4400 y Fg(0)1971 4423 y Fv(\))2025 4465 y Ft(e)2070 4423 y Fm(\000)p Fo( )r Fv(\()p Ff(x)p Fo(;)p Ff(x)2306 4400 y Fg(0)2329 4423 y Fv(\))2360 4465 y Fs(F)10 b Fu(\()p Ft(\013)q(;)17 b Fu(1;)g(2)k Ft( )t Fu(\()p Fw(x)p Ft(;)c Fw(x)3017 4423 y Fm(0)3040 4465 y Fu(\)\))196 b(\(5.39\))324 4720 y(where)39 b Ft( )t Fu(\()p Fw(x)p Ft(;)17 b Fw(x)878 4684 y Fm(0)902 4720 y Fu(\))37 b(=)h Ft(!)t Fs(j)p Fw(x)25 b Fs(\000)i Fw(x)1431 4684 y Fm(0)1454 4720 y Fs(j)1482 4684 y Fv(2)1521 4720 y Ft(=)p Fu(4,)40 b Ft(\013)e Fu(=)g Fs(\000)17 b Fu(\()p Ft(z)t(=!)26 b Fs(\000)d Fu(1\))16 b Ft(=)p Fu(2)37 b Fs(6)p Fu(=)h Fs(\000)p Fu(1)p Ft(;)17 b Fs(\000)p Fu(2)p Ft(;)g(:)g(:)g(:)37 b Fu(,)j Ft(\015)j Fu(is)c(the)324 4840 y(Euler)28 b(function,)h(and)e Fs(F)10 b Fu(\()p Ft(\013)q(;)17 b(\014)6 b Fu(;)17 b Ft(\020)8 b Fu(\))25 b(is)j(the)f(con\015uen)m(t)i(h)m(yp)s(er-geometric)g (function)f([Ab-)324 4961 y(St].)1894 5251 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 470 548 a Fu(T)-8 b(ak)m(e)34 b(an)m(y)f Ft(g)e Fs(2)d Ft(C)1139 512 y Fm(1)1132 573 y Fv(0)1214 548 y Fu(\()p Fp(R)1324 512 y Fv(2)1363 548 y Fu(\).)44 b(Since)1048 661 y Fl(\000)1094 741 y Ft(T)1151 756 y Ff(e)1192 741 y Fu(\()p Ft(h)1286 756 y Fv(0)1348 741 y Fs(\000)23 b Ft(z)t Fu(\))1535 700 y Fm(\000)p Fv(1)1630 741 y Ft(g)1681 661 y Fl(\001)1742 741 y Fu(\()p Fw(x)p Fu(\))28 b(=)2008 661 y Fl(\000)2054 741 y Fu(\()p Ft(h)2148 756 y Fv(0)2210 741 y Fs(\000)22 b Ft(z)t Fu(\))2396 700 y Fm(\000)p Fv(1)2491 741 y Ft(T)2548 756 y Ff(e)2590 741 y Ft(g)2641 661 y Fl(\001)2702 741 y Fu(\()p Fw(x)p Fu(\))475 b(\(5.40\))324 935 y(for)32 b(an)m(y)h Fw(x)28 b Fs(2)g Fp(R)910 898 y Fv(2)982 935 y Fu(and)33 b Fw(e)27 b Fs(2)h Fp(Z)1410 898 y Fv(2)1483 935 y Fu(,)k(one)h(has:)521 1128 y Ft(K)604 1143 y Fv(0)643 1128 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)843 1087 y Fm(0)889 1128 y Fu(+)22 b Fw(e)p Fu(;)17 b Ft(z)t Fu(\))g(exp)h(\()p Ft(i\036)p Fu(\()p Fw(x)1578 1087 y Fm(0)1601 1128 y Ft(;)f Fw(e)p Fu(\)\))27 b(=)h(exp)17 b(\()p Ft(i\036)p Fu(\()p Fw(x)p Ft(;)g Fw(e)p Fu(\)\))p Ft(K)2548 1143 y Fv(0)2588 1128 y Fu(\()p Fw(x)22 b Fs(\000)g Fw(e)p Ft(;)17 b Fw(x)2960 1087 y Fm(0)2984 1128 y Fu(;)g Ft(z)t Fu(\))197 b(\(5.41\))324 1321 y(for)32 b(an)m(y)h Fw(x)716 1285 y Fm(0)767 1321 y Fs(2)28 b Fp(R)933 1285 y Fv(2)1005 1321 y Fu(and)33 b(for)f(eac)m(h)h Fw(e)1614 1285 y Fm(0)1666 1321 y Fs(2)28 b Fp(Z)1826 1285 y Fv(2)1898 1321 y Fu(one)33 b(has)466 1514 y Ft(K)549 1529 y Fv(0)589 1514 y Fu(\()p Fw(x)p Ft(;)17 b(x)p 730 1530 56 4 v -41 x Fm(0)830 1514 y Fu(+)23 b Fw(e)980 1473 y Fm(0)1003 1514 y Fu(;)17 b Ft(z)t Fu(\))g(exp)h(\()p Ft(i\036)p Fu(\()p Ft(x)p 1484 1530 V -41 x Fm(0)1562 1514 y Ft(;)f Fw(e)1657 1473 y Fm(0)1680 1514 y Fu(\)\))28 b(=)f(exp)18 b(\()p Ft(i\036)p Fu(\()p Fw(x)p Ft(;)f Fw(e)2374 1473 y Fm(0)2397 1514 y Fu(\)\))p Ft(K)2556 1529 y Fv(0)2596 1514 y Fu(\()p Fw(x)22 b Fs(\000)g Fw(e)2865 1473 y Fm(0)2889 1514 y Ft(;)17 b(x)p 2933 1530 V -41 x Fm(0)3011 1514 y Fu(;)g Ft(z)t Fu(\))p Ft(;)143 b Fu(\(5.42\))324 1708 y(for)36 b(an)m(y)h Ft(x)p 665 1724 V -36 x Fm(0)778 1708 y Fs(2)e Fu(\004)944 1672 y Fv(2)1020 1708 y Fu(\(notice)i(that)f Ft(\036)p Fu(\()p Fw(x)p Ft(;)17 b Fw(x)p Fu(\))34 b(=)h(0\).)55 b(T)-8 b(ak)m(e)37 b(a)f(smo)s(oth)h Ft(g)h Fs(2)d(H)q Fu(.)55 b(Then)38 b(b)m(y)324 1828 y(\(5.38\))32 b(and)g(\(5.42\),)g(w) m(e)i(get)568 2165 y Fs(f)p Fu([\()p Ft(h)739 2180 y Fv(0)801 2165 y Fs(\000)22 b Ft(z)t Fu(\))987 2124 y Fm(\000)p Fv(1)1082 2165 y Fu(])p Ft(U)1185 2124 y Fm(\003)1226 2165 y Ft(g)t Fs(g)p Fu(\()p Fw(x)p Fu(\))83 b(=)1725 2071 y Fl(X)1704 2288 y Ff(e)1741 2269 y Fg(0)1764 2288 y Fm(2)p Fh(Z)1855 2269 y Fj(2)1906 2030 y Fl(Z)1962 2255 y Fv(\004)2009 2236 y Fj(2)2064 2165 y Ft(dx)p 2115 2181 V -41 x Fm(0)2210 2030 y Fl(Z)2265 2255 y Fv(\004)2312 2236 y Fj(2)2368 2165 y Ft(d\030)2467 2124 y Fm(0)2489 2165 y Ft(e)2534 2124 y Fo(i)p Ff(k)p Fv(\()p Fo(\030)2662 2101 y Fg(0)2685 2124 y Fv(\))p Fm(\001)p Ff(e)2769 2101 y Fg(0)2792 2124 y Fv(+)p Fo(i\036)p Fv(\()p Ff(x)p Fo(;)p Ff(e)3040 2101 y Fg(0)3062 2124 y Fv(\)+)p Fo(ib)p Fv(\()p Ff(e)3262 2101 y Fg(0)3286 2124 y Fv(\))1544 2413 y Fs(\002)g Ft(K)1787 2428 y Fv(0)1827 2413 y Fu(\()p Fw(x)22 b Fs(\000)g Fw(e)2096 2372 y Fm(0)2120 2413 y Ft(;)17 b(x)p 2164 2429 V -41 x Fm(0)2242 2413 y Fu(;)g Ft(z)t Fu(\))p Ft(g)t Fu(\()p Ft(\030)2510 2372 y Fm(0)2533 2413 y Ft(;)g(x)p 2577 2429 V -41 x Fm(0)2655 2413 y Fu(\))28 b Ft(:)564 b Fu(\(5.43\))324 2606 y(Then)54 b(with)g(the)f(help)h(of)e(\(5.37\))h (in)g(the)g(ab)s(o)m(v)m(e)h(equation,)59 b(the)54 b(expression)h(for) 324 2727 y Fs(f)p Ft(U)10 b Fu([\()p Ft(h)571 2742 y Fv(0)633 2727 y Fs(\000)23 b Ft(z)t Fu(\))820 2691 y Fm(\000)p Fv(1)915 2727 y Fu(])p Ft(U)1018 2691 y Fm(\003)1058 2727 y Ft(g)t Fs(g)p Fu(\()p Ft(\030)5 b(;)17 b(x)p 1289 2743 V -2 w Fu(\))33 b(reads)g(as:)723 2874 y Fl(X)674 3091 y Ff(e)p Fo(;)p Ff(e)768 3072 y Fg(0)790 3091 y Fm(2)p Fh(Z)881 3072 y Fj(2)933 2833 y Fl(Z)988 3059 y Fv(\004)1035 3040 y Fj(2)1091 2969 y Ft(dx)p 1142 2985 V -42 x Fm(0)1237 2969 y Fu(exp)17 b([)p Fs(\000)p Ft(i\036)p Fu(\()p Ft(x)p 1635 2985 V 1 w(;)g Fw(e)p Fu(\))22 b Fs(\000)h Ft(ib)p Fu(\()p Fw(e)p Fu(\))g(+)f Ft(i\036)p Fu(\()p Ft(x)p 2397 2985 V 22 w Fu(+)g Fw(e)p Ft(;)17 b Fw(e)2718 2927 y Fm(0)2741 2969 y Fu(\))22 b(+)g Ft(ib)p Fu(\()p Fw(e)3062 2927 y Fm(0)3087 2969 y Fu(\)])160 b(\(5.44\))513 3283 y Fs(\002)84 b Ft(K)757 3298 y Fv(0)796 3283 y Fu(\()p Ft(x)p 834 3299 V 23 w Fu(+)22 b Fw(e)g Fs(\000)h Fw(e)1234 3242 y Fm(0)1257 3283 y Ft(;)17 b(x)p 1301 3299 V -41 x Fm(0)1379 3283 y Fu(;)g Ft(z)t Fu(\))g(exp)h([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)e Fw(e)p Fu(])2180 3147 y Fl(Z)2236 3373 y Fv(\004)2283 3354 y Fj(2)2338 3283 y Ft(d\030)2437 3242 y Fm(0)2476 3283 y Fu(exp)d([)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)2847 3242 y Fm(0)2870 3283 y Fu(\))k Fs(\001)g Fw(e)3031 3242 y Fm(0)3054 3283 y Fu(])p Ft(g)t Fu(\()p Ft(\030)3218 3242 y Fm(0)3240 3283 y Ft(;)17 b(x)p 3284 3299 V -41 x Fm(0)3363 3283 y Fu(\))27 b Ft(:)324 3529 y Fu(Changing)h(the)g(summation)g(o)m(v)m(er)h(the)f(v)-5 b(ariable)28 b Fw(e)f Fu(to)g Fw(f)39 b Fu(=)27 b Fw(e)12 b Fs(\000)g Fw(e)2719 3492 y Fm(0)2743 3529 y Fu(,)28 b(one)g(gets)g(for)f(\(5.43\))324 3649 y(that:)788 3771 y Fl(X)767 3988 y Ff(e)804 3969 y Fg(0)827 3988 y Fm(2)p Fh(Z)918 3969 y Fj(2)976 3771 y Fl(X)969 3988 y Ff(f)8 b Fm(2)p Fh(Z)1093 3969 y Fj(2)1144 3730 y Fl(Z)1200 3956 y Fv(\004)1247 3937 y Fj(2)1302 3866 y Ft(dx)p 1353 3882 V -41 x Fm(0)1448 3866 y Fu(exp)18 b([)p Fs(\000)p Ft(i\036)p Fu(\()p Ft(x)p 1847 3882 V 1 w(;)f Fw(f)10 b Fu(\))22 b Fs(\000)h Ft(ib)p Fu(\()p Fw(e)2314 3825 y Fm(0)2360 3866 y Fu(+)f Fw(f)10 b Fu(\))22 b(+)g Ft(i\036)p Fu(\()p Fw(f)10 b Ft(;)17 b Fw(e)2928 3825 y Fm(0)2952 3866 y Fu(\))22 b(+)g Ft(ib)p Fu(\()p Fw(e)3273 3825 y Fm(0)3297 3866 y Fu(\)])606 4114 y Fs(\002)84 b Ft(K)850 4129 y Fv(0)889 4114 y Fu(\()p Ft(x)p 927 4130 V 23 w Fu(+)22 b Fw(f)10 b Ft(;)17 b(x)p 1191 4130 V -42 x Fm(0)1270 4114 y Fu(;)g Ft(z)t Fu(\))g(exp)g([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)f Fw(f)10 b Fu(])17 b(exp)g([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)f Fw(e)2673 4072 y Fm(0)2696 4114 y Fu(])606 4307 y Fs(\002)767 4172 y Fl(Z)822 4397 y Fv(\004)869 4378 y Fj(2)925 4307 y Ft(d\030)1024 4266 y Fm(0)1063 4307 y Fu(exp)17 b([)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)1433 4266 y Fm(0)1456 4307 y Fu(\))23 b Fs(\001)e Fw(e)1617 4266 y Fm(0)1641 4307 y Fu(])p Ft(g)t Fu(\()p Ft(\030)1805 4266 y Fm(0)1827 4307 y Ft(;)c(x)p 1871 4323 V -41 x Fm(0)1949 4307 y Fu(\))p Ft(:)1298 b Fu(\(5.45\))324 4553 y(Since)34 b(b)m(y)h(the)e(magnetic)i(\015ux)f(rationalit)m(y)g (\(5.32\))e(one)i(has)g Fs(\000)p Ft(b)p Fu(\()p Fw(e)2818 4517 y Fm(0)2865 4553 y Fu(+)22 b Fw(f)10 b Fu(\))23 b(+)g Ft(\036)p Fu(\()p Fw(f)10 b Ft(;)17 b Fw(e)3402 4517 y Fm(0)3425 4553 y Fu(\))23 b(+)324 4674 y Ft(b)p Fu(\()p Fw(e)454 4637 y Fm(0)478 4674 y Fu(\))f(+)g Ft(b)p Fu(\()p Fw(f)10 b Fu(\))28 b Fs(2)g Fu(2)p Ft(\031)t Fp(Z)p Fu(,)33 b(and)g(since)562 4821 y Fl(X)542 5038 y Ff(e)579 5019 y Fg(0)601 5038 y Fm(2)p Fh(Z)692 5019 y Fj(2)744 4915 y Fu(exp)17 b([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)f Fw(e)1353 4874 y Fm(0)1376 4915 y Fu(])1420 4780 y Fl(Z)1475 5005 y Fv(\004)1522 4986 y Fj(2)1578 4915 y Ft(d\030)1677 4874 y Fm(0)1716 4915 y Fu(exp)17 b([)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)2086 4874 y Fm(0)2109 4915 y Fu(\))23 b Fs(\001)e Fw(e)2270 4874 y Fm(0)2294 4915 y Fu(])p Ft(g)t Fu(\()p Ft(\030)2458 4874 y Fm(0)2480 4915 y Ft(;)c(x)p 2524 4931 V -41 x Fm(0)2602 4915 y Fu(\))28 b(=)f Ft(g)t Fu(\()p Ft(\030)5 b(;)17 b(x)p 2952 4931 V 3006 4874 a Fm(0)3029 4915 y Fu(\))p Ft(;)218 b Fu(\(5.46\))1894 5251 y(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 324 548 a Fu(w)m(e)35 b(conclude)g(b)m(y)g (\(5.45\))e(that)g(the)h(resolv)m(en)m(t)i(\()p Ft(h)2177 563 y Fv(0)2240 548 y Fs(\000)23 b Ft(z)t Fu(\))2427 512 y Fm(\000)p Fv(1)2556 548 y Fu(is)34 b(is)h(decomp)s(osable,)g(and) 324 668 y(its)e(\014b)s(ers)g(are)g(in)m(tegral)g(op)s(erators)f(on)h Ft(L)1874 632 y Fv(2)1913 668 y Fu(\(\004)2016 632 y Fv(2)2056 668 y Fu(\))f(with)h(k)m(ernels:)374 890 y Ft(K)457 905 y Fv(0)496 890 y Fu(\()p Ft(\030)5 b Fu(;)17 b Ft(x)p 626 906 56 4 v(;)g(x)p 725 906 V -42 x Fm(0)803 890 y Fu(;)g Ft(z)t Fu(\))28 b(=)1073 795 y Fl(X)1066 1012 y Ff(f)8 b Fm(2)p Fh(Z)1190 993 y Fj(2)1241 890 y Fu(exp)17 b([)p Fs(\000)p Ft(i\036)p Fu(\()p Ft(x)p 1639 906 V 1 w(;)g Fw(f)10 b Fu(\))23 b Fs(\000)f Ft(ib)p Fu(\()p Fw(f)10 b Fu(\))23 b Fs(\000)g Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Fw(f)10 b Fu(])p Ft(K)2702 905 y Fv(0)2742 890 y Fu(\()p Ft(x)p 2780 906 V 13 w Fu(+)j Fw(f)d Ft(;)17 b(x)p 3025 906 V 3081 848 a Fm(0)3104 890 y Fu(;)g Ft(z)t Fu(\))p Ft(:)50 b Fu(\(5.47\))470 1193 y(In)24 b(order)f(to)g(obtain)h(decomp)s(osition)g(of)f(the)h(op)s (erator)f Ft(h)2567 1208 y Fv(0)2630 1193 y Fu(in)m(to)h(\014b)s(ers)g Ft(h)3129 1208 y Fv(0)3168 1193 y Fu(\()p Ft(\030)5 b Fu(\))23 b(whic)m(h)324 1313 y(ha)m(v)m(e)35 b(a)e(common)h(domain)g (\(i.e.)47 b(indep)s(enden)m(t)36 b(of)d Ft(\030)5 b Fu(\),)33 b(one)h(has)g(to)g(\\rotate")e Fs(H)j Fu(with)324 1433 y(the)d(unitary)g(op)s(erator)f Fs(V)40 b Fu(de\014ned)33 b(on)f(eac)m(h)h(\014b)s(er)f(b)m(y)g(the)g(m)m(ultiplication)h Fs(V)8 b Fu(\()p Ft(\030)d(;)17 b(x)p 3365 1449 V Fu(\))28 b(=)324 1554 y(exp)17 b([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)f Ft(x)p 882 1570 V Fu(].)63 b(Let)39 b Fs(U)49 b Fu(:=)38 b Fs(V)8 b Ft(U)i Fu(.)64 b(Then)40 b Fs(U)10 b Fu(\()p Ft(h)2150 1569 y Fv(0)2216 1554 y Fs(\000)27 b Ft(z)t Fu(\))2407 1518 y Fm(\000)p Fv(1)2502 1554 y Fs(U)2574 1518 y Fm(\003)2652 1554 y Fu(=)2767 1473 y Fl(R)2814 1588 y Fv(\004)2861 1569 y Fj(2)2916 1554 y Ft(d\030)5 b Fu([)p Ft(h)3098 1569 y Fv(0)3137 1554 y Fu(\()p Ft(\030)g Fu(\))26 b Fs(\000)g Ft(z)t Fu(])3466 1518 y Fm(\000)p Fv(1)324 1674 y Fu(and)441 1878 y([)p Ft(h)524 1893 y Fv(0)563 1878 y Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\000)g Ft(z)t Fu(])884 1837 y Fm(\000)p Fv(1)979 1878 y Fu(\()p Ft(x)p 1017 1894 V 1 w(;)17 b(x)p 1117 1894 V -41 x Fm(0)1195 1878 y Fu(\))28 b(=)f(exp)18 b([)p Fs(\000)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\001)e Ft(x)p 1922 1894 V 1 w Fu(])p Ft(K)2088 1893 y Fv(0)2127 1878 y Fu(\()p Ft(\030)5 b Fu(;)17 b Ft(x)p 2257 1894 V(;)g(x)p 2356 1894 V -41 x Fm(0)2434 1878 y Fu(;)g Ft(z)t Fu(\))g(exp)h([)p Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Ft(x)p 3063 1894 V -41 x Fm(0)3141 1878 y Fu(])p Ft(:)117 b Fu(\(5.48\))324 2082 y(One)35 b(can)g(see)h(that)e(the)h(range)g([)p Ft(h)1607 2097 y Fv(0)1647 2082 y Fu(\()p Ft(\030)5 b Fu(\))23 b Fs(\000)h Ft(z)t Fu(])1971 2046 y Fm(\000)p Fv(1)2066 2082 y Ft(C)2143 2046 y Fm(1)2136 2106 y Fv(0)2218 2082 y Fu(\(\004)2321 2046 y Fv(2)2360 2082 y Fu(\))35 b(is)g(con)m(tained)h(in)f(the)g(restric-)324 2202 y(tion)f(to)p 647 2119 105 4 v 34 w(\004)712 2173 y Fv(2)785 2202 y Fu(of)g(all)g Ft(C)1112 2166 y Fm(1)1187 2202 y Fu(\()p Fp(R)1297 2166 y Fv(2)1336 2202 y Fu(\)-functions)h Ft( )j Fu(with)d(the)f(prop)s(ert)m(y)h(that)f(b)s(oth)g Ft( )t Fu(\()p Fw(x)p Fu(\))g(and)324 2323 y([)p Fs(\000)p Ft(i)p Fs(r)544 2338 y Ff(x)601 2323 y Fs(\000)9 b Fw(a)p Fu(\()p Fw(x)p Fu(\)])p Ft( )t Fu(\()p Fw(x)p Fu(\))27 b(are)f(in)m(v)-5 b(arian)m(t)27 b(with)g(resp)s(ect)g(to)f(the)h(mo)s(di\014ed)g (magnetic)g(trans-)324 2443 y(lations)40 b(\(5.36\).)65 b(W)-8 b(e)40 b(sa)m(y)h(that)f(the)g(functions)h(with)f(suc)m(h)i (prop)s(ert)m(y)e(v)m(erify)i(\\mag-)324 2563 y(netic")29 b(p)s(erio)s(dic)f(b)s(oundary)h(conditions.)43 b(Denote)28 b(this)h(restriction)g(b)m(y)g Ft(D)s Fu(.)42 b(Then)29 b(the)324 2684 y(op)s(erator)h Ft(h)771 2699 y Fv(0)810 2684 y Fu(\()p Ft(\030)5 b Fu(\))27 b(:=)h([)p Fs(\000)p Ft(i)p Fs(r)1312 2699 y Fo(x)p 1312 2711 44 3 v 1375 2684 a Fs(\000)19 b Fw(a)p Fu(\()p Ft(x)p 1563 2700 56 4 v 1 w Fu(\))g(+)f Fw(k)p Fu(\()p Ft(\030)5 b Fu(\)])1980 2647 y Fv(2)2050 2684 y Fu(with)31 b(\\magnetic")g(p)s(erio)s(dic)g(b)s (oundary)324 2804 y(conditions)h(is)f(essen)m(tially)j(self-adjoin)m(t) d(on)g Ft(D)s Fu(.)43 b(No)m(w,)32 b(if)f Ft(\017)d(>)f Fu(0,)k(ev)m(erything)j(remains)324 2924 y(true)f(for)f Ft(h)735 2939 y Fo(\017)768 2924 y Fu(,)g(whose)i(\014b)s(ers)f(are)g (de\014ned)h(as)f(op)s(erator)f(sum:)1252 3128 y Ft(h)1308 3143 y Fo(\017)1341 3128 y Fu(\()p Ft(\030)5 b Fu(\))27 b(=)g Ft(h)1651 3143 y Fv(0)1691 3128 y Fu(\()p Ft(\030)5 b Fu(\))21 b(+)h Ft(\017V)g Fu(\()p Ft(x)p 2090 3144 V 1 w Fu(\))p Ft(;)114 b(x)p 2325 3144 V 28 w Fs(2)28 b Fu(\004)2567 3087 y Fv(2)2606 3128 y Ft(:)679 b Fu(\(5.49\))324 3332 y(Then)33 b(from)g(\(5.48\))f(and)g(\(5.49\))g(w)m(e)i(deriv)m(e)g (\(3.17\).)470 3498 y(As)43 b(it)g(is)g(w)m(ell-kno)m(wn)i(\(see)f (e.g.)74 b([B-E-S]\),)43 b(the)g(orthogonal)f(pro)5 b(jectors)44 b(of)e Ft(h)3522 3513 y Fv(0)324 3618 y Fu(corresp)s(onding)25 b(to)g(the)g Ft(n)p Fu(-th)f(Landau)h(eigen)m(v)-5 b(alue)26 b Ft(!)t Fu(\(2)p Ft(n)6 b Fu(+)g(1\))24 b(are)h(in)m(tegral)g(op)s (erators,)324 3738 y(with)33 b(the)g(k)m(ernel)h(\(see)g(also)e (\(5.39\))g(for)g(notations\):)930 3965 y Ft(P)993 3980 y Fv(0)p Fo(;n)1094 3965 y Fu(\()p Fw(x)p Ft(;)17 b Fw(x)1294 3924 y Fm(0)1318 3965 y Fu(\))27 b(=)1518 3898 y Ft(!)p 1497 3942 108 4 v 1497 4034 a Fu(2)p Ft(\031)1614 3965 y(e)1659 3924 y Fo(i\036)p Fv(\()p Ff(x)p Fo(;)p Ff(x)1858 3901 y Fg(0)1881 3924 y Fv(\))1935 3965 y Ft(e)1980 3924 y Fm(\000)p Fo( )r Fv(\()p Ff(x)p Fo(;)p Ff(x)2216 3901 y Fg(0)2238 3924 y Fv(\))2270 3965 y Fs(L)2339 3980 y Fo(n)p Fv(+1)2476 3965 y Fu(\(2)p Ft( )t Fu(\()p Fw(x)p Ft(;)17 b Fw(x)2830 3924 y Fm(0)2853 3965 y Fu(\)\))p Ft(;)356 b Fu(\(5.50\))324 4200 y(where)47 b Fs(L)688 4215 y Fo(m)755 4200 y Fu(\()p Ft(\020)8 b Fu(\))45 b(is)i(the)g Ft(m)p Fu(-th)f(Laguerre)h(p)s(olynomial,)j(with)d Fs(L)2752 4215 y Fo(m)2818 4200 y Fu(\(0\))k(=)g(1,)f(for)c(an)m(y)324 4320 y Ft(m)28 b Fs(\025)g Fu(1.)470 4441 y(Then)34 b(for)e(eac)m(h)i (\014b)s(er)f Ft(h)1375 4456 y Fv(0)1414 4441 y Fu(\()p Ft(\030)5 b Fu(\))32 b(w)m(e)i(ha)m(v)m(e)g Ft(h)1995 4456 y Fv(0)2035 4441 y Fu(\()p Ft(\030)5 b Fu(\))27 b(=)2289 4366 y Fl(P)2395 4470 y Fo(n)p Fm(\025)p Fv(0)2548 4441 y Ft(!)t Fu(\(2)p Ft(n)22 b Fu(+)g(1\))p Ft(P)3028 4456 y Fv(0)p Fo(;n)3129 4441 y Fu(\()p Ft(\030)5 b Fu(\),)32 b(where)324 4561 y(similar)h(to)f(\(5.47\))g(the)h(\\free")g(\014b)s (er)g(pro)5 b(jectors)33 b(ha)m(v)m(e)h(the)f(k)m(ernels:)324 4782 y Ft(P)387 4797 y Fv(0)p Fo(;n)489 4782 y Fu(\()p Ft(\030)5 b Fu(;)17 b Ft(x)p 619 4798 56 4 v -1 w(;)g(x)p 717 4798 V -41 x Fm(0)795 4782 y Fu(\))28 b(=)972 4688 y Fl(X)965 4905 y Ff(f)8 b Fm(2)p Fh(Z)1089 4886 y Fj(2)1140 4782 y Fu(exp)17 b([)p Fs(\000)p Ft(i\036)p Fu(\()p Ft(x)p 1538 4798 V 1 w(;)g Fw(f)10 b Fu(\))23 b Fs(\000)f Ft(ib)p Fu(\()p Fw(f)10 b Fu(\))24 b Fs(\000)e Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Fu(\()p Ft(x)p 2485 4798 V 22 w Fu(+)g Fw(f)33 b Fs(\000)23 b Ft(x)p 2827 4798 V -41 x Fm(0)2905 4782 y Fu(\)])p Ft(P)3033 4797 y Fv(0)p Fo(;n)3135 4782 y Fu(\()p Ft(x)p 3173 4798 V Fu(+)p Fw(f)10 b Ft(;)17 b(x)p 3392 4798 V 3448 4741 a Fm(0)3471 4782 y Fu(;)g Ft(z)t Fu(\))p Ft(:)3312 5002 y Fu(\(5.51\))1894 5251 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 324 548 a Fu(Notice)33 b(that)f(the)h(rank)g Ft(r)i Fu(of)e Ft(P)1488 563 y Fv(0)p Fo(;n)1589 548 y Fu(\()p Ft(\030)5 b Fu(\))32 b(can)h(b)s(e)g(easily)g(obtained)h (from:)756 772 y Ft(r)87 b Fu(=)1047 636 y Fl(Z)1102 862 y Fv(\004)1149 843 y Fj(2)1204 772 y Ft(dx)p 1255 788 56 4 v 28 w(P)1401 787 y Fv(0)p Fo(;n)1503 772 y Fu(\()p Ft(\030)5 b Fu(;)17 b Ft(x)p 1633 788 V -1 w(;)g(x)p 1731 788 V Fu(\))887 1031 y(=)1078 963 y Ft(!)p 1056 1008 108 4 v 1056 1099 a Fu(2)p Ft(\031)1198 936 y Fl(X)1191 1153 y Ff(f)8 b Fm(2)p Fh(Z)1315 1134 y Fj(2)1366 895 y Fl(Z)1421 1121 y Fv(\004)1468 1102 y Fj(2)1524 1031 y Ft(dx)p 1575 1047 56 4 v 44 w Fu(exp)18 b([)p Fs(\000)p Fu(2)p Ft(i\036)p Fu(\()p Ft(x)p 2122 1047 V 1 w(;)f Fw(f)10 b Fu(\))22 b Fs(\000)g Ft(ib)p Fu(\()p Fw(f)10 b Fu(\))24 b Fs(\000)e Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Fw(f)10 b Fu(])886 1284 y Fs(\002)84 b Fu(exp)17 b([)p Fs(\000)p Ft(!)t Fw(f)1425 1243 y Fv(2)1465 1284 y Ft(=)p Fu(4])27 b Fs(L)1686 1299 y Fo(n)p Fv(+1)1823 1284 y Fu(\()p Ft(!)t Fw(f)1970 1243 y Fv(2)2009 1284 y Ft(=)p Fu(2\))p Ft(:)1140 b Fu(\(5.52\))324 1457 y(Since)40 b Fw(f)50 b Fu(=)39 b Ft(f)832 1472 y Fv(1)871 1457 y Fw(e)922 1472 y Fv(1)988 1457 y Fu(+)27 b Ft(f)1139 1472 y Fv(2)1178 1457 y Fw(e)1229 1472 y Fv(2)1308 1457 y Fs(2)39 b Fp(Z)1479 1421 y Fv(2)1558 1457 y Fu(and)h Ft(x)p 1755 1473 V 39 w Fu(=)f Ft(x)2019 1472 y Fv(1)2059 1457 y Fw(e)2110 1472 y Fv(1)2176 1457 y Fu(+)27 b Ft(x)2334 1472 y Fv(2)2374 1457 y Fw(e)2425 1472 y Fv(2)2503 1457 y Fs(2)40 b Fu(\004)2674 1421 y Fv(2)2713 1457 y Fu(,)h(then)f Fs(\000)p Fu(2)p Ft(i\036)p Fu(\()p Ft(x)p 3265 1473 V 1 w(;)17 b Fw(f)10 b Fu(\))39 b(=)324 1578 y(2)p Ft(i\031)t(N)10 b Fu(\()p Ft(f)639 1593 y Fv(1)679 1578 y Ft(x)734 1593 y Fv(2)796 1578 y Fs(\000)22 b Ft(f)943 1593 y Fv(2)983 1578 y Ft(x)1038 1593 y Fv(1)1078 1578 y Fu(\).)43 b(Therefore,)34 b(the)f(in)m(tegral)1460 1663 y Fl(Z)1516 1889 y Fv(\004)1563 1870 y Fj(2)1618 1799 y Ft(dx)p 1669 1815 V 45 w Fu(exp)17 b([)p Fs(\000)p Fu(2)p Ft(i\036)p Fu(\()p Ft(x)p 2216 1815 V 1 w(;)g Fw(f)10 b Fu(\)])887 b(\(5.53\))324 2028 y(is)46 b(zero)h(except)g(for)f Fw(f)61 b Fu(=)50 b(0,)f(when)f(it)d (is)i(equal)g(to)f Fs(j)p Fu(\004)2471 1992 y Fv(2)2510 2028 y Fs(j)k Fu(=)g Fs(j)p Fw(e)2793 2043 y Fv(1)2864 2028 y Fs(^)32 b Fw(e)3013 2043 y Fv(2)3052 2028 y Fs(j)p Fu(.)84 b(Then)47 b(b)m(y)324 2148 y(virtue)d(of)f(\(5.32\))g(w)m(e)h (get)g Ft(r)k Fu(=)e Ft(N)10 b Fu(,)47 b(for)c(an)m(y)h Ft(n)i Fs(\025)h Fu(0.)76 b(Then)44 b(if)g Ft(\017)i(>)g Fu(0)d(and)h(small)324 2269 y(enough,)f(b)m(y)e(analytic)h(p)s (erturbation)e(theory)i(one)e(obtains)h(that)g Ft(h)2926 2284 y Fo(\017)2959 2269 y Fu(\()p Ft(\030)5 b Fu(\))39 b(has)i(in)g(the)324 2389 y(neigh)m(b)s(orho)s(o)s(d)31 b(of)g(eac)m(h)h(Landau)f(lev)m(el)i Ft(!)t Fu(\(2)p Ft(n)19 b Fu(+)g(1\))31 b(exactly)i Ft(N)42 b Fu(discrete)33 b(eigen)m(v)-5 b(alues)324 2523 y Fs(f)p Ft(\025)431 2472 y Fv(\()p Fo(n)p Fv(\))431 2549 y Fo(j)532 2523 y Fu(\()p Ft(\017;)17 b(\030)5 b Fu(\))p Fs(g)789 2487 y Fo(N)789 2548 y(j)t Fv(=1)915 2523 y Fu(.)83 b(If)46 b Ft(P)1199 2538 y Fo(\017;n)1294 2523 y Fu(\()p Ft(\030)5 b Fu(\))45 b(is)h(the)g(pro)5 b(jector)47 b(corresp)s(onding)f(to)g (eac)m(h)g(group)g(of)324 2671 y(eigen)m(v)-5 b(alues)34 b Fs(f)p Ft(\025)935 2620 y Fv(\()p Fo(n)p Fv(\))935 2696 y Fo(j)1037 2671 y Fu(\()p Ft(\017;)17 b(\030)5 b Fu(\))p Fs(g)1294 2635 y Fo(N)1294 2695 y(j)t Fv(=1)1452 2671 y Fu(and)33 b(if)f Fs(S)1791 2686 y Fo(\017;n)1887 2671 y Fu(\()p Ft(\030)5 b Fu(\))32 b(is)h(the)g(in)m(tert)m(wining)h (unitary:)1286 2843 y Fs(S)1346 2858 y Fo(\017;n)1442 2843 y Fu(\()p Ft(\030)5 b Fu(\))p Ft(P)1629 2858 y Fo(\017;n)1723 2843 y Fu(\()p Ft(\030)g Fu(\))27 b(=)h Ft(P)2041 2858 y Fv(0)p Fo(;n)2142 2843 y Fu(\()p Ft(\030)5 b Fu(\))p Fs(S)2326 2858 y Fo(\017;n)2421 2843 y Fu(\()p Ft(\030)g Fu(\))27 b Ft(;)324 3016 y Fu(then)32 b(after)f(rotation)f(b)m(y)i Fs(S)1340 3031 y Fo(\017;n)1436 3016 y Fu(\()p Ft(\030)5 b Fu(\))30 b(of)h(the)h(\\reduced")g(p)s(erturb)s(ed)g(Hamiltonian)f (giv)m(en)324 3136 y(b)m(y)i Ft(P)522 3151 y Fo(\017;n)617 3136 y Fu(\()p Ft(\030)5 b Fu(\))p Ft(h)797 3151 y Fo(\017)829 3136 y Fu(\()p Ft(\030)g Fu(\))p Ft(P)1016 3151 y Fo(\017;n)1111 3136 y Fu(\()p Ft(\030)g Fu(\),)32 b(one)g(obtains)h(that)g(its)g (eigen)m(v)-5 b(alues)1102 3326 y Fs(f)p Ft(\025)1209 3275 y Fv(\()p Fo(n)p Fv(\))1209 3351 y Fo(j)1310 3326 y Fu(\()p Ft(\017;)17 b(\030)5 b Fu(\))27 b(=:)h Ft(!)t Fu(\(2)p Ft(n)21 b Fu(+)h(1\))g(+)g Ft(\017a)2301 3341 y Fo(n;j)2401 3326 y Fu(\()p Ft(\017;)17 b(\030)5 b Fu(\))p Fs(g)2658 3284 y Fo(N)2658 3350 y(j)t Fv(=1)3312 3326 y Fu(\(5.54\))324 3498 y(are)29 b(lo)s(calized)i(\(up)f(to)f(an)g (error)h(of)f(the)h(order)g Ft(\017)2103 3462 y Fv(2)2172 3498 y Fu(and)g(uniformly)g(in)g Ft(\030)5 b Fu(\))29 b(in)h(the)g(neigh-)324 3619 y(b)s(orho)s(o)s(d)h(of)h(the)h(eigen)m(v) -5 b(alues)35 b(of)d(the)h(op)s(erator)935 3791 y Ft(L)1001 3806 y Fo(n;\017)1096 3791 y Fu(\()p Ft(\030)5 b Fu(\))27 b(:=)h Ft(!)t Fu(\(2)p Ft(n)21 b Fu(+)h(1\))p Ft(P)1857 3806 y Fv(0)p Fo(;n)1959 3791 y Fu(\()p Ft(\030)5 b Fu(\))21 b(+)h Ft(\017P)2304 3806 y Fv(0)p Fo(;n)2406 3791 y Fu(\()p Ft(\030)5 b Fu(\))p Ft(V)21 b(P)2671 3806 y Fv(0)p Fo(;n)2773 3791 y Fu(\()p Ft(\030)5 b Fu(\))27 b Ft(:)324 3964 y Fu(In)35 b(the)h(particular)g(case)g(of)f(Section)h(3.2,)g(when)g Ft(N)43 b Fu(=)32 b(1)j(\(or)g Ft(!)h Fu(=)c(2)p Ft(\031)t Fu(\),)k(one)f(obtains)324 4084 y(that)23 b(the)g(op)s(erator)f Ft(L)1133 4099 y Fo(n;\017)1229 4084 y Fu(\()p Ft(\030)5 b Fu(\))22 b(has)i(only)f(one)g(eigen)m(v)-5 b(alue)25 b(whic)m(h)f(di\013ers)g(from)f(2)p Ft(\031)t Fu(\(2)p Ft(n)s Fu(+)s(1\))324 4205 y(b)m(y)33 b Ft(\017a)549 4220 y Fo(n;N)7 b Fv(=1)770 4205 y Fu(\()p Ft(\017)28 b Fu(=)f(0)p Ft(;)17 b(\030)5 b Fu(\),)32 b(where)i(\(see)f(\(3.19\)\)) 488 4429 y Ft(a)539 4444 y Fo(n)586 4429 y Fu(\()p Ft(\017)28 b Fu(=)g(0)p Ft(;)17 b(\030)5 b Fu(\))82 b(:=)h Ft(a)1293 4444 y Fo(n;N)7 b Fv(=1)1513 4429 y Fu(\()p Ft(\017)28 b Fu(=)f(0)p Ft(;)17 b(\030)5 b Fu(\))27 b(=)2030 4293 y Fl(Z)2086 4519 y Fv(\004)2133 4500 y Fj(2)2188 4429 y Ft(dx)p 2239 4445 V 28 w(V)21 b Fu(\()p Ft(x)p 2438 4445 V 1 w Fu(\))p Ft(P)2595 4444 y Fv(0)p Fo(;n)2696 4429 y Fu(\()p Ft(\030)5 b Fu(;)17 b Ft(x)p 2826 4445 V(;)g(x)p 2925 4445 V Fu(\))1069 4657 y(=)1249 4562 y Fl(X)1242 4779 y Ff(f)8 b Fm(2)p Fh(Z)1366 4760 y Fj(2)1417 4657 y Fu(exp)18 b([)p Fs(\000)p Ft(ib)p Fu(\()p Fw(f)10 b Fu(\))23 b Fs(\000)g Ft(i)p Fw(k)p Fu(\()p Ft(\030)5 b Fu(\))22 b Fs(\001)g Fw(f)10 b Fu(])44 b(exp)18 b([)p Fs(\000)p Ft(\031)t Fw(f)2780 4615 y Fv(2)2820 4657 y Ft(=)p Fu(2])28 b Fs(L)3042 4672 y Fo(n)p Fv(+1)3179 4657 y Fu(\()p Ft(\031)t Fw(f)3320 4615 y Fv(2)3359 4657 y Fu(\))1068 4962 y Fs(\002)1242 4826 y Fl(Z)1297 5052 y Fv(\004)1344 5033 y Fj(2)1399 4962 y Ft(dx)p 1450 4978 V 28 w(V)22 b Fu(\()p Ft(x)p 1650 4978 V Fu(\))17 b(exp)g([)p Fs(\000)p Fu(2)p Ft(i\036)p Fu(\()p Ft(x)p 2207 4978 V 1 w(;)g Fw(f)10 b Fu(\)])p Ft(:)869 b Fu(\(5.55\))1894 5251 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 470 548 a Fu(Denote)33 b(with)1016 781 y(^)-52 b Ft(v)1060 796 y Fo(f)1094 805 y Fj(1)1128 796 y Fo(;f)1182 805 y Fj(2)1248 781 y Fu(:=)1379 645 y Fl(Z)1434 871 y Fv(\004)1481 852 y Fj(2)1537 781 y Ft(dx)1643 796 y Fv(1)1710 781 y Ft(dx)1816 796 y Fv(2)1883 781 y Ft(e)1928 739 y Fm(\000)p Fv(2)p Fo(\031)r(i)p Fv(\()p Fo(f)2146 748 y Fj(1)2181 739 y Fo(x)2221 748 y Fj(1)2256 739 y Fv(+)p Fo(f)2345 748 y Fj(2)2379 739 y Fo(x)2419 748 y Fj(2)2453 739 y Fv(\))2485 781 y Ft(V)21 b Fu(\()p Ft(x)2656 796 y Fv(1)2696 781 y Ft(;)c(x)2795 796 y Fv(2)2835 781 y Fu(\))324 1042 y(the)33 b(F)-8 b(ourier)32 b(comp)s(onen)m(ts)i(of)e Ft(V)54 b Fu(\()p Ft(f)1674 1057 y Fv(1)1714 1042 y Ft(;)17 b(f)1806 1057 y Fv(2)1873 1042 y Fs(2)28 b Fp(Z)p Fu(\).)44 b(Since)33 b Ft(!)e Fu(=)d(2)p Ft(\031)t(N)43 b Fu(and)32 b Ft(N)38 b Fu(=)28 b(1)k(then)1103 1164 y Fl(Z)1159 1390 y Fv(\004)1206 1371 y Fj(2)1261 1300 y Ft(dx)p 1312 1316 56 4 v 28 w(V)21 b Fu(\()p Ft(x)p 1511 1316 V 1 w Fu(\))44 b(exp)18 b([)p Fs(\000)p Fu(2)p Ft(i\036)p Fu(\()p Ft(x)p 2097 1316 V(;)f Fw(f)10 b Fu(\)])28 b(=)59 b(^)-52 b Ft(v)2512 1315 y Fo(f)2546 1324 y Fj(2)2580 1315 y Fo(;)p Fm(\000)p Fo(f)2689 1324 y Fj(1)2755 1300 y Ft(:)324 1562 y Fu(By)33 b(virtue)g(of)f(\(5.55\))g(w)m(e)i(get)e(that)436 1684 y Fl(Z)491 1909 y Fv(\004)538 1890 y Fj(2)593 1819 y Ft(d\030)g Fs(jr)830 1834 y Fo(\030)868 1819 y Ft(a)919 1834 y Fo(n)966 1819 y Fu(\()p Ft(\017)c Fu(=)f(0)p Ft(;)17 b(\030)5 b Fu(\))p Fs(j)1381 1778 y Fv(2)1447 1819 y Fu(=)1558 1724 y Fl(X)1550 1942 y Ff(f)j Fm(2)p Fh(Z)1674 1923 y Fj(2)1726 1819 y Fw(f)1770 1778 y Fv(2)1854 1819 y Fu(exp)18 b([)p 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Fu(\))g(the)g(unp)s(erturb)s(ed)h(\()p Ft(\017)e Fu(=)f(0\))i(Landau)f(fundamen)m(tal)i(state)f(in)m(to)324 2609 y(a)j(simple,)i(absolutely)g(con)m(tin)m(uous)g(sp)s(ectral)g (band.)470 2847 y Fw(Ac)m(kno)m(wledgemen)m(ts)p Fu(.)43 b(The)30 b(pap)s(er)f(is)g(a)g(result)h(of)e(a)h(collab)s(oration)f (whic)m(h)j(im-)324 2968 y(plied)24 b(the)g(authors')g(visits)h(resp)s (ectiv)m(ely)i(to)c(Cen)m(tre)i(de)f(Ph)m(ysique)i(Th)m(\023)-46 b(eorique-CNRS-)324 3088 y(Lumin)m(y)41 b(\(Marseille\))f(and)g(to)f (Departmen)m(t)h(of)f(Mathematical)h(Sciences,)j(Aalb)s(org)324 3209 y(Univ)m(ersit)m(y)-8 b(.)44 b(W)-8 b(e)28 b(are)g(thankful)g(to)f (these)i(institutions)g(for)e(hospitalit)m(y)i(and)f(\014nancial)324 3329 y(supp)s(ort.)324 3660 y Fx(References)324 3879 y Fu([Ab-St])138 b(M.)39 b(Abramo)m(witz)g(and)g(I.)f(Stegun,)80 b Fi(Handb)-5 b(o)g(ok)39 b(of)h(mathematic)-5 b(al)39 b(func-)768 4000 y(tions)33 b(with)h(formulas,)e(gr)-5 b(aphs,)33 b(and)g(mathematic)-5 b(al)32 b(tables)f Fu(\(Do)m(v)m(er)g (Pub-)768 4120 y(lications,)j(Inc.,)f(1992\).)324 4320 y([Ag])268 b(S.)33 b(Agmon,)g Fi(L)-5 b(e)g(ctur)g(es)35 b(on)f(Exp)-5 b(onential)34 b(De)-5 b(c)g(ay)34 b(of)h(Solutions)f(of)h (Se)-5 b(c)g(ond-)768 4441 y(Or)g(der)47 b(El)5 b(liptic)48 b(Equations:)70 b(Bounds)46 b(on)i(Eigenfunctions)e(of)h Ft(N)10 b Fi(-b)-5 b(o)g(dy)768 4561 y(Schr)g(o)g(e)g(dinger)40 b(Op)-5 b(er)g(ators)p Fu(,)40 b(Mathematical)g(Notes)f Fw(29)h Fu(\(Princeton)g(Uni-)768 4681 y(v)m(ersit)m(y)35 b(Press,)f(1982\).)324 4882 y([A-C])214 b(N.)29 b(Angelescu)h(and)f(A.) f(Corcio)m(v)m(ei,)j(On)e(free)f(quan)m(tum)i(gases)f(in)g(a)f(homo-) 768 5002 y(geneous)34 b(magnetic)f(\014eld,)h Fi(R)-5 b(ev.)34 b(R)-5 b(oum.)35 b(Phys)d Fw(20)p Fu(:661-671)f(\(1975\).)1894 5251 y(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 324 548 a Fu([B-E-S])135 b(J.)47 b(Bellissard,)52 b(A.)47 b(v)-5 b(an)47 b(Elst,)k(and)c(H.)g(Sc)m(h)m (ulz-Baldes,)53 b(The)48 b(noncom-)768 668 y(m)m(utativ)m(e)i(geometry) f(of)f(the)g(quan)m(tum)i(Hall)e(e\013ect,)53 b Fi(J.)c(Math.)h(Phys.) 768 789 y Fw(35)p Fu(:5373{5451)31 b(\(1994\).)324 992 y([B-S])234 b(F.A.)65 b(Berezin)i(and)e(M.A.)h(Sh)m(ubin,)149 b Fi(The)64 b(Schr\177)-50 b(odinger)64 b(Equation)768 1112 y Fu(\(Klu)m(w)m(er,)34 b(1991\).)324 1316 y([BeS-M])102 b(K.)43 b(Berg-S\034rensen)h(and)f(K.)f(M\034lmer,)47 b(Bose-Einstein)d(condensates)h(in)768 1436 y(spatially)34 b(p)s(erio)s(dic)f(p)s(oten)m(tials,)g Fi(Phys.)i(R)-5 b(ev.)34 b(A)f Fw(58)p Fu(:1480{1484)e(\(1998\).)324 1640 y([C-N])214 b(H.D.)46 b(Cornean)g(and)g(G.)g(Nenciu,)k(On)c (eigenfunction)h(deca)m(y)g(for)e(t)m(w)m(o-)768 1760 y(dimensional)72 b(magnetic)f(Sc)m(hrdinger)h(op)s(erators,)80 b Fi(Commun.)68 b(Math.)768 1880 y(Phys.)33 b Fw(192)p Fu(:671{685)e(\(1998\).)324 2084 y([deSM-Z])57 b(Ph.)37 b(de)f(Smedt)h(and)f(V.A.)g(Zagrebno)m(v,)i(V)-8 b(an)36 b(de)g(W)-8 b(aals)36 b(limit)g(of)g(an)g(in-)768 2204 y(teracting)28 b(Bose)h(gas)f(in)g(a)g(w)m(eak)h(external)g(\014eld,)h Fi(Phys.)g(R)-5 b(ev.)30 b(A)f Fw(35)p Fu(:4763{)768 2325 y(4769)j(\(1987\).)324 2528 y([D-I-M])125 b(S.I.)40 b(Doi,)h(A.)e(Iw)m(atsuk)-5 b(a,)43 b(and)c(T.)h(Mine,)j(The)d (uniqueness)i(of)d(the)h(in)m(te-)768 2648 y(grated)e(densit)m(y)i(of)e (states)h(for)e(the)i(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erators)f (with)g(mag-)768 2769 y(netic)c(\014eld,)f Fi(Math.)i(Z.)d Fw(237)p Fu(:335-371)f(\(2001\).)324 2972 y([H-S])230 b(B.)34 b(Hel\013er)g(and)g(J.)f(Sj\177)-49 b(ostrand,)34 b(Equation)h(de)f(Sc)m(hr\177)-49 b(odinger)35 b(a)m(v)m(ec)g(c)m(hamp) 768 3093 y(magn)m(\023)-46 b(etique)34 b(et)c(\023)-46 b(equation)33 b(de)g(Harp)s(er,)g Fi(L)-5 b(e)g(ct.)35 b(Notes)g(in)f(Phys.)f Fw(345)p Fu(:118-)768 3213 y(197)f(\(1989\).)324 3416 y([Hu])263 b(K.)33 b(Huang,)65 b Fi(Statistic)-5 b(al)35 b(Me)-5 b(chanics)p Fu(,)31 b(\(Klu)m(w)m(er,)j(1963\).)324 3620 y([J-P])241 b(R.)35 b(Jo)m(yn)m(t)i(and)e(R.)h(Prange,)g (Conditions)h(for)e(the)g(quan)m(tum)i(Hall)f(e\013ect,)768 3740 y Fi(Phys.)f(R)-5 b(ev.)35 b(B)d Fw(29)p Fu(:3303{3320)e (\(1984\).)324 3944 y([K-S])227 b(W.)47 b(Kirsc)m(h)h(and)e(B.)h (Simon,)k(Comparison)c(theorems)h(for)e(the)h(gap)f(of)768 4064 y(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erators,)e Fi(J.)j(F)-7 b(unct.)35 b(A)n(nal.)c Fw(75)p Fu(:396-410)g(\(1987\).) 324 4267 y([G-Kl])177 b(F.)46 b(Germinet)h(and)f(A.)h(Klein,)j(Op)s (erator)c(k)m(ernel)i(estimates)g(for)e(func-)768 4388 y(tions)35 b(of)g(generalized)h(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erators,)f Fi(Pr)-5 b(o)g(c.)37 b(A)n(m.)f(Math.)h(So)-5 b(c.)768 4508 y Fw(131)p Fu(:911-920)31 b(\(2002\).)324 4711 y([P-Z])231 b(Vl.V.)48 b(P)m(ap)s(o)m(y)m(an)h(and)f(V.A.)g (Zagrebno)m(v,)k(The)c(ensem)m(ble)i(equiv)-5 b(alence)768 4832 y(problem)23 b(for)f(Bose)h(systems)i(\(non-ideal)e(Bose)g(gas\),) h Fi(The)-5 b(or.)25 b(Math.)h(Phys.)768 4952 y Fw(69)p Fu(:1240-1253)31 b(\(1986\).)1894 5251 y(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 324 548 a Fu([P-F])227 b(L.A.)45 b(P)m(astur)h(and)f(A.L.)h(Figotin,)95 b Fi(Sp)-5 b(e)g(ctr)g(a)46 b(of)g(R)-5 b(andom)45 b(and)h(A)n(lmost)768 668 y(Perio)-5 b(dic)34 b(Op)-5 b(er)g(ators)32 b Fu(\(Springer)h(V)-8 b(erlag,)33 b(Berlin,)g(Heidelb)s(erg,)h(1992\).)324 872 y([R-S)e(I)s(I])126 b(M.)39 b(Reed)f(and)h(B.)f(Simon,)79 b Fi(Metho)-5 b(d)40 b(of)g(Mo)-5 b(dern)39 b(A)n(nalysis)h(II:)e(F)-7 b(ourier)768 992 y(A)n(nalysis,)34 b(Self-A)-5 b(djointness)31 b Fu(\(Academic)j(Press,)h(New-Y)-8 b(ork,)33 b(1975\).)324 1196 y([R-S)f(IV])91 b(M.)32 b(Reed)f(and)h(B.)f(Simon,)63 b Fi(Metho)-5 b(d)34 b(of)f(Mo)-5 b(dern)33 b(A)n(nalysis)g(IV:)g(A)n (nalysis)768 1316 y(of)i(Op)-5 b(er)g(ators)32 b Fu(\(Academic)i (Press,)g(New-Y)-8 b(ork,)33 b(1978\).)324 1519 y([Ru])264 b(D.)45 b(Ruelle,)99 b Fi(Statistic)-5 b(al)47 b(Me)-5 b(chanics.)46 b(R)n(igor)-5 b(ous)46 b(R)-5 b(esults)45 b Fu(\(W.A.)h(Ben-)768 1640 y(jamin,)33 b(Reading,)g(1991\).)324 1843 y([S)g(1])254 b(B.)28 b(Simon,)h(Sc)m(hr\177)-49 b(odinger)29 b(semigroups,)h Fi(Pr)-5 b(o)g(c.)30 b(A)n(mer.)g(Math.)h (So)-5 b(c.)27 b Fw(7)p Fu(:447-)768 1963 y(526)32 b(\(1982\).)324 2167 y([S)h(2])254 b(B.)47 b(Simon,)103 b Fi(F)-7 b(unctional)47 b(Inte)-5 b(gr)g(ation)47 b(and)h(Quantum)g(Physics)f Fu(\(Aca-)768 2287 y(demic)34 b(Press,)g(New)g(Y)-8 b(ork,)32 b(1979\).)324 2491 y([vdB-L-deSm])50 b(M.)23 b(v)-5 b(an)23 b(den)g(Berg,)i(J.T.)f(Lewis,)i(and)d(Ph.)h(de)f(Smedt,)j(Condensation) 768 2611 y(in)33 b(the)g(imp)s(erfect)h(b)s(oson)e(gas,)h Fi(J.)i(Stat.)g(Phys.)d Fw(37)p Fu(:697-707)f(\(1984\).)324 2814 y([Z-U-K])115 b(R.M.)36 b(Zi\013,)g(G.E.)g(Uhlen)m(b)s(ec)m(k,)j (and)d(M.)g(Kac,)h(The)g(ideal)f(Bose-Einstein)768 2935 y(gas,)d(revisited,)h Fi(Phys.)h(R)-5 b(ep.)32 b Fw(32C)p Fu(:169{248)f(\(1977\).)324 3138 y([Z-B])228 b(V.)33 b(A.)g(Zagrebno)m(v)g(and)f(J.-B.)h(Bru,)f(The)i(Bogoliub)s(o)m(v)f(mo) s(del)g(of)f(w)m(eakly)768 3259 y(imp)s(erfect)i(Bose)f(gas,)g Fi(Phys.)h(R)-5 b(ep.)33 b Fw(350)p Fu(:291{434)e(\(2001\).)1894 5251 y(31)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0309231123226--