This is a multi-part message in MIME format. ---------------0303141029613 Content-Type: text/plain; name="03-116.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-116.keywords" spectral theory, random operators, multiscale analysis, integrated density of states, Lifshitz tails. ---------------0303141029613 Content-Type: application/postscript; name="divergence.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="divergence.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: divergence.dvi %%CreationDate: Fri Mar 14 16:01:31 2003 %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMBX12 CMSY10 CMBX10 CMR10 CMMI10 CMMI8 CMR12 %%+ CMMI12 MSBM10 CMEX10 CMR8 CMMI6 CMSY7 CMSY8 CMTI12 MSBM7 CMR6 CMSY6 %%+ CMBX8 CMBXTI10 CMSL12 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: D:\TEXMF\MIKTEX\BIN\DVIPS.EXE -P pdf divergence %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.03.14:1601 %%BeginProcSet: tex.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/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]{Ci}imagemask restore}B/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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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73dd246784df2c98ec9218350886e30121406cb4d09548c9b9154bfd677cad45 5421f41f72c885d64c25c2d21d840207fd2b8074ca5b13970566c4baeab581e3 1f2651cc9258e203eb8ad95ecda7c3a4a0f0d38afcbb63cbd3b48286bfc7d6b3 434eca899f4b2b2ad0aa88fa20c747ed3a58a8241d9c8f6b94e60248366682df fdfeb030f33628a30f7f350a43347535411c5e90e74102b2ed6f1d68775edffe 5423067b20f08be796230cca114fd28a3149e3c213adb850335083c09250af4e 5e931c5eb63116eea4eb5a296f1775c968140c2cbeba59437a52b9af0d1c0abf a0e93f98c7af71533fe0eb1dcaad3395e05f7901269ed1a779e51d92ea011c01 b6490e1f48df49d343dbfda1d8c5c7f2eb82bd6cd4bd1f8531e67ab84c8acf99 93394a52add4dd71dcd06d8b47b002bbd440dcd634363a2f5efc91d4a2fd8c81 ac24d7e0ffae26851c15428541b0100d179ff9342c7647ee0244df05c7c774b1 b08e8095aa2ad2ac3ed85a86ee41fc0b3e8922a2a54d0b6eb7e172f97ae0a3bb a19732e94ecb389d95df95bc9545fec86a81366578e1f7c3beb18055e3a6fd7b dd7e258a106ec08e1801d620336bc3faa872b03efdc7cb9116d795b4fbd741a6 bd70e278bcd69ce69d6e61c8347d06e4c22c1e6de933db822ffad6ac8089a727 235b8c947b6de97bcede7b4fab16c3d7b8aafaeb49474f8e1a931de792aab241 76bb4e75dbebd5af961a74de5aa4702ec9b6acb63c2a993f68c9bd820bee7358 c44c4596f126a0f3bf8ce6c0c999f1cfe0ab984ab0781172f3928bf2e818adb8 c3053449daf0e7554a6c8d9fdcd38911fcf1f19874cbafcf98f91934fa8b6b4e c27aabdfb75722bc52b14e74ddab86c1c1277c2b5c5ffc1be3a570ffab350b77 b5ee6013662a590510d1412b7eab46d5216ffd59445db5a5d2d13c19ed51628b d00602e2a3eb26b8cac2b3d1ccd447978b4dd51c3c6716e93de27060d003d2b2 6bb6b30b1b52468bfce48a632f114f720f6fd82bef3a2d20c725eadbd588ffec 7471020f2af5be0b427be0b47b0263912ce19d62f1baeb8b8fc64457d3e7d4a6 185722594a355c750b7c8cd9df062e7e6c2bbc3133297a2db9125b60d7a27c33 5b077c66a2d97efc3a685a9be477743958d3812a0b76e11d53c539b43e07ff99 aa61e06a00f984e5d1246507be468f59236fd5ccb96f9baa41ec16d0031b8145 ae16acdaf963f23d2e63fa4ac5bc5b0316dd1feff80683aed6e9fb9b0bc14b0e 37af7121f71041c8c18a5825ffad1be1951a7420c0d4094ebbe31b9b0fc97a15 303f5f008a6e7c35f742cdf92a5c94bf6ec4c5e13ccdf733dcf4705138f8c59e 0a1431b2bb647a588966039e43ad27d66e4f1ca96c3f7dc72c7097ab619edfd7 7750b7e5cac430d6dbdbde8658dc7a130b3251bd9b6becce98b72fe619265b98 9172282c74efcb73cb632bd15ce1ce45f064c53a606605020a2a1dd1274c6bbc 2d313b77d642f944bc9c770a3c0809f6d2b012ce2a30fadabf2c938462c9f0a9 8ac775d802139f554f4b02a93b90a7ca7ef10c40a10604a7e01c16aa38bbf0ba 53ce68674de86858aebaafc142f1d7783b83a5cd8ed2ef300ba06add30af25d9 05729306f792fd03925e086d85f8f0981642799bdd9386de601665c7c05df144 d268b95e4fdce23446b1040706712316c75dc7419fe26a7db90749824998a10b 89d754f059786993f1d1628d49921fb9db30dc6a3b7caad091b98ec7476a9b63 d5a7c6604111d4b576baeab2ce638b69f74ddf9542d5f714faa91e8f54f09956 38a67f4822d434e448a8132b5136f4fe2c44d0801246440121d5a33453049155 677c150268934adf7c5d14eed745f591df4796c0fe19a0f495c8075531928efa 0e8c8b1d75a4a00d87cf8bb849022eb56bef79ce59d7675dcaa9cd7c72aebd21 cc5a6c369e023fb1be8279dcc44192bd8cfddede407cd071176f5196537559e6 c6865e2524c46ea2edc19d541b38c939f453aa892b8d07493322092123083deb 1a3459d38457d9eed5b487735b398db322f7f2c29c2cf69d7339ea799841775c 9cf9d7a30c82cb1f945fd3b4bff2ef30b16f7bbb7f75fc85a4b7cb9d2bdbd51a 556dfae4a4342efa6392972f55294ca54db33699154809fa55e44b1bb5a1fed1 f74f62f5222dbacb347ee2b31c6af3ea33fd047f9bc93fd441001e0b337854c4 de115077caf798df0217f5647be5f730208fddb321f00cd719016d8d041be66b ad8ad3a7eb0bb737fbea79e4ad8d733f29aac5e5e7558a6482481d10619eb85d 476b375887470fec2c29d724903419b6aeb9d054ee4ba45d28d737125082cf32 ff96380be4236e0e0acd7b3652f00202ba7f16f693e69afa2c0b0b934d807537 76e2b5a1626f3d90cb8a2d60f4c83fc1d25f5d3719235b2afda0a4a78ffdffb1 3e5b28d249dbbe13cd2f167452e9c223eee19e3cdd89f57cc0b080cb51b40593 753306dcd1512133aca612d4c5abea367aef1a013a2c3a7b3e55c75248b48b5a 24db57df2f43072ac54e35b2f8547f6f7c9b6ece76047e93253433be729e6883 c9f8721f76c29c0fe6daf4e244d7fae29c7b7bce462ca09a6a374ec8d20af4f3 21b6a8438be93bd3f6c5592ace9b398ff65f0ab4e7e49c3668413670a1d0f0f3 4aacbd5c03249cec134145ab2e5886766cff6e1ef166f1064a7135275ee4cb90 4e67fdb521df7ea6213cd22cd4631e2f4490b2a888699bbdac62e5f2f4c407b6 33da75cc9fe780c72eea94c2bf13caef874f42339620ec6ae460cf3b734d88e1 4901c19b767b9e9571c44fdbb0b08ae9e7cf1e8176cf4a5a0d4518e2ef03e52f 1949d396a85a6a3af9c9fe8c29581c3de4e2f25d6ebe19a3be525f06b3463412 f214d77bfbb70bb9d8665cf5a053f2c72d4dd79cafd01c5d8bf47ed17462416e 42103393784296677ebd481368474fe258b73cba0000ae1738d08539b4db69ba cec84c17c3296f3f23e77178128de7742ec148e899532978138681c600e68a0e 646a58d46dbe2d16780c836b274b935a1cb7e6c0e8db290d2159394af99578d1 f0d35da2779a982afce22feb82db55287aea888a5517be1657d773f98a21ab66 221e1b0eedf4b9dd2b5d64da30e1569db1052ac4b67bb1149e21b7e72233b40b 3079d6089f4a56a47c147fb6e7e6623a5a669758a687f0394d018268081a3861 70eca6cc7f9873ada2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 49 /one put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueXX 5000766 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d38c2fbc0984ce6a04f5d3bc84e2601397499a981b200b307 771141805f1f5cd7b12befe020f7121ec103dc2baf17f0aca1d0740c92a8dd61 12fa9c0d30cdae4f2af757b9f713c127d6ca02ef315205b81eff629104b21462 17d2b0f823133bccc80c9e400d6840db5fdf828f845313290fedee040d34f309 8f1b5a6b8941c81d1e607dcb310b14169cb433329994b4d1ac1b491e7422d518 2a8faf7ed3cbc2e23c58a2e310b56da54f51360556d27d58783694c51aaf62a3 a2db91ac8df80303c35cfd503341005020a5cf501fc0b64ff4f497951df6cd50 f00fafb04725d23bdd36616c7c06cd9052305682f6b6e521882922377b4979a2 e6f031b91b2a86d8631e81de17f90158ca05045c3b823da8b2d2aefd37a32728 45bace7594ceaafaecbf6140212789b3937a24cdd8dee327c934a9b0e328fd83 a1b535d9bbea6f0204ce5ee74cc30769734d4fa3dd0cac3986510c99ba062868 9285146cbe64caa89cc2f18a61d924e7f312fa189e0ae4deb913ed1cdfeae61b 4e0fe25f2775a8b2edabadebab4eb2ecf47258c1a77101c810cccd3b06633907 f765fc80bb1a2527f8a59c52760fffabbbac1d65a11708a3bea4557f24536773 d17dfd0eeee4e22f406e04765cf582d6ca4414f2441b6ff463c56018e44aa2fd c9af3f0480da99856aaa97e56b4bc28d9a74e6d74e5ee199932ba9548c9f5a24 4ad6d8509e006d51a96e77cf6decfe21ff0391441c8b95ace9dcb2d93694eb75 e76e0ef6fcdfb8a73c860e09dc49dcfea499450162b7373cccfe5e771eb05086 9552a2ff03a1c1c6a08405ec3ad93fa9c951b4aa746b4c4bf3f98a916ba7134e 93d777c339423c4fad91e1968b981744bda6f085de1cc197203a9c445611f874 04f536e7380673d2182990e967bc7e504b728a904bbb58f9576caed31495a93e fe7f1b8e33f5bcfb5419fb70481d40400c9b15efdbf9b60f888d18ae36f2f6a8 0a565df9b1badc8b7740b3c33d9fb134ae79daabb791fcd60e3e83f101c9de60 d438b1c6788a71d015041a218363fea5062fbc3523a9d872296fa925e602a08e 4f58e4908e95008019e24405e61944790bda4ef08a9d76666ab4655a1773b621 c4c418c7027361f324a76fcacd9de730a57c94d50a587dbf25e7e91d35df79f3 859bd9316b6c6647181d54cbb568a59907d2ec78dbf282fc03325c7da2b92fca 1eeac03d727987c4439c0ba3474776458e62a1aa3290f706235f04 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /parenleftbig put dup 162 /parenrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 175 /circledot put dup 176 /circlecopyrt put dup 179 /equivasymptotic put dup 180 /equivalence put dup 183 /lessequal put dup 184 /greaterequal put dup 189 /propersubset put dup 190 /propersuperset put dup 48 /prime put dup 54 /negationslash put dup 63 /perpendicular put dup 102 /braceleft put dup 103 /braceright put dup 107 /bardbl put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 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a355ad3ce9e43e777639c18e78f2e74441618d27ce6a7f72a191d23658c91b1f 63450c2793d56c0e5eb4a76d7838e865e4c064ecbf0942e40cf87ca37336a40d d44e6c5bdccd3e42a5bda9e2ed06db71e772ad2a8ef8c29f4bb52aba0bc6965a 01f200353055ab3361596a85f9aabbd5190316ed77884fe9a89f2b35e880249e f895184d6c6800aaee01f9de4a3e91fa2cfb93a59eadbd73db2ac5ff962eee40 1424c9d7039e636606bfafb1e78dc9158d2fd5259bd0550a8b7ca4656471dcf7 a1010f60054bce9067e0b6f12b179ab3f41bdecee1fd700ec6d281f180229317 a734a33294300978ea374c71c3d06bfbe132652401714c21e5cce08e1063ab7b b3d259f2e8a6bf9e93c96f2a913fe30727aad74f025f0efc5d819e3b66a4dcbe 5c38ccab45d4585a6433f021532bd97f5250c148a41bb98b3087e7ae878f5cad 93bfaf3404409ca700e5bee86575682d6dc768e773d57070f4433a07bcbaf6ef dcf94483c1d7cf0f6327963c491b65dcd355300a6f046a730aafb27b59c7bc0e f0dac8f3d1dd47f6e7a4347e2dff047d3418fd9a44d5210d1a4767837bbd5716 65cb095bb7f96cc3bbda325d62ec8e368b216cc331bf8bb0c3d6585b56e062ec f5b8a7bada4f67b4f5876b817f897589cc2bdffdd67bef8cf79e8bca8bad839a 20b42ee2c0bb815d8bdac39241fd040cb7a618200ccbb92db4e2255a3739a80a 5287dc8d8f17120c58da51f66d7fe47e2cefd1ab074fdf043ed53b763f0fffdf 467a295d9f5d942a1b91bab86446812e194d3f7cfa7a79f96840b499f959e685 2d417c6643b8c338ea3a14fdf6bf0652021064dc75848bae34a0a6d706aefee5 761ab024ea230718e7b12304afef35fdd97126a3a67cb07eaf4a94cfc31a1a40 ad5581539e0e3dd2b1b82ddb3768c8d8339d6a0da0c1537e47f132133cd46d94 bb5fcd2500ad535f8d97fa82c5d49c95fe7a59c9e412e8809e1693ecff1276cd 251a0b1ee86c0bfa33b43c7fb3501ba1976e5b3fd8e69487796247a84e0cb810 e8cc71ba9bacfcd93265bf531289fd649de779967cecc3f1d861445e97af3ee5 57103faaf20d1dd33d6be440438d46d0effe72f07338cb6248bd64520f2b9dc9 7059cb49270c5b84571ec4eb74e8315f12e33cb4c9969436e53b8a3a526a6e94 3c04196edc7401100e73a8b214c02a64a5b2925d31ff6beb3e8b64ec6aa6983a ef7457577a1cd4efe486d6707a723709bbd0162fce0167df2d3aa061aaae9cfc d321b0ceb78c4e846183565a69c7723191b09b8daf795321eb29a9884d3f32d8 169da979a7c228a1fd973bc076c698ba36fdefa1a560248cbdd711cadc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 176 /fl put dup 179 /dotlessi put dup 180 /dotlessj put dup 183 /caron put dup 184 /breve put dup 189 /ae put dup 190 /oe 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 59 /semicolon put dup 61 /equal put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dfb4a28be726d326 668de841816f412bc43d643c551b3d47cb7cbc4818ae3fe9ff9dc327868aeb35 0b49a9cd9c4ea904cba6feb29b4083755eeba6a955d759d8a7ef61920ddd16f8 eb076e471a0ed368f9d6de5e1d7473c189e6225bc7db81b6646a3ded9493c9e9 acf9ac7c0f9c0a744f4b05c972959ee524d1f3d2b66538de42bf404ba51a3c08 34cad90c0b841a3ab19fcc22e7828c36c751aec3ad298f2c3b2ae9abfd24f8af 4b85daf87cd926a88abb17f242b420996fb2ab2758808c037c98057a750cb49b 0bb72173d18f2a4ee8eb059b92e5fb814e1efc5a7be6458c8240aad3b3a9ee81 791992c929fd02fe1942e3d231299bdf536abdf587d884098498b795866ba2ff 6f24bf2dfa42fa87c4221136da1d9ab0a0ece93cfaff1f755df84fe4c0134e7c a941a0b822997a20f2f627e8a58e7659d8877a8c0137d4748c77f3b43d421c2a 1408abcbd28964743c76e7f561bf96711e271521f313b55fda7bf6d557a31e4b 8c577aa035037b946d778c30e9426f9e011b48841eae5ac7c90d86d1b6ab834d 242167ade20bf52fb111286789ea605802a1b91c6908c6cd0cbbdc9a3028f95d f50da494d49a7a2fc627428f6a457d406b94426df0e3339ecea75f8508888e76 b6d525de2f62d0a92a7865b44bedde46f3c91ca4c6c516a5cab9c5da7af9664b d2242f61bb3c319cd5de2152f21b6fe624210cd3272c949b92b09e8329f38934 8f1fd5624ab0cd4df9479aae297f6e85298a049ebfce95606f02ff86ecdf9650 21c767e60d11a053bf258d7ecbf199d9272cd26861dbf5d4b5dc1a03d2487dc4 2ca4b4f8df18e9433889ca9bd62c83bfd1bf10f6776bb270604f44b41e1b46ca bd9e9e995e908ecdac94ca1b1088b13e246ec116769b15fabf5c5206ac3a85bf 97f587e81c050f29043f09b5ff770258366097b0ccc09cf1bd57769ac4c585cf a24f2fd01ce0e50ee03ef4de65f9f146b065526cebaf8d578661c42fea7c024f 5f53eae70687397d0b357f8451c927eb9d806846ec23b8963b3e049fe5558aaf 6eede2557d216301d1e95d9b7b6f1d0aa3e29c51c69668919256e86514e9b473 76d1e14a24c42326746c3491c23566ee68e549e448acbb678d12789e796361c6 1b3246a02e8055bd6af71106be7ad93c2dfadfbf1ada25b5421f4f68cda86a06 6e604b5ca5fd2fc85d08df3685677e7d6fd8e7507f64b37b8f7f2a6f1f3888f4 f36ffd3fc8f7448aeec1848e952ad0d54e485593f1942d8e0657aab983ec99b7 49b4bbecca19acde909cf3a6799b15576126fd602a2e610622d7d6518c3f98f7 46007bcc4f89ade94b46cec72cd4787f31673ebf71054e98933bf3dc2345aa4e 3efbd115ed905b33a22303a9e444621972b4c7200fb395c2f92e6e25dcb24ca9 c5eef8681f71bcd37d37bff091edba6f7ee872e61970a953dc9e011cc22994d4 345545f4b10e531f96115204a2f7abcca2dda2c53b1f67cbfcb424f5defdc53f 3cce4b2955cce4d63ec451c2deb39f7b9dbd2a0c565c22def777007f64bc43a1 3dd0cc26714b44e2a49b33bb3c64b0f806b9a3c184054dd97fb0b75704eb0799 e0c2dd497f3e221655c455c282baf9be9005f6ace300329bf68fe1b249100110 6f744857a429d02193cc9bc2fb25c9e46324ba9878ced639e670f9a79524c647 c820d680c0c1b1ac4cd2bec2f2b05e5a6fcf16bd19eda592a54435267a979177 3d1a6df2656f03990d93e8f432d85968a1c9a746cc61a12524be416b2efbcf7a 20d6b726117a29db5e8c2aea1656a5d380a304f68788785113ffb983267b8eb5 e4cba02a817b816375ca24eff4bab8e97f2fc1417a91c9f8228d9107275664aa 6f03e5dbab6f950ff95505c98a02c28b6598a29105a3a1d6d0ebcc289452de71 6329c7c5f19e092af06882d8b7cc6185b4af67f88a3865e638b0852ac7c175dc 760f29ccb7e90667c6ee896ee8c9d3c9c78595d468f9ea9b59466a46a2f09bcd bd08085dafef17ccfca3cdb3d19d24876800ca3b9d921ca33cb5c690ca92a0b4 c5a0e8ea5a2cf38cd91e501e4737f6b96553bf0226b67901d61c1336fec14b8b 4fb1d5e25d8073da377b8aaca692be03f8f94c46f0d2d91b1e95067094f4ed44 f0d19a2c49d111642d479202f359a1539d3bc01ff68b87c1b074cc3438bb2505 cdb079e14654f5e36ec8e690f1bee1b1bf724dce0cfc822b2fc33e76da1520e2 5efcfec76f9ce8e77778d0ac01804852a997be715d2ad9031e76c4bf09510f5a 3383d3d0febb0623711b10e6bccd6cad9d72f20d95db70554f176822a1a2ee10 06ca6ad992f8d7c6c33a0ee514324c96f64266086519cf03136d8a2dc66fa9fb 77c4b270e66f125c675dd4ceae0285dbd9938ac108033acff7e7cb48dd03e98e a3921ce7ab647923324e7c7cd0cf374781e6fac6ff6e1f91b929eaa31ba69b19 9a21d04816211c8da4e49983a85c71e4b8dd54edcc95beed681640cfb46c787d bc552ad8a0d97449080f9170f85666d3d6d9e4ec526a64c1670165afa1c4508f 3612549e60d62e800c87a8ba78a093c0a18b2c990396108de34973439753f4a6 c57d08f434adf944ab92e82b4c68589656e19870391a0b5420dab75a2ab735eb 254c65941d13cda80b30deea04cb81fc2e74a67b8f95b7ed3410c4fec8c86d2b 863662cb83e8fd40c260372a074a1e88ddb8ed1c5349c806ed7f1b45281ee92d df14d896530c962e105348362c0ae4e5673c5d470115996b43990e47df0c719b 3ac20ccff8b9ba96b11fa65f5a2563d95b351aeb2c6e7a5aff2e00827392f726 98fdede0da7649d9f86e4fa27d8ab2f0777ae339d33418a08b81cb660b5cd286 fad75322dd6471be58ff8adb9cc71a55221391848e40905a4d4863a9b51ab055 10bfae4f54c1338341db0288ad49e08a5b503e10c083db557978bcd90ed5cfee 7cc60d5a2109d40def33e293dd99389bc1294d628d79390e7f2c7a6dc3038307 882793cc4df0ce890f9970096f44a1667335d20737d2bcddf84f572f1f796742 1c5d059fcc6516e74bb8abb750e14f52b68d72e9aaea9638f971515901b15d9f e8cf7a4c9cdc11c3db69dcd5c1390384e2a414f38e673fa88f6bab31d582ade4 ed7f3071b812d01a8e018269e60be0bbd5f07f3978b60c37ed0292ce19e8576f 3508822a144d68ea1b833cbbc3323aa76acca1845231acc774ba965864b8c796 adfc9d3e6bdbc00da5de249120bd1bdbdab41b66e0aa6b7a1b91de909c4067e4 93cd76dbcf2d60d6054bf2edfe13bb64876adb5117237c59ad018bd5909bb577 59305956f0298c5fc166793f8965784d01e327124371dd23191caf715f42ab99 789f61d08e8bf646e32bfd931f9f362301fc39744fbfe052c95655de6a23422e 8fd56f45d02c16beba343504b035a24152bdcf184cb5e492b16d67aaf26af385 a73e3eaafc310b1c2a72223c8ca8477d2862499cbeb183894b453dbbe0ca1907 d54332d097ed10722f5ceb8c323169036fe9f08636a3f7ce491aff10110ecb09 73cfe4f1a14547904db2012b98cf58e24d542defc624ffc441cc7b4699020247 9d54c2a95895e6db0b0bf5e2a04e6ad41cca963cae04b3a1a2fa3837fb6d13fb 7f56330f7df3b787ce7503f63ba13279fdbeee45bcdd7e50891247f5f4176293 032ab458025e980583c3feaa35750d171cc4c166e781ff259c96c556c8f72a40 3b8b94265b2bebbb91efe2c69c38a6b5b1c61970c1b1ccd48d6fee745f45b3ae af371e0b95d5fd498cc1933f2682570b6f13d559900d7dffb260d9746ce87c56 a396c594708b2d0e08bc5569a1753d3d26a0e6133a24cc73d97e9dfc7581a2d6 ea4dfe4894e9648755278664c2a9944141fd7a1d433132edb951462cb8b7076d 1e3c3b5dabb9847473d3f2a2cae94816c95bd3a4d3ce9e1c859770fda9af657e 5935b442ecba00fbcbdd0b954101a885b6111b027d9edd63d3950b1d2d80c20a 51b31e846a7dd3b6dc61b7a5806b92edb10f2713874b1b45a48495350e1cb89a 01f1223ba5da12c36ea5fc457798ed43a07aa38a39b72dc55e1e79cca8332ff4 63e6ca5ad9a00cf79dd78c35028f41a305acf41d16d28290455679475a87cd2b 6df9ab8d636306ff53ff3fb251555e897952cfc705636d0ea519708fbd5c83ea 0bcba192dbe00ca97bf9cbbf89e7a7337876250b6c70725164cb3176eaa07eb6 f025f3877b79daa16f3e8a151dc324802a6227e7b2be192cfd12390125f8930d fa1582d50916d769ccb334fbcbea27a3 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 161 /lessornotequal put dup 162 /greaterornotequal put dup 164 /notgreaterequal put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 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dfd4b231fae0ce32d12188ac93b26483b71d4c6f158c5f3b7af7addab8bc3105 e9067d7afd513d8c0f1ec984357055dca17ce8f7f67638d6d8ccc35541f2fbc9 2ab144f82a99b90933a0cb32070a0c34e898ca1b1e10564f44fae6d2a13ad397 fb141a5c1131baa638c4a206ee3d1a6d0596b17057f4ec24696c5efc2c8e315a 9dba3e35ddeb2016388df2e2316307476e930f732f84564d41b310fb8874405c fa5194e32f47f699bdf24e26ea6ad261ea3e1bbfe5c8f4e433e8411db39604f2 6d6af258532c3f5a6131abd99c9b19f6f7580f93059b84f8626af1f3c2f45afa cb50d7e6705f9a90f0219a8a91d9c5abb58c53a0213746bddc4ea9add49ab72a b80fbf61a6eb3f75c78845c44e480080527a78ee1245fc335769e580e3d580b7 fbf15b6f77f4f5993faca0700080ee0068c1e818556b9fff8d0337045d069c22 c4d4b518356588cd473d322c378183a563b8f1e78790aa9223873a756639db36 1b0dba8afa4eb2747aaea0c2d5a49d875d2b07b114f4ce6cc0b68ba18d560b76 0ccf4f580822daf5ee8b2b2a1efce468fa88082011cc8da6879ef0757c7328cc 6db298c80f10d5f5e7a2fa0bf36455f9d4cbb95f192a26586c51ee73a3bac8af efa16b4a13f4970b8cc77c8289ce7d13b923b49c63eaecb64df5d8377db505ab 338c02ecfec03b511842dcea951121f968e6fe416649bf556bdab9a5fbeb2de0 4328db3d86deffff43fe42d42645465a1f5a831f38f2c4d3add5f6ecd06f336c c29382125204c122601737974c1966a9a93f5f7d0ff24a92204fed34b9a21ef1 e6308139d1718c4e55f076a9b4f8e563fb527152443766adb859bda87730fcc6 dc8c9a11fe340c2a02e9b75f380e7e75005395afb3907cce5db1bd078df3d9f6 d93562575cd7a78916aa719465d29e9988cc32d95f85e188af30f2bdb5fdfb65 0b2cba505f6b25acda84fb440ce3894f67cf53ddaed9a539a2deecb8b49de285 88f7ef1e318680445624d09aaad0268b7dbc88621084265ae5ee0ef8947fb15b 0bcfc3858a982909eb3ea613d83dda8b8494c617ed3907fe25361a01aa727c2d 9107d668d4a5e4c9846be13149aaeac2530917e3503dd66472f4f45bbdb4cc8d d425bd40400661bde656d6d88725dfef43032c616b8d74dc8b4fe2d36868edd9 c494c4531d6bd63b9b7cff31520bd801015dd86a894f92fe5850661013b52fe3 1b726fc3f485a92f2d274f2f729fd3fda3c4838c698b40b11e68a2dc3707e298 fba9d3bffbab865b797d2232ce63776cbc6c672c53f0e8fba58061d0c16f0bb7 ea20a31288ff9f3d3206c40979b8579a7cb86d566b3ec416c11b0b1a58df6b58 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500139c3e3356b3f1b4fce49179b4482073bdbcefa4e297e675eeabdc355b454 3990fdcabadf4ba8e591422ec2c595d47533e9533f1198cebdb9382dcc009e1d 7b2add227d3baa2c414322d114bab0ab7b06bd8ff74dbddcab7925c55d1f49f9 56e823552e79e4a9f6d192f72f3e419edb9ca26225b4adc3419bbe5db27f963c 6cb7f25312994ae560ed4718e60a3e26612d403efdc4557fafa52002cf8f9b8d 0d1915dcfc1ea0c87f0ee0549a8351a0819a56167fdcaf5728d1bc149c3e8dc2 018f9bccf5306f776ca29c2dd4aeb76e991d9551625399fbe6ae4d763a8756ba 46de7abffb5a17fec81ec1268d7dd92ab48b20873cbf2df1c278eebf7e6632fa 0a1803f6124ff5570e99cc28f8c628abdaffa878d9c653c2a2217bc7459fc1eb 211c8455bf590b07c566deac76d37052728f151cdd23f58bb02530a17a34b46c 28dde889a838c30b26b0d015e61dd5617da4af1ff81f001b3b5b3ab431728338 65fee2d57638db7fb8222f5da595a99b8f8486a1e2dfd1666613716ade8be7c9 214f051179ca55f039fcf409dff31fbccb4cdbeb4fb756386409b8c698d7580a 75972539786e7a124b5ad2a531e0b7fa94e7d79d499caf94bbd54e7bcdd2ef46 a8c2a39bea4714ae5a079bb912df572331526fedc279cb041466ba14e5e0738a 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422c247cc7bb5bb57d46757c61f669440eda08aab6d51c034e189b8ad5d59f9d 620d0ec7879a7ff2f202a6d73e2604ee2ac2c52eca4ea5c7a8e6bfdf6020200d 38e2c63ce50275c98377ee7055b30a20a676b7498dc4c7c465c3f5676d1bfc1b ba8e58b76cf6d927241eb3f8027ad94bd6cc806925a596e190f00ee5aedf1a99 4abf3e154eb9ff8a8a90da8d010700fd6612abd3b19752b343a7ed4053338c3d 3829226c95b59e1851de3995e5ef14fbc0b0f30016cb61833c93343fcb8e311d 63fe4c91facd94ee756e76595880e9c20b142f53cb070c4cb34276b7d3a38224 b9a143beb4ea119ab7888e99afd346c7179fa5edbe4cdcb2a0dcb0006757ab4e 669e9f1b8850c87c456736b91c20ddc6aef478376ee53933c737f6790c8fe3c0 678b50326d6cec8ee57185dc9de4d971a2b791c114fdb601637e37d446f55967 ced9d52aee44046f3840e5ab7ce99dbd9fff18f74f4cbba0782b0f862c8ddcab 9719cba5db4a0fd9cd93c49fac90f181976a209a490bacf4b668c84a313b07ab 767537698b0f8b954ba53ff39e34f7fab22d942c9ddaa2bf5c159119709566e4 5ca9e751cd61fc504ea7c87ba06a4522863579a005ad68e3f04cb00cfe211caf fa248ee4a9053d202ab38725a4f2598e4b99c1086581106e42ab735059075cd7 9fa3cecf7e59e74c02fd16a33232905f7ae3c1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 176 /fl put dup 179 /dotlessi put dup 180 /dotlessj put dup 183 /caron put dup 184 /breve put dup 189 /ae put dup 190 /oe put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright 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 65 /A put dup 68 /D put dup 69 /E put dup 70 /F 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 80 /P put dup 82 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 179 /dotlessi put dup 180 /dotlessj put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 189 /ae put dup 190 /oe 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 54 /six put dup 56 /eight put dup 57 /nine 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 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M 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 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 122 /z put dup 196 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 169 /circleplus put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 189 /propersubset put dup 190 /propersuperset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 72 /H put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 196 /spade put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b 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04026e9fdfb1a48013969724083310efa820975da8ffa595ea20034b187ba293 083e8b6d055cdb498ad246fb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 169 /Phi put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 189 /ae put dup 190 /oe 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 51 /three put dup 52 /four put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 79 /O put dup 80 /P 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 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p 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 196 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 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2e44d3df235de8ed5fbf2d2fbbf1f3d3f0f1fd4b3ec3603e3e8e573498ff1f6b f1bbf4ce93058a87fbb7088dc46149125bc0fb3f24bbc2c2ba2011af7f7df774 52823dfb2fb54b4ccfbd60131ce398039d4e883ae08a4ca8246affd1aca0b763 94a0813941e740a55616fa9f0d7680934693a301037918a6216bdfabc5644b82 5f29f1963dc05147e6cce21420498ae3bbc34cc4cbd4096d2a1645f062ad373d 51ca46c238d8ad9a472082ad9373987a5aaca8d7db62d34e7710c97084e19ae4 31641a8e4feb9779490de297885f829675dbb4dcf5ab35fb84c0b7594c631584 2432c519ec29d8dfdf04cd4052d49f8fb8946484a99a66cf4ea5949af88fa0bb 2fc02262233de71548f13309b98de863c8a36b285f72cb73561dfc9e8c90a38c 56bbd9270ea8ab4eeca8981b72fb65d77999c27eeb43d513755853c1a6433ed3 0452d3c1f87e8ac9eeb5fd6b548fe04bc14a4f0747adac67038bea551133ca5a 35a508cfc7c9f97690aada564f292dfb9daf35e4532a982384ee53e8a37f2de8 01cea091ef82af07777f2ba461c03f6148fad38d0cc7a60d1f8931ebbb9d8211 61a332add6e527dbf6f5f3fc2a0c4e6bbbafa9f9eae1ffc85a91347a613efc41 afdaf41475014f7394b76b8a3ffad57859df486dd5d430c8e5eecf8d92b0e4bb 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df32d35d5da7d2f3af0a87eb33fb5537819b9542f7ea3eb4a65193196fd2e223 83ae07316967b552387e1a2e19aa028b75c009a90095645451bfb1b93fadd484 b8d317a051cff7537dbff02ef4213e4aae5b56a45b5e92c8f3e3041c0117c800 0919e3647d53903d333028448717601d28018551f59f42c40cb537b0bde2571a ea47309fd25fa23874dfba3d05f1b6ec2ff3325e991d39330058e6950d6c303d 5a48a3fdc402239ffa757d425874d2f2db7e9f88678f91dae899eb6e98e43303 a12a8c3a0fd71f9eff926171c25f740182b69355c941f56ab992e86dd1319784 34817879f75a8268a22f1e1ce8596fd9d28a8d9584e79aeecab1591e92785f94 d1cf628c2b39500bb9acabed2839a2062cb2690f70c7cf245e580903b0671a62 faa08d2a00f7f09b22650288584c2160a9d358f4221ecc4fb92bead92dc74e13 5ccf9c3b48ec213929100b9ce30515671ab6d69afdac721d2cb298530caa15de 66cf62bac8dec4e1e3fb8b5b84df43647deded2797f46371a3dc6ab774b46c86 150350ab925cb3a6400097dd3516f195e630c1f60055059ece9d0c1ca8873a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 169 /circleplus put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 189 /propersubset put dup 190 /propersuperset put dup 196 /spade put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b87123038f7859ead991bc89f56703cf31c393da6f5b9f255cee55ff15 21c2879219ae38967e0c9c9f346fc5c95e953da9b395357b7483a8cc041cf0dc 58ec77355caebd19519bac5c5782035b73401f7a3d5bd27ebf1e0fac587934cb eb3745dc8711ad2553af4316bc0bacb511a630c80594d0741ab596e70ac7ef11 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930e1d755c6a5e36ca207153994b374477273a7b3fde9e0a0254dea8386f4f5c aa63df8a70199b98edcfcc01bd331d557068deb69bf7a80289e7dbbb5739604e 263822ee251132e811f33f74e7f1aab6ce51f0947c8cafb3ba61081ba1371e5e aade3ed640ae2d3fbddb9cfe6779dcf164d5347ff14e7276cc6904e26b85383e f38e035453dc0ff68deeb96bbe1aab13d09e444d7d31afcacde31ef650b3be47 b43ae11da7962f3ccf97112d4045ea7e2b5dc177002a9022df95232e2d098d3f 6bab56176359d1ad14c0d6cc5e4b6ea0818101b1018032e7e412ec849f0beab1 2f782cfe6fad0e3f04c855905215c7b005b47bc8412b406b0013dbc09eddce3a a8b9227c94afbef63055dfe57acdf0c8b153b99a78867dd7b21e46c710017722 e2e44e5ce6fa5cfcb6ba4a8b35277f2e225626881dca423f2f5be97573911c63 b0454d47227be27ff6cf65aad77d6381da47482e34f48f36e06738be1a9e32ea 80c96efe4bb8e0de0dc006191a208a08cb08bd89a933e4b65cb617b2d393fcba 695c3ae80c30fed6304b210fce16c218f682ef07422c16aa2237d2e780080233 18e64e2ec7ea8e1cedefb4e0e489184f7cfc87fdfa39e384526b1aa778b36ff8 ead6ee53b8eb240010a8d1e66d8b37112fd3ecb62b4e8b1d7e60e13185f92d7d 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c842efd511a8c9e653633429aeaa0e90fc69ecde4c3f08b822f9cd4c6541a5c5 9b12a43534a3cf0e98e7205c12c96f083d758d1d145361c3daf4c742ce0fac1c 3a6ef53bc699b480edf17dc5e806a6079b2d206e98d723ac90539e6afb791748 f28cd2bf7d932b23eaa7249a46f007eb7fa1b8b77798575aa7ffe3cd69a2ad0a aaaf52ba1a2325956d8571a44cacf62f07fe187f104d20620ac70f5c63e3e2d1 d70d8da1b8f8403e1a82d0aa92080a0276fd151b0d1345cfcfe022c45d64b8bc 4b6f76109011ff11fa4a8df699fc59ad44e1302d433c5fed5edf88ed82ed5aba 8c3040b254d6c62e445575d0e157e74d0cc18a75d74713b0360d2b285ff3872c bb2861b325dba4703c0ce4939471652dc2d6521144ab1d65d0e0ba9cf3fe86fb 5f206851ea2eef05265d114ed6b938e77dec28426737261a468ff7979cb33e82 8cdae3c66e6d936523cb1e84687ebe468fd2c51ceb3cb0d2e8caef33114fcd29 7f4dc0bf889d8a167fd2e4229f3b8ae028a50ee4bcd53258941b6c44e7fc2422 1198e4e1930668ebf19e1a25b1b6f38049ab88b74d16b33b96bd39dcaef159f1 aa072017b0836af12f62160884d5cddc2cbab3e6796b7c88e1f242411700cbfa aa582c7a5f4b34798a3dad39388d614e6c4b1f7dc670c091cb8dae75cf50848e bdc3b8abb6d9bd876b214d95f0cb0e0f0f41dfe871119cb83a20aee1c3020fc9 33fad34a2a00cad733197f677a52aaf7c20460057c92def6ba27dd34cd8383dd bacb3c4e06dcf4da8c9f84f05ad23db6b701b4b8e750ee0281343fe3b4c9f718 ae2d3ad62f2878c36299a2f05ca30e48076483e1cb02a6a4c6eca326375e40cc 5be5f8f28d442637e422473bf19c25cacd5dddaae94b59e24f129a8970d8d1f3 fd293aabc375dde154ee14a51452636c92c2e5d11db82f7282ba9f11ef644a79 4906753ce8a6b7dd15e6de5e7460ec51dabff715c5bd1e496f344fb567f68fbc b0c6a1022f74625cf1a825a29addeb0d8fcdb6479f7776a976dd2da270551025 79ab7841917e9e3bbc92068ed440cf22d3f75a50f8ce5c2e6b0a8ad1399331bf a75fb97a4116367c279634772a91133f10c4c14c9d7e30dc5febf1c71898335e 231ebd9f569b9764fe6e47e96cb1907fc275bb30d529613c87625721d449ccd5 5b89c9c3297d3997cbe4f529affd784bd717969c2b3fd11063cc0ffd27ff8f9d 195c5feadfc2a6b3b8bdd57be4c81ca34336311dc958442a9e52ebf1dde7785f 2bc20bab871fbf23058b153f9abf84580f74637ccabdd45b9c76db77aec1dcd2 3ba7c6f67a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 169 /Phi put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 186 /nu put dup 187 /xi put dup 189 /rho put dup 190 /sigma put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 65 /A put dup 67 /C put dup 78 /N put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 112 /p put dup 115 /s put dup 120 /x put dup 196 /tie put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 169 /circleplus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 186 /followsequal put dup 187 /similar put dup 189 /propersubset put dup 190 /propersuperset put dup 33 /arrowright put dup 39 /similarequal put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 60 /Rfractur put dup 65 /A put dup 68 /D put dup 70 /F put dup 72 /H put dup 85 /U put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put dup 196 /spade put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 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8ae7e1da0b779a4b71a3e4fb12d9dee166d26b937138fe1178733e8a09ec02b8 a68a54d0c180a9a7c69da784c76d11cb8b356471ca27dcd66d723bc91a4ff6fe 5dbd13d08e1a65f6b41cf52e26103c46c7d5d091e806f0ade421c654dcab29fd 8163db37ed1bf13b88e561b2404ec649fe32a4039eca598bbcef41cc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 168 /Upsilon put dup 169 /Phi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 186 /ring put dup 187 /cedilla put dup 189 /ae put dup 190 /oe put dup 35 /numbersign 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 59 /semicolon put dup 61 /equal put dup 78 /N put dup 91 /bracketleft put dup 93 /bracketright put dup 102 /f put dup 104 /h put dup 105 /i put dup 116 /t put dup 196 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 69 /E put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 90 /Z put dup 196 /epsiloninv put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % 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 (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 168 /Upsilon put dup 169 /Phi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 194 /chi put dup 33 /omega put dup 34 /epsilon put dup 37 /rho1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g 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 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 196 /tie put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % 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 (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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 194 /Oslash put dup 35 /numbersign put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 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a6dcf9e0c521b03df4126a0f8beef3ef5a96f411e2ff7ef4ab4f9bea8fee176f 0408cb16c9bc7d2fff7e2529d279f200f591474b216d26cf47376a2b22d74231 44f179b71541c11ae0655ab52f4c0883430ac6dee7b2fe02e384503e73f8ce52 4cc38599cd 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 194 /chi put dup 195 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 67 /C put dup 69 /E put dup 73 /I put dup 75 /K put dup 76 /L put dup 78 /N put dup 82 /R put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f 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 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 196 /tie put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 194 /chi put dup 195 /psi put dup 65 /A put dup 97 /a put dup 196 /tie put readonly def /FontBBox{-32 -250 1048 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % 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 (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 194 /Oslash put dup 195 /suppress put dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put dup 196 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 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55441dfc3adab7a8ca8d27f49e603962d55a4a60f1a2c6f2fa09a240848e0c68 1962454b57e88fd415c0d67791ef2502bdad997af36b833a2e345f1154ed0825 e81d2cbb69b3522367040444b3424823848a1af8472a9856c70bcdce9a9abd52 3e6bc253ce179a77f251991a2b451e94384ef1894fa04c83fbec50739b3a6399 6683f969c193d2fcb5f83403b0a2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % 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 (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 194 /Oslash put dup 195 /suppress put dup 76 /L 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 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p 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 122 /z put dup 196 /dieresis put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 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Fs(=)369 b Fn(f)p Fr(u)g Fn(2)g Fr(L)18803 45100 y Fo(2)18803 45977 y Fv(l)11 b(oc)20012 45648 y Fs(\()p Fq(R)21477 45100 y Fv(d)22016 45648 y Fs(\);)221 b Fn(8)p Fr(\260)442 b Fn(2)368 b Fq(Z)27099 45100 y Fv(d)27638 45648 y Fr(;)221 b(u)p Fs(\()p Fr(x)297 b Fs(+)d Fr(\260)72 b Fs(\))369 b(=)g Fr(e)35414 45100 y Fv(i\260)52 b(\265)36791 45648 y Fr(u)p Fs(\()p Fr(x)p Fs(\))p Fn(g)p Fr(:)4317 48582 y Fs(As)333 b Fr(A)7113 48781 y Fo(0)7971 48582 y Fs(is)g(elliptic,)h(w)-36 b(e)332 b(kno)-36 b(w)334 b(that,)e Fr(A)22864 48781 y Fo(0)23390 48582 y Fs(\()p Fr(\265)36 b Fs(\))332 b(has)h(a)g(compact)f(resolv)-36 b(en)g(t;)334 b(hence)d(its)i(sp)36 b(ec-)4317 50187 y(trum)378 b(is)i(discrete)f([33)q(].)h(W)-108 b(e)379 b(denote)f(its)h(eigen)-36 b(v)-72 b(alues,)381 b(called)e(Flo)36 b(quet)380 b(eigen)-36 b(v)-72 b(alues)380 b(of)4317 51792 y Fr(A)5292 51991 y Fo(0)5818 51792 y Fs(,)434 b(b)-36 b(y)15719 53397 y Fr(E)16682 53596 y Fo(0)17207 53397 y Fs(\()p Fr(\265)36 b Fs(\))369 b Fn(\267)g Fr(E)21595 53596 y Fo(1)22121 53397 y Fs(\()p Fr(\265)36 b Fs(\))369 b Fn(\267)g(\242)295 b(\242)g(\242)369 b(\267)g Fr(E)29977 53596 y Fv(n)30603 53397 y Fs(\()p Fr(\265)36 b Fs(\))369 b Fn(\267)g(\242)296 b(\242)f(\242)p Fr(:)4317 55722 y Fs(The)333 b(corresp)36 b(onding)333 b(eigenfunctions)g(are)g (denoted)g(b)-36 b(y)333 b(\()p Fr(w)36 b Fs(\()p Fr(x;)221 b Fn(\242)p Fs(\))36301 55921 y Fv(j)36787 55722 y Fs(\))37293 55921 y Fv(j)51 b Fj(2)p Fg(N)39047 55722 y Fs(.)334 b(The)f(functions)4317 57327 y(\()p Fr(\265)405 b Fn(!)370 b Fr(E)8495 57526 y Fv(n)9121 57327 y Fs(\()p Fr(\265)36 b Fs(\)\))11281 57526 y Fv(n)p Fj(2)p Fg(N)13607 57327 y Fs(are)434 b(Lipshitz-con)-36 b(tin)g(uous,)432 b(and)h(w)-36 b(e)433 b(ha)-36 b(v)g(e)13372 60260 y Fr(E)14335 60459 y Fv(n)14961 60260 y Fs(\()p Fr(\265)36 b Fs(\))369 b Fn(!)h Fs(+)p Fn(1)867 b Fs(as)434 b Fr(n)369 b Fn(!)g Fs(+)p Fn(1)867 b Fs(uniformly)453 b(in)867 b Fr(\265)36 b(:)4317 63194 y Fs(The)434 b(sp)36 b(ectrum)432 b Fr(\276)48 b Fs(\()p Fr(A)14938 63393 y Fo(0)15464 63194 y Fs(\))433 b(of)i Fr(A)18860 63393 y Fo(0)19819 63194 y Fs(has)f(a)f(band)g (structure.)g(\(i.e)h Fr(\276)48 b Fs(\()p Fr(A)36915 63393 y Fo(0)37440 63194 y Fs(\))369 b(=)g Fn([)40582 63393 y Fv(n)p Fj(2)p Fg(N)42474 63194 y Fr(E)43437 63393 y Fv(n)44063 63194 y Fs(\()p Fq(T)45455 62712 y Fj(\244)45981 63194 y Fs(\))p Fr(:)p Fs(\))25578 70015 y(5)p eop %%Page: 6 6 6 5 bop 4317 7306 a Fh(1.2.2)1495 b(The)499 b(main)g(assumptions)4317 9769 y Fs(W)-108 b(e)438 b(assume)g(that)f Fr(A)14857 9968 y Fo(0)15821 9769 y Fs(has)h(a)g(sp)36 b(ectral)438 b(gap)g(b)36 b(elo)-36 b(w)439 b Fr(E)31236 9968 y Fo(+)32023 9769 y Fs(.)f(More)g(precisely)g(w)-36 b(e)439 b(assume)4317 11374 y(that:)4317 12979 y(\()p Fh(A)p Fr(:)p Fh(1)p Fs(\))4317 14584 y(There)491 b(exists)g Fr(E)12743 14783 y Fo(+)14020 14584 y Fs(and)f Fr(\261)516 b(>)465 b Fs(0)491 b(suc)-36 b(h)490 b(that)g Fr(\276)48 b Fs(\()p Fr(A)28450 14783 y Fo(0)28976 14584 y Fs(\))333 b Fn(\\)h Fs([)p Fr(E)32359 14783 y Fo(+)33146 14584 y Fr(;)221 b(E)34691 14783 y Fo(+)35813 14584 y Fs(+)333 b Fr(\261)50 b Fs(\))466 b(=)f([)p Fr(E)41557 14783 y Fo(+)42345 14584 y Fr(;)221 b(E)43890 14783 y Fo(+)45011 14584 y Fs(+)334 b Fr(\261)50 b Fs(\))4317 16189 y(and)433 b Fr(\276)48 b Fs(\()p Fr(A)9114 16388 y Fo(0)9640 16189 y Fs(\))295 b Fn(\\)g Fs(\()p Fr(E)13091 16388 y Fo(+)14173 16189 y Fn(\241)g Fr(\261)-22 b(;)221 b(E)17600 16388 y Fo(+)18387 16189 y Fs(])369 b(=)g Fn(;)p Fs(.)4317 19399 y(As,)597 b(\242)p Fr(A)8822 19598 y Fv(!)10140 19399 y Fn(\270)646 b Fs(0)597 b(,)f(\247)h(con)-36 b(tains)596 b(an)g(in)-36 b(terv)-72 b(al)597 b(of)g(the)f(form)g([)p Fr(E)36256 19598 y Fo(+)37044 19399 y Fr(;)221 b(E)38589 19598 y Fo(+)39782 19399 y Fs(+)406 b Fr(a)p Fs(])596 b(\()p Fr(a)646 b(>)f Fs(0\))4317 21005 y([18)q(].)549 b(As)f(w)-36 b(e)547 b(are)h(in)-36 b(terested)547 b(in)g(the)g(b)36 b(eha)-36 b(vior)548 b(of)h(the)e(IDS)g(in)h(the)f(neigh)-36 b(b)36 b(orho)g(o)g(d)4317 22610 y(of)564 b Fr(E)6891 22809 y Fo(+)7678 22610 y Fs(,)f(w)-36 b(e)563 b(require)g(that)f Fr(E)19057 22809 y Fo(+)20407 22610 y Fs(remains)g(alw)-36 b(a)g(ys)565 b(the)d(edge)h(of)g(a)g(gap)g(for)g(\247,)h(when)4317 24215 y(the)498 b(p)36 b(erturbation)496 b(is)i(turned)f(on.)h(More)g (precisely)-108 b(,)499 b(if)g(for)f(all)h Fr(t)478 b Fn(2)g Fs([0)p Fr(;)221 b Fs(1],)500 b(w)-36 b(e)499 b(de\257ne)4317 25820 y Fr(A)5292 26019 y Fv(!)32 b(;t)7207 25820 y Fs(=)641 b Fr(A)9835 26019 y Fo(0)10765 25820 y Fs(+)403 b Fr(t)p Fs(\242)p Fr(A)14709 26019 y Fv(!)15975 25820 y Fs(and)593 b(\247)19603 26019 y Fv(t)20592 25820 y Fs(is)h(the)f(almost)h(sure)f(sp)36 b(ectrum)592 b(of)j Fr(A)40187 26019 y Fv(!)32 b(;t)41460 25820 y Fs(,)594 b(then)f(one)4317 27425 y(requires)434 b(that)f(the)g(follo)-36 b(wing)436 b(assumption)d(holds.)4317 29030 y(\()p Fh(A)p Fr(:)p Fh(2)p Fs(\))4317 30635 y(There)h(exists)g Fr(\261)12292 30153 y Fj(0)12971 30635 y Fr(>)369 b Fs(0)434 b(suc)-36 b(h)433 b(that)g(for)h(all)g Fr(t)369 b Fn(2)f Fs([0)p Fr(;)221 b Fs(1])p Fr(;)658 b Fs(\247)31636 30834 y Fv(t)32326 30635 y Fn(\\)295 b Fs([)p Fr(E)34831 30834 y Fo(+)35914 30635 y Fn(\241)g Fr(\261)37868 30153 y Fj(0)38178 30635 y Fr(;)221 b(E)39723 30834 y Fo(+)40511 30635 y Fs(\))369 b(=)f Fn(;)p Fs(.)4317 32240 y(W)-108 b(e)434 b(assume)f(also)i(the)e (follo)-36 b(wing:)4317 33845 y(\()p Fh(A)p Fr(:)p Fh(3)p Fs(\))17628 36183 y(lim)221 b(sup)18360 37339 y Fv(")p Fj(!)p Fo(0)20205 37087 y Ff(+)21969 35284 y Fs(log)240 b Fn(j)221 b Fs(log)241 b Fq(P)27174 35483 y Fo(0)27700 35284 y Fs(\([0)p Fr(;)221 b(")p Fs(]\))p Fn(j)p 21969 35877 9676 54 v 25072 37094 a(j)g Fs(log)241 b Fr(")p Fn(j)32146 36183 y Fs(=)369 b(0)10638 b(\(1.6\))4317 38508 y(.)4317 43964 y FA(1.3)1793 b(The)598 b(main)h(results)4317 46427 y Fs(Let)433 b(us)g(consider)h(the)f(w)-36 b(a)g(v)g(e)434 b(equation:)15228 49177 y Fr(@)15987 48695 y Fo(2)16513 49177 y Fr(v)p 15228 49770 1962 54 v 15331 50987 a(@)72 b(t)16560 50603 y Fo(2)17691 50076 y Fs(=)369 b Fn(\241)p Fr(A)21080 50275 y Fv(!)21752 50076 y Fr(v)48 b(;)655 b(v)48 b Fs(\(0\))368 b(=)h Fr(v)28162 50275 y Fo(0)28687 50076 y Fr(;)29837 49177 y(@)72 b(v)p 29837 49770 1436 54 v 29940 50987 a(@)g(t)31405 48548 y Fp(\257)31405 49345 y(\257)31405 50142 y(\257)31847 50994 y Fv(t)p Fo(=0)33814 50076 y Fs(=)369 b Fr(v)35824 50275 y Fo(1)36349 50076 y Fr(:)8105 b Fs(\(1.7\))4317 53430 y(The)434 b(solution)g(of)g (the)f(equation)h(\(1.7\))g(is)g(giv)-36 b(en)434 b(b)-36 b(y:)14740 56729 y Fr(v)48 b Fs(\()p Fr(t)p Fs(\))368 b(=)g(cos)20610 55254 y Fp(\263)21403 56729 y Fr(t)21873 55506 y Fp(p)p 23202 55506 1648 54 v 23202 56729 a Fr(A)24177 56928 y Fv(!)24849 55254 y Fp(\264)25642 56729 y Fr(v)26271 56928 y Fo(0)27092 56729 y Fs(+)295 b(sin)30217 55254 y Fp(\263)31010 56729 y Fr(t)31480 55506 y Fp(p)p 32808 55506 V 1223 x Fr(A)33783 56928 y Fv(!)34456 55254 y Fp(\264)35249 56729 y Fr(w)36179 56928 y Fo(1)36704 56729 y Fr(;)4317 60162 y Fs(where)434 b Fr(v)8704 60361 y Fo(1)9598 60162 y Fs(=)10979 59137 y Fn(p)p 12086 59137 V 1025 x Fr(A)13061 60361 y Fv(!)13733 60162 y Fr(w)14663 60361 y Fo(1)15622 60162 y Fs(and)f Fr(v)18780 60361 y Fo(0)19306 60162 y Fr(;)221 b(w)20818 60361 y Fo(1)21778 60162 y Fs(are)433 b(in)h(the)f(domain)g(of)i Fr(A)34784 60361 y Fv(!)35456 60162 y Fs(.)4317 61767 y(The)f(result)f(whic)-36 b(h)433 b(w)-36 b(e)434 b(presen)-36 b(t)433 b(can)g(b)36 b(e)433 b(summarized)h(as)g(follo)-36 b(w:)4317 63373 y Fn(\262)434 b Fs(the)f(sp)36 b(ectrum)432 b(of)j Fr(A)15791 63572 y Fv(!)16897 63373 y Fs(is)e(pure)g(p)36 b(oin)-36 b(t)433 b(in)h(a)g(neigh)-36 b(b)36 b(orho)g(o)g(d)433 b(of)h(the)f(edge)h(of)g(\247)p Fn(nf)p Fs(0)p Fn(g)p Fs(.)4317 64978 y Fn(\262)352 b Fs(the)g(eigenfunctions)g(corresp)36 b(onding)352 b(to)g(eigen)-36 b(v)-72 b(alues)353 b(in)f(this)g(neigh) -36 b(b)36 b(orho)g(o)g(d)352 b(deca)-36 b(y)4317 66583 y(exp)36 b(onen)-36 b(tially)-108 b(.)25578 70015 y(6)p eop %%Page: 7 7 7 6 bop 4317 7306 a Fn(\262)434 b Fs(for)g(a)g(go)36 b(o)g(d)434 b(c)-36 b(hoice)434 b(of)g Fr(v)17641 7505 y Fo(0)18601 7306 y Fs(and)f Fr(w)22060 7505 y Fo(1)22585 7306 y Fs(,)h Fr(v)48 b Fs(\()p Fr(t)p Fs(\))432 b(do)36 b(es)434 b(not)f(disp)36 b(erse)434 b(in)f(the)g(space.)4317 8911 y(The)599 b(\257rst)f(and)h(second)f(assertion)i(are)f(often)g (named)f Fh(Anderson)690 b(lo)42 b(calization)p Fs(.)4317 10516 y(The)434 b(last)g(one)f(is)h(called)933 b Fh(strong)499 b(dynamical)h(lo)42 b(calization)p Fs(.)4317 12121 y(The)434 b(main)f(theorem)h(is:)4317 14363 y Fh(Theorem)499 b(1.1)651 b Fi(L)-66 b(et)497 b Fr(A)16462 14562 y Fv(!)17633 14363 y Fi(de\257ne)-66 b(d)496 b(by)i(\(1.1\).)h(We)e(assume)i(that)f Fh(\(A.1\))p Fi(,)g Fh(\(A.2\))i Fi(and)4317 15968 y Fh(\(A.3\))466 b Fi(hold.)f(Ther)-66 b(e)464 b(exists)h Fr(")19322 16167 y Fo(0)20216 15968 y Fr(>)369 b Fs(0)465 b Fi(such)h(that)4317 17573 y(\(i\))f Fs(\247)296 b Fn(\\)f Fs([)p Fr(E)9983 17772 y Fo(+)10770 17573 y Fr(;)221 b(E)12315 17772 y Fo(+)13398 17573 y Fs(+)295 b Fr(")15314 17772 y Fo(0)15839 17573 y Fs(])369 b(=)g(\247)18889 17772 y Fv(pp)20187 17573 y Fn(\\)295 b Fs([)p Fr(E)22692 17772 y Fo(+)23479 17573 y Fr(;)221 b(E)25024 17772 y Fo(+)26107 17573 y Fs(+)295 b Fr(")p Fs(])p Fi(.)4317 19178 y(\(ii\))453 b(an)h(eigenfunction)c(c)-66 b(orr)g(esp)g(onding) 451 b(to)i(an)g(eigenvalue)f(in)h Fs([)p Fr(E)37264 19377 y Fo(+)38051 19178 y Fr(;)221 b(E)39596 19377 y Fo(+)40679 19178 y Fs(+)295 b Fr(")42595 19377 y Fo(0)43120 19178 y Fs(])454 b Fi(de)-66 b(c)g(ays)4317 20783 y(exp)g(onential)66 b(ly)463 b(.)4317 22388 y(\(iii\))h(ther)-66 b(e)464 b(exist)g Fr(p)369 b(>)g Fs(0)465 b Fi(such)g(that)10144 25168 y Fq(E)11030 23694 y Fp(n)12137 25168 y Fs(sup)12345 26224 y Fv(t>)p Fo(0)14317 23641 y Fp(\257)14317 24438 y(\257)14317 25235 y(\257)14759 23641 y(\257)14759 24438 y(\257)14759 25235 y(\257)15202 23641 y(\257)15202 24438 y(\257)15202 25235 y(\257)15645 25168 y Fr(X)16828 23641 y Fp(\257)16828 24438 y(\257)16828 25235 y(\257)17272 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y Fs(sin)19913 28356 y Fp(\263)20706 29830 y Fr(t)21176 28607 y Fp(p)p 22504 28607 V 1223 x Fr(A)23479 30029 y Fv(!)24151 28356 y Fp(\264)24944 29830 y Fr(P)25782 30037 y Fo([)p Fv(E)26733 30156 y Ff(+)27410 30037 y Fv(;E)28361 30156 y Ff(+)29038 30037 y Fo(+)p Fv(")30204 30160 y Ff(0)30665 30037 y Fo(])30982 29830 y Fs(\()p Fr(A)32463 30029 y Fv(!)33135 29830 y Fs(\))p Fr(\302)34456 30029 y Fv(K)35367 28302 y Fp(\257)35367 29099 y(\257)35367 29896 y(\257)35810 28302 y(\257)35810 29099 y(\257)35810 29896 y(\257)36253 28356 y(o)37507 29830 y Fr(<)g Fs(+)p Fn(1)p Fr(:)4317 32665 y Fi(Her)-66 b(e)434 b Fr(P)8233 32864 y Fv(I)8766 32665 y Fs(\()p Fr(A)10247 32864 y Fv(!)10919 32665 y Fs(\))g Fi(is)h(the)g(sp)-66 b(e)g(ctr)g(al)434 b(pr)-66 b(oje)g(ction)432 b(on)j(the)g(interval)e Fr(I)104 b(;)436 b(\302)36937 32864 y Fv(K)38283 32665 y Fi(is)f(the)g(char)-66 b(acter-)4317 34270 y(istic)464 b(function)f(of)i Fr(K)95 b Fi(,)464 b Fr(K)560 b Fi(is)465 b(a)g(c)-66 b(omp)g(act)463 b(of)i 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b(op)36 b(erators)542 b(\(See)e([18)q(,)i(23)q(,)f(24)q(,)h(28,)g(29)q(,)f(30)q(,)h(32]\).)h (F)-108 b(or)540 b(acoustic)i(op)36 b(erators)4317 65089 y(this)452 b(is)g(pro)-36 b(v)g(ed)452 b(in)g([29)q(])g(under)e(the)i (assumption)f(of)i(non-degeneracy)f(of)g(the)g(IDS)g(of)4317 66694 y(the)433 b(bac)-36 b(k-ground)433 b(op)36 b(erator)434 b Fr(A)20214 66893 y Fo(0)21173 66694 y Fs(whic)-36 b(h)433 b(is)h(relaxed)g(here.)25578 70015 y(7)p eop %%Page: 8 8 8 7 bop 4317 7306 a Fh(Theorem)499 b(1.2)651 b Fi(L)-66 b(et)414 b Fr(A)16379 7505 y Fv(!)17466 7306 y Fi(the)h(op)-66 b(er)g(ator)414 b(de\257ne)-66 b(d)414 b(by)h(\(1.1\).)g(We)f(assume)i (that)f Fh(\(A.1\))p Fi(,)4317 8911 y Fh(\(A.2\))466 b Fi(and)f Fh(\(A.3\))h Fi(hold.)f(Then)12689 12459 y Fs(lim)11721 13256 y Fv(")p Fj(!)p Fo(0)p Fv(;">)p Fo(0)15819 11476 y Fs(log)240 b Fn(j)221 b Fs(log)20212 10400 y Fp(\241)20821 11476 y Fr(N)139 b Fs(\()p Fr(E)23471 11675 y Fo(+)24552 11476 y Fs(+)295 b Fr(")p Fs(\))g Fn(\241)g Fr(N)139 b 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Fs(=)368 b Fn(\241r)296 b(\242)f Fr(a)27943 44079 y Fv(!)32 b(;k)29685 43880 y Fn(\242)295 b(r)p Fr(;)4317 46478 y Fs(where)434 b(the)f(function)g Fr(a)16092 46677 y Fv(!)32 b(;k)17972 46478 y Fs(is)434 b(the)f(matrix)h(with)g(the)f(co)36 b(e\261cien)-36 b(t)10877 49347 y Fr(a)11560 49546 y Fv(i;j)12629 49347 y Fs(\()p Fr(!)48 b(;)221 b(k)45 b Fs(\))369 b(=)f Fr(a)18234 49546 y Fo(0)p Fv(;i;j)20330 49347 y Fs(+)22699 48085 y Fp(X)21637 51013 y Fv(\260)52 b Fj(2)p Fv(C)23472 51169 y Fm(k)23980 51013 y Fj(\\)p Fg(Z)25197 50761 y Fm(d)25901 49347 y Fr(!)26711 49546 y Fv(\260)29160 48085 y Fp(X)27524 51013 y Fv(\257)46 b Fj(2)p Fo(\(2)p Fv(k)24 b Fo(+1\))p Fg(Z)32233 50761 y Fm(d)32937 49347 y Fr(u)33677 49546 y Fv(i;j)34746 49347 y Fs(\()p Fn(\242)295 b(\241)g Fr(\260)367 b Fn(\241)296 b Fr(\257)74 b Fs(\))4317 53393 y Fr(C)5248 53592 y Fv(k)6250 53393 y Fs(is)434 b(the)f(cub)36 b(e)10814 56517 y Fr(C)11745 56716 y Fv(k)12683 56517 y Fs(=)368 b Fn(f)p Fr(x)i Fn(2)e Fq(R)18049 55968 y Fv(d)18589 56517 y Fs(;)221 b Fn(8)p Fs(1)370 b Fn(\267)f Fr(j)444 b Fn(\267)369 b Fr(d;)655 b Fn(\241)27573 55618 y Fs(2)p Fr(k)341 b Fs(+)295 b(1)p 27573 56211 3624 54 v 29060 57428 a(2)31699 56517 y Fr(<)368 b(x)33818 56716 y Fv(j)34674 56517 y Fn(\267)36209 55618 y Fs(2)p Fr(k)341 b Fs(+)295 b(1)p 36209 56211 V 37696 57428 a(2)39966 56517 y Fn(g)p Fr(:)4317 60720 y(A)5292 60919 y Fv(!)32 b(;k)7282 60720 y Fs(is)542 b(\(2)p Fr(k)415 b Fs(+)369 b(1\))p Fq(Z)14368 60238 y Fv(d)14907 60720 y Fs(-p)36 b(erio)g(dic)542 b(and)g(essen)-36 b(tially)544 b(self)f(adjoin)-36 b(t)543 b(op)36 b(erator.)543 b(Let)f Fq(T)45354 60238 y Fj(\244)45354 61065 y Fv(k)46477 60720 y Fs(=)4317 62325 y(\()p Fq(R)5782 61843 y Fv(d)6322 62325 y Fs(\))p Fr(=)p Fs(2\(2)p Fr(k)341 b Fs(+)294 b(1\))p Fr(\274)48 b Fq(Z)14435 61843 y Fv(d)14974 62325 y Fs(.)434 b(W)-108 b(e)434 b(de\257ne)e Fr(N)22808 62524 y Fv(!)32 b(;k)24688 62325 y Fs(the)433 b(IDS)h(of)g Fr(A)32003 62524 y Fv(!)32 b(;k)33884 62325 y Fs(b)-36 b(y)13979 65608 y Fr(N)15021 65807 y Fv(!)32 b(;k)16467 65608 y Fs(\()p Fr(E)78 b Fs(\))369 b(=)21570 64709 y(1)p 20402 65302 2987 54 v 20402 66519 a(\(2)p Fr(\274)48 b Fs(\))22850 66135 y Fv(d)23743 64346 y Fp(X)23784 67170 y Fv(n)p Fj(2)p Fg(N)25883 63800 y Fp(Z)26621 66807 y Fj(f)p Fv(\265)26 b Fj(2)p Fg(T)28773 66502 y Fe(\244)28773 67123 y Fm(k)29283 66807 y Fv(;)434 b(E)30668 66963 y Fm(!)24 b(;k)14 b(;n)32732 66807 y Fo(\()p Fv(\265)26 b Fo(\))p Fj(\267)p Fv(E)50 b Fj(g)36148 65608 y Fr(d\265)36 b(:)6338 b Fs(\(2.10\))25578 70015 y(8)p eop %%Page: 9 9 9 8 bop 4317 7306 a Fs(Let)402 b Fr(dN)8334 7505 y Fv(!)32 b(;k)10183 7306 y Fs(the)403 b(deriv)-72 b(ativ)-36 b(e)403 b(of)g Fr(N)20815 7505 y Fv(!)32 b(;k)22665 7306 y Fs(in)402 b(the)g(distribution)g(sense.)h(As)g Fr(N)40020 7505 y Fv(!)32 b(;k)41869 7306 y Fs(is)403 b(increas-)4317 8911 y(ing,)381 b Fr(dN)8511 9110 y Fv(!)32 b(;k)10338 8911 y Fs(is)380 b(a)h(p)36 b(ositiv)-36 b(e)381 b(measure;)g(it)f(is)h (the)e(densit)-36 b(y)380 b(of)h(states)g(of)g Fr(A)39356 9110 y Fv(!)32 b(;k)40803 8911 y Fs(.)381 b(W)-108 b(e)380 b(denote)4317 10516 y(b)-36 b(y)355 b Fr(dN)493 b Fs(the)354 b(densit)-36 b(y)355 b(of)g(states)g(of)h Fr(A)22227 10715 y Fv(!)22899 10516 y Fs(.)g(F)-108 b(or)354 b(all)i Fr(')369 b Fn(2)f Fr(C)31096 10034 y Fj(1)31001 10845 y Fo(0)32093 10516 y Fs(\()p Fq(R)p Fs(\))p Fr(;)221 b(dN)36364 10715 y Fv(!)32 b(;k)38166 10516 y Fs(v)-36 b(eri\257es)355 b(\(see)g([23]\))12900 14110 y Fn(h)p Fr(';)221 b(dN)16569 14309 y Fv(!)32 b(;k)18016 14110 y Fn(i)1107 b Fs(=)23059 13211 y(1)p 21891 13804 2987 54 v 21891 15021 a(\(2)p Fr(\274)48 b Fs(\))24339 14637 y Fv(d)25232 12301 y Fp(Z)25970 15309 y Fv(\265)26 b Fj(2)p Fg(T)27652 15004 y Fe(\244)27652 15624 y Fm(k)28438 14110 y Fs(tr)29450 14309 y Fj(H)30251 14465 y Fm(\265)30775 13034 y Fp(\241)31383 14110 y Fr(')p Fs(\()p Fr(A)33716 14309 y Fv(!)32 b(;k)24 b(;\265)35890 14110 y Fs(\))36396 13034 y Fp(\242)37004 14110 y Fr(d\265)36 b(;)19640 17822 y Fs(=)23653 16923 y(1)p 21891 17516 4174 54 v 21891 18733 a(v)-36 b(ol)q(\()p Fr(C)24990 18932 y Fv(k)25558 18733 y Fs(\))26197 17822 y(tr)27209 16347 y Fp(\263)28002 17822 y Fr(\302)28817 18021 y Fv(C)29488 18177 y Fm(k)30051 17822 y Fr(')p Fs(\()p Fr(A)32384 18021 y Fv(!)32 b(;k)33831 17822 y Fs(\))p Fr(\302)35152 18021 y Fv(C)35823 18177 y Fm(k)36386 16347 y Fp(\264)37179 17822 y Fr(;)6625 b Fs(\(2.11\))4317 21576 y(where)445 b(for)g(\244)388 b Fn(\275)g Fq(R)13755 21094 y Fv(d)14295 21576 y Fr(;)666 b(\302)16137 21775 y Fo(\244)17290 21576 y Fs(will)446 b(design)e(the)h(c)-36 b(haracteristic)445 b(function)f(of)i(\244)e (and)g(tr\()p Fr(A)p Fs(\))4317 23181 y(is)465 b(the)g(trace)g(of)g Fr(A)g Fs(\(w)-36 b(e)465 b(index)g(b)-36 b(y)465 b Fn(H)23111 23380 y Fv(\265)24096 23181 y Fs(if)h(the)e(trace)h(is)g(tak)-36 b(en)465 b(in)g Fn(H)38456 23380 y Fv(\265)38976 23181 y Fs(\).)g(The)g(pro)36 b(of)466 b(of)4317 24786 y(\(2.11\))435 b(is)f(giv)-36 b(en)434 b(in)f([23)q(].)4317 27477 y Fh(Theorem)499 b(2.1)651 b Fi([23,)562 b(28])f(F)-100 b(or)563 b(any)f Fr(')548 b Fn(2)g Fr(C)27158 26995 y Fj(1)27063 27805 y Fo(0)28155 27477 y Fs(\()p Fq(R)p Fs(\))561 b Fi(and)h(for)g(almost)g(al)66 b(l)563 b Fr(!)596 b Fn(2)548 b Fs(\255)562 b Fi(we)4317 29082 y(have)18966 30687 y Fs(lim)18672 31523 y Fv(k)24 b Fj(!1)21067 30687 y Fn(h)p Fr(';)221 b(dN)24736 30886 y Fv(!)32 b(;k)26183 30687 y Fn(i)368 b Fs(=)h Fn(h)p Fr(';)221 b(dN)139 b Fn(i)p Fr(:)4317 33775 y Fs(In)527 b(what)g(follo)-36 b(w)529 b(w)-36 b(e)528 b(pro)-36 b(v)g(e)527 b(that)f(the)h(IDS)g(of)h Fr(A)29346 33974 y Fv(!)30545 33775 y Fs(is)f(exp)36 b(onen)-36 b(tially)529 b(w)-36 b(ell)528 b(appro)-36 b(xi-)4317 35380 y(mated)462 b(b)-36 b(y)462 b(the)f(exp)36 b(ectation)463 b(of)g(the)e(IDS)h(of)h(the)e(p)36 b(erio)g(dic)463 b(op)36 b(erators)462 b Fr(A)41434 35579 y Fv(!)32 b(;k)43343 35380 y Fs(when)462 b Fr(k)4317 36985 y Fs(is)434 b(p)36 b(olynomial)435 b(in)f Fr(")14472 36503 y Fj(\241)p Fo(1)15729 36985 y Fs(.)g(More)f(precisely)i(w)-36 b(e)433 b(pro)-36 b(v)g(e)4317 39675 y Fh(Lemma)499 b(2.2)651 b Fi(F)-100 b(or)513 b(any)f Fr(\264)17825 39874 y Fo(0)18807 39675 y Fr(>)456 b Fs(0)p Fi(,)513 b(ther)-66 b(e)511 b(exists)h Fr(\272)29366 39874 y Fo(0)30348 39675 y Fr(>)456 b Fs(0)513 b Fi(and)f Fr(")36159 39874 y Fo(0)37141 39675 y Fr(>)457 b Fs(0)512 b Fi(such)h(that,)f(for)4317 41281 y Fs(0)369 b Fr(<)g(")g(<)f(")9684 41480 y Fo(0)10675 41281 y Fi(and)465 b Fr(k)414 b Fn(\270)369 b Fr(k)16367 41480 y Fo(1)17262 41281 y Fs(=)g Fr(")19252 40798 y Fj(\241)p Fv(\272)20447 40921 y Ff(0)20962 41281 y Fi(,)465 b(we)g(have)5424 44521 y Fq(E)p Fs([)p Fr(N)7713 44720 y Fv(!)32 b(;k)9160 44521 y Fs(\()p Fr(E)372 b Fs(+)295 b Fr("=)p Fs(2\))g Fn(\241)h Fr(N)17389 44720 y Fv(!)32 b(;k)18835 44521 y Fs(\()p Fr(E)373 b Fn(\241)295 b Fr("=)p Fs(2\)])h Fn(\241)f Fr(e)27008 43972 y Fv(")27442 43660 y Fe(\241)p Fm(\264)28510 43859 y Ff(0)20237 46458 y Fn(\267)369 b Fr(N)139 b Fs(\()p Fr(E)372 b Fs(+)295 b Fr(")p Fs(\))f Fn(\241)i Fr(N)139 b Fs(\()p Fr(E)78 b Fs(\))18227 48668 y Fn(\267)369 b Fq(E)p Fs([)p Fr(N)21918 48867 y Fv(!)32 b(;k)23364 48668 y Fs(\()p Fr(E)373 b Fs(+)295 b(2)p Fr(")p Fs(\))g Fn(\241)g Fr(N)30943 48867 y Fv(!)32 b(;k)32390 48668 y Fs(\()p Fr(E)373 b Fn(\241)295 b Fs(2)p Fr(")p Fs(\)])g(+)g Fr(e)39891 48119 y Fj(\241)p Fv(")41057 47807 y Fe(\241)p Fm(\264)42125 48006 y Ff(0)42696 48668 y Fr(:)1108 b Fs(\(2.12\))4317 51576 y Fh(Pro)42 b(of.)515 b Fs(The)g(last)g(result)f(is)h(w)-36 b(ell)515 b(kno)-36 b(wn)515 b(for)g(op)36 b(erators)515 b(with)g(compact)f(single)h(sit) 4317 53181 y(p)36 b(oten)-36 b(tials)527 b([24)q(,)g(26)q(].)h(F)-108 b(or)526 b(this)h(w)-36 b(e)527 b(need)f(to)h(de\257ne)f(another)h(op) 36 b(erator.)528 b(More)f(pre-)4317 54786 y(cisely)426 b(a)f(function)g Fr(f)567 b Fs(on)424 b Fq(R)17945 54304 y Fv(d)18910 54786 y Fs(,)h(one)g(de\257ne)e Fr(f)26603 54304 y Fv(")27093 54786 y Fs(\()p Fr(x)p Fs(\))369 b(=)g Fr(f)142 b Fs(\()p Fr(x)p Fs(\))p 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y(\260)31043 65388 y Fr(u)31783 64840 y Fv(")31783 65717 y(i;j)32852 65388 y Fs(\()p Fn(\242)295 b(\241)g Fr(\260)72 b Fs(\))p Fr(:)25578 70015 y Fs(9)p eop %%Page: 10 10 10 9 bop 4317 7306 a Fs(The)560 b(p)36 b(erio)g(dic)560 b(appro)-36 b(ximations)560 b(of)h Fr(A)24122 7505 y Fv(";!)26050 7306 y Fs(it)e(is)h(de\257ned)f(analogously)i(to)f Fr(A)43386 7505 y Fv(!)32 b(;k)45393 7306 y Fs(and)4317 8911 y(will)500 b(b)36 b(e)499 b(denoted)e(b)-36 b(y)499 b Fr(A)16500 9110 y Fv(";!)32 b(;k)18643 8911 y Fs(.)499 b(Let)g Fr(N)22941 9110 y Fv(")23929 8911 y Fs(\()f(resp)36 b(ectiv)-36 b(ely)500 b Fr(N)33129 9110 y Fv(";!)32 b(;k)35770 8911 y Fs(\))499 b(b)36 b(e)498 b(the)g(IDS)h(of)h Fr(A)46121 9110 y Fv(";!)4317 10516 y Fs(\(resp)36 b(ectiv)-36 b(ely)438 b(of)g Fr(A)14375 10715 y Fv(";!)32 b(;k)16518 10516 y Fs(\).)438 b(Notice)f(that)g(from)h(the)f(deca)-36 b(ying)437 b(assumption)g(\(1.4\))h(and)4317 12121 y(the)379 b(fact)h(that)f(the)g(random)g(v)-72 b(ariables)380 b(are)g(b)36 b(ounded)378 b(uniformly)i(in)f Fr(k)424 b Fs(and)379 b Fr(!)48 b Fs(,)380 b(w)-36 b(e)379 b(get)4317 13726 y(that)339 b(there)g(exits)h Fr(C)464 b(>)368 b Fs(0)340 b(suc)-36 b(h)339 b(that)g(for)g(an)-36 b(y)340 b(\252)369 b(=)f(\()p Fn(\241)p Fs(1)103 b Fn(\241)g Fs(\242\))34618 13244 y Fj(\241)p Fo(1)35876 13726 y Fr(')p Fs(,)339 b(where)h Fr(')369 b Fn(2)f Fr(L)44452 13244 y Fo(2)44978 13726 y Fs(\()p Fq(R)46443 13244 y Fv(d)46983 13726 y Fs(\))12964 16660 y(0)h Fn(\267)g(h)p Fr(A)16877 16859 y Fv(!)17549 16660 y Fs(\252)p Fr(;)221 b Fs(\252)p Fn(i)295 b(\241)g(h)p Fr(A)23787 16859 y Fv(";!)25155 16660 y Fs(\252)p Fr(;)221 b Fs(\252)p Fn(i)368 b(\267)h Fr(C)391 b Fn(\242)295 b Fr(")32643 16111 y Fv(\272)58 b Fj(\241)p Fv(d)34729 16660 y Fn(\242)295 b(k)q Fs(\252)p Fn(k)37733 16014 y Fo(2)38481 16660 y Fr(:)4317 19593 y Fs(The)401 b(same)g(inequalit)-36 b(y)402 b(holds)e(for)h(the)g(p)36 b(erio)g(dic)401 b(appro)-36 b(ximations.)401 b(This)g(yields)h(that,) 4317 21198 y(uniformly)434 b(in)g Fr(k)478 b Fs(and)434 b Fr(!)481 b Fs(and)433 b(lo)36 b(cally)435 b(uniformly)g(in)e(the)g (energy)h Fr(E)78 b Fs(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)12753 24132 y Fr(N)13795 24331 y Fv(")14284 24132 y Fs(\()p Fr(E)373 b Fn(\241)295 b Fr(C)390 b Fn(\242)295 b Fr(")20048 23583 y Fv(\272)58 b Fj(\241)p Fv(d)21840 24132 y Fs(\))369 b Fn(\267)g Fr(N)139 b Fs(\()p Fr(E)78 b Fs(\))367 b Fn(\267)i Fr(N)30162 24331 y Fv(")30652 24132 y Fs(\()p Fr(E)j Fs(+)295 b Fr(C)390 b Fn(\242)296 b Fr(")36395 23583 y Fv(\272)58 b Fj(\241)p Fv(d)38186 24132 y Fs(\))p Fr(;)5112 b Fs(\(2.13\))4317 27065 y(and)8783 29999 y Fr(N)9825 30198 y Fv(";!)32 b(;k)11968 29999 y Fs(\()p Fr(E)373 b Fn(\241)295 b Fr(C)390 b Fn(\242)295 b Fr(")17732 29450 y Fv(\272)58 b Fj(\241)p Fv(d)19524 29999 y Fs(\))368 b Fn(\267)i Fr(N)22843 30198 y Fv(!)32 b(;k)24289 29999 y Fs(\()p Fr(E)78 b Fs(\))368 b Fn(\267)i Fr(N)29155 30198 y Fv(";!)32 b(;k)31297 29999 y Fs(\()p Fr(E)373 b Fs(+)295 b Fr(C)390 b Fn(\242)295 b Fr(")37040 29450 y Fv(\272)58 b Fj(\241)p Fv(d)38831 29999 y Fs(\))p Fr(:)4467 b Fs(\(2.14\))4317 32932 y(This)370 b(tells)g(us)f(that)h (the)f(IDS)g(of)i Fr(A)21395 33131 y Fv(!)22436 32932 y Fs(is)f(w)-36 b(ell)371 b(appro)-36 b(ximated)370 b(b)-36 b(y)369 b(that)g(of)i Fr(A)41317 33131 y Fv(";!)42685 32932 y Fs(.)f(But)f(for)4317 34537 y(the)408 b(last)h(op)36 b(erator)408 b(the)g(singles)h(sites)f(p)36 b(oten)-36 b(tials)409 b(are)f(compactly)i(supp)36 b(orted)406 b(and)i(so)4317 36142 y(man)-36 b(y)389 b(tec)-36 b(hniques)388 b(and)g(results)g(are)h (a)-36 b(v)-72 b(ailable)390 b([24)q(].)g(W)-108 b(e)388 b(need)g(the)g(follo)-36 b(wing)391 b(lemma)4317 38854 y Fh(Lemma)499 b(2.3)651 b Fi([24])533 b(We)h(assume)h(that)e(the)h (single)f(sit)h(p)-66 b(otential)532 b(is)i(supp)-66 b(orte)g(d)534 b(in)f(a)4317 40459 y(b)-66 b(al)66 b(l)598 b(of)f(r)-66 b(adius)598 b Fr(R)13526 40658 y Fv(")14016 40459 y Fi(.)g(L)-66 b(et)597 b Fr(I)104 b Fi(,)597 b(a)h(r)-66 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49387 y Fn(\242)2420 b Fs(sup)24979 50477 y Fv(x)p Fj(2)p Fg(R)p Fv(;)p Fo(0)p Fj(\267)p Fv(j)51 b Fj(\267)p Fv(n)p Fo(+)p Fv(\275)31408 47859 y Fp(\257)31408 48656 y(\257)31408 49453 y(\257)31851 49387 y Fs(\()p Fn(j)p Fr(x)p Fn(j)295 b Fs(+)g(1\))36592 48839 y Fv(\275)p Fo(+)p Fv(n)38432 49387 y Fr(')39284 48839 y Fo(\()p Fv(j)51 b Fo(\))40503 49387 y Fs(\()p Fr(x)p Fs(\))42254 47859 y Fp(\257)42254 48656 y(\257)42254 49453 y(\257)42696 49387 y Fr(:)1108 b Fs(\(2.15\))4317 53328 y Fh(Remark)499 b(2.4)650 b Fi(This)560 b(lemma)e(is)h(pr)-66 b(oven)558 b(in)h([24)o(])g(for)g(the)g(Schr\304)-664 b(odinger)557 b(c)-66 b(ase.)559 b(It)g(is)4317 54933 y(stil)66 b(l)548 b(true)g(for)f(our)h(c)-66 b(ase.)547 b(The)h(pr)-66 b(o)g(of)547 b(is)g(b)-66 b(ase)g(d)547 b(on)h(the)f(Hel\256er)g(Sj\304)-664 b(ostr)-66 b(and)547 b(formula)4317 56538 y(and)424 b(the)g(r)-66 b(esolvent)423 b(e)-66 b(quation)423 b(with)h(the)g(exp)-66 b(onential)422 b(de)-66 b(c)g(ay)423 b(of)h(the)f(r)-66 b(esolvent)423 b(kernels)4317 58143 y(\(the)465 b(Comb)-66 b(es-Thomas)465 b(ar)-66 b(gument\).)4317 61187 y Fs(Let)457 b Fr(')h Fs(b)36 b(e)458 b(a)g(Gevrey)g(class)g(function,)g(of)h(Gevrey)f(exp)36 b(onen)-36 b(t)457 b Fr(\256)419 b(>)410 b Fs(1)458 b(\(see)g([15)q (]\))f(suc)-36 b(h)4317 62792 y(that)434 b Fr(')g Fs(is)h(supp)36 b(orted)433 b(in)h([)p Fn(\241)p Fs(1)p Fr(;)221 b Fs(2],)437 b(0)371 b Fn(\267)f Fr(')h Fn(\267)f Fs(1)435 b(and)f Fr(')370 b Fn(\264)h Fs(1)434 b(on)h([0)p Fr(;)221 b Fs(1].)436 b(F)-108 b(or)434 b(0)371 b Fr(<)f(")g(<)g Fs(1,)4317 64397 y(w)-36 b(e)434 b(set)19449 66314 y Fr(')20301 66513 y Fv(E)20991 66632 y Ff(+)21667 66513 y Fv(;")22418 66314 y Fs(\()p Fn(\242)p Fs(\))369 b(=)g Fr(')26401 64840 y Fp(\263)27327 65416 y Fn(\242)295 b(\241)g Fr(E)30282 65615 y Fo(+)p 27327 66009 3743 54 v 28894 67226 a Fr(")31202 64840 y Fp(\264)31995 66314 y Fr(:)25253 70015 y Fs(10)p eop %%Page: 11 11 11 10 bop 4317 7306 a Fs(F)-108 b(rom)502 b(Lemma)g(2.3)h(and)e(prop)36 b(erties)502 b(of)h(Gevrey)f(class)h(functions,)g(w)-36 b(e)502 b(deduce)f(that)4317 8911 y(there)433 b(exists)h Fr(C)464 b(>)369 b Fs(1)434 b(suc)-36 b(h)432 b(that)i(for)g(all)g Fr(k)414 b(>)369 b(R)28098 9110 y Fv(")28588 8911 y Fs(,)434 b Fr(n)369 b Fn(\270)g Fs(1)434 b(and)f(0)369 b Fr(<)g(")g(<)f Fs(1)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)5424 10940 y Fp(\257)5424 11737 y(\257)5424 12534 y(\257)5867 12468 y Fq(E)p Fs(\()p Fn(h)p Fr(')8628 12667 y Fv(E)9318 12786 y Ff(+)9993 12667 y Fv(;")10744 12468 y Fr(;)221 b(dN)13044 12667 y Fv(!)32 b(;k)14491 12468 y Fn(i)p Fs(\))295 b Fn(\241)g(h)p Fr(')18506 12667 y Fv(E)19196 12786 y Ff(+)19872 12667 y Fv(;")20623 12468 y Fr(;)221 b(dN)139 b Fn(i)23579 10940 y Fp(\257)23579 11737 y(\257)23579 12534 y(\257)25029 14987 y Fn(\267)369 b Fr(")27040 14438 y Fj(\241)p Fv(n)p Fj(\241)p Fv(\275)29611 14987 y Fs(\()p Fr(n)296 b Fs(+)e Fr(\275)p Fs(\))33672 14438 y Fo(2)p Fv(\256)p Fo(\()p Fv(n)p Fo(+)p Fv(\275)p Fo(\))37319 14987 y Fs(\()p Fr(k)340 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b(as)g(in)e([28)q(])i(and)e (consists)h(on)g(pro)-36 b(ving)4317 65360 y(the)433 b(follo)-36 b(wing)436 b(theorem.)25253 70015 y(11)p eop %%Page: 12 12 12 11 bop 4317 7306 a Fh(Theorem)499 b(2.5)651 b Fi(L)-66 b(et)374 b Fr(A)16339 7505 y Fv(!)17011 7306 y Fi(,)h(the)g(op)-66 b(er)g(ator)374 b(de\257ne)-66 b(d)374 b(by)h(\(1.1\).)h(We)e(assume)i (that)f Fh(\(A.1\))p Fi(,)4317 8911 y Fh(\(A.2\))466 b Fi(and)f Fh(\(A.3\))h Fi(hold.)f(Then,)f(we)h(have)11535 13160 y Fs(lim)221 b(inf)12075 14058 y Fv(")p Fj(!)p Fo(0)13920 13806 y Ff(+)15491 11778 y Fs(log)17393 10251 y Fp(\257)17393 11048 y(\257)17393 11845 y(\257)18057 11778 y Fs(log)19958 10304 y Fp(\263)20752 11778 y Fr(N)139 b Fs(\()p Fr(E)23402 11977 y Fo(+)24483 11778 y Fs(+)295 b Fr(")p Fs(\))g Fn(\241)g Fr(N)139 b Fs(\()p Fr(E)31178 11977 y Fo(+)31964 11778 y Fs(\))32470 10304 y Fp(\264)33263 10251 y(\257)33263 11048 y(\257)33263 11845 y(\257)p 15491 12854 18215 54 v 23344 14071 a Fs(log)240 b Fr(")34208 13160 y Fn(\270)369 b(\241)37938 12261 y Fr(d)p 36776 12854 3001 54 v 36776 14071 a(\272)378 b Fn(\241)296 b Fs(2)39909 13160 y Fr(:)3895 b Fs(\(2.17\))4317 16260 y Fh(Pro)42 b(of.)373 b Fs(As)g(b)-36 b(y)373 b(assumption,)g(there)f (is)h(a)g(sp)36 b(ectral)373 b(gap)g(b)36 b(elo)-36 b(w)374 b Fr(E)36857 16459 y Fo(+)38017 16260 y Fs(of)f(length)g(at)g(least) 4317 17865 y Fr(\261)4943 17383 y Fj(0)5623 17865 y Fr(>)368 b Fs(0.)434 b(Th)-36 b(us,)434 b(for)g Fr(")368 b(<)h(\261)17076 17383 y Fj(0)17820 17865 y Fs(w)-36 b(e)434 b(ha)-36 b(v)g(e)12013 20367 y Fr(N)139 b Fs(\()p Fr(E)14663 20566 y Fo(+)15745 20367 y Fs(+)294 b Fr(")p Fs(\))h Fn(\241)g Fr(N)139 b Fs(\()p Fr(E)22439 20566 y Fo(+)23226 20367 y Fs(\))368 b(=)h Fr(N)139 b Fs(\()p Fr(E)28131 20566 y Fo(+)29212 20367 y Fs(+)295 b Fr(")p Fs(\))g Fn(\241)g Fr(N)139 b Fs(\()p Fr(E)35907 20566 y Fo(+)36988 20367 y Fn(\241)296 b Fr(")p Fs(\))p Fr(:)4317 22869 y Fs(T)-108 b(o)552 b(pro)-36 b(v)g(e)552 b(Theorem)g(2.5,)h(w)-36 b(e)551 b(will)i(lo)-36 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b(will)i(construct)d(a)h(family)i(of)4317 29289 y(appro)-36 b(ximate)438 b(eigen)-36 b(v)g(ectors)439 b(asso)36 b(ciated)439 b(to)e(appro)-36 b(ximate)439 b(eigen)-36 b(v)-72 b(alues)438 b(of)h Fr(A)43612 29488 y Fv(!)32 b(;)p Fo(\244)45143 29637 y Fm(N)46405 29289 y Fs(in)4317 30894 y([)p Fr(E)5641 31093 y Fo(+)6487 30894 y Fn(\241)59 b Fr(";)221 b(E)9733 31093 y Fo(+)10579 30894 y Fs(+)59 b Fr(")p Fs(].)318 b(These)g(functions)g(will)h(b)36 b(e)317 b(constructed)g(from)h(an)g (eigen)-36 b(v)g(ector)318 b(of)h Fr(A)46963 31093 y Fo(0)4317 32499 y Fs(asso)36 b(ciated)377 b(to)f Fr(E)12947 32698 y Fo(+)13734 32499 y Fs(.)g(Lo)36 b(cating)376 b(this)g(eigen)-36 b(v)g(ector)376 b(in)g Fr(\265)36 b Fs(,)376 b(one)g(obtains)g(an)f(appro)-36 b(ximate)4317 34104 y(eigenfunction)402 b(of)g Fr(A)14694 34303 y Fv(!)32 b(;)p Fo(\244)16225 34452 y Fm(N)17050 34104 y Fs(.)402 b(Lo)36 b(cating)402 b(this)f(eigenfunction)h(in)g Fr(x)g Fs(in)f(sev)-36 b(eral)403 b(disjoin)-36 b(ted)4317 35709 y(places,)434 b(w)-36 b(e)434 b(get)g(sev)-36 b(eral)435 b(eigenfunctions)e(t)-36 b(w)g(o)434 b(b)-36 b(y)434 b(t)-36 b(w)g(o)433 b(orthogonal.)4317 37315 y(In)530 b(order)g(to)g(simplify)h(the)f(notations,)h(in)f(what)g(follo)-36 b(ws)532 b(w)-36 b(e)530 b(assume)g(that)g Fr(\265)44234 36833 y Fo(0)45293 37315 y Fs(=)j(0)4317 38920 y(is)h(a)h(p)36 b(oin)-36 b(t)533 b(where)h Fr(E)15227 39119 y Fo(1)15753 38920 y Fs(\()p Fr(\265)36 b Fs(\))534 b(reac)-36 b(hes)534 b Fr(E)23528 39119 y Fo(+)24315 38920 y Fs(.)g(F)-108 b(rom)534 b([23)q(,)g(28)q(])g(there)f(exists)i Fr(C)635 b(>)539 b Fs(1,)c Fr(V)823 b Fs(a)4317 40525 y(neigh)-36 b(b)36 b(orho)g(o)g(d)383 b(of)g(0)g(and)f Fr(f)511 b Fs(:)370 b Fr(\265)405 b Fn(2)368 b Fr(V)659 b Fn(!)369 b Fr(f)142 b Fs(\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))383 b(a)g(real)g(analytic)h(function)f(suc)-36 b(h)382 b(that,)4317 42130 y Fn(jj)p Fr(f)142 b Fs(\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))p Fn(jj)9181 42352 y Fv(L)9821 42100 y Ff(2)10283 42352 y Fo(\()p Fv(C)11320 42475 y Ff(0)11780 42352 y Fo(\))12571 42130 y Fs(=)368 b(1)434 b(and)15658 44632 y Fn(jj)p Fs(\()p Fr(A)17877 44831 y Fo(0)18403 44632 y Fs(\()p Fr(\265)36 b Fs(\))295 b Fn(\241)g Fr(E)22643 44831 y Fo(+)23430 44632 y Fs(\))p Fr(f)142 b Fs(\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))p Fn(jj)28062 44855 y Fv(L)28702 44602 y Ff(2)29164 44855 y Fo(\()p Fv(C)30201 44978 y Ff(0)30662 44855 y Fo(\))31452 44632 y Fn(\267)369 b Fr(C)95 b Fn(j)p Fr(\265)36 b Fn(j)35260 44083 y Fo(2)35786 44632 y Fr(:)8018 b Fs(\(2.18\))4317 47134 y(Let)507 b(0)496 b Fr(<)e(\273)560 b(<)494 b Fs(1)508 b(b)36 b(e)508 b(a)g(small)g(constan)-36 b(t.)508 b(Let)f Fr(\302)495 b Fn(2)g Fr(C)31448 46652 y Fj(1)31353 47462 y Fo(0)32444 47134 y Fs(\()p Fq(R)p Fs(\))508 b(suc)-36 b(h)506 b(that)i(it)f(is)h (p)36 b(ositif,)4317 49385 y(supp)g(orted)432 b(in)i([)12453 48801 y Fv(\273)p 12442 49079 471 54 v 12442 49843 a Fo(2)13045 49385 y Fr(;)221 b(\273)64 b Fs(])435 b(and)17584 47577 y Fp(Z)18322 50584 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Fv(f)17580 9906 y(")18294 9577 y Fs(\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))868 b(and)e Fn(W)26150 9029 y Fo(f)26041 9906 y Fv(\256;";\257)46 b(;\263)28967 9577 y Fs(\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))370 b(=)f(e)33901 9029 y Fj(\241)p Fo(i)p Fv(\257)46 b Fj(\242)p Fv(\265)36249 9577 y Fs(\(\246)37730 9784 y Fo(\244)38383 9895 y Fm(\256)38972 9784 y Fo(\()p Fv(\263)61 b Fo(\))40234 9577 y Fn(W)41655 9029 y Fo(f)41546 9906 y Fv(")42068 9577 y Fs(\)\()p Fn(\242)p Fr(;)221 b(\265)36 b Fs(\))p Fr(;)4317 11849 y Fs(where)434 b(\244)8978 12048 y Fv(\256)9637 11849 y Fs(\()p Fr(\263)100 b Fs(\))434 b(is)g(the)f(cub)36 b(e)433 b(de\257ned)f(b)-36 b(y)11422 14330 y(\244)12325 14529 y Fv(\256)12984 14330 y Fs(\()p Fr(\263)100 b Fs(\))369 b(=)g Fn(f)p Fr(\260)441 b Fn(2)368 b Fq(Z)20334 13782 y Fv(d)20873 14330 y Fs(;)656 b(for)434 b(1)369 b Fn(\267)g Fr(j)444 b Fn(\267)369 b Fr(d)p Fs(;)655 b Fn(j)p Fr(\260)31418 14529 y Fv(j)31905 14330 y Fn(j)369 b(\267)g Fr(\263)34713 13748 y Fj(\241)p Fo(\()36503 13386 y Ff(1)p 35944 13547 1525 40 v 35944 14095 a Fm(\272)45 b Fe(\241)p Fm(d)37601 13748 y Fo(+)p Fv(\256)p Fo(\))39359 14330 y Fn(g)p Fr(;)4317 16602 y Fs(and)433 b(\246)7821 16808 y Fo(\244)8474 16919 y Fm(\256)9063 16808 y Fo(\()p Fv(\263)61 b Fo(\))10758 16602 y Fs(is)434 b(the)f(orthogonal)i(pro)72 b(jection)434 b(on)f(\244)29697 16801 y Fv(\256)30357 16602 y Fs(\()p Fr(\263)100 b Fs(\).)4317 18207 y(W)-108 b(e)434 b(set)7968 20792 y Fn(U)8931 20162 y Fv(f)8799 21162 y(";\257)10124 20792 y Fs(\()p Fr(x)p Fs(\))369 b(=)13624 18984 y Fp(Z)14362 21991 y Fg(T)14952 21739 y Fe(\244)15709 20792 y Fn(W)17130 20162 y Fv(f)17021 21162 y(";\257)18346 20792 y Fs(\()p Fr(x;)221 b(\265)36 b Fs(\))p Fr(d\265)470 b Fs(and)433 b Fn(U)26565 20162 y Fv(f)26433 21162 y(\256;";\257)46 b(;\263)29360 20792 y Fs(\()p Fr(x)p Fs(\))368 b(=)32860 18984 y Fp(Z)33598 21991 y Fg(T)34188 21739 y Fe(\244)34945 20792 y Fn(W)36366 20162 y Fv(f)36257 21162 y(\256;";\257)46 b(;\263)39183 20792 y Fs(\()p Fr(x;)221 b(\265)36 b Fs(\))p Fr(d\265)g(:)4317 24013 y Fs(F)-108 b(or)526 b Fr(N)664 b Fs(large)526 b(and)g(w)-36 b(ell)527 b(c)-36 b(hosen)525 b Fr(\257)600 b Fs(and)525 b(\()p Fr(!)26642 23531 y Fv(i;j)26594 24341 y(\260)27710 24013 y Fs(\))28216 24269 y Fv(\260)52 b Fj(2)p Fg(Z)29970 24017 y Fm(d)30453 24269 y Fv(;)p Fo(1)p Fj(\267)p Fv(i;j)f Fj(\267)p Fv(d)34201 24013 y Fs(,)526 b Fn(U)36051 23383 y Fv(f)35919 24383 y(\256;";\257)46 b(;\263)39372 24013 y Fs(will)527 b(b)36 b(e)525 b(an)h(ap-)4317 25618 y(pro)-36 b(ximate)377 b(eigenfunction)f(of)h Fr(A)20728 25817 y Fv(!)32 b(;)p Fo(\244)22259 25966 y Fm(N)23460 25618 y Fs(asso)k(ciated)378 b(with)e(an)g(appro)-36 b(ximate)377 b(eigen)-36 b(v)-72 b(alue)4317 27223 y(in)434 b(the)f(in)-36 b(terv)-72 b(al)433 b([)p Fr(E)14095 27422 y Fo(+)15178 27223 y Fn(\241)295 b Fr(";)221 b(E)18660 27422 y Fo(+)19743 27223 y Fs(+)295 b Fr(")p Fs(].)4317 28828 y(As)543 b(in)f([23)q(])h(using)g(the)f(non-stationary)g(phase)h (w)-36 b(e)543 b(see)f(that)h Fn(U)37081 28198 y Fv(f)36949 29198 y(\256;";\257)46 b(;\263)40418 28828 y Fs(and)542 b Fn(U)44019 28198 y Fv(f)43887 29198 y(";\257)45754 28828 y Fs(are)4317 30578 y(close)458 b(to)f(eac)-36 b(h)457 b(others.)g(More)g(precisely)-108 b(,)458 b(for)f(an)-36 b(y)457 b Fr(n)410 b Fn(2)e Fq(N)457 b Fs(and)g Fr(\257)483 b Fn(2)408 b Fq(Z)39626 30096 y Fv(d)40165 30578 y Fs(,)458 b(there)e(exists)4317 32183 y Fr(C)5248 32382 y Fv(n)6243 32183 y Fr(>)369 b Fs(0)433 b(suc)-36 b(h)433 b(that)4939 34455 y(\(v)-36 b(ol)q(\()p Fq(T)8499 33906 y Fj(\244)9024 34455 y Fs(\)\))295 b Fn(\242)g(jjU)12696 33825 y Fv(f)12564 34824 y(\256;";\257)46 b(;\263)15786 34455 y Fn(\241)295 b(U)18077 33825 y Fv(f)17945 34824 y(";\257)19270 34455 y Fn(jj)20008 34677 y Fv(L)20648 34425 y Ff(2)21109 34677 y Fo(\()p Fg(R)p Fo(\))22905 34455 y Fs(=)368 b Fn(jjW)26444 33825 y Fv(f)26335 34824 y(\256;";\257)46 b(;\263)29557 34455 y Fn(\241)295 b(W)32306 33825 y Fv(f)32197 34824 y(";\257)33522 34455 y Fn(jj)34260 34654 y Fj(H)35486 34455 y Fn(\267)369 b Fr(C)37819 34654 y Fv(n)38445 34455 y Fr(")39054 33906 y Fj(\241)p Fv(n=)p Fo(2)41353 34455 y Fr(\263)42021 33872 y Fv(n)p Fo(\()43650 33510 y Ff(1)p 43091 33671 V 43091 34219 a Fm(\272)45 b Fe(\241)p Fm(d)44748 33872 y Fo(+)p Fv(\256)p Fo(\))46505 34455 y Fr(:)44165 36060 y Fs(\(2.20\))4317 37665 y(So,)434 b(for)g Fr(\263)469 b Fs(=)369 b Fr(")433 b Fs(small)i(enough,)e(w)-36 b(e)434 b(get)20390 39936 y Fn(jjU)22091 39306 y Fv(f)21959 40306 y(\256;";\257)46 b(;\263)24885 39936 y Fn(jj)25623 40192 y Fv(L)26263 39940 y Ff(2)26724 40192 y Fo(\()p Fg(R)27730 39940 y Fm(d)28212 40192 y Fo(\))29002 39936 y Fn(\270)369 b Fs(1)p Fr(:)4317 42207 y Fs(No)-36 b(w)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(to)g(lo)36 b(ok)435 b(to)e(the)g (conditions)h(under)e(whic)-36 b(h)434 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)17939 43205 y Fp(\257)17939 44002 y(\257)17939 44799 y(\257)18381 43205 y(\257)18381 44002 y(\257)18381 44799 y(\257)18824 43258 y(\263)19617 44732 y Fr(A)20592 44931 y Fv(!)21560 44732 y Fn(\241)295 b Fr(E)23851 44931 y Fo(+)24638 43258 y Fp(\264)25432 44732 y Fn(U)26395 44103 y Fv(f)26263 45102 y(\256;";\257)46 b(;\263)29189 43205 y Fp(\257)29189 44002 y(\257)29189 44799 y(\257)29632 43205 y(\257)29632 44002 y(\257)29632 44799 y(\257)30074 43555 y Fo(2)30969 44732 y Fn(\267)369 b Fr(")32980 44184 y Fo(2)33506 44732 y Fr(:)4317 47295 y Fs(Notice)434 b(that)5424 48624 y Fp(\257)5424 49421 y(\257)5424 50218 y(\257)5867 48624 y(\257)5867 49421 y(\257)5867 50218 y(\257)6310 48678 y(\263)7103 50152 y Fr(A)8078 50351 y Fv(!)32 b(;)p Fo(\244)9609 50500 y Fm(N)10729 50152 y Fn(\241)295 b Fr(E)13020 50351 y Fo(+)13807 48678 y Fp(\264)14600 50152 y Fn(U)15563 49522 y Fv(f)15431 50522 y(\256;";\257)46 b(;\263)18358 48624 y Fp(\257)18358 49421 y(\257)18358 50218 y(\257)18800 48624 y(\257)18800 49421 y(\257)18800 50218 y(\257)19243 48975 y Fo(2)20138 50152 y Fn(\267)21540 48624 y Fp(\257)21540 49421 y(\257)21540 50218 y(\257)21983 48624 y(\257)21983 49421 y(\257)21983 50218 y(\257)22426 48678 y(\263)23219 50152 y Fr(A)24194 50351 y Fv(!)25161 50152 y Fn(\241)296 b Fr(E)27453 50351 y Fo(+)28240 48678 y Fp(\264)29033 50152 y Fn(U)29996 49522 y Fv(f)29864 50522 y(\256;";\257)46 b(;\263)32790 48624 y Fp(\257)32790 49421 y(\257)32790 50218 y(\257)33233 48624 y(\257)33233 49421 y(\257)33233 50218 y(\257)33676 48975 y Fo(2)17091 53206 y Fn(\267)369 b Fs(2)19143 51678 y Fp(\257)19143 52475 y(\257)19143 53272 y(\257)19586 51678 y(\257)19586 52475 y(\257)19586 53272 y(\257)20029 51732 y(\263)20822 53206 y Fr(A)21797 53405 y Fo(0)22618 53206 y Fn(\241)296 b Fr(E)24910 53405 y Fo(+)25697 51732 y Fp(\264)26490 53206 y Fn(U)27453 52576 y Fv(f)27321 53576 y(\256;";\257)46 b(;\263)30247 51678 y Fp(\257)30247 52475 y(\257)30247 53272 y(\257)30690 51678 y(\257)30690 52475 y(\257)30690 53272 y(\257)31133 52029 y Fo(2)31954 53206 y Fs(+)295 b(2)33911 51678 y Fp(\257)33911 52475 y(\257)33911 53272 y(\257)34354 51678 y(\257)34354 52475 y(\257)34354 53272 y(\257)34797 53206 y Fs(\242)p Fr(A)36856 53405 y Fv(!)37528 53206 y Fn(U)38491 52576 y Fv(f)38359 53576 y(\256;";\257)46 b(;\263)41285 51678 y Fp(\257)41285 52475 y(\257)41285 53272 y(\257)41728 51678 y(\257)41728 52475 y(\257)41728 53272 y(\257)42171 52029 y Fo(2)42696 53206 y Fr(:)1108 b Fs(\(2.21\))4317 55843 y(Equations)426 b(\(2.19\))g(and)f(\(2.20\))h(giv)-36 b(e)426 b(the)f(b)36 b(ound)424 b(on)h(the)g(\257rst)f(mem)-36 b(b)36 b(er)425 b(of)h(\(2.21\),)g(it)4317 57448 y(just)434 b(remains)f(to)h(us)f(to)h (con)-36 b(trol)433 b(the)g(second)h(term.)f(W)-108 b(e)434 b(ha)-36 b(v)g(e)5424 59035 y Fp(\257)5424 59832 y(\257)5424 60629 y(\257)5867 59035 y(\257)5867 59832 y(\257)5867 60629 y(\257)6310 59089 y(\263)7103 60563 y Fs(\242)p Fr(A)9162 60762 y Fv(!)9834 59089 y Fp(\264)10627 60563 y Fn(U)11590 59933 y Fv(f)11458 60933 y(\256;";\257)46 b(;\263)14385 59035 y Fp(\257)14385 59832 y(\257)14385 60629 y(\257)14828 59035 y(\257)14828 59832 y(\257)14828 60629 y(\257)15270 59386 y Fo(2)16165 60563 y Fn(\267)369 b Fs(2)18217 59035 y Fp(\257)18217 59832 y(\257)18217 60629 y(\257)18660 59035 y(\257)18660 59832 y(\257)18660 60629 y(\257)21330 59301 y(X)19325 62228 y Fv(\260)52 b Fj(2)p Fg(Z)21079 61976 y Fm(d)21561 62228 y Fv(;)p Fo(1)p Fj(\267)p Fv(i;j)f Fj(\267)p Fv(d)25476 60563 y Fr(!)26334 60015 y Fv(i;j)26286 60891 y(\260)27402 60563 y Fs(\()p Fr(@)28595 60762 y Fv(x)29125 60897 y Fm(i)29530 60563 y Fr(u)30270 60762 y Fv(i;j)31339 60563 y Fs(\)\()p Fn(\242)295 b(\241)h Fr(\260)72 b Fs(\))294 b Fn(\242)36556 59089 y Fp(\263)37349 60563 y Fr(@)38036 60762 y Fv(x)38566 60897 y Fm(j)39054 60563 y Fn(U)40017 59933 y Fv(f)39885 60933 y(\256;";\257)46 b(;\263)42812 59089 y Fp(\264)43605 59035 y(\257)43605 59832 y(\257)43605 60629 y(\257)44048 59035 y(\257)44048 59832 y(\257)44048 60629 y(\257)44490 59386 y Fo(2)45016 60563 y Fs(+)19547 64724 y(2)20197 63196 y Fp(\257)20197 63993 y(\257)20197 64790 y(\257)20640 63196 y(\257)20640 63993 y(\257)20640 64790 y(\257)23310 63462 y(X)21304 66389 y Fv(\260)52 b Fj(2)p Fg(Z)23058 66137 y Fm(d)23541 66389 y Fv(;)p Fo(1)p Fj(\267)p Fv(i;j)f Fj(\267)p Fv(d)27455 64724 y Fr(!)28313 64175 y Fv(i;j)28265 65052 y(\260)29381 64724 y Fr(u)30121 64923 y Fv(i;j)31190 64724 y Fs(\()p Fn(\242)295 b(\241)h Fr(\260)72 b Fs(\))p Fr(@)35701 64175 y Fo(2)35629 65052 y Fv(x)36159 65187 y Fm(i)36509 65052 y Fv(x)37039 65187 y Fm(j)37528 64724 y Fn(U)38491 64094 y Fv(f)38359 65094 y(\256;";\257)46 b(;\263)41285 63196 y Fp(\257)41285 63993 y(\257)41285 64790 y(\257)41728 63196 y(\257)41728 63993 y(\257)41728 64790 y(\257)42171 63547 y Fo(2)42696 64724 y Fr(:)1108 b Fs(\(2.22\))25253 70015 y(13)p eop %%Page: 14 14 14 13 bop 4317 7306 a Fs(T)-108 b(o)434 b(estimate)g(\(2.22\),)h(one)e (needs)g(the)g(follo)-36 b(wing)436 b(lemmas.)4317 9637 y Fh(Lemma)499 b(2.6)651 b Fi([29,)496 b(30]Ther)-66 b(e)495 b(exists)h Fr(K)522 b(>)427 b Fs(0)p Fi(,)496 b(such)h(that)f(for)g(any)g Fs(1)428 b Fn(\267)f Fr(i;)221 b(j)503 b Fn(\267)427 b Fr(d)496 b Fi(we)4317 11242 y(have)4317 12457 y Fp(\257)4317 13254 y(\257)4317 14051 y(\257)4760 12457 y(\257)4760 13254 y(\257)4760 14051 y(\257)5714 12722 y(X)5424 15650 y Fv(\260)52 b Fj(2)p Fg(Z)7178 15398 y Fm(d)7661 15650 y Fv(;)8143 13984 y Fr(!)9001 13436 y Fv(i;j)8953 14313 y(\260)10070 13984 y Fs(\()p Fr(@)11263 14183 y Fv(x)11793 14318 y Fm(i)12198 13984 y Fr(u)12938 14183 y Fv(i;j)14007 13984 y Fs(\)\()p Fn(\242\241)p Fr(\260)72 b Fs(\))p Fn(\242)18043 12510 y Fp(\263)18836 13984 y Fr(@)19523 14183 y Fv(x)20053 14318 y Fm(j)20541 13984 y Fn(U)21504 13355 y Fv(f)21372 14354 y(\256;";\257)46 b(;\263)24299 12510 y Fp(\264)25092 12457 y(\257)25092 13254 y(\257)25092 14051 y(\257)25535 12457 y(\257)25535 13254 y(\257)25535 14051 y(\257)25978 12807 y Fo(2)26872 13984 y Fn(\267)370 b Fr(")28884 13436 y Fo(4)29409 13984 y Fs(+)p Fr(K)31623 12510 y Fp(\263)32416 13984 y Fr(")33025 13436 y Fv(s\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))36588 13984 y Fn(\242)p Fr(")p Fs(+)1827 b(sup)38711 15128 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)42304 15239 y Fm(\256)42892 15128 y Fo(\()p Fv(")p Fo(\))39865 15935 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)44413 13984 y Fr(!)45271 13436 y Fv(i;j)45223 14313 y(\260)46339 12510 y Fp(\264)47132 12807 y Fo(2)47658 13984 y Fr(:)44165 17309 y Fs(\(2.23\))4317 19846 y Fh(Lemma)499 b(2.7)651 b Fi([29,)496 b(30]Ther)-66 b(e)495 b(exists)h Fr(K)522 b(>)427 b Fs(0)p Fi(,)496 b(such)h(that)f(for)g(any)g Fs(1)428 b Fn(\267)f Fr(i;)221 b(j)503 b Fn(\267)427 b Fr(d)496 b Fi(we)4317 21451 y(have)4856 22665 y Fp(\257)4856 23462 y(\257)4856 24259 y(\257)5299 22665 y(\257)5299 23462 y(\257)5299 24259 y(\257)6122 22931 y(X)5963 25858 y Fv(\260)52 b Fj(2)p Fg(Z)7717 25606 y Fm(d)8421 24193 y Fr(!)9279 23644 y Fv(i;j)9231 24521 y(\260)10348 24193 y Fr(u)11088 24392 y Fv(i;j)12156 24193 y Fs(\()p Fn(\242)295 b(\241)h Fr(\260)72 b Fs(\))p Fr(@)16667 23644 y Fo(2)16595 24521 y Fv(x)17125 24656 y Fm(i)17475 24521 y Fv(x)18005 24656 y Fm(j)18494 24193 y Fn(U)19457 23563 y Fv(f)19325 24563 y(\256;";\257)46 b(;\263)22251 22665 y Fp(\257)22251 23462 y(\257)22251 24259 y(\257)22694 22665 y(\257)22694 23462 y(\257)22694 24259 y(\257)23137 23016 y Fo(2)24032 24193 y Fn(\267)369 b Fr(")26043 23644 y Fo(4)26863 24193 y Fs(+)295 b Fr(K)29372 22718 y Fp(\263)30165 24193 y Fr(")30774 23644 y Fv(s\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))34632 24193 y Fn(\242)295 b Fr(")g Fs(+)2123 b(sup)37640 25337 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)41233 25448 y Fm(\256)41822 25337 y Fo(\()p Fv(")p Fo(\))38795 26144 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)43343 24193 y Fr(!)44201 23644 y Fv(i;j)44153 24521 y(\260)45269 22718 y Fp(\264)46062 23016 y Fo(2)46588 24193 y Fr(:)44165 27518 y Fs(\(2.24\))4317 30054 y(No)-36 b(w,)572 b(com)-36 b(bining)571 b(\(2.23\),)i(\(2.24\))f(and)f(taking)h (\(2.22\))g(in)-36 b(to)571 b(accoun)-36 b(t)571 b(w)-36 b(e)572 b(get)f(that)4317 31659 y(there)433 b(exists)h Fr(K)464 b(>)369 b Fs(0)434 b(suc)-36 b(h)433 b(that)7719 32909 y Fp(\257)7719 33706 y(\257)7719 34503 y(\257)8162 32909 y(\257)8162 33706 y(\257)8162 34503 y(\257)8604 34437 y Fs(\(\242)p Fr(A)11169 34636 y Fv(!)11841 34437 y Fs(\))p Fn(U)13310 33807 y Fv(f)13178 34807 y(\256;";\257)46 b(;")16065 32909 y Fp(\257)16065 33706 y(\257)16065 34503 y(\257)16508 32909 y(\257)16508 33706 y(\257)16508 34503 y(\257)16950 33260 y Fo(2)17845 34437 y Fn(\267)369 b Fr(")19856 33888 y Fo(3)20677 34437 y Fs(+)295 b Fr(K)23186 32962 y Fp(\263)23979 34437 y Fr(")24588 33888 y Fv(s\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))28446 34437 y Fn(\242)295 b Fr(")g Fs(+)2123 b(sup)31454 35581 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)35047 35692 y Fm(\256)35636 35581 y Fo(\()p Fv(")p Fo(\))32609 36388 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)37156 34437 y Fr(!)38014 33888 y Fv(i;j)37966 34765 y(\260)39082 32962 y Fp(\264)39876 33260 y Fo(2)40402 34437 y Fr(:)3402 b Fs(\(2.25\))4317 38646 y(By)434 b(\(2.19\),\(2.20\))i(and)d (\(2.25\),)i(it)e(follo)-36 b(ws)436 b(that:)6154 40396 y Fp(\257)6154 41193 y(\257)6154 41990 y(\257)6597 40396 y(\257)6597 41193 y(\257)6597 41990 y(\257)7039 40449 y(\263)7833 41924 y Fr(A)8808 42123 y Fv(!)9775 41924 y Fn(\241)295 b Fr(E)12066 42123 y Fo(+)12853 40449 y Fp(\264)13647 41924 y Fn(U)14610 41294 y Fv(f)14478 42294 y(\256;";\257)46 b(;")17364 40396 y Fp(\257)17364 41193 y(\257)17364 41990 y(\257)17807 40396 y(\257)17807 41193 y(\257)17807 41990 y(\257)18250 40747 y Fo(2)19145 41924 y Fn(\267)20680 41025 y Fr(")21289 40543 y Fo(2)p 20680 41618 1135 54 v 20922 42835 a Fs(2)22242 41924 y(+)295 b Fr(K)24751 40449 y Fp(\263)25544 41924 y Fr(")26153 41375 y Fv(s\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))30011 41924 y Fn(\242)295 b Fr(")g Fs(+)2123 b(sup)33019 43068 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)36612 43179 y Fm(\256)37201 43068 y Fo(\()p Fv(")p Fo(\))34174 43875 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)38721 41924 y Fr(!)39579 41375 y Fv(i;j)39531 42252 y(\260)40648 40449 y Fp(\264)41441 40747 y Fo(2)41967 41924 y Fr(:)1837 b Fs(\(2.26\))4317 46133 y(No)-36 b(w,)373 b(for)f Fr(N)510 b Fs(large,)373 b(w)-36 b(e)372 b(ma)-36 b(y)372 b(divide)g(\244)23851 46332 y Fv(N)25119 46133 y Fs(in)-36 b(to)372 b Fr(N)139 b Fs(\()p Fr(")p Fs(\))370 b(disjoin)-36 b(ts)372 b(cub)36 b(es)372 b(of)g(size)g(2\244)44847 46332 y Fv(\256)45507 46133 y Fs(\()p Fr(")p Fs(\).)4317 47738 y(One)433 b(has)20434 49844 y Fr(N)139 b Fs(\()p Fr(")p Fs(\))368 b Fn(')26317 48946 y Fs(\(2)p Fr(N)139 b Fs(\))29160 48463 y Fv(d)p 25139 49539 5739 54 v 25139 51071 a Fr(")25748 50488 y Fj(\241)p Fv(d)p Fo(\()28022 50126 y Ff(1)p 27462 50287 1525 40 v 27462 50835 a Fm(\272)45 b Fe(\241)p Fm(d)29120 50488 y Fo(+)p Fv(\256)p Fo(\))31010 49844 y Fs(;)12794 b(\(2.27\))4317 52633 y(and)19135 54238 y Fn([)20021 53559 y Fv(N)94 b Fo(\()p Fv(")p Fo(\))20021 54578 y Fv(j)51 b Fo(=1)22083 54238 y Fs(\()p Fr(\257)23323 54437 y Fv(j)24105 54238 y Fs(+)295 b(\244)26315 54437 y Fv(\256)26975 54238 y Fs(\()p Fr(")p Fs(\)\))368 b Fn(\275)h Fs(\244)31775 54437 y Fv(N)4317 56372 y Fs(and)433 b(for)h Fr(j)444 b Fn(6)p Fs(=)369 b Fr(j)11809 55890 y Fj(0)12119 56372 y Fs(,)15960 57977 y(\()p Fr(\257)17200 58176 y Fv(j)17983 57977 y Fs(+)294 b(2\244)20842 58176 y Fv(\256)21502 57977 y Fs(\()p Fr(")p Fs(\)\))g Fn(\\)h Fs(\()p Fr(\257)26344 58176 y Fv(j)26776 57924 y Fe(0)27425 57977 y Fs(+)f(2\244)30284 58176 y Fv(\256)30944 57977 y Fs(\()p Fr(")p Fs(\)\))368 b(=)h Fn(;)p Fr(:)4317 60233 y Fs(This)434 b(implies)g(that)f(for)h Fr(j)444 b Fn(6)p Fs(=)369 b Fr(j)19483 59751 y Fj(0)19793 60233 y Fr(;)434 b Fn(U)21551 59603 y Fv(f)21419 60603 y(\256;";\257)23507 60738 y Fm(j)23942 60603 y Fv(;")25127 60233 y Fs(and)f Fn(U)28619 59603 y Fv(f)28487 60603 y(\256;";\257)30575 60857 y Fm(j)30954 60666 y Fe(0)31309 60603 y Fv(;")32494 60233 y Fs(are)h(orthogonal.)4317 62020 y(W)-108 b(e)589 b(denote)g(the)g(coun)-36 b(ting)589 b(function)g(of)h(the)f(eigen)-36 b(v)-72 b(alues)590 b(of)g Fr(A)37700 62219 y Fv(!)32 b(;)p Fo(\244)39231 62368 y Fm(N)40645 62020 y Fs(b)k(elo)-36 b(w)590 b Fr(E)667 b Fs(b)-36 b(y)4317 63625 y(\243)5329 63824 y Fo(\244)5982 63973 y Fm(N)6806 63625 y Fs(\()p Fr(E)78 b Fs(\),)433 b(then)4317 66495 y Fq(E)5203 65021 y Fp(\263)5996 66495 y Fs(\243)7008 66694 y Fo(\244)7661 66843 y Fm(N)8485 66495 y Fs(\()p Fr(E)78 b Fs(+)p Fr(")p Fs(\))p Fn(\241)p Fs(\243)14204 66694 y Fo(\244)14857 66843 y Fm(N)15679 66495 y Fs(\()p Fr(E)g Fn(\241)p Fr(")p Fs(\))19374 65021 y Fp(\264)20536 66495 y Fs(=)369 b Fq(E)22803 65021 y Fp(\263)23595 66495 y Fs(#)24679 65021 y Fp(n)25565 66495 y Fs(eigen)-36 b(v)-72 b(alues)868 b(of)382 b(\246)35130 66694 y Fo(N)35890 66495 y Fs(A)36865 66694 y Fv(!)37538 66495 y Fs(\246)38513 66694 y Fo(N)39562 66495 y Fs(in)289 b([E)42181 66694 y Fo(+)42968 66495 y Fn(\241)p Fr(";)221 b Fs(E)46077 66694 y Fo(+)46865 66495 y Fs(+)p Fr(")p Fs(])48847 65021 y Fp(o)o(\264)25253 70015 y Fs(14)p eop %%Page: 15 15 15 14 bop 4742 8538 a Fn(\270)370 b Fq(E)7031 7064 y Fp(\263)7823 8538 y Fs(#)8907 7064 y Fp(n)9793 8538 y Fs(1)f Fn(\267)g Fr(j)444 b Fn(\267)369 b Fr(N)139 b Fs(\()p Fr(")p Fs(\);)221 b Fn(jj)p Fs(\()p Fr(A)20201 8737 y Fv(!)21168 8538 y Fn(\241)295 b Fr(E)23459 8737 y Fo(+)24246 8538 y Fs(\))p Fn(U)25715 7909 y Fv(f)25583 8908 y(\256;";\257)27671 9043 y Fm(j)28106 8908 y Fv(;")28858 8538 y Fn(jj)29596 8794 y Fv(L)30236 8542 y Ff(2)30697 8794 y Fo(\()p Fg(R)31703 8542 y Fm(d)32184 8794 y Fo(\))32975 8538 y Fn(\267)369 b Fr(")34986 7064 y Fp(o)o(\264)37033 8538 y Fn(\270)g Fq(E)39321 7064 y Fp(\263)40336 6775 y Fv(N)94 b Fo(\()p Fv(")p Fo(\))40380 7276 y Fp(X)40522 10076 y Fv(j)51 b Fo(=1)42564 8538 y Fr(B)43552 8737 y Fv(j)44040 8538 y Fs(\()p Fr(!)d Fs(\))45910 7064 y Fp(\264)46702 8538 y Fr(;)44165 11488 y Fs(\(2.28\))4317 13093 y(where)8833 17786 y Fr(B)9821 17985 y Fv(j)10308 17786 y Fs(\()p Fr(!)g Fs(\))368 b(=)13927 14266 y Fp(8)13927 15462 y(>)13927 15860 y(>)13927 16259 y(<)13927 18650 y(>)13927 19048 y(>)13927 19447 y(:)15661 17014 y Fs(1)1107 b(if)1201 b Fr(K)20800 14344 y Fp(0)20800 16735 y(@)21962 17014 y Fr(")22571 16466 y Fv(s\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))26429 17014 y Fn(\242)295 b Fr(")g Fs(+)2123 b(sup)29437 18158 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)33030 18269 y Fm(\256)33619 18158 y Fo(\()p Fv(")p Fo(\))30592 18965 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)35140 17014 y Fr(!)35998 16466 y Fv(i;j)35950 17343 y(\260)37066 14344 y Fp(1)37066 16735 y(A)38228 14642 y Fo(2)39123 17014 y Fn(\267)40658 16116 y Fr(")41267 15634 y Fo(2)p 40658 16709 1135 54 v 40900 17926 a Fs(2)41925 17014 y Fr(:)15661 20356 y Fs(0)1107 b(if)1201 b(not)p Fr(:)4317 22812 y Fs(The)433 b(\()p Fr(B)8484 23011 y Fv(j)8971 22812 y Fs(\))9477 23019 y Fo(1)p Fj(\267)p Fv(j)51 b Fj(\267)p Fv(N)94 b Fo(\()p Fv(")p Fo(\))14338 22812 y Fs(are)432 b(indep)36 b(enden)-36 b(t,)431 b(iden)-36 b(tically)434 b(distributed,)d (Bernoulli)i(random)4317 24417 y(v)-72 b(ariables.)370 b(So)e(equations)h(\(2.27\))g(and)e(\(2.28\),)j(imply)e(that)g(there)f (exists)i Fr(C)464 b(>)369 b Fs(0,)f(suc)-36 b(h)4317 26022 y(that)433 b(one)h(has)5424 28827 y Fr(N)6466 29026 y Fo(\244)7119 29175 y Fm(N)7943 28827 y Fs(\()p Fr(E)373 b Fs(+)294 b Fr(")p Fs(\))h Fn(\241)g Fr(N)14871 29026 y Fo(\244)15524 29175 y Fm(N)16348 28827 y Fs(\()p Fr(E)373 b Fn(\241)295 b Fr(")p Fs(\))369 b(=)10466 30607 y(1)p 7469 31200 6646 54 v 7469 32417 a(\(\(2)p Fr(N)433 b Fs(+)295 b(1\)\))13575 32033 y Fv(d)14247 31506 y Fq(E)15133 30031 y Fp(\263)15926 31506 y Fs(#)17010 30031 y Fp(n)17895 31506 y Fs(eigen)-36 b(v)-72 b(alues)868 b(of)527 b(\246)27605 31705 y Fo(N)28365 31506 y Fs(A)29340 31705 y Fv(!)30013 31506 y Fs(\246)30988 31705 y Fo(N)32181 31506 y Fs(in)434 b([E)34945 31705 y Fo(+)36027 31506 y Fn(\241)296 b Fr(";)221 b Fs(E)39432 31705 y Fo(+)40514 31506 y Fs(+)295 b Fr(")p Fs(])42791 30031 y Fp(o\264)17922 35087 y Fn(\270)20874 34188 y Fr(N)139 b Fs(\()p Fr(")p Fs(\))p 19457 34781 5635 54 v 19457 35998 a(\(2)p Fr(N)433 b Fs(+)295 b(1\))24551 35614 y Fv(d)25224 35087 y Fq(P)p Fs(\()p Fr(B)27530 35286 y Fo(1)28424 35087 y Fs(=)369 b(1\))g Fn(\270)33053 34188 y Fs(1)p 32865 34781 1026 54 v 32865 35998 a Fr(C)34024 35087 y(")34633 34538 y Fv(d)p Fo(\(1)p Fv(=)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\)+)p Fv(\256)40283 35087 y Fq(P)p Fs(\()p Fr(B)42589 35286 y Fo(1)43484 35087 y Fs(=)368 b(1\))p Fr(:)4317 38231 y Fs(Hence,)434 b(taking)g(the)f(limit)h Fr(N)508 b Fn(!)369 b(1)p Fs(,)434 b(w)-36 b(e)434 b(get)g(that,)f(for)h Fr(")369 b(>)g Fs(0)434 b(small,)g(w)-36 b(e)434 b(obtain)13773 41222 y Fr(N)139 b Fs(\()p Fr(E)16423 41421 y Fo(+)17504 41222 y Fs(+)295 b Fr(")p Fs(\))g Fn(\241)g Fr(N)139 b Fs(\()p Fr(E)24199 41421 y Fo(+)25280 41222 y Fn(\241)296 b Fr(")p Fs(\))368 b Fn(\270)29815 40323 y Fs(1)p 29627 40917 V 29627 42133 a Fr(C)30786 41222 y(")31395 40674 y Fv(d)31934 41222 y Fq(P)p Fs(\()p Fr(B)34240 41421 y Fo(1)35135 41222 y Fs(=)g(1\))p Fr(:)6133 b Fs(\(2.29\))4317 44108 y(It)536 b(just)f(remains)g(to)h(estimate)g Fq(P)p Fs(\()p Fr(B)22762 44307 y Fo(1)23829 44108 y Fs(=)542 b(1\).)537 b(If)f(for)g(1)543 b Fn(\267)f Fr(j)617 b Fn(\267)543 b Fr(N)139 b Fs(\()p Fr(")p Fs(\),)534 b(and)h(an)-36 b(y)536 b Fr(\260)614 b Fn(2)4317 46021 y Fr(\257)5051 46220 y Fv(j)5834 46021 y Fs(+)294 b(\244)8043 46220 y Fv(\256)8703 46021 y Fs(\()p Fr(")p Fs(\);)221 b(1)369 b Fn(\267)g Fr(i;)221 b(j)444 b Fn(\267)370 b Fr(d)p Fs(,)433 b(one)h(has)f Fr(!)23770 45473 y Fv(i;j)23722 46349 y(\260)25208 46021 y Fn(\267)27364 45122 y Fr(")p 26743 45716 1853 54 v 26743 46932 a Fs(2)p Fr(K)28728 46021 y Fs(,)g(then)g(for)h Fr(")f Fs(rather)g(small)7827 49862 y Fr(K)9029 48387 y Fp(\263)9823 49862 y Fr(")10432 49313 y Fv(\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))13854 49862 y Fn(\242)295 b Fr(")g Fs(+)2123 b(sup)16862 51006 y Fv(\260)52 b Fj(2)p Fv(\257)46 b Fo(+2\244)20455 51117 y Fm(\256)21044 51006 y Fo(\()p Fv(")p Fo(\))18017 51813 y Ff(1)p Fe(\267)p Fm(i;j)39 b Fe(\267)p Fm(d)22565 49862 y Fr(!)23423 49313 y Fv(i;j)23375 50190 y(\260)24491 48387 y Fp(\264)25284 48685 y Fo(2)26179 49862 y Fn(\267)369 b Fr(")28190 49313 y Fo(2)29011 49862 y Fn(\242)295 b Fr(K)30877 48387 y Fp(\263)31670 49862 y Fr(")32279 49313 y Fv(\256)p Fo(\()p Fv(\272)58 b Fj(\241)p Fv(d)p Fo(\))35702 49862 y Fs(+)37743 48963 y(1)p 37142 49556 V 37142 50773 a(2)p Fr(K)39127 48387 y Fp(\264)39920 48685 y Fo(2)40815 49862 y Fn(\267)42350 48963 y Fr(")42959 48481 y Fo(2)p 42350 49556 1135 54 v 42592 50773 a Fs(2)43617 49862 y Fr(:)4317 53981 y Fs(As)384 b(the)e(random)h(v)-72 b(ariables)385 b(are)e(indep)36 b(enden)-36 b(t)382 b(iden)-36 b(tically)384 b(distributed,)e(one)i(has)f(the)4317 55586 y(estimate)15184 57614 y Fq(P)p Fs(\()p Fr(B)17490 57813 y Fv(j)18345 57614 y Fs(=)369 b(1\))g Fn(\270)22653 56140 y Fp(\263)23446 57614 y Fq(P)p Fs(\()p Fr(!)25622 56984 y Fo(1)p Fv(;)p Fo(1)25574 57940 y(0)27248 57614 y Fn(\267)29405 56715 y Fr(")p 28783 57309 1853 54 v 28783 58525 a Fs(2)p Fr(K)30768 57614 y Fs(\))31274 56140 y Fp(\264)32067 56437 y Fv(d)32551 56124 y Ff(2)33013 56437 y Fo(#\244)34450 56548 y Fm(\256)35039 56437 y Fo(\()p Fv(")p Fo(\))36261 57614 y Fr(:)4317 60154 y Fs(So,)482 b(taking)h(the)f(double)f (logarithm)i(of)f(\(2.29\),)h(using)f(assumption)g Fh(\(A.3\))h Fs(and)e(the)4317 61970 y(fact)434 b(that)f(#\244)11687 62169 y Fv(\256)12347 61970 y Fs(\()p Fr(")p Fs(\))368 b(=)h Fr(")16326 61387 y Fj(\241)p Fv(d)p Fo(\()18599 61025 y Ff(1)p 18040 61186 1525 40 v 18040 61734 a Fm(\272)45 b Fe(\241)p Fm(d)19697 61387 y Fo(+)p Fv(\256)p Fo(\))21455 61970 y Fs(,)434 b(w)-36 b(e)433 b(get)h(that)9191 66189 y(lim)8833 67087 y Fv(")p Fj(!)p Fo(0)10678 66835 y Ff(+)11709 64808 y Fs(log)13611 63280 y Fp(\257)13611 64077 y(\257)13611 64874 y(\257)14275 64808 y Fs(log)16176 63333 y Fp(\263)16970 64808 y Fr(N)139 b Fs(\()p Fr(E)19620 65007 y Fo(+)20701 64808 y Fs(+)295 b Fr(")p Fs(\))g Fn(\241)g Fr(N)139 b Fs(\()p Fr(E)27396 65007 y Fo(+)28182 64808 y Fs(\))28688 63333 y Fp(\264)29481 63280 y(\257)29481 64077 y(\257)29481 64874 y(\257)p 11709 65884 18215 54 v 19562 67100 a Fs(log)240 b Fr(")30426 66189 y Fn(\270)369 b(\241)34169 65290 y Fr(d)p 32994 65884 3026 54 v 32994 67100 a(\272)378 b Fn(\241)296 b Fr(d)36448 66189 y Fn(\241)f Fr(d\256)8 b(:)4517 b Fs(\(2.30\))25253 70015 y(15)p eop %%Page: 16 16 16 15 bop 4317 7306 a Fs(The)356 b(equation)h(\(2.30\))g(is)f(true)f (for)i(an)-36 b(y)356 b Fr(\256)378 b(>)369 b Fs(0,)356 b(b)-36 b(y)356 b(taking)h Fr(\256)365 b Fs(small)357 b(w)-36 b(e)356 b(end)f(the)h(pro)36 b(of)4317 8911 y(of)434 b(Theorem)g(2.5)h(and)e(hence)g(the)g(subsection.)p 46724 8911 54 832 v 46778 8106 744 54 v 47465 8911 54 832 v 46747 8911 744 54 v 4317 12735 a FA(2.3)1793 b(The)598 b(upp)50 b(er)599 b(b)50 b(ound)4317 15198 y Fs(The)306 b(pro)36 b(of)307 b(of)g(the)e(upp)36 b(er)305 b(b)36 b(ound)305 b(in)g([29)q(])i(is)f(giv)-36 b(en)307 b(in)e(the)h(case)g (of)h(the)f(non)f(degeneracy)4317 16803 y(of)434 b(the)f(IDS)h(of)h (the)e(bac)-36 b(k)433 b(ground)g(op)36 b(erator)434 b(whic)-36 b(h)433 b(is)h(relaxed)g(here.)4317 18408 y(F)-108 b(rom)477 b(Lemma)f(2.2)i(and)f(for)g Fr(\264)19844 18607 y Fo(0)20812 18408 y Fr(>)443 b Fs(1)p Fr(=)p Fs(\()p Fr(\272)408 b Fn(\241)325 b Fr(d)p Fs(\))477 b(and)f Fr(k)488 b Fn(\273)443 b Fr(")33962 17926 y Fj(\241)p Fv(\261)35675 18408 y Fs(suc)-36 b(h)476 b(that)h Fr(\261)492 b(>)443 b(\272)44680 18607 y Fo(0)45682 18408 y Fs(the)4317 20013 y(pro)36 b(of)434 b(of)h(the)e(upp)36 b(er)432 b(b)36 b(ound)432 b(is)i(reduced)f(to)g(pro)-36 b(v)g(e)434 b(that)7884 23418 y(lim)221 b(sup)8616 24574 y Fv(")p Fj(!)p Fo(0)10461 24322 y Ff(+)12224 22519 y Fs(log)240 b Fn(j)221 b Fs(log)20 b(\()p Fq(E)p Fs(\()p Fr(N)19336 22718 y Fv(!)32 b(;k)20782 22519 y Fs(\()p Fr(E)22251 22718 y Fo(+)23333 22519 y Fs(+)294 b Fr(")p Fs(\))h Fn(\241)g Fr(N)28419 22718 y 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34811 y Fi(has)i(a)f(pr)-66 b(ob)g(ability)463 b(at)i(le)-66 b(ast)464 b Fs(1)296 b Fn(\241)f Fq(P)27834 35010 y Fv(")28789 34811 y Fi(wher)-66 b(e)464 b Fq(P)33270 35010 y Fv(")34225 34811 y Fi(satis\257es)17011 38329 y Fs(lim)221 b(sup)17743 39485 y Fv(")p Fj(!)p Fo(0)19588 39233 y Ff(+)21351 37430 y Fs(log)240 b Fn(j)221 b Fs(log)20 b(\()p Fq(P)26841 37629 y Fv(")27330 37430 y Fs(\))p Fn(j)p 21351 38023 6855 54 v 23523 39240 a Fs(log)240 b Fr(")28707 38329 y Fn(\267)369 b(\241)32450 37430 y Fr(d)p 31275 38023 3026 54 v 31275 39240 a(\272)379 b Fn(\241)295 b Fr(d)34434 38329 y(:)9370 b Fs(\(2.32\))4317 41967 y Fh(Pro)42 b(of.)434 b Fs(Using)f(equation)h(\(1.4\))h(w)-36 b(e)433 b(get)h(that)f(for)h(an)-36 b(y)434 b Fr(')369 b Fn(2)g Fr(C)35710 41484 y Fj(1)35615 42295 y Fo(0)36706 41967 y Fs(\()p Fq(R)38171 41484 y Fv(d)38711 41967 y Fs(\))433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)4317 44945 y Fn(h)p Fs(\242)p Fr(A)6893 45144 y Fv(!)32 b(;k)8340 44945 y Fr(';)221 b(')p Fn(i)1118 b 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b(ha)-36 b(v)g(e)9631 61408 y Fq(P)p Fs(\()p Fn(f)p Fs(\242)p Fr(A)13672 61607 y Fv(!)32 b(;k)15488 61408 y Fn(\270)369 b(\241)p Fr(")p Fs(\242)p Fn(g)p Fs(\))1107 b Fn(\270)g Fq(P)p Fs(\()p Fn(f8)p Fr(\260)441 b Fn(2)368 b Fq(Z)30009 60859 y Fv(d)30009 61736 y Fo(2)p Fv(k)24 b Fo(+1)32250 61408 y Fs(;)221 b Fr(g)33455 61607 y Fj(\241)34657 60146 y Fp(X)34464 63073 y Fv(\256)p Fj(2)p Fg(Z)36285 62821 y Fm(d)36990 61408 y Fr(A)37965 61607 y Fv(\256)38994 61408 y Fn(\270)369 b Fr(")p Fn(g)p Fs(\))21893 65100 y Fn(\270)1107 b Fs(1)296 b Fn(\241)27195 63838 y Fp(X)26307 66818 y Fv(\260)52 b Fj(2)p Fg(Z)28061 66513 y Fm(d)28061 67133 y Ff(2)p Fm(k)14 b Ff(+1)30224 65100 y Fq(P)p Fs(\()p Fn(f)p Fr(A)33181 65299 y Fv(\260)33773 65100 y Fs(\()p Fr(!)48 b Fs(\))368 b Fn(\267)h Fr(")p Fn(g)p Fs(\))p Fr(:)25253 70015 y Fs(16)p eop %%Page: 17 17 17 16 bop 4317 7306 a Fs(As)434 b(the)f(random)g(v)-72 b(ariables)434 b(are)g(i.i.d,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)11084 10239 y Fq(P)p Fs(\()p Fn(f)p Fs(\242)p Fr(A)15125 10438 y Fv(!)32 b(;k)16941 10239 y Fn(\270)369 b(\241)p Fr(")p Fs(\242)p Fn(g)p Fs(\))g Fn(\270)g Fs(1)295 b Fn(\241)h Fs(\(2)p Fr(k)340 b Fs(+)295 b(1\))30919 9691 y Fv(d)31459 10239 y Fq(P)p Fs(\()p Fn(f)p Fr(A)34416 10438 y Fo(0)34941 10239 y Fs(\()p Fr(!)48 b Fs(\))368 b Fn(\267)i Fr(")p Fn(g)p Fs(\))p Fr(:)3443 b Fs(\(2.33\))4317 13173 y(T)-108 b(o)422 b(estimate)g Fq(P)p Fs(\()p Fn(f)p Fr(A)14375 13372 y Fo(0)14901 13173 y Fs(\()p Fr(!)48 b Fs(\))368 b Fn(\267)h Fr(")p Fn(g)p Fs(\))422 b(it)f(su\261ces)h(to)g (follo)-36 b(w)423 b(the)f(same)g(computation)f(done)4317 14778 y(in)531 b([24)q(])g(and)f(based)h(on)g(the)f(Mark)-36 b(o)g(v's)532 b(inequalit)-36 b(y)-108 b(,)532 b(and)e(the)h(T)-108 b(a)-36 b(ylor)532 b(expansion)f(of)4317 16383 y Fr(e)4920 15901 y Fj(\241)p Fv(x)6670 16383 y Fs(to)434 b(get)g(that)17782 18553 y Fq(P)p Fs(\()p Fn(f)p Fr(A)20739 18752 y Fo(0)21264 18553 y Fs(\()p Fr(!)48 b Fs(\))369 b Fn(\267)g Fr(")p Fn(g)p Fs(\))f Fn(\267)i Fr(e)29058 18005 y Fj(\241)30036 17643 y Ff(1)p 29922 17804 633 40 v 29922 18352 a Fm(C)30688 18005 y Fv(")31122 17474 y Fe(\241)32444 17213 y Fm(d)p 31895 17324 1525 33 v 31895 17817 a(\272)45 b Fe(\241)p Fm(d)33663 18553 y Fr(:)10141 b Fs(\(2.34\))4317 20878 y(This)434 b(complete)g(the)f(pro)36 b(of)434 b(of)g(Lemma)g(2.8.)p 46724 20878 54 832 v 46778 20073 744 54 v 47465 20878 54 832 v 46747 20878 744 54 v 4317 23590 a Fh(Lemma)499 b(2.9)651 b Fi(Ther)-66 b(e)411 b(exists)g Fr(C)464 b(>)368 b Fs(0)412 b Fi(and)g Fr(")26272 23789 y Fo(0)27166 23590 y Fr(>)369 b Fs(0)412 b Fi(\(uniform)f(in)f Fr(k)457 b Fi(and)411 b Fr(!)48 b Fi(\))412 b(such)g(that,)4317 25195 y(if)464 b Fs(0)369 b Fr(<)g(")g(<)f(")10946 25394 y Fo(0)11937 25195 y Fi(and)465 b Fr(!)512 b Fi(satis\257es)465 b Fs(\242)21749 25394 y Fv(!)32 b(;k)23565 25195 y Fn(\270)369 b(\241)p Fr(C)95 b(")p Fs(\242)p Fi(,)464 b(for)h Fr(k)414 b Fn(2)368 b Fq(N)p Fi(,)465 b(one)f(has)18492 28129 y Fr(N)19534 28328 y Fv(!)32 b(;k)20980 28129 y Fs(\()p Fr(E)22449 28328 y Fo(+)23236 28129 y Fs(\))369 b(=)g Fr(N)26534 28328 y Fv(!)32 b(;k)27980 28129 y Fs(\()p Fr(E)29449 28328 y Fo(+)30531 28129 y Fs(+)295 b Fr(")p Fs(\))p Fr(:)4317 31062 y Fs(Lemma)581 b(2.9)g(sa)-36 b(ys)582 b(that)e(if)i(\242)p Fr(A)20621 31261 y Fv(!)32 b(;k)22687 31062 y Fn(\270)620 b(\241)p Fr(C)95 b(")p Fs(\242,)580 b(then)g(the)g(sp)36 b(ectrum)580 b(of)i Fr(A)42961 31261 y Fv(!)32 b(;k)44988 31062 y Fs(do)k(es)4317 32667 y(not)552 b(in)-36 b(tersect)552 b(\()p Fr(E)13582 32866 y Fo(+)14369 32667 y Fr(;)221 b(E)15914 32866 y Fo(+)17078 32667 y Fs(+)376 b Fr(")p Fs(\))551 b(for)i Fr(")g Fs(small)g(.)g(This)g(lemma)g(will)h(b)36 b(e)552 b(pro)-36 b(v)g(ed)552 b(in)g(the)4317 34272 y(app)36 b(endix.)4317 35877 y(No)-36 b(w)585 b(estimate)f(\(2.31\))h(is)g(an)f (immediate)g(consequence)g(of)h(Lemma)f(2.8)h(and)e(2.9.)4317 37482 y(Indeed,)433 b(pic)-36 b(king)434 b Fr(C)529 b Fs(as)434 b(in)f(Lemma)h(2.9;)g(one)g(computes)4317 40707 y Fq(E)5203 39233 y Fp(\263)5996 40707 y Fr(N)7038 40906 y Fv(!)32 b(;k)8485 40707 y Fs(\()p Fr(E)9954 40906 y Fo(+)11036 40707 y Fs(+)294 b Fr(")p Fs(\))h Fn(\241)g Fr(N)16122 40906 y Fv(!)32 b(;k)17569 40707 y Fs(\()p Fr(E)19038 40906 y Fo(+)19825 40707 y Fs(\))20331 39233 y Fp(\264)22242 40707 y Fs(=)1117 b Fq(E)25257 39233 y Fp(\263)26050 40707 y Fs([)p Fr(N)27453 40906 y Fv(!)32 b(;k)28900 40707 y Fs(\()p Fr(E)30369 40906 y Fo(+)31451 40707 y Fs(+)295 b Fr(")p Fs(\))f Fn(\241)i Fr(N)36538 40906 y Fv(!)32 b(;k)37984 40707 y Fs(\()p Fr(E)39453 40906 y Fo(+)40240 40707 y Fs(\)])41107 40906 y Fc(1)41649 41099 y Fe(f)p Fm(!)24 b Ff(;\242)p Fm(A)44127 41332 y(!)g(;k)45429 41099 y Fe(\270\241)p Fm(C)46 b(")p Ff(\242)p Fe(g)48930 39233 y Fp(\264)22242 43541 y Fs(+)1117 b Fq(E)25257 42067 y Fp(\263)26050 43541 y Fs([)p Fr(N)27453 43740 y Fv(!)32 b(;k)28900 43541 y Fs(\()p Fr(E)30369 43740 y Fo(+)31451 43541 y Fs(+)295 b Fr(")p Fs(\))f Fn(\241)i Fr(N)36538 43740 y Fv(!)32 b(;k)37984 43541 y Fs(\()p Fr(E)39453 43740 y Fo(+)40240 43541 y Fs(\)])41107 43740 y Fc(1)41649 43933 y Fe(f)p Fm(!)24 b Ff(;\242)p Fm(A)44127 44166 y(!)g(;k)45429 43933 y(<)p Fe(\241)p Fm(C)46 b(")p Ff(\242)p Fe(g)48930 42067 y Fp(\264)22231 45844 y Fn(\267)1107 b Fr(C)25302 46043 y Fo(0)25827 45844 y Fq(P)p Fs(\()p Fn(f)p Fr(!)48 b Fs(;)221 b(\242)p Fr(A)31308 46043 y Fv(!)32 b(;k)33124 45844 y Fr(<)369 b Fn(\241)p Fr(C)95 b(")p Fs(\242)p Fn(g)p Fs(\))22242 47781 y(=)1117 b Fr(C)25302 47980 y Fo(0)25827 47781 y Fs(\(1)296 b Fn(\241)f Fq(P)p Fs(\()p Fh(E)30906 47980 y Fv(C)63 b Fj(\242)p Fv(";!)33268 47781 y Fs(\)\))369 b(=)f Fr(C)36960 47980 y Fo(0)37486 47781 y Fq(P)38298 47980 y Fv(C)63 b Fj(\242)p Fv(")39782 47781 y Fr(:)4317 50714 y Fs(Here,)341 b(w)-36 b(e)341 b(ha)-36 b(v)g(e)342 b(used)e(the)g(fact)h(that)g Fr(N)23649 50913 y Fv(!)32 b(;k)25436 50714 y Fs(is)341 b(b)36 b(ounded,)340 b(lo)36 b(cally)343 b(uniformly)f(in)e(energy)-108 b(,)4317 52319 y(uniformly)571 b(in)f Fr(!)48 b Fs(,)570 b Fr(k)615 b Fs(b)-36 b(y)570 b Fr(C)17987 52518 y Fo(0)18512 52319 y Fs(.)h(T)-108 b(aking)571 b(\(2.32\))g(in)-36 b(to)570 b(accoun)-36 b(t,)570 b(w)-36 b(e)570 b(end)g(the)f(pro)36 b(of)571 b(of)4317 53925 y(\(2.31\),)435 b(and)e(consequen)-36 b(tly)434 b(the)f(pro)36 b(of)434 b(of)g(the)f(upp)36 b(er)433 b(b)36 b(ound.)p 46724 53925 54 832 v 46778 53120 744 54 v 47465 53925 54 832 v 46747 53925 744 54 v 4317 58362 a Ft(3)2152 b(The)716 b(pro)60 b(of)716 b(of)h(Theorem)f(1.1.)4317 61282 y Fs(All)535 b(the)f(pro)36 b(ofs)535 b(of)h(lo)36 b(calization,)537 b(except)d(in)h(the)f (discrete)g(case)h([1])g(for)g(the)f(m)-36 b(ulti-)4317 62887 y(dimensional)566 b(case,)g(use)f(the)g(metho)36 b(d)565 b(of)h(the)f(m)-36 b(ultiscale)566 b(analysis.)i(This)d(metho) 36 b(d)4317 64492 y(w)-36 b(as)444 b(used)f(for)h(the)f(\257rst)f(time) i(b)-36 b(y)443 b(F)-108 b(r\304)-650 b(ohlic)-36 b(h)443 b(and)g(Sp)36 b(encer,)442 b(at)i(the)f(b)36 b(eginning)443 b(of)h(the)4317 66097 y(eigh)-36 b(ties)455 b([12)q(])f(and)g(it)h (knew)g(man)-36 b(y)454 b(extensions)h(and)f(simpli\257cations)g(to)h (lead)g(to)f(the)25253 70015 y(17)p eop %%Page: 18 18 18 17 bop 4317 7306 a Fs(form)465 b(describ)36 b(ed)463 b(in)h(the)g(section)g(b)36 b(elo)-36 b(w.)465 b(This)g(analysis)g(mak) -36 b(es)465 b(p)36 b(ossible)464 b(to)h(obtain)4317 8911 y(information)449 b(on)f(the)g(op)36 b(erator)449 b(in)f(the)g(hole)h(space,)f(starting)h(from)g(information)g(on)4317 10516 y(the)418 b(op)36 b(erator)419 b(restricted)f(to)h(cub)36 b(es)418 b(of)h(\257nished)e(size)i([7)q(].)g(F)-108 b(or)418 b(sak)-36 b(e)419 b(of)h(completeness)4317 12121 y(in)524 b(the)f(follo)-36 b(wing)526 b(section)e(w)-36 b(e)525 b(brie\260y)e(recall)i(the)f(tec)-36 b(hnique)523 b(of)i(m)-36 b(ultiscale)524 b(analy-)4317 13726 y(sis.)515 b(W)-108 b(e)513 b(refer)g(the)g(reader)g(to)h([7,)g(17)q(,)f(20)q(,)g (34)q(])h(for)g(detailed)f(description)g(of)h(results)4317 15331 y(men)-36 b(tioned)433 b(b)36 b(elo)-36 b(w.)4317 19183 y FA(3.1)1793 b(Multiscale)599 b(analysis)g(.)4317 21646 y Fs(W)-108 b(e)550 b(start)g(b)-36 b(y)549 b(in)-36 b(tro)36 b(ducing)549 b(some)i(notations.)g(Let)e(\244)31798 21845 y Fv(k)32367 21646 y Fs(\()p Fr(x)p Fs(\),)h(b)36 b(e)549 b(the)g(cub)36 b(e)550 b(of)g(cen)-36 b(ter)4317 23251 y Fr(x)369 b Fn(2)g Fq(Z)7566 22769 y Fv(d)8453 23251 y Fs(and)349 b(with)f(side)h(length)f(2)p Fr(k)45 b Fs(.)350 b(The)e(restriction)h(of)g Fr(A)33366 23450 y Fv(!)34387 23251 y Fs(on)g(\244)37012 23450 y Fv(k)37580 23251 y Fs(\()p Fr(x)p Fs(\))f(with)h(Diric)-36 b(hlet)4317 24856 y(b)36 b(oundary)498 b(condition)g(is)h(noted)f(b)-36 b(y)498 b Fr(A)23905 25055 y Fv(x;k)25265 24856 y Fs(.)h(Eac)-36 b(h)499 b Fr(A)30399 25055 y Fv(x;k)32257 24856 y Fs(is)g(a)g(p)36 b(ositiv)-36 b(e)499 b(op)36 b(erator)499 b(with)4317 26461 y(compact)481 b(resolv)-36 b(en)g(t)481 b Fr(R)16067 26660 y Fv(x;k)17427 26461 y Fs(\()p Fr(z)59 b Fs(\))449 b(=)h(\()p Fr(A)22494 26660 y Fv(x;k)24181 26461 y Fn(\241)327 b Fr(z)59 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b(estimate:)409 b Fh(\(P1\))320 b Fs(\()p Fr(k)19777 65836 y Fo(0)20303 65637 y Fr(;)221 b(m)22023 65836 y Fo(0)22549 65637 y Fr(;)g(p)p Fs(\))320 b(for)f Fr(E)27445 65836 y Fo(0)28340 65637 y Fn(2)369 b Fs([)p Fr(E)30919 65836 y Fo(+)31706 65637 y Fr(;)221 b(E)33251 65836 y Fo(+)34100 65637 y Fs(+)62 b Fr(")35783 65836 y Fo(0)36308 65637 y Fs(])p Fr(;)221 b(p)370 b(>)e(d)319 b Fs(and)g Fr(m)44202 65836 y Fo(0)45096 65637 y Fr(>)369 b Fs(0,)25253 70015 y(18)p eop %%Page: 19 19 19 18 bop 4317 7306 a Fs(there)433 b(exists)h Fr(k)11908 7505 y Fo(0)12804 7306 y Fn(2)368 b Fq(N)434 b Fs(suc)-36 b(h)433 b(that)12990 10663 y Fq(P)13802 10862 y Fo(\244)14455 11018 y Fm(k)14894 11217 y Ff(0)15465 9188 y Fp(n)16351 10663 y Fs(\244)17254 10862 y Fv(k)17744 10985 y Ff(0)18260 10663 y Fs(\(0\))h(is)g(\()p Fr(m)23308 10862 y Fo(0)23833 10663 y Fr(;)221 b(E)25378 10862 y Fo(0)25904 10663 y Fs(\))295 b Fn(\241)h Fs(regular)32008 9188 y Fp(o)33262 10663 y Fn(\270)370 b Fs(1)295 b Fn(\241)37371 9764 y 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y Fr(R)26949 59710 y Fv(y)g(;k)28276 59511 y Fs(\()p Fr(E)78 b Fs(\))p Fr(\302)31144 59710 y Fv(y)31695 59511 y Fn(jj)32433 59710 y Fv(y)32 b(;k)33760 59511 y Fn(jj)p Fs(\241)35311 59710 y Fv(x;k)36616 59458 y Fe(0)36969 59511 y Fr(R)37959 59710 y Fv(x;k)39264 59458 y Fe(0)39617 59511 y Fs(\()p Fr(z)59 b Fs(\))p Fr(\302)42107 59710 y Fv(y)42604 59458 y Fe(0)42958 59511 y Fn(jj)43696 59710 y Fv(x;k)45001 59458 y Fe(0)45354 59511 y Fr(:)4317 62408 y Fi(F)-100 b(or)466 b(some)f Fr(y)10746 61926 y Fj(0)11425 62408 y Fn(2)368 b Fs(\250)13691 62607 y Fv(k)14260 62408 y Fs(\()p Fr(y)48 b Fs(\))464 b Fi(and)h Fr(C)19873 62607 y Fv(z)20772 62408 y Fs(=)369 b Fr(C)95 b Fs(\()p Fr(z)59 b(;)221 b(d;)g(a)26871 62607 y Fo(+)27659 62408 y Fr(;)g(a)28924 62607 y Fj(\241)29712 62408 y Fs(\))p Fi(.)4317 65089 y Fs(T)-108 b(o)434 b(mak)-36 b(e)434 b(this)f(dev)-36 b(elopmen)g(t)434 b(successfully)g(one)g (needs)f(t)-36 b(w)g(o)433 b(main)h(ingredien)-36 b(ts:)4317 66694 y(1\))484 b(A)f(succession)g(of)h(regular)g(cub)36 b(es)482 b(\244)24153 66893 y Fv(k)24643 67028 y Fm(j)25132 66694 y Fs(\()p Fr(x)p Fs(\))h(whic)-36 b(h)483 b(join)h(the)f(edges)g (of)h(\244)42001 66893 y Fv(k)42491 67028 y Fm(j)39 b Ff(+1)44007 66694 y Fs(\(0\).)484 b(It)25253 70015 y(19)p eop %%Page: 20 20 20 19 bop 4317 7306 a Fs(is)434 b(sho)-36 b(wn)433 b(that)g(this)h(is)g (the)f(case)h(with)f(a)h(large)h(probabilit)-36 b(y)-108 b(.)4317 8911 y(2\))451 b(A)f(fundamen)-36 b(tal)450 b(estimate)h(of)g Fr(R)22596 9110 y Fo(0)p Fv(;k)23817 9245 y Fm(j)39 b Ff(+1)25334 8911 y Fs(\()p Fr(E)78 b Fs(\))449 b(for)i(eac)-36 b(h)451 b Fr(E)475 b Fn(2)397 b Fs([)p Fr(E)36829 9110 y Fo(+)37617 8911 y Fr(;)221 b(E)39162 9110 y Fo(+)40244 8911 y Fs(+)295 b Fr(")42160 9110 y Fo(0)42686 8911 y Fs(],)451 b(with)f(a)4317 10516 y(su\261cien)-36 b(tly)535 b(large)h(probabilit)-36 b(y)-108 b(.)536 b(This)f(is)h(a)f(consequence)g(of)h(the)f(uniform)g(W)-108 b(egner)4317 12121 y(estimate)868 b Fh(\(P2\))p Fs(.)4317 13726 y(Then,)481 b(w)-36 b(e)481 b(obtain)f(an)h(estimate)g(on)g(the)f (deca)-36 b(y)481 b(of)h Fr(R)31608 13925 y Fo(0)p Fv(;k)32829 14060 y Fm(j)39 b Ff(+1)34345 13726 y Fs(\()p Fr(z)59 b Fs(\))481 b(with)g(a)g(rate)g(sligh)-36 b(tly)4317 15331 y(smaller)434 b(compared)g(to)f(that)g(of)i Fr(R)21609 15530 y Fo(0)p Fv(;k)22830 15665 y Fm(j)23320 15331 y Fs(,)e(but)g(with)h(a)f(higher)h(probabilit)-36 b(y)-108 b(.)4317 16936 y(Here)319 b(w)-36 b(e)319 b(use)f(the)g Fi(uniform)417 b Fs(W)-108 b(egner)319 b(estimate)g([17)q(])g(instead)f (of)i(the)e(W)-108 b(egner)318 b(estimate)4317 18542 y([8)q(],)364 b(to)f(o)-36 b(v)g(ercome)365 b(the)e(di\261cult)-36 b(y)363 b(caused)g(b)-36 b(y)363 b(the)g(long)h(range)g(p)36 b(oten)-36 b(tial.)364 b(Notice)g(that)4317 20147 y(the)429 b(crucial)h(probabilistic)g(estimates)f(in)h([7])g(\(See)f(Lemma)g (4.1\))h(can)g(b)36 b(e)429 b(done)g(using)4317 21752 y(the)k(fact)h(that)f(for)i(an)-36 b(y)433 b(\244)17287 21951 y Fo(1)17813 21752 y Fr(;)221 b Fs(\244)19298 21951 y Fo(2)20258 21752 y Fs(in)433 b Fq(Z)22661 21270 y Fo(2)23187 21752 y Fs(,)h(w)-36 b(e)433 b(ha)-36 b(v)g(e)17009 24685 y Fq(P)17821 24884 y Fo(\244)18474 25007 y Ff(2)18990 24685 y Fs(\()p Fr(A)p Fs(\))369 b Fn(\267)g Fq(P)23560 24884 y Fo(\244)24213 25007 y Ff(1)24729 24685 y Fs(\()p Fr(A)p Fs(\))1300 b(if)1394 b(\244)31071 24884 y Fo(1)31966 24685 y Fn(\275)369 b Fs(\244)34271 24884 y Fo(2)4317 27619 y Fs(and)11347 29224 y Fq(P)12159 29423 y Fo(\244)12812 29546 y Ff(1)13272 29423 y Fj([)p Fo(\244)14552 29546 y Ff(2)15068 29224 y Fs(\()p Fr(A)295 b Fn(\\)g Fr(B)67 b Fs(\))369 b Fn(\267)g Fq(P)22169 29423 y Fo(\244)22822 29546 y Ff(1)23338 29224 y Fs(\()p Fr(A)p Fs(\))295 b Fn(\242)g Fq(P)27096 29423 y Fo(\244)27749 29546 y Ff(2)28265 29224 y Fs(\()p Fr(B)67 b Fs(\))867 b(if)1394 b(\244)34254 29423 y Fo(1)35075 29224 y Fn(\\)294 b Fs(\244)37158 29423 y Fo(2)38053 29224 y Fs(=)369 b Fn(;)p Fr(:)4317 33075 y FA(3.2)1793 b(Lo)50 b(calization)598 b(for)f(the)i(div)-50 b(ergence)600 b(op)50 b(erator)4317 35538 y Fs(Our)411 b(strategy)-108 b(,)413 b(in)f(pro)-36 b(ving)413 b(lo)36 b(calization,)415 b(consists)d(in)g(pro)-36 b(ving)413 b(that)f(from)g(Theorem)4317 37143 y(1.2)501 b(w)-36 b(e)500 b(are)f(able)h(to)g(obtain)g(the)f(necessary)h(standard)e (ingredien)-36 b(ts)499 b(for)i(recurrence)4317 38748 y(pro)36 b(cedure)373 b(and,)h(th)-36 b(us,)374 b(to)g(start)g(the)f (mac)-36 b(hinery)375 b(of)f(m)-36 b(ultiscale)375 b(analysis,)h(and)e (pro)-36 b(v)g(e)4317 40353 y(the)438 b(existence)h(lo)36 b(cated)439 b(states)f(on)g(edge)h(of)g(the)e(almost)i(sure)f(sp)36 b(ectrum)437 b(\247)p Fn(nf)p Fs(0)p Fn(g)p Fs(.)j(In-)4317 41958 y(deed,)532 b(p)36 b(oin)-36 b(ts)532 b(\()p Fr(i)p Fs(\))f(and)h(\()p Fr(ii)p Fs(\))f(of)i(Theorem)f(1.1)h(are)g(pro)-36 b(v)g(en)532 b(in)g([8])h(and)e(in)h([34)q(])h(as)f(a)4317 43563 y(consequence)390 b(of)g Fh(\(P1\))g Fs(and)f Fh(\(P2\))p Fs(.)h(F)-108 b(or)389 b(the)g(strong)g(dynamical)i(lo)36 b(calization)391 b(\(p)36 b(oin)-36 b(t)4317 45169 y(\()p Fr(iii)p Fs(\))444 b(of)i(Theorem)f(1.1\);)h(Damanik)h(and)d(Stollmann) h(in)g([6)q(],)h(sho)-36 b(w)445 b(that)g(this)g(can)g(b)36 b(e)4317 46774 y(seen)464 b(as)h(a)g(result)f(of)h(Anderson)f(lo)36 b(calization)466 b(\(h)-36 b(yp)36 b(othesis)465 b Fh(EDI)f Fs(in)g([34)q(]\);)h(whic)-36 b(h)464 b(is)4317 48379 y(guaran)-36 b(teed)433 b(b)-36 b(y)434 b(p)36 b(oin)-36 b(ts)433 b(\()p Fr(i)p Fs(\))g(and)g(\()p Fr(ii)p Fs(\))f(of)j(Theorem) e(1.1.)4317 49984 y(Recen)-36 b(tly)-108 b(,)493 b(Strong)f(dynamical)i (lo)36 b(calization)494 b(up)e(to)h(an)-36 b(y)493 b(order,)f(is)h(pro) -36 b(v)g(ed)492 b(b)-36 b(y)493 b(Ger-)4317 51589 y(minet)480 b(and)g(Klein)h(in)f([13)q(].)h(It)f(is)h(giv)-36 b(en)481 b(for)g(Sc)-36 b(hr\304)-650 b(odinger)479 b(op)36 b(erators,)481 b(but)e(it)i(is)g(also)4317 53194 y(true)433 b(for)h(our)f(mo)36 b(del.)4317 54799 y(F)-108 b(rom)627 b(what)g(it)g(w)-36 b(as)627 b(said)h(previously)g(to)f(pro)-36 b(v)g(e)627 b(Theorem)g(1.1,)h(it)g(is)f(enough)f(to)4317 56404 y(c)-36 b(hec)g(k)564 b(the)f(necessary)h(assumptions)f(to)h(start)f(the)g(m) -36 b(ultiscale)564 b(analysis)h(i.e;)g Fh(\(P1\))4317 58009 y Fs(and)433 b Fh(\(P2\))p Fs(.)4317 61861 y FA(3.3)1793 b(Initial)598 b(estimate)4317 64324 y Fs(W)-108 b(e)493 b(start)g(b)-36 b(y)493 b(v)-36 b(erifying)513 b Fh(\(P1\))p Fs(.)494 b(This)f(already)h(has)f(b)36 b(een)493 b(done)f(in)h([3)q(,)g (4)q(,)g(6,)h(19,)g(20])4317 65929 y(for)366 b(Sc)-36 b(hr\304)-650 b(odinger)365 b(op)36 b(erators)366 b(and)f(in)h([34)q(]) g(for)g(the)f(mo)36 b(del)366 b(giv)-36 b(en)367 b(b)-36 b(y)365 b(the)g(form)i(of)f(\(1.1\))25253 70015 y(20)p eop %%Page: 21 21 21 20 bop 4317 7306 a Fs(under)603 b(the)g(assumption)h(of)h(a)f(sp)36 b(ecial)605 b(mo)36 b(de)604 b(of)g(disorder.)h(More)f(precisely)h (their)4317 8911 y(approac)-36 b(h)507 b(requires)h(a)f(su\261cien)-36 b(tly)508 b(fast)g(deca)-36 b(y)508 b(of)g(the)f(probabilit)-36 b(y)508 b(distribution)e Fr(\271)4317 10516 y Fs(at)365 b(the)f(approac)-36 b(h)363 b(to)i(the)f(edges)g(of)h(the)f(supp)36 b(ort.)364 b(Indeed,)g(they)g(assumed)g(that)g(there)4317 12121 y(exists)456 b(some)f Fr(\277)553 b(>)405 b Fs(0)455 b(\()p Fr(\277)553 b(>)18023 11598 y Fo(3)p Fv(d)p 18023 11816 955 54 v 18265 12579 a Fo(2)19566 12121 y Fs(in)454 b([3)q(];)h Fr(\277)554 b(>)404 b(d)455 b Fs(in)f([19)q(])h(and)g Fr(\277)553 b(>)36194 11598 y Fv(d)p 36194 11816 485 54 v 36201 12579 a Fo(2)37265 12121 y Fs(in)455 b([20)q(]\):)g(suc)-36 b(h)454 b(that)4317 13726 y Fn(8)p Fr(")369 b(>)f Fs(0)p Fr(;)17290 15934 y Fq(P)18102 16133 y Fo(0)18627 15934 y Fs(\([0)p Fr(;)221 b(")p Fs(]\))370 b(=)23953 14126 y Fp(Z)25281 14477 y Fv(")24691 17133 y Fo(0)25992 15934 y Fr(g)48 b Fs(\()p Fr(s)p Fs(\)\()p Fr(s)p Fs(\))p Fr(ds)367 b Fn(\267)i Fr(")33580 15386 y Fv(\277)34155 15934 y Fr(;)4317 18535 y Fs(or)16659 20674 y Fq(P)17471 20873 y Fo(0)17996 20674 y Fs(\([1)296 b Fn(\241)f Fr(";)221 b Fs(1]\))370 b(=)25596 18866 y Fp(Z)26924 19217 y Fo(1)26334 21874 y(1)p Fj(\241)p Fv(")28247 20674 y Fr(g)48 b Fs(\()p Fr(s)p Fs(\))p Fr(ds)368 b Fn(\267)h Fr(")34211 20126 y Fv(\277)34786 20674 y Fr(:)4317 23719 y Fs(According)573 b(to)g(whether)g(one)g(considers)g(the)f(higher)h(or)g(lo)-36 b(w)g(er)574 b(edges)f(of)h(the)e(gap.)4317 25324 y(Those)434 b(conditions)g(are)f(relaxed)i(here.)4317 26929 y(In)523 b(what)g(follo)-36 b(ws,)526 b(to)d(get)g Fh(\(P1\))h Fs(one)f(will)h(adapt)f(an)g(other)f(approac)-36 b(h)523 b(whic)-36 b(h)523 b(uses)4317 28534 y(the)367 b(fact)i(that)e(at)h (the)g(in)-36 b(ternal)367 b(band)g(edges,)h(the)g(IDS)g(of)g Fr(A)34256 28733 y Fv(!)35296 28534 y Fs(exhibits)g(Lifshitz)h(tails.) 4317 30139 y(See)433 b([21)q(,)h(36)q(,)g(29,)g(30)q(].)4317 31744 y(Let)f(us)g(consider)h(the)f(follo)-36 b(wing)436 b(assumption)4317 33641 y Fh(\(P')498 b(1\))434 b Fq(P)9406 33840 y Fo(\244)10059 33996 y Fm(k)10498 34195 y Ff(0)11069 32166 y Fp(n)11955 33641 y Fr(d)p Fs(\()p Fr(\276)48 b Fs(\()p Fr(A)15405 33840 y Fo(0)p Fv(;k)16626 33963 y Ff(0)17143 33641 y Fs(\))p Fr(;)221 b Fs([)p Fr(E)19555 33840 y Fo(+)20343 33641 y Fr(;)g(E)21888 33840 y Fo(+)22970 33641 y Fs(+)295 b Fr(")24886 33840 y Fo(0)25412 33641 y Fs(]\))369 b Fr(<)28161 33118 y Fo(1)p 28161 33335 471 54 v 28161 34099 a(2)28765 33641 y Fr(k)29486 33091 y Fj(\241)p Fv(\256)29441 33966 y Fo(0)30877 32166 y Fp(o)32132 33641 y Fn(\267)33929 33118 y Fo(1)p 33667 33335 996 54 v 33667 34108 a Fv(k)34181 33674 y Fm(p)34157 34391 y Ff(0)34795 33641 y Fs(,)4317 35611 y(with)434 b Fr(\256)377 b Fn(2)p Fs(]0)p Fr(;)221 b Fs(1[)436 b(and)d Fr(")15548 35810 y Fo(0)16442 35611 y Fr(>)369 b Fs(0.)4317 37216 y(In)434 b(the)f(app)36 b(endix)433 b(w)-36 b(e)434 b(pro)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(statemen)-36 b(t:)4317 39928 y Fh(Lemma)499 b(3.3)651 b(\(P'1\))465 b Fi(implies)929 b Fh(\(P1\))466 b Fi(for)e(c)-66 b(ertain)463 b Fr(m)31873 40127 y Fo(0)32768 39928 y Fn(\270)369 b Fr(C)95 b(k)35917 39378 y Fj(\241)p Fo(2)35872 40254 y(0)37174 39928 y Fi(.)4317 42640 y Fs(So)434 b(the)f(pro)36 b(of)434 b(of)g(the)f(Theorem)h(1.1)g(is)g(reduced)f(to)g(the)g(pro)36 b(of)435 b(of:)4317 45352 y Fh(Prop)42 b(osition)500 b(3.4)651 b Fi(Under)464 b(the)g(assumptions)i(of)e(The)-66 b(or)g(em)464 b(1.1,)g Fh(\(P'1\))h Fi(holds.)4317 48816 y Fh(3.3.1)1495 b(The)499 b(pro)42 b(of)498 b(of)g(the)g(Prop)42 b(osition)500 b(3.4.)4317 51279 y Fs(T)-108 b(o)486 b(get)g Fh(\(P'1\))p Fs(,)g(in)f(an)h(in)-36 b(terv)-72 b(al)485 b(of)h(energy)g(whic)-36 b(h)486 b(con)-36 b(tains)485 b Fr(E)36502 51478 y Fo(+)37774 51279 y Fs(one)h(will)g(compare)4317 52884 y(the)362 b(n)-36 b(um)g(b)36 b(er)361 b(of)i(states)g(of)g Fr(A)18572 53083 y Fv(!)19607 52884 y Fs(to)f(that)g(of)h Fr(A)26257 53083 y Fo(0)p Fv(;k)27558 52884 y Fs(.)g(This)g(will)h(b)36 b(e)362 b(done)g(b)-36 b(y)363 b(a)f(comparison)4317 54489 y(b)36 b(et)-36 b(w)g(een)620 b Fr(N)139 b Fs(\()p Fr(A)12188 54688 y Fv(!)12859 54489 y Fr(;)221 b(E)78 b Fs(\))621 b(and)f Fr(N)139 b Fs(\()p Fr(A)20987 54688 y Fo(0)p Fv(;k)22287 54489 y Fr(;)221 b(E)78 b Fs(\).)621 b(A)-36 b(t)621 b(the)e(b)36 b(ottom)621 b(of)g(the)f(sp)36 b(ectrum,)619 b(suc)-36 b(h)4317 56094 y(tec)g(hnique)512 b(w)-36 b(as)514 b(used)e(with)h(the)21383 55759 y(\267)21239 56094 y(Cebi)-69 b(\267)-582 b(sev)-36 b(e)515 b(inequalit)-36 b(y)513 b(b)-36 b(y)513 b(Klopp,)g(in)g([21)q(],)g(and)g(b)-36 b(y)4317 57699 y(Kirsc)g(h,)388 b(Stollman)f(and)g(Stolz)g(in)g([19)q (].)h(In)f(the)g(medium)f(of)i(the)f(sp)36 b(ectrum)386 b(one)h(refers)4317 59305 y(to)548 b(the)f(w)-36 b(ork)548 b(of)g(V)-108 b(eseli)-36 b(\266)-614 b(c)548 b(in)g([27)q(,)g(36].)h (All)f(results)f(quoted)h(here)f(are)h(giv)-36 b(en)548 b(in)f(the)4317 60910 y(Sc)-36 b(hr\304)-650 b(odinger)433 b(case.)4317 62515 y(Notice)571 b(that)e(\(1.9\))i(is)f(a)h(limit)f (when)g(one)g(approac)-36 b(hes)570 b(the)f(band)g(edges.)j(W)-108 b(e)570 b(ask)4317 64120 y(for)494 b(an)f(estimate)g(to)h(hold)f(on)g (an)g(in)-36 b(terv)-72 b(al.)494 b(This)f(is)h(the)f(purp)36 b(ose)492 b(of)i(the)e(follo)-36 b(wing)4317 65725 y(computation.)25253 70015 y(21)p eop %%Page: 22 22 22 21 bop 4317 7306 a Fh(Lemma)499 b(3.5)651 b Fi(F)-100 b(or)552 b(al)66 b(l)553 b Fr(n)530 b Fn(2)f Fq(N)553 b Fi(and)e Fr(\256)539 b Fn(2)p Fs(]0)p Fr(;)221 b Fs(1[)p Fi(;)553 b(ther)-66 b(e)551 b(exists)g Fr(k)36923 7505 y Fo(2)37979 7306 y Fn(2)529 b Fq(N)553 b Fi(such)f(that)g(if)4317 8911 y Fr(k)414 b Fn(\270)369 b Fr(k)7485 9110 y Fo(2)8477 8911 y Fi(we)465 b(have)14034 12077 y Fq(E)14920 12276 y Fo(\244)15573 12432 y Fm(k)16135 10602 y Fp(\263)16929 12077 y Fr(N)17971 12276 y Fv(!)32 b(;k)19417 12077 y Fs(\()p Fr(E)20886 12276 y Fo(+)21968 12077 y Fs(+)295 b Fr(k)23996 11528 y Fj(\241)p Fv(\256)25388 12077 y Fs(\))g Fn(\241)g Fr(N)28559 12276 y Fv(!)32 b(;k)30005 12077 y Fs(\()p Fr(E)31474 12276 y Fo(+)32261 12077 y Fs(\))32767 10602 y Fp(\264)33929 12077 y Fn(\267)370 b Fr(k)36053 11528 y Fj(\241)p Fv(n)37411 12077 y Fr(:)6393 b Fs(\(3.39\))4317 15397 y Fh(Pro)42 b(of.)385 b Fs(By)g(the)f(use)g (of)h(the)f(result)g(of)h(Theorem)g(1.2,)h(and)e(Lemma)g(2.2;)i(for)f Fr(")368 b Fs(=)h Fr(k)46097 14914 y Fj(\241)p Fv(\256)4317 17002 y Fs(and)446 b Fr(k)435 b(>)390 b(k)10048 17201 y Fo(1)11020 17002 y Fs(\()p Fr(k)12202 17201 y Fo(1)13175 17002 y Fs(of)446 b(Lemma)g(2.2\))h(w)-36 b(e)447 b(get)f(that)g(for)g (all)h Fr(n)391 b Fn(2)f Fq(N)p Fs(,)446 b(there)g(exists)h Fr(k)44520 17201 y Fo(2)45436 17002 y Fr(>)390 b Fs(0)4317 18607 y(suc)-36 b(h)433 b(that)g(if)h Fr(k)414 b Fn(\270)369 b Fr(k)14429 18806 y Fo(2)15325 18607 y Fr(>)f(k)17381 18806 y Fo(1)17908 18607 y Fs(,)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)14034 21832 y Fq(E)14920 22031 y Fo(\244)15573 22187 y Fm(k)16135 20357 y Fp(\263)16929 21832 y Fr(N)17971 22031 y Fv(!)32 b(;k)19417 21832 y Fs(\()p Fr(E)20886 22031 y Fo(+)21968 21832 y Fs(+)295 b Fr(k)23996 21283 y Fj(\241)p Fv(\256)25388 21832 y Fs(\))g Fn(\241)g Fr(N)28559 22031 y Fv(!)32 b(;k)30005 21832 y Fs(\()p Fr(E)31474 22031 y Fo(+)32261 21832 y Fs(\))32767 20357 y Fp(\264)33929 21832 y Fn(\267)370 b Fr(k)36053 21283 y Fj(\241)p Fv(n)37411 21832 y Fr(:)6393 b Fs(\(3.40\))4317 25057 y(So)434 b(the)f(pro)36 b(of)434 b(of)g(Lemma)g(3.5)g(is)g(ended)p 46724 25057 54 832 v 46778 24252 744 54 v 47465 25057 54 832 v 46747 25057 744 54 v 4317 26662 a(W)-108 b(e)434 b(need)f(also)h(the)f(follo) -36 b(wing)436 b(lemma.)4317 29374 y Fh(Lemma)499 b(3.6)651 b Fi([36])464 b(F)-100 b(or)466 b(al)66 b(l)465 b Fr(")369 b(>)g Fs(0)p Fi(,)464 b(we)i(have)4317 31145 y Fp(Z)5055 34152 y Fg(T)5645 33847 y Fe(\244)5645 34468 y Fm(k)6430 32953 y Fq(P)7242 33152 y Fo(\244)7895 33308 y Fm(k)8458 31478 y Fp(\263)9251 32953 y Fn(f)p Fr(!)48 b Fn(j)p Fr(\276)g Fs(\()p Fr(A)13410 33152 y Fv(!)32 b(;k)14857 32953 y Fs(\()p Fr(\265)k Fs(\)\))p Fn(\\)p Fs([)p Fr(E)19227 33152 y Fo(+)20014 32953 y Fr(;)221 b(E)21559 33152 y Fo(+)22346 32953 y Fs(+)p Fr(")p Fs(])369 b Fn(6)p Fs(=)f Fn(;g)27405 31478 y Fp(\264)28199 32953 y Fr(d\265)405 b Fn(\267)369 b Fs(\(2)p Fr(\274)48 b Fs(\))33736 32404 y Fv(d)34275 32953 y Fq(E)35161 33152 y Fo(\244)35814 33308 y Fm(k)36377 31478 y Fp(\263)37170 32953 y Fr(N)38212 33152 y Fv(!)32 b(;k)39659 32953 y Fs(\()p Fr(E)41128 33152 y Fo(+)41915 32953 y Fs(+)p Fr(")p Fs(\))p Fn(\241)p Fr(N)46117 33152 y Fv(!)g(;k)47562 32953 y Fs(\()p Fr(E)49031 33152 y Fo(+)49818 32953 y Fs(\))50324 31478 y Fp(\264)51117 32953 y Fr(:)4317 37004 y Fs(Let)407 b(us)h(note)f(the)g Fr(j)13956 36522 y Fo(th)15307 37004 y Fs(eigen)-36 b(v)-72 b(alue)409 b(of)f Fr(A)23926 37203 y Fv(!)32 b(;k)25373 37004 y Fs(\()p Fr(\265)k Fs(\)b)-36 b(y)408 b Fr(E)29771 37203 y Fv(j)30257 37004 y Fs(\()p Fr(A)31738 37203 y Fv(!)32 b(;k)33185 37004 y Fs(\()p Fr(\265)k Fs(\)\).)408 b(In)f(the)g(app)36 b(endix)408 b(one)4317 38609 y(sho)-36 b(ws)434 b(the)f(follo)-36 b(wing)436 b(lemma.)4317 41321 y Fh(Lemma)499 b(3.7)651 b Fi(F)-100 b(or)465 b(al)66 b(l)466 b Fr(\265)36 b(;)221 b(\265)18323 40839 y Fj(0)19004 41321 y Fn(2)368 b Fq(T)21144 40839 y Fj(\244)21144 41666 y Fv(k)22178 41321 y Fi(and)465 b Fr(j)444 b Fn(2)368 b Fq(N)465 b Fi(ther)-66 b(e)464 b(exists)h Fr(M)36412 41520 y Fv(j)o(;k)37990 41321 y Fr(>)369 b Fs(0)465 b Fi(such)h(that)14257 44254 y Fn(j)p Fr(E)15589 44453 y Fv(j)16076 44254 y Fs(\()p Fr(A)17557 44453 y Fv(!)32 b(;k)19004 44254 y Fs(\()p Fr(\265)k Fs(\))295 b Fn(\241)g Fr(E)23244 44453 y Fv(j)23731 44254 y Fs(\()p Fr(A)25212 44453 y Fv(!)32 b(;k)26659 44254 y Fs(\()p Fr(\265)27807 43706 y Fj(0)28117 44254 y Fs(\))p Fn(j)369 b(\267)g Fr(M)32021 44453 y Fv(j)o(;k)33231 44254 y Fn(j)p Fr(\265)331 b Fn(\241)296 b Fr(\265)36508 43706 y Fj(0)36819 44254 y Fn(j)p Fr(:)6616 b Fs(\(3.41\))4317 47188 y(No)-36 b(w)317 b(using)f(Lemma)g(3.6)i(and)e(Lipshitz)f(con)-36 b(tin)g(uit)g(y)316 b(in)g Fr(\265)353 b Fs(of)317 b(the)f(Flo)36 b(quet)317 b(eigen)-36 b(v)-72 b(alues,)4317 48793 y(w)-36 b(e)468 b(will)i(b)36 b(e)467 b(able)i(to)f(estimate)g(the)g (probabilit)-36 b(y)468 b(to)g(\257nd)f(an)h(eigen)-36 b(v)-72 b(alue)468 b(of)h Fr(A)44008 48992 y Fo(0)p Fv(;k)45309 48793 y Fs(\()p Fr(\265)46457 48311 y Fo(0)46983 48793 y Fs(\))4317 50398 y(in)599 b(the)f(in)-36 b(terv)-72 b(al)599 b([)p Fr(E)14591 50597 y Fo(+)15378 50398 y Fr(;)221 b(E)16923 50597 y Fo(+)18118 50398 y Fs(+)408 b Fr(")p Fs(].)600 b(Here)e Fr(A)25679 50597 y Fo(0)p Fv(;k)26980 50398 y Fs(\()p Fr(\265)28128 49916 y Fo(0)28654 50398 y Fs(\))650 b(=)f Fr(A)32446 50597 y Fo(0)p Fv(;k)33747 50398 y Fn(j)34116 49916 y Fv(\265)34581 49603 y Ff(0)34116 50734 y Fo(\244)34769 50890 y Fm(k)35931 50398 y Fs(is)599 b(the)f(op)36 b(erator)599 b Fr(A)46188 50597 y Fo(0)p Fv(;k)4317 52003 y Fs(restricted)433 b(to)h(\244)12598 52202 y Fv(k)13600 52003 y Fs(with)g Fr(\265)17205 51521 y Fo(0)17731 52003 y Fs(-p)36 b(erio)g(dic)433 b(conditions.)h(More)g (precisely)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)4317 54715 y Fh(Lemma)499 b(3.8)651 b Fi(L)-66 b(et)464 b Fr(\265)15057 54233 y Fo(0)15952 54715 y Fn(2)368 b Fq(T)18092 54233 y Fj(\244)18092 55061 y Fv(k)18661 54715 y Fi(.)464 b(F)-100 b(or)466 b(al)66 b(l)465 b Fr(")369 b(>)g Fs(0)465 b Fi(we)g(have)5424 58359 y Fq(P)6236 58558 y Fo(\244)6889 58714 y Fm(k)7452 56885 y Fp(\263)8245 58359 y Fn(f)p Fr(!)48 b Fn(j)p Fr(\276)g Fs(\()p Fr(A)12404 58558 y Fv(!)32 b(;k)13851 58359 y Fs(\()p Fr(\265)14999 57811 y Fo(0)15525 58359 y Fs(\)\))294 b Fn(\\)h Fs([)p Fr(E)19336 58558 y Fo(+)20124 58359 y Fr(;)221 b(E)21669 58558 y Fo(+)22751 58359 y Fs(+)295 b Fr(")p Fs(])369 b Fn(6)p Fs(=)g Fn(;g)28106 56885 y Fp(\264)14737 61661 y Fn(\267)16843 60763 y Fs(\(2)p Fr(\274)48 b Fs(\))19291 60281 y Fv(d)p 16272 61356 4129 54 v 16272 62572 a Fs(v)-36 b(ol)q(\()p Fq(T)19326 62115 y Fj(\244)19326 62942 y Fv(k)19894 62572 y Fs(\))20533 61661 y Fq(E)21419 61860 y Fo(\244)22072 62016 y Fm(k)22635 60187 y Fp(\263)23428 61661 y Fr(N)24470 61860 y Fv(!)32 b(;k)25917 61661 y Fs(\()p Fr(E)27386 61860 y Fo(+)28468 61661 y Fs(+)294 b Fr(")h Fs(+)g Fr(ck)33266 61113 y Fj(\241)p Fo(1)34524 61661 y Fs(\))g Fn(\241)g Fr(N)37695 61860 y Fv(!)32 b(;k)39141 61661 y Fs(\()p Fr(E)40610 61860 y Fo(+)41397 61661 y Fs(\))41903 60187 y Fp(\264)42696 61661 y Fr(:)1108 b Fs(\(3.42\))25253 70015 y(22)p eop %%Page: 23 23 23 22 bop 4317 7306 a Fh(Pro)42 b(of.)492 b Fs(F)-108 b(rom)491 b(Lemma)h(3.7)g(w)-36 b(e)492 b(kno)-36 b(w)493 b(that)e(there)g(exists)i Fr(M)35768 7505 y Fv(j)o(;k)37445 7306 y Fr(>)468 b Fs(0)492 b(suc)-36 b(h)491 b(that)g(for)4317 8911 y(all)435 b Fr(j)443 b Fn(2)369 b Fq(N)434 b Fs(w)-36 b(e)434 b(ha)-36 b(v)g(e)13897 11844 y Fn(j)p Fr(E)15229 12043 y Fv(j)15716 11844 y Fs(\()p Fr(A)17197 12043 y Fo(0)p Fv(;k)18498 11844 y Fs(\()p Fr(\265)36 b Fs(\)\))295 b Fn(\241)g Fr(E)23244 12043 y Fv(j)23731 11844 y Fs(\()p Fr(A)25212 12043 y Fo(0)p Fv(;k)26512 11844 y Fs(\()p Fr(\265)27660 11296 y Fj(0)27971 11844 y Fs(\)\))p Fn(j)368 b(\267)i Fr(M)32381 12043 y Fv(j)o(;k)33590 11844 y Fn(j)p Fr(\265)332 b Fn(\241)295 b Fr(\265)36867 11296 y Fj(0)37178 11844 y Fn(j)p Fr(:)4317 14778 y Fs(As)414 b(w)-36 b(e)415 b(are)f(in)-36 b(terested)413 b(only)i(in)f(the)g(eigen)-36 b(v)-72 b(alues)415 b(in)f([)p Fr(E)32296 14977 y Fo(+)33083 14778 y Fr(;)221 b(E)34628 14977 y Fo(+)35671 14778 y Fs(+)256 b Fr(")p Fs(],)414 b(the)g Fr(M)42163 14977 y Fv(j)o(;k)43787 14778 y Fs(can)g(b)36 b(e)4317 16383 y(c)-36 b(hosen)433 b(uniformly)i(in)e Fr(j)508 b Fs(and)433 b Fr(k)45 b Fs(.)434 b(So)g(there)f(exists)h Fr(M)508 b(>)369 b Fs(0)434 b(suc)-36 b(h)432 b(that)19597 19316 y Fr(M)508 b Fn(\270)369 b Fr(M)24023 19515 y Fv(j)o(;k)25233 19316 y Fr(;)1089 b Fn(8)p Fr(j)r(;)221 b(k)415 b Fn(2)368 b Fq(N)p Fr(:)4317 22250 y Fs(No)-36 b(w)434 b(w)-36 b(e)434 b(will)h(use)e(this)h(to)f(estimate)14345 25183 y Fq(P)15157 25382 y Fo(\244)15810 25538 y Fm(k)16373 25183 y Fs(\()p Fn(f)p Fr(!)48 b Fn(j)p Fr(\276)g Fs(\()p Fr(A)21038 25382 y Fo(0)p Fv(;k)22338 25183 y Fs(\()p Fr(\265)23486 24635 y Fo(0)24012 25183 y Fs(\)\))295 b Fn(\\)f Fs([)p Fr(E)27823 25382 y Fo(+)28611 25183 y Fr(;)221 b(E)30156 25382 y Fo(+)31239 25183 y Fs(+)294 b Fr(")p Fs(])369 b Fn(6)p Fs(=)g Fn(;g)p Fs(\))p Fr(:)4317 28117 y Fs(F)-108 b(or)433 b(this)h(it)f(is)h(enough)f(to)h(estimate) 11946 31050 y Fq(P)12758 31249 y Fo(\244)13411 31405 y Fm(k)13973 31050 y Fs(\()p Fn(f)p Fr(!)48 b Fn(j9)433 b Fr(j)444 b Fn(2)369 b Fq(N)p Fs(;)655 b Fr(E)22716 31249 y Fv(j)23203 31050 y Fs(\()p Fr(A)24684 31249 y Fo(0)p Fv(;k)25985 31050 y Fs(\()p Fr(\265)27133 30502 y Fo(0)27658 31050 y Fs(\)\))369 b Fn(2)g Fs([)p Fr(E)31618 31249 y Fo(+)32405 31050 y Fr(;)221 b(E)33950 31249 y Fo(+)35033 31050 y Fs(+)295 b Fr(")p Fs(])p Fn(g)p Fs(\))369 b(=)8642 33770 y(1)p 6903 34363 4129 54 v 6903 35580 a(v)-36 b(ol)q(\()p Fq(T)9957 35122 y Fj(\244)9957 35950 y Fv(k)10525 35580 y Fs(\))11385 32861 y Fp(Z)12123 35868 y Fg(T)12713 35563 y Fe(\244)12713 36184 y Fm(k)13498 34669 y Fq(P)14310 34868 y Fo(\244)14963 35024 y Fm(k)15526 34669 y Fs(\()p Fn(f)p Fr(!)48 b Fn(j9)433 b Fr(j)444 b Fn(2)369 b Fq(N)p Fs(;)655 b Fr(E)24269 34868 y Fv(j)24756 34669 y Fs(\()p Fr(A)26237 34868 y Fo(0)p Fv(;k)27537 34669 y Fs(\()p Fr(\265)28685 34120 y Fo(0)29211 34669 y Fs(\)\))369 b Fn(2)f Fs([)p Fr(E)33170 34868 y Fo(+)33958 34669 y Fr(;)221 b(E)35503 34868 y Fo(+)36586 34669 y Fs(+)294 b Fr(")p Fs(])p Fn(g)p Fs(\))p Fr(d\265)36 b(:)2454 b Fs(\(3.43\))4317 38067 y(Th)-36 b(us)615 b(if)h Fr(E)10131 38266 y Fv(j)10618 38067 y Fs(\()p Fr(A)12099 38266 y Fo(0)p Fv(;k)13400 38067 y Fs(\()p Fr(\265)14548 37585 y Fo(0)15073 38067 y Fs(\)\))679 b Fn(2)f Fs([)p Fr(E)19652 38266 y Fo(+)20439 38067 y Fr(;)221 b(E)21984 38266 y Fo(+)23191 38067 y Fs(+)418 b Fr(")p Fs(],)616 b(the)f(Lipshitz)g(con) -36 b(tin)g(uit)g(y)615 b(implies)h(that)4317 39672 y Fr(E)5280 39871 y Fv(j)5767 39672 y Fs(\()p Fr(A)7248 39871 y Fo(0)p Fv(;k)8548 39672 y Fs(\()p Fr(\265)36 b Fs(\)\))479 b Fn(2)g Fs([)p Fr(E)13876 39871 y Fo(+)14663 39672 y Fr(;)221 b(E)16208 39871 y Fo(+)17335 39672 y Fs(+)339 b Fr(")g Fs(+)g Fr(M)139 b Fs(diam\()p Fq(T)26592 39190 y Fj(\244)26592 40017 y Fv(k)27161 39672 y Fs(\)])p Fr(;)720 b Fn(8)p Fr(\265)515 b Fn(2)479 b Fq(T)33219 39190 y Fj(\244)33219 40017 y Fv(k)34286 39672 y Fs(with)498 b(diam\()p Fq(T)41523 39190 y Fj(\244)41523 40017 y Fv(k)42091 39672 y Fs(\))479 b Fn(\267)h Fr(ck)45870 39190 y Fj(\241)p Fo(1)47127 39672 y Fs(.)4317 41277 y(So)434 b(w)-36 b(e)433 b(get)h(that)4317 44782 y(\(3)p Fr(:)p Fs(43\))1108 b Fn(\267)12760 43884 y Fs(1)p 11021 44477 V 11021 45693 a(v)-36 b(ol)q(\()p Fq(T)14075 45236 y Fj(\244)14075 46063 y Fv(k)14643 45693 y Fs(\))15503 42974 y Fp(Z)16241 45981 y Fg(T)16831 45676 y Fe(\244)16831 46297 y Fm(k)17616 44782 y Fq(P)18428 44981 y Fo(\244)19081 45137 y Fm(k)19644 44782 y Fs(\()p Fn(f)p Fr(!)48 b Fn(j9)433 b Fr(j)444 b Fn(2)369 b Fq(N)p Fs(;)655 b Fr(E)28387 44981 y Fv(j)28874 44782 y Fs(\()p Fr(A)30355 44981 y Fo(0)p Fv(;k)31656 44782 y Fs(\()p Fr(\265)36 b Fs(\)\))368 b Fn(2)h Fs([)p Fr(E)36763 44981 y Fo(+)37550 44782 y Fr(;)221 b(E)39095 44981 y Fo(+)40178 44782 y Fs(+)295 b Fr(")g Fs(+)f Fr(ck)44976 44234 y Fj(\241)p Fo(1)46234 44782 y Fs(])p Fn(g)p Fs(\))p Fr(d\265)8759 48548 y Fs(=)12760 47649 y(1)p 11021 48242 V 11021 49459 a(v)-36 b(ol)q(\()p Fq(T)14075 49001 y Fj(\244)14075 49829 y Fv(k)14643 49459 y Fs(\))15503 46740 y Fp(Z)16241 49747 y Fg(T)16831 49442 y Fe(\244)16831 50063 y Fm(k)17616 48548 y Fq(P)18428 48747 y Fo(\244)19081 48903 y Fm(k)19644 48548 y Fs(\()p Fn(f)p Fr(!)48 b Fn(j)p Fs(;)221 b Fr(\276)48 b Fs(\()p Fr(A)24891 48747 y Fo(0)p Fv(;k)26192 48548 y Fs(\()p Fr(\265)36 b Fs(\)\))295 b Fn(\\)g Fs([)p Fr(E)31152 48747 y Fo(+)31939 48548 y Fr(;)221 b(E)33484 48747 y Fo(+)34567 48548 y Fs(+)295 b Fr(")f Fs(+)h Fr(ck)39365 47999 y Fj(\241)p Fo(1)40623 48548 y Fs(])369 b Fn(6)p Fs(=)g Fn(;g)p Fs(\))p Fr(d\265)8748 52501 y Fn(\267)11592 51602 y Fs(\(2)p Fr(\274)48 b Fs(\))14040 51120 y Fv(d)p 11021 52196 V 11021 53412 a Fs(v)-36 b(ol)q(\()p Fq(T)14075 52955 y Fj(\244)14075 53782 y Fv(k)14643 53412 y Fs(\))15282 52501 y Fq(E)16168 52700 y Fo(\244)16821 52856 y Fm(k)17383 51027 y Fp(h)18011 52501 y Fr(N)19053 52700 y Fv(!)32 b(;k)20499 52501 y Fs(\()p Fr(E)21968 52700 y Fo(+)23050 52501 y Fs(+)295 b Fr(")g Fs(+)g Fr(ck)27849 51953 y Fj(\241)p Fo(1)29106 52501 y Fs(\))g Fn(\241)h Fr(N)32278 52700 y Fv(!)32 b(;k)33724 52501 y Fs(\()p Fr(E)35193 52700 y Fo(+)35980 52501 y Fs(\))36486 51027 y Fp(i)37113 52501 y Fr(:)4317 56144 y Fs(This)434 b(pro)-36 b(v)g(es)434 b(Lemma)f(3.8.)p 46724 56144 54 832 v 46778 55339 744 54 v 47465 56144 54 832 v 46747 56144 744 54 v 4317 59354 a(F)-108 b(or)384 b Fr(\256)378 b(<)369 b Fs(1,)385 b(\()p Fr(\256)393 b Fs(giv)-36 b(en)385 b(in)f(Lemma)g(3.5\))i(and)e Fr(")368 b Fs(=)h Fr(k)29826 58872 y Fj(\241)p Fv(\256)31586 59354 y Fr(>)g Fs(0)385 b(there)f(exists)h Fr(k)41495 59553 y Fo(3)42390 59354 y Fn(2)369 b Fq(N)384 b Fs(suc)-36 b(h)4317 60959 y(that)408 b(if)h Fr(k)414 b(>)369 b(k)11424 61158 y Fo(3)11950 60959 y Fs(,)409 b(w)-36 b(e)408 b(ha)-36 b(v)g(e)409 b Fr(k)18304 60477 y Fj(\241)p Fv(\256)19939 60959 y Fs(+)243 b Fr(ck)22475 60477 y Fj(\241)p Fo(1)24102 60959 y Fn(\267)369 b Fs(2)p Fr(k)26875 60477 y Fj(\241)p Fv(\256)28675 60959 y Fs(where)409 b Fr(k)33084 61158 y Fo(3)34018 60959 y Fs(dep)36 b(ends)407 b(on)i Fr(\256)417 b Fs(and)407 b Fr(c)p Fs(.)i(By)4317 62564 y(using)434 b(equation)g(\(3.40\))g(w)-36 b(e)434 b(obtain)f(that)h(for)g Fr(k)414 b(>)368 b(k)30702 62763 y Fo(3)31229 62564 y Fs(;)13429 65863 y Fq(E)14315 66062 y Fo(\244)14968 66218 y Fm(k)15530 64388 y Fp(\263)16324 65863 y Fr(N)17366 66062 y Fv(!)32 b(;k)18812 65863 y Fs(\()p Fr(E)20281 66062 y Fo(+)21363 65863 y Fs(+)295 b(2)p Fr(k)24041 65314 y Fj(\241)p Fv(\256)25433 65863 y Fs(\))g Fn(\241)g Fr(N)28604 66062 y Fv(!)32 b(;k)30051 65863 y Fs(\()p Fr(E)31520 66062 y Fo(+)32307 65863 y Fs(\))32813 64388 y Fp(\264)33975 65863 y Fn(\267)369 b Fr(ck)36658 65314 y Fj(\241)p Fv(n)38016 65863 y Fr(:)25253 70015 y Fs(23)p eop %%Page: 24 24 24 23 bop 4317 7306 a Fs(So)434 b(Lemma)f(3.8)i(giv)-36 b(es:)10493 10531 y Fq(P)11305 10730 y Fo(\244)11958 10886 y Fm(k)12521 9056 y Fp(\263)13314 10531 y Fn(f)p Fr(!)48 b Fn(j)p Fr(\276)g Fs(\()p Fr(A)17473 10730 y Fo(0)p Fv(;k)18773 10531 y Fs(\()p Fr(\265)19921 9982 y Fo(0)20447 10531 y Fs(\)\))295 b Fn(\\)g Fs([)p Fr(E)24259 10730 y Fo(+)25046 10531 y Fr(;)221 b(E)26591 10730 y Fo(+)27674 10531 y Fs(+)294 b Fr(k)29701 9982 y Fj(\241)p Fv(\256)31093 10531 y Fs(])369 b Fn(6)p Fs(=)g Fn(;g)34532 9056 y Fp(\264)35695 10531 y Fn(\267)g Fr(ck)38378 9982 y Fj(\241)p Fv(n)p Fo(+)p Fv(d)40952 10531 y Fr(:)4317 13756 y Fs(Let)363 b Fr(p)369 b(>)f(d)c Fs(and)f Fr(n)g Fs(giv)-36 b(en)364 b(in)g(Lemma)f(3.5)h(suc)-36 b(h)363 b(that)g Fr(n)369 b(>)g(p)152 b Fs(+)g Fr(d)p Fs(.)363 b(There)g(exists)h Fr(k)44379 13955 y Fo(4)45275 13756 y Fn(2)k Fq(N)4317 15361 y Fs(suc)-36 b(h)478 b(that)g(if)i Fr(k)491 b Fn(\270)446 b Fr(k)14719 15560 y Fo(4)15724 15361 y Fs(w)-36 b(e)479 b(ha)-36 b(v)g(e)479 b Fr(ck)22009 14879 y Fj(\241)p Fv(n)p Fo(+)p Fv(d)25029 15361 y Fn(\267)446 b Fr(k)27229 14879 y Fj(\241)p Fv(p)28490 15361 y Fr(:)479 b Fs(This)h(implies)f(that)f(for)i Fr(k)491 b(>)445 b(k)45002 15560 y Fo(4)46007 15361 y Fs(w)-36 b(e)4317 16966 y(ha)g(v)g(e)11429 18571 y Fq(P)12241 18770 y Fo(\244)12894 18926 y Fm(k)13457 17097 y Fp(\263)14250 18571 y Fn(f)p Fr(!)48 b Fn(j)p Fr(\276)g Fs(\()p Fr(A)18409 18770 y Fo(0)p Fv(;k)19710 18571 y Fs(\()p Fr(\265)20858 18023 y Fo(0)21383 18571 y Fs(\)\))295 b Fn(\\)g Fs([)p Fr(E)25195 18770 y Fo(+)25982 18571 y Fr(;)221 b(E)27527 18770 y Fo(+)28610 18571 y Fs(+)295 b Fr(k)30638 18023 y Fj(\241)p Fv(\256)32030 18571 y Fs(])369 b Fn(6)p Fs(=)f Fn(;g)35468 17097 y Fp(\264)36631 18571 y Fn(\267)h Fr(k)38754 18023 y Fj(\241)p Fv(p)40015 18571 y Fr(:)3789 b Fs(\(3.44\))4317 21261 y(This)434 b(ends)f(the)g(pro)36 b(of)434 b(of)g(the)f(Prop)36 b(osition)434 b(3.4)h Fh(\(P'1\))p Fs(.)p 46724 21261 54 832 v 46778 20456 744 54 v 47465 21261 54 832 v 46747 21261 744 54 v 4317 25112 a FA(3.4)1793 b(W)-149 b(egner)598 b(estimate)4317 27575 y Fs(As)472 b(it)g(w)-36 b(as)473 b(men)-36 b(tioned)471 b(previously)i(the)f(estimate)g(w)-36 b(e)473 b(are)f(aiming)h(at)f(is)g(one)g(of)h(the)4317 29180 y(basic)431 b(ingredien)-36 b(ts)429 b(of)j(the)d(m)-36 b(ultiscale)431 b(analysis.)h(Roughly)f(sp)36 b(eaking)432 b(it)e(is)h(an)f(upp)36 b(er)4317 30786 y(b)g(ound)525 b(on)i(the)f(probabilit)-36 b(y)527 b(that)f(the)g(sp)36 b(ectrum)525 b(of)j(the)e(lo)36 b(cal)528 b(op)36 b(erator)526 b Fr(A)43788 30985 y Fv(x;k)45675 30786 y Fs(lies)4317 32391 y(within)398 b(an)h Fr(")p Fs(-neigh)-36 b(b)36 b(orho)g(o)g(d)397 b(of)i(a)f(giv)-36 b(en)399 b(energy)-108 b(.)399 b(This)g(probabilit)-36 b(y)398 b(v)-72 b(anishes)398 b(as)h(the)4317 33996 y(size)455 b(of)g(the)f(energy)h(neigh)-36 b(b)36 b(orho)g(o)g(d)454 b(shrinks)g(to)g(zero.)h(Suc)-36 b(h)453 b(an)i(estimate)f(is)h(pro)-36 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Fs(\()p Fr(!)48 b Fs(\))p 21169 64301 4439 54 v 21747 65711 a Fr(@)72 b(!)23364 65082 y Fv(i)23684 64829 y Fe(0)23982 65082 y Fv(;j)24675 64829 y Fe(0)23316 66087 y Fv(\260)23853 65835 y Fe(0)26109 64607 y Fn(\270)369 b Fr(g)28134 64806 y Fj(\241)29216 64607 y Fn(\242)296 b Fr(a)30564 64058 y Fj(\241)p Fo(1)30564 64935 y(+)32116 64607 y Fn(\242)f Fr(E)33743 64806 y Fo(+)34530 64607 y Fr(;)25253 70015 y Fs(25)p eop %%Page: 26 26 26 25 bop 4317 7306 a Fs(and)433 b(consequen)-36 b(tly)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)15976 10424 y(1)370 b Fn(\267)21715 9526 y Fs(1)p 18531 10119 7020 54 v 18531 11399 a Fr(g)19154 11598 y Fj(\241)20236 11399 y Fn(\242)295 b Fr(a)21583 10849 y Fj(\241)p Fo(1)21583 11689 y(+)23136 11399 y Fn(\242)g Fr(E)24763 11598 y Fo(+)25978 10424 y Fn(\242)27633 9162 y Fp(X)27281 12058 y Fv(\260)27818 11745 y Fe(0)28116 12058 y Fj(2)p Fo(\244)29396 12214 y Fm(k)26775 12909 y Ff(1)p Fe(\267)p Fm(i)28117 12718 y Fe(0)28415 12909 y Fm(;j)29046 12718 y Fe(0)29343 12909 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48674 y Fp(Z)10968 49025 y Fo(1)10378 51681 y(0)13781 49220 y Fp(Y)12006 52049 y Fv(\260)52 b Fj(2)p Fo(\2441)p Fj(\267)p Fv(i;j)f Fj(\267)p Fv(d)11848 52901 y Fm(\260)43 b Fe(6)p Ff(=)p Fm(\260)13421 52710 y Fe(0)13720 52901 y Fm(;i)p Fe(6)p Ff(=)p Fm(i)15185 52710 y Fe(0)15483 52901 y Fm(;j)c Fe(6)p Ff(=)p Fm(j)17114 52710 y Fe(0)17765 50482 y Fr(g)48 b Fs(\()p Fr(!)19800 49934 y Fv(i;j)19752 50811 y(\260)20868 50482 y Fs(\))p Fr(d!)22908 49934 y Fv(i;j)22860 50811 y(\260)22816 55753 y Fn(\243)24144 53945 y Fp(Z)25472 54296 y Fo(1)24882 56952 y(0)26220 55753 y Fr(g)g Fs(\()p Fr(!)28255 55123 y Fv(i)28575 54811 y Fe(0)28872 55123 y Fv(;j)29565 54811 y Fe(0)28207 56129 y Fv(\260)28744 55877 y Fe(0)29919 55753 y Fs(\))30558 54734 y Fr(@)72 b(\275)p Fs(\()p Fr(E)33457 54933 y Fv(n)34303 53658 y Fp(\241)34912 54734 y Fr(!)35770 54252 y Fv(i;j)35722 55063 y(\260)36838 53658 y Fp(\242)37742 54734 y Fn(\241)296 b Fr(E)373 b Fs(+)294 b Fr(t)p Fs(\))p 30558 55448 V 34982 56858 a Fr(@)72 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y Fv(;j)14187 6364 y Fe(0)12827 7682 y Fv(\260)13364 7430 y Fe(0)14954 7306 y Fs(=)413 b Fr(\256)8 b Fs(\))460 b(is)g(the)f Fr(n)p Fs(-th)g(eigen)-36 b(v)-72 b(alue)461 b(of)f Fr(A)33401 7505 y Fo(0)p Fv(;k)35161 7306 y Fs(when)g Fr(!)39442 6676 y Fv(i)39762 6364 y Fe(0)40060 6676 y Fv(;j)40753 6364 y Fe(0)39394 7682 y Fv(\260)39931 7430 y Fe(0)41566 7306 y Fs(is)g(c)-36 b(hanged)4317 8911 y(b)g(y)434 b Fr(\256)8 b Fs(.)434 b(So)5424 11910 y Fq(P)6236 12109 y Fo(\244)6889 12265 y Fm(k)7452 11910 y Fn(f)p Fr(dist)p Fs(\()p Fr(\276)48 b Fs(\()p Fr(A)13093 12109 y Fo(0)p Fv(;k)14392 11910 y Fs(\))p Fr(;)221 b(E)78 b Fs(\))369 b Fn(\267)g Fr(")p Fn(g)g(\267)8997 14863 y Fr(C)390 b Fn(\242)295 b(k)p Fr(g)48 b Fn(k)12981 15062 y Fj(1)14198 13601 y Fp(X)14872 16390 y Fv(n)17329 13601 y Fp(X)16977 16496 y Fv(\260)17514 16184 y Fe(0)17812 16496 y Fj(2)p Fo(\244)19092 16652 y Fm(k)16471 17348 y Ff(1)p Fe(\267)p Fm(i)17813 17157 y Fe(0)18111 17348 y Fm(;j)18742 17157 y Fe(0)19040 17348 y(\267)p Fm(d)20460 13055 y Fp(Z)21789 13405 y Fo(2)p Fv(")21198 16062 y Fj(\241)p Fo(2)p Fv(")23112 14863 y Fr(dt)24479 13055 y Fp(Z)25807 13405 y Fo(1)25217 16062 y(0)26554 14863 y Fn(\242)295 b(\242)h(\242)28473 13055 y Fp(Z)29801 13405 y Fo(1)29211 16062 y(0)32614 13601 y Fp(Y)30708 16430 y Fv(\260)52 b Fj(2)p Fo(\244)p Fv(;)p Fo(1)p Fj(\267)p Fv(i;j)f Fj(\267)p Fv(d)30681 17281 y Fm(\260)43 b Fe(6)p Ff(=)p Fm(\260)32254 17090 y Fe(0)32553 17281 y Fm(;i)p Fe(6)p Ff(=)p Fm(i)34018 17090 y Fe(0)34316 17281 y Fm(;j)c Fe(6)p Ff(=)p Fm(j)35947 17090 y Fe(0)36599 14863 y Fr(g)48 b Fs(\()p Fr(!)38634 14314 y Fv(i;j)38586 15191 y(\260)39701 14863 y Fs(\))p Fr(d!)41741 14314 y Fv(i;j)41693 15191 y(\260)8839 19369 y Fn(\243)295 b(f)p Fr(\275)p Fs(\()p Fr(E)12971 19568 y Fv(n)13597 19369 y Fs(\()p Fr(!)14961 18820 y Fv(i;j)14913 19697 y(\260)16029 19369 y Fr(;)221 b(!)17469 18739 y Fv(i)17789 18427 y Fe(0)18088 18739 y Fv(;j)18781 18427 y Fe(0)17421 19745 y Fv(\260)17958 19493 y Fe(0)19504 19369 y Fs(=)368 b(1\))295 b Fn(\241)h Fr(E)373 b Fs(+)294 b Fr(t)p Fs(\))h Fn(\241)h Fr(\275)p Fs(\()p Fr(E)31046 19568 y Fv(n)31671 19369 y Fs(\()p Fr(!)33035 18820 y Fv(i;j)32987 19697 y(\260)34103 19369 y Fr(;)221 b(!)35543 18739 y Fv(i)35863 18427 y Fe(0)36162 18739 y Fv(;j)36855 18427 y Fe(0)35495 19745 y Fv(\260)36032 19493 y Fe(0)37578 19369 y Fs(=)368 b(0\))296 b Fn(\241)f Fr(E)373 b Fs(+)295 b Fr(t)p Fs(\))p Fn(g)p Fr(:)4317 22036 y Fs(F)-108 b(or)433 b(our)h(c)-36 b(hoice)433 b(of)i Fr(\275)e Fs(w)-36 b(e)434 b(get)g(that)11491 25495 y Fq(P)12303 25694 y Fo(\244)12956 25850 y Fm(k)13519 25495 y Fn(f)p Fr(dist)p Fs(\()p Fr(\276)48 b Fs(\()p Fr(A)19160 25694 y Fo(0)p Fv(;k)20459 25495 y Fs(\))p Fr(;)221 b(E)78 b Fs(\))369 b Fn(\267)g Fr(")p Fn(g)g(\267)g Fr(C)390 b Fn(\242)29895 24233 y Fp(X)30568 27023 y Fv(n)33025 24233 y Fp(X)32673 27129 y Fv(\260)33210 26816 y Fe(0)33508 27129 y Fj(2)p Fo(\244)34788 27285 y Fm(k)32168 27980 y Ff(1)p Fe(\267)p Fm(i)33510 27789 y Fe(0)33807 27980 y Fm(;j)34438 27789 y Fe(0)34736 27980 y(\267)p Fm(d)36156 23687 y Fp(Z)37485 24038 y Fo(2)p Fv(")36894 26695 y Fj(\241)p Fo(2)p Fv(")38808 25495 y Fr(dt:)4317 30343 y Fs(Notice)440 b(that)f(the)g(n)-36 b(um)g(b)36 b(er)438 b(of)j(terms)e(in)g(the)g(sum)g(o)-36 b(v)g(er)440 b Fr(\260)72 b(;)221 b(i)440 b Fs(and)f Fr(j)514 b Fs(can)440 b(b)36 b(e)439 b(b)36 b(ounded)4317 31948 y(b)-36 b(y)355 b Fr(d)6721 31466 y Fo(2)7247 31948 y Fn(j)p Fs(\244)8519 32147 y Fv(k)9087 31948 y Fn(j)h Fs(and)e(b)-36 b(y)355 b(the)g(W)-108 b(eyl's)356 b(la)-36 b(w)356 b(w)-36 b(e)355 b(kno)-36 b(w)356 b(that)f(there)f(are)h(at)g(most)h Fr(C)316 b Fn(j)p Fs(\244)42872 32147 y Fv(k)43441 31948 y Fn(j)355 b Fs(eigen-)4317 33553 y(v)-72 b(alues)338 b(of)g Fr(A)10454 33752 y Fo(0)p Fv(;k)12092 33553 y Fs(b)36 b(elo)-36 b(w)339 b(some)e(\257xed)g(energy)-108 b(.)339 b(This)e(ends)g(the)g(pro)36 b(of)338 b(of)g(the)f(Prop)36 b(osition)4317 35158 y(3.9.)p 46724 35158 54 832 v 46778 34353 744 54 v 47465 35158 54 832 v 46747 35158 744 54 v 4317 39556 a Ft(4)2152 b(App)60 b(endix)715 b(.)4317 42476 y Fh(The)533 b(pro)42 b(of)532 b(of)g(Lemma)h(2.9.)464 b Fs(Let)e(us)h(tak)-36 b(e)464 b(\242)p Fr(A)30184 42675 y Fv(!)32 b(;k)32050 42476 y Fr(>)419 b Fn(\241)p Fr(C)95 b(")p Fs(\242.)463 b(Let)g Fr(')419 b Fn(2)g(D)37 b Fs(\()p Fr(A)45536 42675 y Fv(!)32 b(;k)46983 42476 y Fs(\))4317 44081 y(suc)-36 b(h)433 b(that)g Fn(h)p Fr(A)11561 44280 y Fv(!)32 b(;k)13008 44081 y Fr(';)221 b(')p Fn(i)369 b Fr(>)g(E)18524 44280 y Fo(+)19311 44081 y Fs(,)433 b(then)14397 46748 y Fn(h)p Fr(A)15889 46947 y Fv(!)32 b(;k)17335 46748 y Fr(';)221 b(')p Fn(i)1118 b Fs(=)g Fn(h)p Fr(A)24878 46947 y Fo(0)p Fv(;k)26178 46748 y Fr(';)221 b(')p Fn(i)295 b Fs(+)g Fn(h)p Fs(\242)p Fr(A)33159 46947 y Fv(!)32 b(;k)34606 46748 y Fr(';)221 b(')p Fn(i)21246 48686 y(\270)1107 b(h)p Fr(A)24878 48885 y Fo(0)p Fv(;k)26178 48686 y Fr(';)221 b(')p Fn(i)295 b Fs(+)g Fr(C)95 b(")p Fn(jr)p Fr(')p Fn(j)34915 48137 y Fo(2)21256 50623 y Fr(>)1118 b 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62057 y Fs(\)\))22710 61146 y Fn(\242)295 b Fr(e)23977 60597 y Fj(\241)p Fv(c)25117 60285 y Fe(0)25415 60597 y Fv(d)p Fo(\()p Fv(E)50 b(;\276)32 b Fo(\()p Fv(A)28907 60753 y Ff(0)p Fm(;k)30004 60952 y Ff(0)30519 60597 y Fo(\)\)\()31750 60117 y Fm(k)32189 60316 y Ff(0)p 31749 60396 901 40 v 31997 60945 a(2)32782 60597 y Fj(\241)p Fo(1\))34775 61146 y Fr(>)369 b(e)36759 60597 y Fj(\241)p Fv(m)38323 60720 y Ff(0)38917 60117 y Fm(k)39356 60316 y Ff(0)p 38917 60396 V 39164 60945 a(2)40005 59671 y Fp(o)41260 61146 y Fn(\267)43094 60247 y Fs(1)p 42794 60840 1251 54 v 42794 62057 a Fr(k)43515 61427 y Fv(p)43470 62383 y Fo(0)44177 61146 y Fr(:)4317 64790 y Fs(This)434 b(completes)g(the)f(pro)36 b(of)434 b(of)g(Lemma)g(3.2.)p 46724 64790 54 832 v 46778 63985 744 54 v 47465 64790 54 832 v 46747 64790 744 54 v 25253 70015 a(28)p eop %%Page: 29 29 29 28 bop 4317 7306 a Fh(The)519 b(pro)42 b(of)519 b(of)g(Lemma)h(3.7.) 452 b Fs(The)g(pro)36 b(of)452 b(is)g(based)f(on)g(the)h(v)-72 b(ariational)453 b(form)-36 b(ula.)4317 8911 y(Indeed)433 b(from)h(the)f(Theorem)h(XI)36 b(I)g(I.2)435 b(of)f([33)q(])g(for)g Fr(')369 b Fn(2)g Fr(C)32674 8361 y Fo(+)p Fj(1)32579 9237 y Fo(0)34402 8911 y Fs(\()p Fq(R)35867 8429 y Fv(d)36406 8911 y Fs(\))434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8962 11844 y Fr(E)9925 12043 y Fv(j)10412 11844 y Fs(\()p Fr(A)11893 12043 y Fv(!)32 b(;k)13339 11844 y Fs(\()p Fr(\265)k Fs(\)\))369 b(=)1628 b(sup)17249 12939 y Fv(\303)17864 13062 y Ff(1)18325 12939 y Fv(;:::;\303)20245 13074 y Fm(j)39 b Fe(\241)p Ff(1)26068 11844 y Fs(inf)22080 12840 y Fv(')p Fj(2)p Fo(\()p Fv(v)32 b(ect)p Fj(h)p Fv(\303)26341 12963 y Ff(1)26802 12840 y Fv(;:::;\303)28722 12975 y Fm(j)39 b Fe(\241)p Ff(1)30203 12840 y Fj(i)p Fo(\))30935 12527 y Fe(?)25648 13721 y(k)p Fm(')p Fe(k)p Ff(=1)31763 11844 y Fn(h)p Fr(A)33255 12043 y Fv(!)32 b(;k)34701 11844 y Fs(\()p Fr(\265)k Fs(\))p Fr(';)221 b(')p Fn(i)p Fr(:)4646 b Fs(\(4.49\))4317 16367 y(As)13596 17972 y Fn(h)p Fr(A)15088 18171 y Fv(!)32 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b(;k)16507 61998 y Fs(\()p Fr(\265)17655 61449 y Fj(0)17965 61998 y Fs(\)\))295 b Fn(\241)h Fr(E)21564 62197 y Fv(j)22050 61998 y Fs(\()p Fr(A)23531 62197 y Fv(!)32 b(;k)24978 61998 y Fs(\()p Fr(\265)k Fs(\)\))369 b Fn(\267)g(j)p Fr(\265)332 b Fn(\241)295 b Fr(\265)32186 61449 y Fj(0)32497 61998 y Fn(j)p Fs(\(1)g(+)g Fr(E)36587 62197 y Fv(k)37156 61998 y Fs(\()p Fr(\265)36 b Fs(\)\))p Fr(:)4317 64931 y Fs(This)434 b(ac)-36 b(hiev)g(e)434 b(the)f(pro)36 b(of)434 b(of)h(Lemma)e(3.7.)p 46724 64931 54 832 v 46778 64127 744 54 v 47465 64931 54 832 v 46747 64931 744 54 v 25253 70015 a(29)p eop %%Page: 30 30 30 29 bop 4317 7306 a Ft(References)4317 10559 y Fs([1])3795 b(M.)428 b(Aizenman)h(and)f(Molc)-36 b(hano)g(v:)429 b Fi(L)-66 b(o)g(c)g(alization)458 b(at)i(lar)-66 b(ge)460 b(disor)-66 b(der)459 b(and)i(at)9484 12164 y(extr)-66 b(eme)536 b(ener)-66 b(gies:)536 b(an)i(elementary)e(derivation)p Fa(.)512 b Fs(Comm)-36 b(u.)513 b(Math.)g(Ph)-36 b(ys.)9484 13769 y Fh(157)932 b Fs(\(1993\))434 b(p)g(245-278.)4317 16481 y([2])3795 b(P)-108 b(.W.)657 b(Anderson:)h Fi(A)-33 b(bsenc)-66 b(e)668 b(of)j(di\256usion)f(in)g(c)-66 b(ertain)668 b(r)-66 b(andom)670 b(lattic)-66 b(es.)9484 18086 y Fs(Ph)-36 b(ysical)434 b(Reviews)h Fh(109)932 b Fs(\(1959\))435 b(p)e(1492-1501.)4317 20798 y([3])3795 b(J.)452 b(M.)g(Barbaroux,)g(J.) g(M)g(Com)-36 b(b)36 b(es)452 b(and)f(P)-108 b(.D)452 b(Hislop)g(:)g Fi(L)-66 b(o)g(c)g(alization)480 b(ne)-66 b(ar)9484 22403 y(b)g(and)436 b(e)-66 b(dges)436 b(for)h(r)-66 b(andom)436 b(Schr\304)-664 b(odinger)435 b(op)-66 b(er)g(ators.)402 b Fs(Helv.)j(ph)-36 b(ys.)403 b(Acta.)g Fh(70)9484 24008 y Fs(\(1997\))434 b(N)14505 23526 y Fj(\261)15465 24008 y Fs(1-2)f(p)h(16-43.)4317 26720 y([4])3795 b(J.)567 b(M.)g(Com)-36 b(b)36 b(es)568 b(and)e(P)-108 b(.D.)568 b(Hislop:)g Fi(L)-66 b(o)g(c)g(alization)585 b(for)j(some)f(c)-66 b(ontinuous)9484 28325 y(r)g(andom)382 b(Hamiltonian)e(in)h Fr(d)p Fi(-dimensions.)342 b Fs(Jour.)i(F)-108 b(un)-36 b(t.)343 b(Anal.)h Fh(124)h Fs(\(1994\))9484 29930 y(p)433 b(149-180.)4317 32642 y([5])3795 b(J.)290 b(M.)g(Com)-36 b(b)36 b(es)290 b(P)-108 b(.D.)290 b(Hislop)g(and)f(A.)h(Tip:)h Fi(Band)333 b(e)-66 b(dge)332 b(lo)-66 b(c)g(alization)331 b(and)i(the)9484 34247 y(density)515 b(of)h(sates)h(for)f(ac)-66 b(oustic)515 b(and)h(ele)-66 b(ctr)g(omagnetic)513 b(waves)k(in)e(r)-66 b(andom)9484 35853 y(me)g(dia.)432 b Fs(Ann.)h(I.H.P)h Fh(70)g Fs(\(1999\))h(N)27287 35371 y Fj(\261)28246 35853 y Fs(4)f(p)f(381-428.)4317 38565 y([6])3795 b(D.)500 b(Damanik)g(and)f(P)-108 b(.)499 b(Stollmann:)g Fi(Multisc)-66 b(ale)525 b(A)-33 b(nalysis)525 b(implies)f(Str)-66 b(ong)9484 40170 y(Dynamic)g(al)444 b(L)-66 b(o)g(c)g(alization.)409 b Fs(Geom.)k(F)-108 b(unct.)411 b(Anal.)i Fh(26)g Fs(\(2001\))g(N)41968 39688 y Fj(\261)42906 40170 y Fs(3)g(p)f(411-)9484 41775 y(428.)4317 44487 y([7])3795 b(H.)627 b(v)-36 b(on)628 b(Dreifus)f(and)g(A.)h(Klein:)g Fi(A)642 b(new)h(pr)-66 b(o)g(of)642 b(of)h(lo)-66 b(c)g(alization)641 b(in)h(the)9484 46092 y(A)-33 b(nderson)502 b(tight)h(binding)e(mo)-66 b(del.)475 b Fs(Comm)-36 b(u.)476 b(Math.)f(Ph)-36 b(ys.)476 b Fh(124)g Fs(\(1989\))h(p)9484 47697 y(285-299.)4317 50409 y([8])3795 b(A.)317 b(Figotin)g(and)g(A.)g(Klein:)h Fi(L)-66 b(o)g(c)g(alization)355 b(of)j(Classic)-66 b(al)357 b(Waves)h(I)f(:)g(A)-66 b(c)g(oustic)9484 52014 y(Waves.)432 b Fs(Comm)-36 b(u.)434 b(Math.)g(Ph)-36 b(ys.)433 b Fh(180)i Fs(\(1996\))f(p)f(439-482.)4317 54726 y([9])3795 b(A.)331 b(Figotin)h(and)e(A.)i(Klein:)g Fi(L)-66 b(o)g(c)g(alization)369 b(of)h(Classic)-66 b(al)371 b(Waves)f(II)g(:)g(ele)-66 b(ctr)g(o-)9484 56331 y(magnetic)463 b(Waves.)432 b Fs(Comm)-36 b(u.)434 b(Math.)g(Ph)-36 b(ys.)433 b Fh(184)h Fs(N)35870 55849 y Fj(\261)36396 56331 y Fs(2)g(\(1997\))h(p)e(411-441.)4317 59043 y([10])3145 b(A.)408 b(Figotin)h(and)f(A.)h(Klein:)g Fi(L)-66 b(o)g(c)g(alization)439 b(of)j(Classic)-66 b(al)442 b(Waves)f(cr)-66 b(e)g(ate)g(d)440 b(by)9484 60648 y(defe)-66 b(cts.)431 b Fs(J.)j(Stat.)g(Ph)-36 b(ys.)434 b Fh(86)f Fs(\(1997\))i(p)e(165-177.)4317 63361 y([11])3145 b(A.)445 b(Figotin)g(and)g(A.)g(Klein:)h Fi(L)-66 b(o)g(c)g(alization)473 b(of)i(light)g(in)f(lossless)j(inhomo)-66 b(ge-)9484 64966 y(ne)g(ous)356 b(diele)-66 b(ctrics.)312 b Fs(J.)k(Opt.)e(So)36 b(c.)316 b(Amer.)f Fh(A)363 b(15)678 b Fs(N)34719 64484 y Fj(\261)35560 64966 y Fs(6)315 b(\(1998\))h(p)f(1748-1773.)25253 70015 y(30)p eop %%Page: 31 31 31 30 bop 4317 7306 a Fs([12])3145 b(J.)537 b(F)-108 b(r\304)-650 b(ohlic)-36 b(h)536 b(and)g(T.)h(Sp)36 b(encer:)537 b Fi(A)-33 b(bsenc)-66 b(e)557 b(of)j(di\256usion)f(in)g(the)h(A)-33 b(nderson)9484 8911 y(tight)463 b(binding)g(mo)-66 b(del.)433 b Fs(Comm)-36 b(u.)434 b(Math.)f(Ph)-36 b(ys.)434 b(\(1983\))g Fh(88)932 b Fs(p)434 b(151-184.)4317 11623 y([13])3145 b(F.)529 b(Germinet)g(and)g(A.)h(Klein:)h Fi(Bo)-66 b(otstr)g(ap)553 b(Multisc)-66 b(ale)552 b(A)-33 b(nalysis)553 b(and)g(L)-66 b(o-)9484 13228 y(c)g(alization)439 b(in)g(r)-66 b(andom)439 b(me)-66 b(dia.)406 b Fs(Comm)-36 b(u.)406 b(Math.)h(Ph)-36 b(ys.)407 b(N)39238 12746 y Fj(\261)40170 13228 y Fs(2)g Fh(222)h Fs(\(2001\))9484 14833 y(p)433 b(415-448.)4317 17545 y([14])3145 b(F.)670 b(Germinet)g(and)g(S.)g(De)h(Bi)-36 b(\265)-614 b(evre:)1355 b Fi(Dynamic)-66 b(al)681 b(lo)-66 b(c)g(alization)681 b(for)i(dis-)9484 19150 y(cr)-66 b(ete)344 b(and)i(c)-66 b(ontinuous)345 b(r)-66 b(andom)345 b(Schr\304)-664 b(odinger)344 b(op)-66 b(er)g(ators.)304 b Fs(Comm)-36 b(u.)304 b(Math.)9484 20756 y(Ph)-36 b(ys.)433 b Fh(194)i Fs(\(1998\))f(p)f(323-341.)4317 23468 y([15])3145 b(L.)512 b(H\304)-650 b(ormander:)512 b Fi(The)537 b(A)-33 b(nalysis)537 b(of)g(Line)-66 b(ar)536 b(Partial)h(Di\256er)-66 b(ential)535 b(Op)-66 b(er)g(a-)9484 25073 y(tors.)433 b Fs(Springer)g(V)-108 b(erlag)434 b(Heidelb)36 b(erg,)434 b(1983.)4317 27785 y([16])3145 b(W.)356 b(Kirsc)-36 b(h:)356 b Fi(R)-66 b(andom)392 b(Schr\304)-664 b(odinger)392 b(op)-66 b(er)g(ators)748 b Fs(A)356 b(Course)g(Lecture)e(Notes)9484 29390 y(In)433 b(Ph)-36 b(ys.)434 b Fh(345)g Fs(\(1989\)Springer-V)-108 b(erlag,)434 b(Berlin)g(p)f(264-370.)4317 32102 y([17])3145 b(W.)511 b(Kirsc)-36 b(h:)512 b Fi(We)-66 b(gner)534 b(estimates)i(and)g(A)-33 b(nderson)535 b(lo)-66 b(c)g(alization)535 b(for)h(al)66 b(loy-)9484 33707 y(typ)-66 b(e)464 b(p)-66 b(otentials.)432 b Fs(Math.)h(Z.)h Fh(221)g Fs(\(1996\))h(p)e(507-512.) 4317 36419 y([18])3145 b(W.)480 b(Kirsc)-36 b(h)480 b(and)f(F.)h (Martinelli:)h Fi(On)508 b(the)f(the)g(Sp)-66 b(e)g(ctrum)506 b(Of)h(Schr\304)-664 b(odinger)9484 38024 y(op)-66 b(er)g(ators)369 b(with)g(a)h(R)-66 b(andom)369 b(Potential.)328 b Fs(Comm)-36 b(u.)331 b(Math.)e(Ph)-36 b(ys.)330 b Fh(85)h Fs(\(1982\))9484 39629 y(p)433 b(329-350.)4317 42341 y([19])3145 b(W.)383 b(Kirsc)-36 b(h,)383 b(P)-108 b(.)383 b(Stollmann)g(and)f(G.)h(Stolz:)g Fi(L)-66 b(o)g(c)g(alization)416 b(for)i(r)-66 b(andom)418 b(p)-66 b(er-)9484 43946 y(turb)g(ation)374 b(of)i(p)-66 b(erio)g(dic)374 b(Schr)-66 b(o)g(dinger)374 b(op)-66 b(er)g(ators)p Fa(.)337 b Fs(Random)g(Op)36 b(er.)336 b(Sto)36 b(c)-36 b(has-)9484 45551 y(tic)433 b(Equations)h Fh(6)g Fs(\(1998\))h(p)e(241-268.)4317 48263 y([20])3145 b(W.)599 b(Kirsc)-36 b(h,)600 b(P)-108 b(.)599 b(Stollmann)g(and)f(G.)h (Stolz:)h Fi(A)-33 b(nderson)616 b(lo)-66 b(c)g(alization)616 b(for)9484 49869 y(r)-66 b(andom)338 b(Schr)-66 b(o)g(dinger)337 b(op)-66 b(er)g(ators)339 b(with)g(long)g(r)-66 b(ange)338 b(inter)-66 b(actions.)293 b Fs(Comm)-36 b(u.)9484 51474 y(Math.)433 b(Ph)-36 b(ys.)434 b Fh(195)932 b Fs(\(1998\))435 b(p)e(495-507.)4317 54186 y([21])3145 b(F.)464 b(Klopp:)h Fi(L)-66 b(o)g(c)g(alization)491 b(for)i(some)g(c)-66 b(ontinus)492 b(R)-66 b(andom)493 b(Schr\304)-664 b(odinger)491 b(op-)9484 55791 y(er)-66 b(ators.)432 b Fs(Comm)-36 b(u.)434 b(Math.)g(Ph)-36 b(ys.)433 b Fh(167)i Fs(\(1995\))f(p)f (553-569.)4317 58503 y([22])3145 b(F.)371 b(Klopp:)h Fi(A)407 b(short)h(intr)-66 b(o)g(duction)405 b(to)j(the)g(the)-66 b(ory)407 b(of)g(r)-66 b(andom)407 b(Schr\304)-664 b(odinger)9484 60108 y(op)-66 b(er)g(ators.)432 b Fs(Vietnam)i(Jour.)g(Math.)f Fh(26)h Fs(\(1998\))h(N)p Fn(\261)e Fs(3)h(p)f(189-216.)4317 62820 y([23])3145 b(F.)405 b(Klopp:)g Fi(Internal)438 b(Lifshits)g(T)-100 b(ails)439 b(F)-100 b(or)439 b(R)-66 b(andom)438 b(Perturb)-66 b(ations)438 b(Of)h(Pe-)9484 64425 y(rio)-66 b(dic)508 b(Schr\304)-664 b(odinger)509 b(Op)-66 b(er)g(ators.)482 b Fs(Duk)-36 b(e)483 b(Math.)g(Jour.)g Fh(98)g Fs(\(1999\))g(N)44624 63943 y Fj(\261)45633 64425 y Fs(2)g(p)9484 66030 y(335-396.)25253 70015 y(31)p eop %%Page: 32 32 32 31 bop 4317 7306 a Fs([24])3145 b(F.)532 b(Klopp:)i Fi(Internal)554 b(Lifshits)h(T)-100 b(ails)556 b(F)-100 b(or)557 b(long)e(r)-66 b(ange)555 b(single)g(site)h(p)-66 b(oten-)9484 8911 y(tials.)433 b Fs(Jour.)g(Math.)h(Ph)-36 b(ys.)434 b Fh(43)f Fs(\(2002\))i(N)30396 8429 y Fj(\261)31355 8911 y Fs(6)f(p)f(2948-2958.)4317 11489 y([25])3145 b(F.)424 b(Klopp:)h Fi(We)-66 b(ak)455 b(disor)-66 b(der)457 b(lo)-66 b(c)g(alization)455 b(and)h(Lifshitz)g(T)-100 b(ails:)456 b(c)-66 b(ontinuous)9484 13094 y(hamiltonian.)431 b Fs(Anna.)j(I.H.P)g Fh(3)g Fs(\(2002\))g(N)30610 12612 y Fj(\261)31570 13094 y Fs(4)f(p)h(711-735.)4317 15673 y([26])3145 b(F.)621 b(Klopp)g(and)g(L.)g(P)-36 b(astur:)621 b Fi(Lifshits)636 b(T)-100 b(ails)638 b(F)-100 b(or)638 b(R)-66 b(andom)636 b(Schr\304)-664 b(odinger)9484 17278 y(op)-66 b(er)g(ators)512 b(with)g(ne)-66 b(gative)510 b(singular)i(Poisson)g(p)-66 b(otential.)483 b Fs(Comm)-36 b(u.)486 b(Math.)9484 18883 y(Ph)-36 b(ys.)433 b Fh(206)i Fs(\(1999\))f(p)f(57-103.)4317 21461 y([27])3145 b(F.)691 b(Klopp)g(and)f(T.)i(W)-108 b(ol\256:)693 b Fi(Lifshitz)700 b(tails)i(for)f(2-dimensional)f(r)-66 b(andom)9484 23066 y(Schr)g(o)g(dinger)462 b(op)-66 b(er)g(ators.)433 b Fs(to)h(app)36 b(ear.)4317 25645 y([28])3145 b(H.)334 b(Na)72 b(jar:)336 b Fi(Asymptotique)372 b(de)h(la)h(densit)-33 b(\266)-631 b(e)372 b(d')-33 b(\266)-631 b(etats)373 b(int)-33 b(\266)-631 b(egr)-33 b(\266)-631 b(ee)371 b(des)j(op)-33 b(\266)-631 b(er)-66 b(ateurs)9484 27250 y(ac)g(oustiques)671 b(al)-33 b(\266)-631 b(eatoir)-66 b(es.)658 b Fs(C.)h(R.)g(Acad.)g(Sci.)g(P)-36 b(aris,)659 b Fh(333)h Fs(I,)f(p)f(191-194)9484 28855 y(\(2001\).)4317 31433 y([29])3145 b(H.)452 b(Na)72 b(jar:)453 b Fi(Lifshitz)481 b(tails)g(for)h(r)-66 b(andom)481 b(ac)-66 b(oustic)481 b(op)-66 b(er)g(ators.)451 b Fs(Jour.)h(Math.)9484 33038 y(Ph)-36 b(ys.)433 b Fh(44)h Fs(N)15952 32556 y Fj(\261)16912 33038 y Fs(4)g(\(2003\))4317 35616 y([30])3145 b(H.)440 b(Na)72 b(jar:)441 b Fi(Asymptotic)469 b(of)h(the)g(inte)-66 b(gr)g(ate)g(d)468 b(density)h(of)h(states)h(of)f(ac)-66 b(oustic)9484 37222 y(op)g(er)g(ator)464 b(with)h(long)f(r)-66 b(ange)464 b(r)-66 b(andom)464 b(p)-66 b(erturb)g(ations.)p Fs(,)432 b(2002)j(submitted)d(.)4317 39800 y([31])3145 b(H.)602 b(Na)72 b(jar:)605 b Fi(2-Dimensional)619 b(lo)-66 b(clisation)619 b(of)h(ac)-66 b(oustic)619 b(waves)i(in)e(r)-66 b(andom)9484 41405 y(p)g(erturb)g(ation)463 b(of)h(p)-66 b(erio)g(dic)463 b(me)-66 b(dia.)432 b Fs(in)i(preparation)f(\(2003\).) 4317 43983 y([32])3145 b(L.)460 b(P)-36 b(astur)461 b(and)f(A.)h (Figotin:)h Fi(Sp)-66 b(e)g(ctr)g(a)488 b(of)i(R)-66 b(andom)489 b(and)h(A)-33 b(lmost-Perio)-66 b(dic)9484 45588 y(Op)g(er)g(ators.)432 b Fs(Springer-V)-108 b(erlag.)4317 48167 y([33])3145 b(M.)513 b(Reed)f(and)g(B.)i(Simon:)f Fi(Metho)-66 b(ds)537 b(of)g(Mo)-66 b(dern)537 b(Mathematic)-66 b(al)536 b(vol)i(IV:)9484 49772 y(A)-33 b(nalysis)464 b(of)h(Op)-66 b(er)g(ators.)432 b Fs(Academic)i(Press,)g(New)g(Y)-108 b(ork,)434 b(1978.)4317 52350 y([34])3145 b(P)-108 b(.)502 b(Stollmann:)g Fi(L)-66 b(o)g(c)g(alization)526 b(for)i(ac)-66 b(oustic)527 b(waves)i(in)e(r)-66 b(andom)528 b(p)-66 b(erturb)g(a-)9484 53955 y(tion)464 b(of)g(p)-66 b(erio)g(dic)463 b(me)-66 b(dia.)432 b Fs(Israel)j(Jour.)e(Math.)h Fh(107)g Fs(\(1998\))g(p)g(125-139.)4317 56533 y([35])3145 b(J.)1191 b(Sj\304)-650 b(ostrand:)1193 b Fi(Micr)-66 b(olo)g(c)g(al)1160 b(analysis)h(for)g(the)g(p)-66 b(erio)g(dic)1160 b(magnetic)9484 58138 y(Schr\304)-664 b(odinger)458 b(e)-66 b(quation)458 b(and)h(r)-66 b(elate)g(d)459 b(questions.)427 b Fs(Lecture)f(Notes)i (in)g(Math)9484 59744 y(1495,)435 b(Springer-V)-108 b(erlag,)433 b(Berlin)h(\(1991\))g(p)f(237-332.)4317 62322 y([36])3145 b(I.)1178 b(V)-108 b(eseli)-36 b(\266)-614 b(c:)1180 b Fi(L)-66 b(o)g(c)g(alization)1147 b(for)i(r)-66 b(andom)1148 b(p)-66 b(erturb)g(ation)1147 b(of)h(p)-66 b(erio)g(dic)9484 63927 y(Schr\304)-664 b(odinger)615 b(op)-66 b(er)g(ators)616 b(with)g(r)-66 b(e)g(gular)616 b(Flo)-66 b(quet)616 b(eigenvalues.)597 b Fs(T)-108 b(ec)-36 b(hnical)9484 65532 y(rep)36 b(ort)433 b(Univ.)h(Bo)36 b(c)-36 b(h)g(um.)25253 70015 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303141029613 Content-Type: application/pdf; name="divergence.pdf" Content-Transfer-Encoding: base64 Content-Disposition: inline; 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