Content-Type: multipart/mixed; boundary="-------------9910241410845" This is a multi-part message in MIME format. ---------------9910241410845 Content-Type: text/plain; name="99-405.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-405.keywords" Spectral theory, self-adjoint operator, generalized eigenfunctions ---------------9910241410845 Content-Type: application/postscript; name="cklf.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cklf.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: cklf.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMR6 CMBX12 CMR10 CMCSC10 CMR8 CMMI8 CMBX10 CMMI10 %%+ CMSY10 CMMI7 CMSY7 CMR7 CMTI8 CMSY6 msbm10 CMEX10 CMFI10 CMTI10 %%+ msam10 CMMI5 CMR5 CMBX8 CMTT8 %%EndComments %DVIPSCommandLine: G:\PUBLIC~1\TEX\BIN\WIN32\DVIPS.EXE -Palltype1 cklf %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 1999.10.01:0332 %%BeginProcSet: texc.pro %! /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup 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 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot} imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}B /d{ -3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w} B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont cvx]cvx def}def /ObliqueSlant {dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end %%EndProcSet %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % 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. 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531c83c249a11ac40947de725a7a4bf15ea91f69ab 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 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 readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 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 93d777c339423c4fad91e1968b981744bda6f085de1cc1972039dff074addff0 8356c3be69c9fcfae25c8eb5206ab487be6c32a1f3b205854ff0a75a7c54c20a 525737bfe15dabc1593b54a13423545749338f0ba6f58dbf834201388f8fe32c 33697cd886c19246e23373e0fcbf8c127a9a416f07b96b89adf966cf65bf126d d39966b5e551d0c2e704e1f1e353f87596c702134af724879d25030e18aa3729 5bf44904beaeb5be523c962bbd6c50468fea94a694f71686bb8e2b1a010ff0ed 0a8fe03fe4420b0a570a838346d27803d3c702d3ddd12da3eed572ffd32f3b54 dd788a12b95bb9ce2029c1ab7fa883afa123fc8dfaa417099907aa389e7f736e 49c9b5551cbbb070518dcf1a7cdd464b92cbe87ee2d9595afdef2fef111eff4d 42a9962974b6d713751214691edf0f86281e64b0ef387b589b0f08cf20f1958d 73bf012e1346e2c831d6b9c8e83eade251a5bd42ea1ed6f8169162c32a9e8d1d 6bd2c4360e834fe54f1a3bbc0076b26b9cd0d2a15d6ddf196583cc02627c968e 3460968836eadbaad07ad1f8962c17e660958fca8f4b4c54a33dbf68ca384ce6 6007bb2aab608edd2653de8d49ad529fc3cafc85d0f1a04ae08d8694b40e4575 e0524258c42fc8e7b40892621e2fda4e04a8fb87cf41a3e94d1f3c55998fbd38 3dfff7c2d3db547c0f9e97352c8f89f621a6f12df52fe742abd3a228684f0085 0be20686170a457832aa3e18640310f431c2e30dfd08a80bb8266527a09e7896 337fa141e27e3a0280099333d2a8a657f608365c3524f03151a2506458fd34e2 12c6107deff1abad8621c809dc78ce7f3b26eb331731a7ec3a4d5165ce182b68 b39604921a0b9c1fdaba7c4f47fba21b83953e02c13f3ce4aa086c66d8f8cef1 b0de400598542b63beb428af6cfc07bd427217f5d471bb08b7887feab9c7b0e9 7a79a576b36d776fe353864eb08831960b253aaf29279904c24db1fb7b15d175 642efcec8742e8912dd2102ffa49ef166a29e557ff6a4a9480550d0f13a86742 d2571e861f4152bd716ba6d10efa5cdc9ac282c960329b089a9b63517417a15e 661a2f5259ebabaaa2c2fe39993ea16dedc3c31f253dd84ab961e2177798f8b9 4ac6f182aa36c7a189bec838de1d6e69ffbe70e3272d1bcd01328130966814f2 f4925dcf78f39422f382585c48cba69cb051b1f48724f2949db34a315c69327f 696d8274f293cbffcae40b1d4aa67bda2006ff70d30e90ed90db29e97829fe51 1928ae11c2d7cc1c26f7e75510a82af1806d29c359ea5417bfab6d9d068c72be 39eea04663f723707d252af0d30a59cc9905e85e6d500d941e0e77f1d2796a52 eb925540c6f35eab19af09237c79fa8c5d860c679c2d6832c765a2fea8cd3462 22cb0d07e0632e254e18420bbf11c326bf38c1421c1bd7ceabc3f1d94fb01706 efcee04fc59f9b2a58ca141e2428e4b8d438348524015b6abfb85fa918deb987 5bbd54a0e89aceceade566553ca578a786c23f28ef6713c178ca8a04809a67c0 fd44d0fafe1d01d7d26fc281031503b1a00245ddaf322d0b1df65fc00947b58d 69b534aec0052d47adcccb7fede6e41b02ae15ef38bd1a49813fb8c864abc620 1093da01ff2b083b277fb41accb0ac163e03f9123711551d520d679c22899b7d 6498c17f9b8a6bdd72c7b52def37fdeef6f99578041fe493a78459581cff4c03 abf70c054acad70c49ee4e4977f81d8964cb26d0992965b18e61fba023672abe 02cb53532f3fad432903b11ae6ddc9d98948a7cda30380955e48d4e948497403 c6ac1c5447226a1153b27b3d535acacf604ee32e49e33cf3d6aff96dd249a55b c550eba1398932463d17806d2abf0b7e620c626588b8b30d80eea3e8ff50c503 beeb57cf9d2655b7361b016e9c7ef9639288b3db077b0107b959a172ebaa292d 1f7b0fec0d7f19d2de06fcc5977e791c8dc7d64a56ccd2f7734d85095586cbae b45107d65af4958e047a33d5144a3199ba2bedfaffee0561fde00fbc0154dad6 833d09e1ea445a687ad6225b41e13280f2450158d20a77b69fd17f73f1a9e75a b36a014eea9fbb6361810d655575684049595dfac43e8e71a40b89cfcc3551f6 4b31db8bf5156ee9856328ce5770133f24d456856706e316e5c4af28a2ae55dc a0957db7eecc284e10bf34507e0b503b4f58096720e6dd3149e213c2ce6a568d 6130385b2aa835a3d27127d1c2d1173b5589ef09fd03ae2e214ce4b658022a43 ddf865cb8769c93c1267cef1667397820197ef34c13aff25d1ec4ba91c155b14 68229bd1359f5002117374d4563583848cc736a0eca2b7702097c05530768ecb 05c84d6c725f17a70e3b9456bb4ff68919293eece99ea271c8870c6135d49229 d0a04f6b9aba1ffc28aa64735af8047a32160e6236c4555fd3c4f831c17e89c4 9fe92cb436857fd42ee7b3b2eece2f0a9f5bc29cb1c4b3691dd7c5c811f63b95 9a117f186d8e0e69d3330538bbad0e3863825dabd2bab7ab767b4e93f2f41196 c64d816248a740ac966a4ea02e7870a95c4dd169361e402e4208378b892361c8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b24c5e6c ef3ce751da62120371d4edc733042e2a6661dc6a3111f38f0baec0eb2410e8dc 1605312e5cfd3f3c89cfe4913fe33f71f1546730d31261296f78a1c156b4af7b b27af1a0706f57d0fe0d5c7b18aef611b69c67ea9ad325f159b826404c1e6f08 89105b670bbbef3e3dc50835c71964a2e992354d26571566eb6d7bd7938acb12 486336dcb8ab542223c6811d37c95679b796e22cf608d05b3e0ab856c5fad35d 990d789bef73c3d17dd66b02adca2aae2fef64976a04310219579a91aea4404a e87f66aa06ba008dfbe8c02a0a74e1d0aeb5b3feeb1e5f20d8136c0020a88677 cb72b8e0e7d5c8c2b230baecaea5aecab78e9bd97bb59018a30aaa15cfd0a1f6 fdb877479ac66003d3f3a62f76469169f467078712ae99862e398a6fba3a149f 80aebd858e6482dd5dc8cdae093b74ca98e10683b08b501d21bb5576e6cfad67 a360b4758832c14a056ad4a6528597cd9a09b4048546c50861918d3bca7e47c2 07efce4574d42b7cc18f61b6d57bc513f6a659921daa707c3ae900b85603258b b6ac6343e09f271e5c2194be84d3140266c66c6abd21a461846ce317312274ae 3e6b0e61a432c18dc19a68fb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 30 /phi put dup 82 /R put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a3acb19152e5f4315 727ce914a0482f158a14c9bd5bb071e06d59c6f4244401ad9bde7f391a009e37 fa148f0be23b9386aa2413e375bd8d16623f6541797ac8d4e9842a2030337a28 a16a1470fe9725c829603744facecc830bf9fc5b1bbb62b8ddf0fc50231acae9 53ac22110e3366baf2588c1949c720a8d3d3cab5207a71929b3c19c094373f1d 21688130735f6ff4b213f41e4f15a2cd2facce01cb8aa4796761b5d7eabe30e9 243332bc32035535f30fb2deff89ed45ce25a2b7a90f9b2112f40082ab2571b8 7ff2738b168b3dcae128f254a9814b2e1fb8352f12e250fd730ea8bcebec676b 3e11edd6ceb9827ba46fde588576ed7c2f8a65d4a275c56c6102a8290df14429 d3e4847c5e2e8e111487ecc2a4b1efe56106ae93ffb6a42b5d4c81a89c849f12 ee25610d6921ee8fcbd1c69e1f073e63ea459aaefc123311310a101033b3ed1f f41f38a371efadad6723b5d863e9080efb547b1e65e3df988eeb8908a157ce2d ce339decd8d7ceec289def2c80d2ca486e783f46541d0e012a413a03753560ff ed011c996845d9535e24b1bc6df90e6efb1350b80668ebf19e38a7f8b5665adb e9c56679926335fd061f5a56edfffd70dea023de38cc29c80606ae9832c61c26 ee46e4b447f2f4a311dd492d378f6a8d30f4a7d2397259caf2094779f63682a8 68b6153ddcafae390bfcb6ece4ebdb63af10770e648eb10d7006fba8bd07aa67 3189013d435d674486a3d250471b5b34c9d74699c5a734a43f09c1ce32400550 054e9875b7a3e3732e532646ee7b00b20e799a4f978bcc59cc189ec83bae2ad6 ec1b8c00c5de33fe1362022bd5b83ec1bce1183802b7e6ee1e57f0d44fd6bc96 c67ea624d35c5cef7e7cf8f93b01a231d9f1f8713547691ffafef80167dceb72 27b9e594583d0d8529d305a9bf7f6be632ab83634dbc834fecbbdab64457ae15 78b504524814e93064f7ce55bc1360996faa4d5aa2d9d6e9d213d9f319240f24 51f3cf90ae121af6692c5b795c7c91e72739100b835c362a93b7f738f333b1fa f64e00ad91320d789d273ccf9df76a00cbe8ad507c31cc4be03be0f6c394b910 9542d3f2f9e5ab57110844e21344a0ce8234e57ced6ce0c66c38bcabd962169a 2b28e73f73fafafdaa98494b5018b97594d60205e86cbdd13b382f50b04d253e 57a55c75 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msam10 %!FontType1-1.0: msam10 1.2/19-Jan-95 %%VMusage: 23308 26649 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msam10) readonly def /FamilyName (msam10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /CapHeight 514 def /Descender -194 def /XHeight 322 def /Ascender 694 def end readonly def /FontName /msam10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 185 /harpoonupright put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [0 -463 1332 1003] readonly def /UniqueID 4702303 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb c799d90205a21291254fa633cc0c2ed88781ef21a5d45b72445c284fa44f8f39 cf566976075a92e9e2947a6fc93d879c29eb26809ecb4409cbc8666526cced92 65de661a2f8b0a16be45a9da17eeaf89b0e023de2b0373dcab997d60b7d7dc1b 4f7e650a2a9f13ca0543f3b080ae33d65f2802bb566f15aa33d40dcee9a893e6 a592fa929c824d90c8bd67b84b3a769791df9d605b1bb2314a936e5e1510957c a6fa3a43d11c77c309dedb8c4929e0bccc3e6c3ba45ca0e4234b8b8c8c390797 78fba7f0d08d6cc8d10faab67e9a67083cc498c2eef3762231cd631cae113c60 d755cc2be3148cdbc43ac27c7fc2574aba0f81bd8358a35fdfbb2b377d47815e 3a06f4b714a23c0e2516696b12a08e19dc10a2c3d6971f897304e0954bce7399 8dd65ffc80d0576429068d48e3af42bd6751cfc9a3d4915b8d67fc2626338ba5 3eab50500c6f66df3af8fd0b21e504f37302f2e039c5220b3d4d1a140fc5ca60 b1076299b00c2c9509e1e21de35128ced5814d075ff837ef9b70bd11ac36794e 8ace3782d6cdcb9e66b63e650c0e9d173d13261158c7e86be58cc108df7c26a0 b7dc31ec9f745cbc02e3e5a36b2f41bbf13a0e1f16a3108462d4dd65c2cc3321 034404023fd48268952139dc897c8444e39aab0f1236880dc35138767bb5a6b3 b56775515fc8a21aff51f75ddd08a557494e50d21ee97c41ddb7bdbbdf05ef1e 9a6052fa14c89c14ef24d9948a38483eed609d1a336985db710bb0a2cc05014d 85098bb25f54926fb6a59b25e92be25f7f23bc103ea7b5a337b511dab50161d0 ec364a5ef73d86de39b4dbafd5f7323f3938bb141012bce326ff7a23eab72034 06a806aaa150a1d1761f86d4547cfea31fce73eff9e367915c81fbadf5264a8a 279ef58224b7c5ca83babe92896f8d6d4f905edf7e98f2a7097e023ef60f1b12 c683f56b3d25616c00e5b35268b84a46ab66751cfdf567ea83f0986eb2bd7468 636f31a9a982dbf03c5cc7fb8bdcc416043be17de661fd3bb17688fa1c38c35a 5a84281813261215fbe3c1b5b647a28909542a416d433b9e2cbd3d51df3fbbe7 bbaef5b49265ad83f90414ed36a305d9cfb19ca1b82d7a6ae994e95e654f5aba a05472568754a594b433dd8abb0908fbf10965e64b6cdb1b2b985f0a868c2eb0 208938f2eb8f37a3452f86d9510f601b4abc4218c4868571b380b569960539a6 da691af6d3ad8c0e5e7d062576a941921b1ed7e1b81ef5126badb451704b9560 54dbbd1b3ddad3d346c3bd6fa774a641ee39e7836226e412ab8ff8ad6044a393 7fd0bfc56c1354eccbdc333cabf2141208e46d70355aa3ec111f43afeaa57a34 1a0e8f82e0e9b45061fdcc36ca71d5da855c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca1744adc0e2605cfc5cf8028f4a1acb9dd2350c61e6070f9 693748698b56e743c435e5f6a1679797b804ec360e546c35779c747bca08fc1d 45d12c97f277de4c2e11a49152bcbbe250bdc6e244c61b95f7e063b0a1c3418d aa73695e306ff8e4bb723ad6570f80a8d78a3cbecfeafa9d4419e0b70bccc4f8 3f4b1c30cc0492ca105c4d59bab7a5df576caa31f9aaa971e86b2a6883a4d81b cf7f4936812cbed8ec4ca911def2f6b335c520ee2157d265e0269bc4794a1cf7 3bdbe52031b5802a8914284febd0f52cda2edc3b1859b9b5c8744decf54d2f40 c407d578a0dabde67bd6710e0a3bc0b0a08112fac7f5c8b6adcf9e826c12ea08 0811b0913d9f0cf445b0e0c48fe596b8abab1a8621d23c8cdd755a89d48dbf5d b3c8df6c892a6d979674923cb7f6dfbc55bb065dca9d5846165945ec1d0fd3b3 d15ccdc8501be486738729c54f832aa43dca0beb3762e86c3d9e3416d5edbe7c 00a8e352df8ec8fa6ec9d2beff696193dfeab15b646366848b8ae7278258e75c fa9ffa4a9975814082b3d0179565e0e4c614748e8189e355886cb34ec963ea73 0a7442aea758f9cb8270c569055afa0cab4c82ce56325baf82fc450fb908096b 2c0572dfe70fe04aacc9f3fbc74aeffd6afc2f6267c84f16c8a251ac17d6b481 57ae312c0c1995c6c4832bc3ca2812a071a4764e95910a9d0448b29df56476ea 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1686172da60cad4ab5eca08c6338c6b3d1b23f3d608ac7fd818e3203c196bf90 42ba00301088c597ebf237829a87a66481ef66fe81a1b54c179cc74284cea764 80bd5ec91588c95d604b9035e192a04f711ebfae2dc9631586824ade56969271 fbcaf3fb01ea564618d53453df9e5236aa2b313a9e87ee7cf72e19e13158d474 fb28facea65dd78429cf3a9067b875dd3140042c57d18843f061111cc6b42a91 c9309ab908465854addf28e4c979e3b6035baadef021900fad900905860ced18 556f5f8fa27af516e721c8c6981c83eb77daeaccafed4da165fbd8937096b438 c6e63793913d94fd5e39c930adbbdfe35d88484fe3d59da0f6f3014e4373f079 50884a4035bccd3c94879e5034a72d055e26eb531cd6461a9d5538625133b29b 5133c7f67012d3efbeb2d2008023dadb04ed22d5d86703c3aec55f871c85d21e 9d05585aed2e2c102e7f3f766282eb1e6a45e98c9e382e2d04a1efa87264d6ac 544f48870228fc26de643e142280c80bc152352ca228398af39511a187dc8286 7a3c944b756e5d50481942d7453ee2770be5ed7ed9566357dd9f6cbb5bd6fdb3 9ab883e71ed091b29e5ad9a8b9bf6553c0506480059b3e291db457022c570b86 c011b4c90d5061601b1dbecf07ea65351d9cafb56ba6a9f580a4fe3733978967 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d75bb75eb44ee85fdcdf84d6e0ef37baf644c61b0f67eb52e7f8d90866d3dede f24bd29d31c81249e0f7636424c74d3151ab474693cd027ec68f3a56fec20483 4b312399d39437a8b87e9536f28cffdbc81d92d9a6f1b7cfcc221c70996382b6 62d354eeed2644b6ffcc163276dec2b11af54f30ae6da65949690abf33a5ba7b c7714bdc07f92da5333c333b204bef31cae4f7d05944e147358681217864ddff 0e86803b48d464640417f60fa93e2ee4553fa4436a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 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985d71420ba2a8b205dd300d4eb8635907e1fd1fd581e685a2f87f1890b096c5 99f5a6ada4ed6542fbe60c3f0913973f4cc40ffc8439f874c42b0be58d516b46 19e0a8cebc3468da7f483d945f92bd064c0662099cc7aa2c460b39336dea2711 3e336144319e54076bfbeef14f82dddc43d4cf7cddff5681c6b84d27b958bb39 e83f2f569f44590dfdb514fb1832e6c9bbfd153ffcc956f9d75480bef5c8a5bc 9ae70acfe229d1c02007d93dc73611ff7b00f391e3f0f738ce731833d263de3f 02ae92aea0365b5b6d2a2872da3dece4a9398dd42dcdf2b44c6b4c199aec48c5 abf14014ac21d614cf23ad9ea3badee5b9b768be77b1b584544680753b8ff605 f152b04d7164dfa45bc15fba05acf43fa29bbca4aa3c906f22d377358b4c4180 ca6239316bee79b028133b1b9bee81649be23c20caac4ada9007684574f93a5b c84819d072e234b81a47bb99a5a4e211eabf47a466158550436c1cf33c70cda2 62e2ef3c5923f103c36434a392cc94d7394c57e8864f51cef7006bdecaa892b5 26e2a903cc2e67525d1acdddbdcda294bf8fd5cbb523a438affc29975798e2de 61b61e89cd47125baa3b60a9e4b17d5a7a4dfdcd6142dd8877295d68c3fa10b0 a1eaf2de802fbfc756d89637d340b1cc3ee225dcf98251068784c208faec6635 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2f9cb926148e88c3988fea0c3add04038551a550d90531c3085aae1be78e3aea ca14548732ac3c5d98dce73f97d5a56c282974a4c241a78bcdf488e58d994d4c 69773c7150d059c386b7dfb75eb2849c9c6c7c6730756507b18f0316f4b26073 d4b5d669e2864263046a53b4e194db37e317326fc3c9ea7de8d6d9ce3a27cae9 eb5dc30e89913da49d83ab8ca014ae94159a2efd583136a67a71c5b48ab43d80 ec29faecde91bf68eb0e6cf8ebe8b8db9de36890f2fbaa395a3e38aa2e3ee90e ead357b22dfb2a4b1a027b30cb816d96088d3a56b2d10be3eb0a92297dae0f9a 1fbf174c65c9386bcb80f7b6a4b59c3c1b9c8335f2db57fd520c1d4189002ac9 b6f6689c605f758ed28617ce64d0f481e3d53e0ba1721ff6a42032acd412a137 9bd7656a5277cf9b19d79022b4c7f1a3851c6a50574cb89dc428b9ffda8915fb d057e7808f526143d965f9187b2f2bf774dbb083423acea4419fc7a404596d38 d3229f60695d6d735b96f2a51c93946836b0536bea92de3bd77391a03e6cde4a 06558d75cdf4ace0fccd25410d8e4e4360381600cb1aa746c71570352a8dbfef fa574b9d368e7daafbb050776ef9a653c3cf497246b68369ed242191ac321b53 98015d6284d5bbbc8ab0587619fa4e4abd1ccb4e38b14221c37ec53292730834 6ae4d272988279bae098216581bfa7a19419030826dd04f3eea246b62fc1a8d3 93712ce8e16666c791c9236caccdb0de45afafe14e7023aaa2d97e3029a2a158 c62dfeee786778e334337efd536313acf06ab34f3b5348886804e6411687da57 09eb677b51bea49ec2bd426a922167b58086d0797f366e3c71bb8a86fa512c5c b798c95629668864cd58c66451c323c70c810899bc9931c92fc76d7da5bf882e e0198dcf9f9e1a7276b35f8a936b7e4d17447f6e396e7e2b09c65ddc0a3a58e6 7074dd27b5d58170eb98824ddcd15e949b3b7c9698a00c6b3b86cb438ede899b cc2abff782b78b0d0b893c60ca03af893afc38e1a294b2ccefd021d1423800e7 c648488130cd75fb38e6dfbfa0c4dd3561346e936048e48fa7fc5f51c34d90e0 8c6d3a932f99c4a3239a6cb861b6510de32e770124e70db4dceda71a0ddcace5 e96c96c3b21436fa70b02e0e7eeb5007ab102b7728cc42b367a0d5d2e705471d 4c1a493ed8edea0570f6ee04953eb3cda3a45dc1d60c20566c6bb9cdb1ee6995 f51e115d53ff41c91ccc74b401406f080fb547f005e0bdb23fa44b5452e2c67c cefcdbb6bb650d2fdfe16d27ee3a96ca50eb43d7f220544e628dfb7c2a0eaac1 f72bbc9873522de8ceb2099287d6b92d0a05c8a82f154675e75de6c957724f3f cfc6b50a9031f32e02504f4266a070db55bc65b7bb8cdf88d00e0511a6a3b682 3bce573b6d7f0d93d2d68331f09976c26cfc57ea7ee41719f3ba06bc7ccaeffc 7aeb1a9e70c1c2f2fcea9241c776570a159d937732f0fd3ee15ca718f3cd76ec bd69538b2ceaeafe91f7563386b6acab3f2b121a43422ae60a5201dbfb636c95 7b03092d01785f1286ade29f730855de901714890d37bf166c310721751fbe50 e2f9b42d9ec09cdc88f75b36af88b3b64e8ff72adf414005a3376cf1449e926b d78d38693f70eb568dc8cd2d378912f6e613af79bc31d99336284aaa588b23c1 aeab96ed0e061669087ae101ca79fff42af4878c5e2bdbb6a86a3d9408184a82 11e7d87e7c87398242b8f419600498f420e0c5822092c26cc2f89993c81c265f 550bac5b758b8a64883d89cc022439164819aca341a74887a9572fad3cafcb2c 212ad30bdeec3454fd9b2d06e6c6e2a3f6f8e4fe45acd91451204d106b80fd96 bc94c097c1af30d304ca8f94a07eb96daa44be599fb25d602adff69dfbe534f4 626eb4b8c84a146ca3c55cace73e17aa197b80a2f21c697f14c13cb17d8ad574 9cd5c70ce80122f8c12b72d47d9644667cec1b5c4637afa8906ae7e8375c58ac c2ae8ac19306cee4ce06bb08c1d8a1c8eb769739fd29b26d123e2faafde66126 964aca9ee4a0afc5a4dffd7544ef05cca15ee56eae2d061488b235cb59aa727c 8ae3a9c78333c5b463da7ed21fc0e97bef5d5eb5bc36bd29cd8eb6cbfe350c69 54eb9d9625cff024684e8366fba2bb246fc7a4ea7cfdf09ad04caa52e675e4fa 8d35ac3caba243bb6f797961e1c1d8809f622c9bb432c36588dbe2d275a0f87b c2eb51d5eed1a5647e954132fc8378b1f0fd5fb629afb91d8153c66ead56a4ed 9e16b9d5a3ea7a2ce330ea0a56259312cf30b2c10fc2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMFI10 %!PS-AdobeFont-1.1: CMFI10 1.0 %%CreationDate: 1991 Aug 18 17:48:10 % 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 (CMFI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -5.71 def /isFixedPitch false def end readonly def /FontName /CMFI10 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 109 /m put readonly def /FontBBox{-45 -361 1093 712}readonly def /UniqueID 5000776 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cdbafa8977dcfabee0cb28b63783883c7272b44 e2db6c00e664eeb0eb5f340f836d03733fa601f28dcaedfc92433862410a9922 7f823d20e4cc9ad6160b0ca5121e7691e8a11648bb25d15dd77ac1d0cee511c1 70cded7630398fecb3092e33060f3e8e09c7864125b12d4c6bb3970970c55e0a 750d503f3570b133f690b8c83496e8189057b248662f642ff77ef157a2eb6e24 36a713ad7c7a2307aa4b487f02baae2039fd1540550aa7e65f0d0474d9c1eb63 eddbc50af19b03a177206c70bf436aef88ae1e2459a514aebb9dc5110c6415b2 7047b209477f68b7bb54cda68f643773b660b15813c6d5113f337616bb5a5874 946e5e4c529f1f196238faea4b35589607c7c1eb3ae0cc6992d492d0acc8784c 33a9bf8a86801ec5ac8afc0ad50b8caa009059a7010718d5aa3241eb90d481c1 a8dd382668a43de2d5328508a222cf9261f6dda5470a5edce202b30139328469 435aa9b8df35770a2a2f6b8a9600f470a758428400012f7f8a4f91da9c33d360 0d71df42de4b49ef5dbf4112a9394d3c415fd08b58e820cb58031cb1ba577f81 d87f0d1088fe0f152de48fe72c1f2562aac0fba533ff915ba376a6b69ec68ddb 8e74ba0431f9c9b11f1b9d8a10e442ede19591ddbd45e7a963eb08f417eb76b6 92b7e32574558e826d483e7a5f7ecb590f82283bb8a9b5638794ee7813a2e6ed c0ce64a061f6ab3b24acecf9275f1908f28e739b709c2e97c69cd3a9ad293665 5388d6bd0c5be985467463cb4b29cddea31c5ab8e430d2e63c80d648918f1ceb 94e06e67f64b16634289edf3e35e8a6aa3dc26d685111102271bb84ac9aaf11d 6dba8aa6330aace6f629581ea1729737045cf7914dd7bad1354ad88a865beeee 645d071b46a4572c3042c19da57e9484454684a0e2e6b0548b81c733ba1d0ee1 e03187f3de87c00772a237a862e3676e58a4604b776c3edc14efe5e857d99186 155f7e96867856f04a376d5d97056e8a175402fd7e3a1dcc658c92812f683318 58b8d1dc4831593997385aec70a8fb44b24ffd794d3b0c2f69d38e7ac2b12b2e 7f09f00394b5390fabf149fadad4bee364a45c025d7b50e608962ca355ece52d d2fdc07c708f07b9867134788aa04e6b3de15a05f72f53e9554a29e86540040e 2fb4a1a60921d0ae3859907aaaf5a64042d8e9a9bd5e56e53526dd42ccc1720a 986e23fec3de818e4618c01bde1650882a47c4fdadfbe67eb0837e68b3d8911c 230c685ae46322198bf516c068cdd0b4504704b258ca2c5ac6f75cc904dfacfe 7317aeea60c8e0c820ca5bdf2cb6a75563199d1520ca883ac62efb6a3cf5a28e 9f1bc72939649ddd4a25ad063c6913dfea4f5a74594755f6a1427ffadfe246e9 d23f009c13e8a4b5f7490d5f6f50d800ff9ece50ce4835d122b94258ef470f97 78745975271f7761c56b161d11ead91676d87bfacf723149ac857c9e3b6765f1 c0f260 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 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd079edb34a91f29fb5cfe125a87634c8e13a562ba4dbee5c3167a51e916a7 b0f3b5c46b0497a23689d39ea181977092a0c321b40f3feb8dd74ef62930e937 77ec7ebf809e8dd9a379971a729e0dbe24b90e5fd72d105126dc499328e2bd53 dc2bd6d0b5d78300142fc50afa6fba77b53ecd1692ca33e81c263140ba263c0e 32137d3d2520f34f117fe6890d1b15b9b6742c1124471e615411283951ab0eb2 cd1ffeb6495bcaa83145d8bb9ac047699c6c958c401aa3173dcfce27e0adaaf1 5b3d7bd851ed82074f6afc08afbb678d126afab4b4ec6d15682f3a61a5668a6f c644d07f4bae227a2f5ded3819b06b5d89481e4752bce5602924b1942cffd887 cd572b06fbbb5dab17b6be3f740362515fd56d780c3c5b393580802b28e911ad 4a4d00690eacb875c41b905e6b1338b51ff0c049611092a612f712697a488a5e daef00eb0fc39370550dc6d7abbd3b69bef39c3b2a6af379262ffefc428a3bc3 ff63ce2b03d216e85ef7dbaf1050b8a7181f1d6deec35e39db62ec2b180e1d70 b4703622e86bd15a8fa76c5297484b495a5a50230ba4d81c12a508bb46a2ba48 e3e6a975b1e63d3f3699dd1bfe5571a7cae2515c0acadbb10841d2ea2a1480c6 62665d502c2d4c4246887fda123bfd26d5eac8c80b9815286e4333cdbdfe1614 a1609f3f1c83e0ad38d2b4962cff1c2511b0b4cd16008eb26365573e8cfbad61 79519b1127b758199a9bc27dfc38aad6003f1829366fc5ba22c380e253250fa6 38e6af5026c80c3e84da628c61e448f6a789a5959d892f1e9707feebc8da8cdd cb126280b9ffc304cf9e90199d3455f603c4bab74a0f5b4b528ba90d9c708fd8 ff6b2717d9b1a9d84f12f38132ce4cd26ee90fb8fe9b36c56aa082f6d7aa97f8 b61606fc5512af393466a1f3848f89be4ee5d739134ddc34b96a2700f704cef1 36f8452c23ba1c99c2b9ff973c4e6440add088950822a262d8a849abed56f521 6217b22faadec902a52d6639e0c42560c2e88d5ac4daae1e7ec23082a512e342 d23016ac2e8ab7a1b5782959439c4aceb578ecdb402e765e9249a8ccd28c8ca9 c4ca977ae0ae3e8dad0a50629676d7a9774cfab6bc334e3350d65178c993fb84 ba8a38ddd2e9e060b91f2d66ddcb60f4a5f0b93d24f49f0980806b2f23f10a7e 41100880178c64019744d9df78ce81b0309f6479724d0236e2a58c3f9ea280c1 8e818e3dfc1149c3b907487e471b3d262be1e99a881ed5a58e3e50f8ad22ff31 eee932603d7689c11180111b2040fb27f7333f698ceaef49f82f56ac0130bb61 09e35d09c45c797fa7491d979eb098c047942d5e0c494f54028807ee899d0eb1 abdaaf9921b3bfb44dc0287d8430e1f552c31f67cf5adbbdfb5564d4d5365e6c e76a25fc8bf8b01a76660be0328c43824e8f48a6478acbdb380b5516e1f5b7e0 710cd220095b1356fade3ee39dac11ac1191c33ebef86ec1a2ed505d499262bb b99cf9cbc7df843fb3fb88322ade45f12f463f5808ab9587a3fbb5904662136a 07606f89073396bf25e48974f26b427aa1a76826a8461b157b831a93a0d30d10 713f018da217b13e7209367f2c6fc31a47b2f402b4f786e852e6015930f767e0 98c015cadbb7799b8336dec738ee83a2f79b0923c56e985b1c8ebad2b96c51bb ec5c88d750829d237be8e6282f76b1e49e246dbefc7d0ced092659891ea23b1e e19a08da63ae2f0000abbfe3bd92ef0a1271dffb4804d5958a21e9273e9cda1d a2f0426bda1237dab320ec6d852c933434f6be43088f40c5864832770ae1aea7 70c4a4db1391c8be360f5e733dcc8297b4a96ecd8c72e4fcfa35fbdd654774ee 87304d78452fc1042d3515794b176925aad9c383fb4724a6077525df5b7c5b00 592600364187705ca7fe542d3c79bc84e9105f195bd39bb7b3f221e42237241c bd61ad74c15cfa736d7c9fbf6b9e292a39961ba00b364cc8fcbcf4dbdb8e4bdf abb0d57c7c7426d5946352dd3a2d26e081360ad80759f16dd63f2c883235ed4b 839e6adc5ead306b97c84f1f47af80aae01033dcc5a1e2d8d940ee2e17059e41 7c4b29720f6acdeaff832696867f9a581585db6818e9e15f0cb974e288b066b1 5500a40eb85c58897ffb2c3706b7b3ae879cb1effd60138701300dcd929d1966 0191ca6da299867d0381b86e541d7c4d0bdb956094882c5c9daa3ca34e17516f 1975b29d5bd8d1da29ef4d2c5b4d5af0d78fbf3c01f769837ba8008d5bb523 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msbm10 %!FontType1-1.0: msbm10 1.2/19-Jan-95 %%VMusage: 30885 32362 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm10) readonly def /FamilyName (msbm10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /CapHeight 683 def /Ascender 694 def /Descender -194 def end readonly def /FontName /msbm10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-56 -420 2332 920] readonly def /UniqueID 4702309 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb c799d90205a21291254fa633cc0c2ed88781ef21a5d45b72445c284fa44f8f39 cf566976075a92e9e2947a6fc93d879c29eb26809ecb4409cbc8666526cced92 65de661a2f8b0a16be45a9da17eeaf89b0e023de2b0373dcab997d60b7d7dc1b 4f7e650a2a9f13ca0543f3b080ae33d65f2802b8c5214d5b60d0d29560968999 7b955e43a4d99acff7bf2fbf2cc4d0f447c1d54fb3d42b7e474427b6dd55a053 08d91c3d62c926f6e6ae84ea2b0934aebc9f41694a4ec2254c5863450a4027a9 7b85652572e1d5de078b35903d1ba6b429da60c39e1ee8b63b34d4c322a15f4d febd46cddcb76b6803739385543a9dec631907147831ea9f0409662fc88ba080 d44dab2bf09232410b1b186408f7fe8cb979241e0833677023980fc352a35814 e765cc5e6cbeb302a629dff1dbf485f8e732ef95e0da866951f1dfeba772c4a4 4bf281611a018ffb9b3ec355d316e871c2a37c24b316267019ffb1177d45d7ea 3978be9296382b4dc59f9caf16e0b53829c1c594dde91c2e34137d29bd1df58c 7a70b1cb03ea2b84da2f3d4f3dd76fca795c125419963303b0eb5d8b8dbd625d 0d1488f1a238d9280760bac27b873ce7fe1b2dda9ddf428a42c9050635922291 d41c025385e37e21ec40811469871787cd8be85af60e312f91d3e1fe80845be0 d42c49c6490ae14269e7bfd313f31c9b0b0d4882db0d7addba36f29d510c102e d4594a4fc8cbc328c0710938db2a0ed84addcefebfac17c74e3ebde0f1386996 53f78f247493d6bf1ac356b4129202377a993a6ec1faed4735bd5e7cc5d2019c d4cd290d7d70635370a9b8337b7b655d9d5f0f995eca7b2bb2caa6e63d4bc1c5 5faa12c4e491d11c10c1310da685577f6293e512356d92e1c4cafc82f7c429ad 3dfbe56fc71602d48b7f718c765b7637f3133e8bb794fb94c5167b5cd35fba30 079243348b54e644642799cc334bc671e4e811983cfa2e5911a27b7bc5e520ef 03a767c09d985e869830505440ba3d3035f1344c3703d012f2bbc64df29300b3 b7aabbaa077f451033cebaabd8550cd1252cdba40a66904232a2a93c028a3949 6c8cd2f47a585c0b07a801fea3c3cb0cb502923e61e4a8042df4ab19138144f6 57a13558354263649d5b112ded46151c6356e97847c859b552503b6a12fd15c0 1a6d57df9fa4f19f695e8b33f1ef6981c556aaa3b9a3d1fad4d2e3ecd4be7743 5c9023c76d09baf41b4be962af65714c18e2f3e0d910977f36cc988d642b7b72 61608cef5b3a9213cf3bbb8ffd7f09756ec0fedb34f5bc19afe7b173e3d923f2 70ad959d9b8d440e811a0f7a24768732797ceec5e7ad8e1a8ad9ed6aeea3ffed 9dde03409147d79cbc5a288518c0c1cb2bc1decdc25d693f6c4a6351060c3c2e f11655682450544b6b71fa7a03720d81205d7bc8d7bd25d6a35a0fa2e8d7cbfb 952360145913755944b448faa88419a055a233bbc06a925ef0b90c88384d50da a22d2470160d398fb019875583c8926d6f070ef032013987ac40f31e9b52f8f9 cad2e821b6ef5a9d05acb895f3d3aa309896efa074063fd94e1a8a62b7841625 63a86df2580d5e50634a55919abc8846d8fd5e65a31e4b3eed0996111aa992fc ae684b2bc84df9ca4464ae4865b539fe2d225c821d29b51862208fab4b4f602b 208acde0aa4138 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /circlecopyrt put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e7424d938721814fd c89886b7a70ffd505fe91e1413a69870eda55016a759501582cb5cd256b5c768 d169b2c489ce0f4cec321537b4304cfeac5e643267bf31fca2da3b15f6dc5024 6967731ef35eadf6582eb9e586dfc47eac19033c0b53197cccea0df81b6ae054 c33f5fbbcf070b6dee6516846c19a5fc1df9999eea0a9aa42d741942b0369662 6d91f2d1171fc9af6eb791b40f5ce713f0d6eb554fc281e4c0863f816700cdc3 5d6ed7a0445c2cb76bc74e742533be8d27f512a871966d0da82106a792fee71d b7aaf91eca4bcf5e73f73fd168fe468ec4fcb1302f2452a5accf1e6c04da36ac d8e49cd6705da1d19c604679e5ceeb0da76aa3882e50b1ccf8d070efea84 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W 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 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36f6372968dbd0df8505aef5b22e57f2b9834fef2c2798f9d715aa323 3e09b183f8d92797bbd3a990980982018887612cdc5507a47581f07a1cf99a5c 587021d6b67b0524e780517e7a088a6388097ced01e5a59aa8779c6ad82af704 f6283c0a8b62d57cc5d24220e4d52a7c736389ff9d77df48c0de3d3db6482774 72a5e726bf4d83d29751b70eca9ee9d04a18fb7d011542a2b2737eff052a4eff 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f17efa3c92480fd44c50a6512a6f0b89c8d1b2793956b761bd9754dbe09b2784 453bb80c59ad1d2d05344a31bf1c1006a7990a9800468110 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 10 /Omega put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 88 /X put dup 89 /Y put dup 99 /c put dup 101 /e put dup 108 /l put dup 111 /o put dup 114 /r put dup 115 /s put dup 118 /v put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a 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9e9c858d65c779f71bddc19620f02321406a5e496b7a157c616d02c4f35d527c 6db2f2758f1b040f586b399e0694f41dbe27f9e95c68752b435c5987a180c3f0 ff3a72924a456a08f64b256d8b8023e477d55a702d7185720868cf67b652de75 2bb2287c9db4623020c4ae02e639416263ddcbd2b98551ab8e19ef87ef5daffd 525e9867d676ebe317db449f4f546d681377fde7414f4e43f8b9784099707d90 553f4f14c75207d0b5ef32e68751e7a63ad9e2417127d6a33b485c5f3744e672 b0a8b253dc189c16bfdee74c6b77f2bb69b618d9b3d32d661194285c4065f08d dd05109d923f0fcd9fb871048fa18402bb84be70fa765942a0b1c22fa16e928a e290de9029995bb652078165c6da5edf915441075bf3ccc5b9bebe4727d4a02f bbbeed2a3920d09b7da0a3b6125b86e07d50c6512c20da4cfe3de7ad67a6f265 84e12f59a234f1fb138a698b756cd9fd9be920dbf0f516a0d1d4b33a063d63ac 63ec8cfc7dce05470c79704866a052a688fcac2de8fcdd42e7efa4931c0c1233 39 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 0 /minus put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 92 /intersection put dup 106 /bar put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b877ca4bc5cd9c6c6dad91024aa3cd58a7ec276a45510cad80f10b9356 1e62c497ea0eea8e11387891398b3c063d850c9c655fdd2b7985026629ba6f33 bb860326eef1d0cdecf8ba2b07d1a332c60c36deda038f2a5a0bd40c5307437b 5a2855e30be321ff1ed39b7101de9db62c1e6f9fb133c7279f0eac28d217df10 1f041855bf1cf0c8b7b730a4282c7c322a64e12058a2b6933fcb3d0dfd881891 12809f2cb9c4afd2e167478b0f84282326bf3d077e704c222bc5c58dabdce759 59ceee73850cc435521a0e94c6fa6269ad9994b839d4f36e87b1b521b633cb5f 45ee9b080bdb888bbd65fde1ba2fd69b071fba4c695d84ef3f59598d4e270b0f fbe643ac5c88ea1edcdd548e538f1b1fdd6f42ea9460b66078ee5aee6b3e669b 5c110821b5eb078f2a377e0f15120610961f623b7e863a4582961a900685da18 591e3b925b15728fb25fae5eb4fe32b218e047c823d453c30d42f9acc93ec4c9 cce8c4f1a0463a349734005713efeed9d229c550bdd51a93854331e1368dc5d0 8327c8ad4e7939232be24d02a5c4c474f876f71d106c7cd87edb452fc5d95fd6 f129b57e1ce2bc2f84fa02a4aee013f443e6867a083c4b63854bb032493f7307 88cf39ae01ad7d3a8707efe2dbf85777e6e0181aae156d996dddfccd72f98426 814ff7d6c5537aeab8364f8a332e8dc0bad91348738501a8f04065efac68c96b c6c16394ed04f8bfab12fe1774f0ba59b83af166b8fa29ac16a051db8969d0d1 83912ccb326bb67e9459b1dd3134ede9cde5676f5f8b43d92f8ae1b3edeefe48 3730054757aa4fc15bcea6989a76573a05787389b19adc70bfd4752dab51c42f a87dcfc8d1862ba42f27427de985fe4bbd7bf3a7148e15548d694337d538dfcd 381e4d4399c2e8b8ed448d6a7bb0b97b1bedfd01a163ed482544b1e2bd7cc9fd 87491b602b609dafca43157d62626781030f52110929fe8c40ff444d03dd9f17 4850fb404befd2c9ad48c101632ae3775fa079a17f560ffc2240ce9d33e4df26 7c3c5c3f9f148b51341b01b5d7381e0e7400b4f9813f8d85992f332dbdba3a69 94f7eaf53cc7ad45989060ea82e9ec21e2eaaebfd5ca4f81c1299d942f662c43 8f9a49ee79cad018d2d8c2ad61d8b01c3b9a79b3c8a7f13327bc4975c0354c66 d20ae488236b2adcfc7c4e27d2970959bd8842ec2fed3d462e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 22 /mu put dup 23 /nu put dup 30 /phi put dup 59 /comma put dup 61 /slash put dup 66 /B put dup 68 /D put dup 69 /E put dup 78 /N put dup 82 /R put dup 86 /V put dup 100 /d 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 118 /v put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 a805e5f76578222b9c4a8a09b13189a16ab746ddef7875f1ecf83e568f493d82 ff729baa1e0834dbec30a35d3c49c9b10c5e1d90c6e3c8fb737df5ceb3299d96 0fb79632f91f257753b4d2e34e3f54a26c7b950981fb7fe4dee3315db63e75b0 24b4d318baec8aac9cde186f6d65767df1dba35124287c2e805bc660a3a22772 0ae4fa097f1d75107441c0a0fbb9eec17d5516fa03b98756b7df7ac7eff9c7a2 364b9f691ed5cb692dfff9dd80c761e9c4d5d6061782b327302c053ce1cd09ba 0605ebf761cf9b730524e1994e7dfaa956799d522ec53d5acba27a35c4d28279 617771938ece58c5ab664aa00c1bc4efba2d15504247a68e3af0f9bd59a12ac1 d8764421dfef5dfa13fbee58f5ecd4deeb4cda9fe9483c229ab29460b05559b0 35c73d6b88f6a384de0e2dfae4b1886f91e522d800b0ad40eae57f680490fea4 a2eda9f8f57aea1a4315674be45989c2293362dfbd0b5d90a523a7b79de04d6f 7b9ffa9f9ff2f2f7a1645e6b6e199c88c8430208a7df1f8f8839769a44b3922b 18294062582dfb8eb65d087635f1b602efbd5aa1d63676616822ac40947e58d4 49f3226143efd388f810db12077f5cbbd8d6cfee0ad3ef2426bdd29bc2cff5ef 5a3b32b1d15456d79148f6307272cbb1edba76f2294a969b03127d11e78060db 98fd83f046fb4492732b25da4971016bc4a1af351feee2a5874771eb28b0b836 68a4c40f2c487249ec5c41defb3934074cceb60e5bcf7db0e472bbc802663359 430b374c65b645f0421887afc3037e2d980429c832cc89e4b7d092ae6a8696c1 5aaee8469bcc6a718511e9b52e7ba0cc1213d8ca13f07e78e4d0c9b7ce43cdca 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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 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 72 /H 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 readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 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0000000000000000000000000000000000000000000000000000000000000000 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 11 /alpha put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 30 /phi put dup 32 /psi 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 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 100 /d put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 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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 1 /Delta put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft 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 124 /emdash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 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f701c6ea452c35ca22ad0f9516e72ba1e31cfc28385a35fbbd0a640d3ea2792a ac1226764ca2ba82e7bd7f3e4dbdfaedb191b957dceb9cca13b6bc87a441e3ab 9a5e282f96ec57cef081e87da73f74 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 44 /comma 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 65 /A put dup 73 /I put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T 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 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def 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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 11 /alpha put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 69718c0c5912d3167bb0971546d733e735aa9bcd89f5aa154656031ab3f6fe41 94ec225ad03ab8b67a9da43ab365c4df17c23e79f685c3b6fd5b654e4a6f2603 6cec3a8641b0b57f93eb95f9e238832aeac2061f3153e0f375513c974f1c38f7 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2eb1aff 52c190f50fc981dccb64bb2c1f7a62c216f211c162c2ed1c554ef49fbaf12b31 aa3b2a80fee259dc0a2a4242869ddec7c29e2941792b28bba7f2c6f77ef69638 12b07d6843f33b51e01c8626b26edc34e1e0fffadede5fb468e8cd5de7e5a64c 366967578f415484d122834de4a647541d66989f7d8b25d1712d8d16729bd9a6 472df1ef13518758e7ab8ce8429d80f39ebd0dd390edeed21a9eab589b6e1045 2c32712b2190d387402873fa0b41cf538ad4e86ff74473979177ce7700591842 0c62f82c5c8ec45492f7f7de2e5cd7bd3909a3b8bf6ee5dc1641615b09157d8a 1180258c8a066922c926801203d22be099bc7e9865dee9b0010afc293d153369 e98027c15707fe3a621efafd424f97c4519afdec25536310d4b885bd7557d3f2 8785cde769504fe16452822390f364f2105e20a7fdd7a701b8b9a3e3ca70aa39 ac4eff3f9f4b87fd5aaaf7edd6cfc0ab48f18240ea8d0c075a1fe0ab5537e8d5 379c48dfc1465aed57d914a8ff0f53872bd0912c36b7b9bbc90c74909815da4b decd69d9a5c3dc818bafb870272c0fcdd4d6c8a6181257e28614472fda8915ad 35bc69571b9368849d98d4bfc7e18cd8178844abb0cc74c8729bc8a57b5ef3db 5c498442c1e2192c4a85407d16950fbca05aa401ca0390fb99ff3e 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 12 /fi put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven 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 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 89 /Y 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 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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4e4dff6f54233d053794cc5f7fa2120938900eec6307f86962db58167c955be8 0772fa4f2be26ebc7752e163365c589a28ea3a65e64da65fd6742f1e42652591 8240c5445c3e5e2720c1d8ef9bcce0d9d722baf60d8245f7a4e175e466206c3b 53e801df70d8cbc75bd4e6d09570088f486e1409ecbdfa57b381d9f56cd2f48c 0a79ccaf05eebdbb016fb683c1146fbf414bbcbe49651b34d2bf6eda02363172 e9423d9457e4eb75eb02704135b746bcb4bf4bc61acedf8a925c2ffe29c03aab 974a5147264a35807a1f7a2db2de8eb39457ba2a3ab71fdef2e8b2cdde48cdcd bcb799adaf2e4920db47e581693f38f3b16471fbd8f09a5bf350a458c8bd336c dc1b5bfc4b17ba2b46b995bf27329bfdaf2c2352e25bb84f976b5b0f402a92b9 e8ff7150d7af4678c4714d7f366cc51c86fed4fdd7afa1aa82d3063c57ac8075 136e1731adcb931e72da2d6c6fa61b4d8ee7033dd98cf3d3208d0b1fb1ac5e2d 88045a4181d875d858afd09f3e100e8faaab84818e225f15f15f9d3448a98cfb 79b072d8781bb088935529d6781664afac01038d6023caaf6485968e2978025b 964d2de41c765ec1 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 65 /A put dup 69 /E put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e 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 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 44f6ae92facec57658999c8bf01f60611560043962c0695e1bb87d090e0fdaab 659d6557208a660be9aeef8ae55f71b8bc76394d792967e41bd8023b6e7f8c3d 9987d5e54c53bf2bbb32855b946473b44833bd78873df8e4c7e668ffff25aabb 8da4148f744b9d0524d6cb8281247b3052cc5d66f3f1bcf8a56a3fb446ea587b d8be30257652cee035628aeddfd8283b5ad21b8b1480ad24d898970ac8a1e33e 543ce4e3c48f9ee621c5ce3e8ae930f890011b630866c988606068d9e92396eb f51bb3db204505fc3d09d4701ac703a14d92bbfe27bd7411b363a88e33a8e9a8 025ac31f21e73dc52b333b918d0b1270c54873a490222d041a66e0192612be6d 2c88958a8f70c10390146ee5091df020ab97675e8e6b9e705fbe5d4a561453fc 99aacd038d0c9d92ce58c764a1a6cd497ec3cef52975ae84cc7e1d0401a7fcd4 d5e847133c58d736e373745cbfa06237a1ee4ff0706bcc02e316bfe85ec2f0b4 2dc3c3bb232c4045115c86f37c3bf3aa17b765020265aa22553707335d7fcd8d 9f619e844ae579bedcd8326b4e97b5ec744e172232ba7d77527bf062d8e3087c a2ca2acef377661d9105d7e8ed2ac5980ccc358a8f64ae5360003f6a947b68ae b82a2f42a76ba3d1c06e4cc296876dd4ecebbc3822267c34f82ee0ba50ba458f ed70df67c1bd89fb861967fc840f79ebae720c315045d8550bf37412cd11939f eb15fbda3f23581cdfb1784e05abb0d61d586d0ceb1550c8b72dc7779c793b59 623e7eb69ce8f31b7cca7d98b3bc8b7fe0b0d138d2920f6e1dd37ac059950e74 131f839392acde350927bb4a5c26ff92a7c355df28f54e5817c6cd286bf169b5 61f142e9edc8cd353b608d27deecc922b4ff4a1e35f166370a12ef619b7d9acc a77627eb60bd696a801e32de3190c3be838a53993ac54d89cc61ee92fc10be47 071e45ab9eb7fe1a06688d2f83e167731924c6f111c2552b700b7b67ad84b682 09753b1c746208b6f48295481734cf85ececbf34a6dcb1645f5aa9e4f00880cc c1c22cf9a00f6bcea9e3ecc93fcd610e61e71ba8e5ea343b5abd2db4754efc4a 4f3ccf582e3c66b8ecdf93300c3d166317faa2b538c9bd9c6ac4ecb784bd8ec7 a0c74e631c55951f1ce47716d0ade6588453a1963eee84e16ee5e105bf41366b ca29407b1826e4c98843a4877cd0df7962c509292dcc6354fb56272e61bfe380 1716f34a405032342035ac42d14e45b9257224d2aee20adb568ec794d3402fd7 e91d3b8a3d50e5027753681f3d041852de163ab5a9f468a8df0317592cabdbd1 49afbcf67126d91ae4bc75f04f99736b76f63e54d402b0e76aec3018db12ab5b 0eaf2dbfb0fe9b406461ba3a10590bd86c626128310cc268bdb7c418d9603c01 63290c62b18ff3a3c2040490551df4d79ccc6dacab2c81592fc5cf802655bbfc 9e24c4a313a65bcb0dcdc601b32da89fb909fa075c818e00d292be85b1924896 fff0879845589e294ffbe72dfe994020c707fc4f28feec63441c12c8bbc6bbd0 4536c41ddbe9e8d760784f4b1f75a777f691b980e39773491cff09aaf0b55aa1 759a81ba69d87f8229f2cccc12c5e0bd57eeceecb7145768889c6b8fee6e5198 a0d670edf0fabec135a93122cfebdba816613ca15022e28aaa91892d5112fdc9 f448a0443b72db94100449141a51529b348f1f5a49d5a65a3b095d25cce5acc1 f0dee87a9b23fa48605b5ef27c1b0d8150c5d6d0a13cfa5f8cdbeae40d7f8156 fb9e17a55d8e9e832b314ef82776106a61fa2702467a57ef07df7b67fd9872a6 9b7bb687c362f259115c7e3f10b0b4bb0ad74d8e16c7b08b3a78bddfabe31446 560f2ac2a3a91af8d5f34f071f323fc43c6c6faed92fa20b21c9767a647b4805 c501c455249fa471b380865f6b436b4de15d6e10a4ca6dd06206bf2e48bb3526 a8ade5ec7b7a574067e159249e0cf9378cb7db3197e246af8b7db3feec557b12 9bd5e4ad260c0e019880eecf348365c61727559425bac901972bdecad0f3959e 32fed946ee74f46ff1bf2421d828030d4d025c03e94e9295047d8cbb10e58251 0198737682ae6ca07a0a0862218a711abf1ac34479b4d1c517506ad9ace694dd a08585df000c6d8b8822d5691b37e64fb28259c98aa8a4f72ec186a75940e378 c0139b5d87f6972858586600943d72f233eaa6935c404f4473e6f8ff661f8adb 2f51588940ac984834b0c6e4fdb4fab408c16700e97057758da517abd60d007c e0998911718328102f9e7b9942cb83afcdedde0ca7dbebd3fa6eaf20a0932b00 5c69b5cbf7b5f53aee3e4d5edf567b8d39f1dfd8d0594e3bff33db695c2d5cba 8aca950e74aa220f05f024d19af52a6c587c3ec805a959b2b563706b52316844 227b9ad76d42495939b412453bbd6fe6311e6325b5c4285cdce2d4a8820ef857 893dbd1a38ed46ed62ed2d408aec92e3db4e0bbe0f26be143d943ec2b5ff8d5d 16514e92da6c84564cea481f5b45fb4194850602a280d5d02ff237a13c97bce0 2ffbd7e92d9032d15da1e7f9ebb9f9e44fd387b2e976c85a22ca72342c5c5d87 ad082630b578e5f327603f9c241696f6591642a6c170536df1ec8fb89b4b5a04 cef677215670fc14c0d31e6d8e79115fb00a899aad725e7f47ce5236836b1098 0fc04b4ac074490f666eaf9ba1bf8f92951043f3a6cd1601c407942abc57d38d d9f7e58966dc1e3b757aef625bf757777749c0978c709524ff9fc821b52cb251 ff5e242eae9cb9273948dfba13edb234aa48a572b89e4773e761208556c3ee5a 87f7efba1cf6aa351012affd866dcd6e63e159d5e6cceed3202156dab281574d 8f7d063be069331a29b956b177b10d61add3a9d0a655c710925e5cc7219df3f3 3b9eb05e877c0c2cac02a49dca448f30d5544f2b4d8adfa1bed4ec15dbbf9b53 56db5af5489c26ee951637a63960b641f764b7b13c9de7e2c854dd5c687fef85 bca1b06dab29412ba62a8a4e8ff7334b57187d6250d1c2b7110b7d90c4084d2e 03c42de1298e1111cd2f3ee507fa999bf109d396424042ec8036f1d8b8733074 62d74b6fd6bc9f2e2f7104ecda1ac6a6e2fd2af610d2d42c32e8a29aa463250c 081234f198a61bb5e02ecf4fbfc62c46b7ae25a7dda9d9d4b880981d308659e6 c762505d843850af63f84e3c533f8f25bae874b7b7b8c48ba81d645d830fbecd f025fbab100dcb2ad675c3ec58c387c9eb2ecd2dfb6ce4a670f876728cfeb309 744ce0bdd36e53c5c542aece4ad00bde7daaf58f3868a3e3a28490f08503f8b6 b97f69511f83bd2da1bc93aa5539755bc12271ba143c7e09967de3193248cecf 60327b654f02f8dfca8933a29c53b01938d0969ea49ff772208028ada97f0c3b 46b6ee4512bd17d96beda86c8f06d5202be34d425e1dae069f69caae905fe91d bdf7cc4965c30e56b596a00e110d526512f6e61d296046e23d427b9e41f6811d 4de270f019466b08d9b7972762a164eb284517f06bf4c99057683c7a77239883 353270e55e7bfeb2c6075826b82aa5cf0a6d0ef976ed8bb7ac73cc993a97291b 9d78db12d02b3cac0ed16f7676c8b3b353a429f54b5915c22ddce7fd4c3dac13 0c9bacedb86747a4229740370b9f71f47c1360e00aa95989d5558c24618c771a a71b00daebe53546aecb99ed0d3a4e871f693052a635e1650cf7e2752aaa34e0 413217cbae46e3c8de045d42d1484c3515cf7b87a4a3471d9238575eb32a7eee b401720be3d9a0f157ffd64069b4ef665237a217ef8b059fc331bd2b15c6aa10 6f9b619bf8d4ccbb09768316dec9bc8051d4e0ab63f245b028875d0924b2a0c6 e61580cf786421ae140e28c3edab01257544aa5792111f08417ae9a88ccab840 101499982dc47e39f729a5e105de11b306b34c0cbcab8df7f9753ade94ec2df2 4b485fc557b3d9822bc47b0429919faf648ec2c404c8bfdfdab58bfae2fa05bb 0a6b4cde132028faa130d4018936f330ad619279c8a4456055fd4ebb775d78a6 4c45fe16a1bbe1a30c852ed211269ac723cb5dbd190a24479519c8c9c0f0e2e2 869b97afd4192591ea92b34b6bc9590b4d13da0e38f566e5f4784cabcb2cc245 0af27e451194dd08c6f6eb68ce330b45ecf8046d6a081c25cba2d2a3fc72ee90 9cd4a4ce96321d02126a057f2a24b3fdec1a16e02bab25acf50d4dac82f1077d a5e46d4c9325ab819a4809ba912240e5bb6e2c3d892c92213f4e42194155e0b5 1229339d76cc5050b024d57abd38333fabb3ebf7c447e513a386b2f60d7f9477 52ac093d7d367d4617dc60d346f2073bdcfde6cff8f7757257c9f8ddcd1604ef 3fdd77e2fd1605145ec95f49931ed89bfa4043e92af9a92304f918763b7eeefe 8bb805138045a412cb07026b37385c0a797d46a51d431dbf6c1789f198652424 4d7ed9ea92ecf0b5f1bef77a95210e59b5e4e96af5fece6ce19450b824e8f379 135c5afec294def3b575d0c4aff3430c1ca1513ab5f8d52710870db798dc8e5a 04fb15a4d195ba1f4ed71be078ad30a10120cea5b5d1dc5d8ce8fd7f0e35315b 7f7ad24b5410e5c94c165d56c5c0995663b7808847476e40a14212552e7564bb ee8736c6b982fa5da416848d3ad1f3c30be8d7bac23c5b2c5a3528a4e76303fc a6168b07c431f4f1cb926ad38d2743b39fd70d4b6a9f6e3accc6fa6447b532bd 1f475d83fbed4cb422929def7ad5c938a0dbafbd8e9d0c5c98bac1f7a232c511 4a797fde422af4c153f2f8b6cb060df6c7d624876c80e6b4f79ef5eec609d5a3 cdc0eaa2c0f366e1af309e82d9773778c07aedb7652ba57c4a1d3d0b3945a015 30b2039746e1878a79f6d6d3b293481eac1f072192058589ae4c7b1b15191b23 a3de4689d9685db61e1114d2047e0e8f171dcf894421dfebebb5a295ecc1b1eb c64f0ed5448c13266052457b81780e69af4fd6adf93b10d10dd3c649e092acaa 833c371869761ef52cd4cdd10668e58b52427f0fccb46a8e7428c400814f538c 83f8c9275d8fb2107e828aaf4516d82c3a52871d33da02f9e8332855cd9d355f 98d3cc22b0599d031593255ee10cb167da173a28db3383609656b86a1bddd096 2fc8d0f0a19729b5cdfaab447a9ecb4c74b2f164ce07e5f57642d1f18954fb34 126d97f5c06d7293647bbe946337df7a7125e0d898278f8f305fb4a8d1de0a68 991d28bae32f24b6d89abc01cb9376fc8893d9b59e571fe13cd89e1690f52d79 516453f8871e8a65c19c69a2cccc7a1cd37bde83b9f5f45aea8cfc48cfdb79ea a29dde95298e98ab7d05bb13f05cc5ae83b47227e4416a2e5ee9619afc24286b 7c4c6686834e20beb67284b400f3988b28a189cfaaf5b7015b4c63ecbda1df07 18a351bfa72f5d5cd62d10ec1be1c13ae898f432de8d0c13955c3b8971837c00 59c70d1f3322c07dfcbbf0a9013d43af47ff46876707a19b 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 66 /B put dup 80 /P put dup 83 /S put dup 85 /U put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l 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 121 /y put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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da36f35ef2e6ef79b1981d6ef6250b7708e07716d13361a802c47feda41132a3 456db4ce9ff069cf82dc46251995f6ac47cb304d36d6b458b899453415d07b1d 5fe796d13a5e4f6a2ae73acdae9d63737c2805290ceb8897464607710bc0dd3b 981d883b41d9be6c0d4bef7b14825c0803c817624a59dd060ebcfb7f9307aeb1 514937358cbf1c9816a23545df46c7eb32f61f3494ea23e13fcbfed461f6b1b8 9f9fb7b2cc0d7c7caf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 300 300 (cklf.dvi) @start /Fa 134[18 2[18 18 18 18 18 2[18 1[18 18 18 1[18 18 18 1[18 18 18 18 18 32[18 17[18 46[{}20 33.2088 /CMTT8 rf /Fb 198[20 20 20 20 20 20 20 20 20 20 48[{}10 33.2088 /CMBX8 rf /Fc 205[14 14 49[{}2 20.7555 /CMR5 rf /Fd 173[21 51[17 30[{}2 20.7555 /CMMI5 rf /Fe 233[17 22[{}1 41.511 /msam10 rf /Ff 128[21 4[17 20 19 28 19 22 14 17 18 19 21 21 23 34 11 19 13 13 21 19 13 19 21 19 19 21 10[31 1[30 23 30 1[28 32 31 1[26 2[16 2[27 28 31 1[29 31 18[13 15 13 4[13 26[23 12[{}46 41.511 /CMTI10 rf /Fg 139[25 1[25 2[26 25 31 3[13 25 1[22 23 16[33 15[34 21[13 46[{}12 41.511 /CMCSC10 rf /Fh 146[44 109[{}1 49.8132 /CMFI10 rf /Fi 165[23 1[60 5[20 40[33 33 18[22 22 7[14 8[17 17 2[{}10 41.511 /CMEX10 rf /Fj 173[30 82[{}1 41.511 /msbm10 rf /Fk 242[30 13[{}1 24.9066 /CMSY6 rf /Fl 128[18 4[15 17 1[24 16 19 12 15 15 16 18 18 20 29 9 16 11 11 18 16 11 16 18 16 16 18 9[36 26 1[26 20 1[27 24 27 1[32 22 1[19 14 26 28 23 24 27 26 25 26 6[11 11[11 13 11 31[20 22 11[{}51 33.2088 /CMTI8 rf /Fm 137[17 2[13 13 2[17 2[9 6[15 1[15 9[25 25 1[25 1[24 18 24 1[22 25 25 30 21 25 1[12 25 26 1[22 25 24 1[25 3[25 3[17 17 17 17 17 17 17 17 17 17 1[9 11 9 25 32[24 10[{}42 29.0577 /CMR7 rf /Fn 149[10 13[22 41[22 33 15[33 32[26{}6 29.0577 /CMSY7 rf /Fo 137[17 7[21 29 10 18 14 12 4[17 13[20 3[25 3[26 8[24 27 1[25 4[17 1[10 28[20 6[17 20 6[14 15 17 1[22 11[{}23 29.0577 /CMMI7 rf /Fp 145[21 2[21 12 16 16 21 21 29[35 15[23 1[0 3[28 42 15[42 6[32 4[32 32 2[32 14[32 12 32{}20 41.511 /CMSY10 rf /Fq 133[19 20 24 1[20 24 6[25 36 1[22 1[14 24 1[20 1[22 12[39 24 28 1[25 32 1[27 1[33 1[28 2[18 35 2[31 34 30 31 31 22 1[32 21 32 12 12 25[27 1[25 2[24 3[21 25 4[21 1[17 18 2[27 11[{}42 41.511 /CMMI10 rf /Fr 128[21 21 1[42 1[18 22 22 30 22 23 16 16 16 22 23 21 23 35 12 22 13 12 23 21 13 18 23 18 23 21 3[12 21 12 3[43 31 31 30 23 31 1[28 32 31 38 26 32 21 15 31 33 27 28 32 30 29 31 3[32 2[12 21 21 21 21 21 21 21 21 21 21 1[12 14 12 32 1[16 16 12 4[21 19[35 1[23 24 30 8[35 1[{}79 41.511 /CMR10 rf /Fs 138[27 19 19 20 25 27 24 27 40 13 2[13 27 24 15 22 27 21 1[23 12[33 27 36 1[33 6[18 7[36 7[24 24 24 24 24 24 24 24 24 24 1[13 1[13 44[{}36 41.511 /CMBX10 rf /Ft 244[23 11[{}1 33.2088 /CMMI8 rf /Fu 128[18 3[18 16 19 19 25 19 20 14 14 14 19 20 18 20 29 10 19 11 10 20 18 11 16 20 16 20 18 3[10 1[10 1[26 1[36 26 1[25 20 26 27 24 27 26 32 22 27 18 13 1[28 23 24 27 25 25 26 5[10 10 18 18 18 18 18 18 18 18 18 18 1[10 12 10 2[14 14 10 24[29 1[20 12[{}72 33.2088 /CMR8 rf /Fv 134[20 1[28 20 20 20 15 20 1[19 21 20 25 17 21 1[10 20 21 18 19 21 20 19 20 11[27 26 20 2[25 2[33 23 1[19 13 27 3[28 26 26 27 7[18 1[18 18 18 18 18 18 18 18 1[11 13 11 44[{}47 33.2088 /CMCSC10 rf /Fw 137[24 1[18 18 18 3[25 1[13 2[13 25 2[20 25 2[23 7[34 11[42 28 35 7[33 1[34 18[13 1[13 44[{}19 45.4545 /CMR10 rf /Fx 134[30 30 1[30 1[22 22 23 1[31 28 31 47 16 2[16 31 28 1[26 31 25 1[27 12[39 31 13[37 3[42 65[{}22 49.8132 /CMBX12 rf /Fy 134[16 4[12 12 12 1[17 15 17 1[9 16 1[9 17 15 1[14 17 14 13[22 1[17 2[20 13[21 17[15 6[12 12 27[18 12[{}23 24.9066 /CMR6 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 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y Fq(\017)487 2190 y Fm(1)p Fn(\000)p Fo(\013)553 2205 y Fr(Im)g Fh(m)654 2211 y Fo(\026)676 2205 y Fr(\()p Fq(E)15 b Fr(+)e Fq(i\017)p Fr(\))20 b(=)h Fp(1)p Fr(,)e(and)g (similarly)e Fq(D)1272 2190 y Fo(\013)1271 2215 y(\026)1296 2205 y Fr(\()p Fq(E)r Fr(\))j(=)g(0)f(if)f(and)h(only)f(if)228 2259 y(lim)7 b(sup)355 2269 y Fo(\017)p Fn(!)p Fm(0)427 2259 y Fq(\017)444 2244 y Fm(1)p Fn(\000)p Fo(\013)510 2259 y Fr(Im)g Fh(m)611 2265 y Fo(\026)633 2259 y Fr(\()p Fq(E)14 b Fr(+)f Fq(i\017)p Fr(\))19 b(=)g(0.)30 b(Borel)18 b(transforms)h(of)e(sp)q(ectral)i(measures)h(arise)228 2309 y(v)o(ery)d(often)f(in)g(sp)q(ectral)i(theory)m(,)f(since)g(they)g (coincide)g(with)f(resolv)o(en)o(t)h(matrix)f(elemen)o(ts.)228 2359 y(Let)11 b Fh(m)343 2365 y Fo(\036)375 2359 y Fr(denote)g Fh(m)550 2365 y Fo(\026)582 2359 y Fr(for)f Fq(\026)i Fr(=)f Fq(\026)747 2344 y Fo(\036)769 2359 y Fr(,)g(then)g(b)o(y)f(the) h(sp)q(ectral)h(theorem)f([)p Fs(24)o Fr(],)f(w)o(e)h(ha)o(v)o(e)f Fh(m)1603 2365 y Fo(\036)1625 2359 y Fr(\()p Fq(z)r Fr(\))i(=)228 2409 y Fp(h)p Fr(\()p Fq(A)d Fp(\000)h Fq(z)r Fr(\))379 2393 y Fn(\000)p Fm(1)424 2409 y Fq(\036;)d(\036)p Fp(i)p Fr(.)303 2458 y(Our)14 b(\014rst)g(\(and,)f(in)g(a)g(sense,)i(main\))e (result)h(in)f(this)h(pap)q(er)g(sho)o(ws)g(that)f(for)g(energies)i Fq(E)228 2508 y Fr(in)g(the)h(essen)o(tial)g(supp)q(ort)h(of)e(the)h (sp)q(ectral)h(measure)f Fq(\026)1142 2493 y Fo(\036)1164 2508 y Fr(,)g(Im)6 b Fh(m)1292 2514 y Fo(\036)1314 2508 y Fr(\()p Fq(E)13 b Fr(+)e Fq(i\017)p Fr(\))16 b(can)f(b)q(e)i(essen-) 228 2558 y(tially)10 b(determined)k(b)o(y)e(supremizing)g(the)h(scalar) g(pro)q(duct)g Fp(h)p Fq( )q(;)7 b(\036)p Fp(i)12 b Fr(using)g(v)o (ectors)i Fq( )f Fr(that)g(are)228 2608 y(appro)o(ximate)g(eigen)o(v)o (ectors)i(\\up)f(to)f(an)h(error)h Fq(\017)p Fr(.")p eop %%Page: 3 3 3 2 bop 488 113 a Fm(APPR)o(O)o(XIMA)m(TE)15 b(EIGENVECTORS)f(AND)g (SPECTRAL)g(THEOR)m(Y)244 b(3)303 213 y Fg(Theorem)16 b Fr(1.1)p Fg(.)k Ff(L)n(et)15 b Fq(E)j Ff(b)n(e)d(such)h(that)945 179 y Fi(R)972 213 y Fr(\()p Fq(x)10 b Fp(\000)g Fq(E)r Fr(\))1113 198 y Fn(\000)p Fm(4)1165 213 y Fq(d\026)1212 198 y Fo(\036)1233 213 y Fr(\()p Fq(x)p Fr(\))j(=)g Fp(1)p Ff(,)i(then)g(for)g(any)i Fr(0)12 b Fq(<)228 262 y(\017)f(<)h Fp(k)p Fr(\()p Fq(A)d Fp(\000)h Fq(E)r Fr(\))p Fq(\036)p Fp(k)p Fq(=)p Fp(k)p Fq(\036)p Fp(k)p Ff(,)364 331 y Fq(\017)d Fr(Im)g Fh(m)489 337 y Fo(\036)511 331 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)p Fr(\))18 b Fq(<)h Fr(sup)q Fp(fjh)p Fq( )q(;)7 b(\036)p Fp(ij)939 314 y Fm(2)968 331 y Fp(j)k(k)p Fq( )q Fp(k)g Fr(=)h(1)p Fq(;)7 b Fp(k)p Fr(\()p Fq(A)h Fp(\000)i Fq(E)r Fr(\))p Fq( )q Fp(k)i Fr(=)g Fq(\017)p Fp(g)-1238 b Fr(\(1.2\))451 396 y Fp(\024)19 b Fr(sup)p Fp(fjh)p Fq( )q(;)7 b(\036)p Fp(ij)713 378 y Fm(2)742 396 y Fp(j)k(k)p Fq( )q Fp(k)h Fr(=)g(1)p Fq(;)7 b Fp(k)p Fr(\()p Fq(A)h Fp(\000)i Fq(E)r Fr(\))p Fq( )q Fp(k)i(\024)f Fq(\017)p Fp(g)18 b(\024)h Fr(2)p Fq(\017)7 b Fr(Im)f Fh(m)1454 402 y Fo(\036)1476 396 y Fr(\()p Fq(E)12 b Fr(+)d Fq(i\017)p Fr(\))p Fq(:)303 489 y Ff(R)n(emarks.)18 b Fr(1.)g(The)c(middle)f(inequalit)o(y)g(is,)g(of)h(course,)g(trivial.) 303 539 y(2.)i(The)c(theorem)g(is)f(empt)o(y)g(if)g Fq(\036)g Fr(happ)q(ens)h(to)f(b)q(e)h(an)f(eigen)o(v)o(ector)h(of)e Fq(A)i Fr(with)f(eigen)o(v)n(alue)228 589 y Fq(E)19 b Fr(\(in)e(whic)o(h)g(case)h Fp(k)p Fr(\()p Fq(A)11 b Fp(\000)h Fq(E)r Fr(\))p Fq(\036)p Fp(k)k Fr(=)i(0\).)27 b(Throughout)17 b(the)h(rest)g(of)f(this)g(pap)q(er)h(w)o(e)f(often)228 639 y(assume)h(that)g(this)g(is)g(not)g(the)g(case,)i(and)d(w)o(e)h(do) g(not)g(b)q(other)g(sa)o(ying)g(so)g(o)o(v)o(er)f(and)h(o)o(v)o(er)228 689 y(again.)f(\(This)d(degenerate)h(case)g(is)f(easy)g(to)g(treat)g (separately)m(.\))303 782 y Fg(Definition)h Fr(1.2)p Fg(.)21 b Fr(Let)15 b Fp(f)p Fq( )748 788 y Fo(n)770 782 y Fp(g)791 767 y Fn(1)791 793 y Fo(n)p Fm(=1)868 782 y Fp(\032)e(H)h Fr(with)h Fp(k)p Fq( )1106 788 y Fo(n)1128 782 y Fp(k)d Fr(=)h(1)25 b Fp(8)p Fq(n)p Fr(.)20 b(W)m(e)14 b(sa)o(y)g(that)g Fp(f)p Fq( )1614 788 y Fo(n)1637 782 y Fp(g)1658 767 y Fn(1)1658 793 y Fo(n)p Fm(=1)228 832 y Fr(is)g(a)f(SO)o(AEV)i(\(whic)o(h)f(stands)h(for)f(Sequence)i(Of) e(Appro)o(ximate)f(EigenV)m(ectors\))i(for)f Fq(A)g Fr(and)228 882 y Fq(E)i Fr(if)d Fp(k)p Fr(\()p Fq(A)c Fp(\000)g Fq(E)r Fr(\))p Fq( )507 888 y Fo(n)530 882 y Fp(k)j(!)f Fr(0)i(as)h Fq(n)e Fp(!)f(1)p Fr(.)228 957 y(As)j(p)q(oin)o(ted)g(out)g (ab)q(o)o(v)o(e,)f(the)h(sp)q(ectrum)i(of)d Fq(A)h Fr(is)g(w)o(ell)f (kno)o(wn)g(to)h(b)q(e)h(giv)o(en)e(b)o(y)521 1026 y Fq(\033)q Fr(\()p Fq(A)p Fr(\))f(=)g Fp(f)p Fq(E)h Fp(j)e Fr(there)16 b(exists)e(a)g(SO)o(AEV)g(for)g Fq(A)g Fr(and)g Fq(E)r Fp(g)p Fq(;)-1195 b Fr(\(1.3\))228 1094 y(and)14 b(moreo)o(v)o(er,)g(the)h(essen)o(tial)f(sp)q(ectrum)i(of)e Fq(A)g Fr(coincides)h(with)f(the)h(set)g(of)f(energies)h Fq(E)h Fr(for)228 1144 y(whic)o(h)11 b(there)h(exist)g(SO)o(AEV's)g (whic)o(h)f(con)o(v)o(erge)h(w)o(eakly)e(to)h(zero.)18 b(SO)o(AEV's)12 b(that)f(con)o(v)o(erge)228 1194 y(w)o(eakly)h(to)h (zero)h(are)g(often)f(called)f(W)m(eyl)h(sequences.)20 b(In)13 b(general,)g(a)g(SO)o(AEV)g(need)h(not)f(sa)o(y)228 1243 y(an)o(ything)e(ab)q(out)h(con)o(tin)o(uit)o(y)f(prop)q(erties)j (of)d(sp)q(ectral)j(measures)f(of)e Fq(A)i Fr(or)f(ab)q(out)g(the)g (nature)228 1293 y(of)i(its)h(eigenfunctions.)22 b(W)m(e)14 b(are)i(in)o(terested)g(in)f(isolating)e(t)o(yp)q(es)j(of)e(SO)o(AEV's) i(that)f(w)o(ould)228 1343 y(encapsulate)g(this)f(t)o(yp)q(e)g(of)f (information.)j(T)m(o)e(this)g(end,)f(w)o(e)i(de\014ne)303 1418 y Fg(Definition)g Fr(1.3)p Fg(.)21 b Fr(Let)11 b Fp(f)p Fq( )744 1424 y Fo(n)766 1418 y Fp(g)787 1403 y Fn(1)787 1429 y Fo(n)p Fm(=1)862 1418 y Fr(b)q(e)g(a)g(SO)o(AEV)g (for)f Fq(A)h Fr(and)f Fq(E)r Fr(.)17 b(W)m(e)10 b(sa)o(y)h(that)f Fp(f)p Fq( )1614 1424 y Fo(n)1637 1418 y Fp(g)1658 1403 y Fn(1)1658 1429 y Fo(n)p Fm(=1)228 1468 y Fr(is)k(ro)q(oted)g(at)g(a)f (v)o(ector)i Fq(\036)c Fp(2)g(H)p Fr(,)j(if)746 1561 y(lim)7 b(inf)760 1586 y Fo(n)p Fn(!1)909 1533 y Fp(jh)p Fq( )964 1539 y Fo(n)986 1533 y Fq(;)g(\036)p Fp(ij)1058 1518 y Fm(2)p 873 1551 239 2 v 873 1589 a Fp(k)p Fr(\()p Fq(A)i Fp(\000)h Fq(E)r Fr(\))p Fq( )1068 1595 y Fo(n)1091 1589 y Fp(k)1128 1561 y Fq(>)i Fr(0)p Fq(:)-977 b Fr(\(1.4\))228 1658 y(W)m(e)13 b(also)h(de\014ne)303 1733 y Fg(Definition)h Fr(1.4)p Fg(.)21 b Fr(Let)13 b Fp(f)p Fq( )746 1739 y Fo(n)769 1733 y Fp(g)790 1718 y Fn(1)790 1744 y Fo(n)p Fm(=1)868 1733 y Fr(b)q(e)h(a)f(SO)o(AEV)h(for)f Fq(A)h Fr(and)f Fq(E)r Fr(,)g(let)g Fq(\036)f Fp(2)f(H)p Fr(,)i(and)h(let)f Fq(\026)228 1783 y Fr(b)q(e)i(a)f(\014nite)g(p)q(ositiv)o(e)g(Borel)h (measure)g(on)f Fj(R)p Fr(.)20 b(W)m(e)14 b(sa)o(y)g(that)g Fp(f)p Fq( )1255 1789 y Fo(n)1278 1783 y Fp(g)1299 1768 y Fn(1)1299 1794 y Fo(n)p Fm(=1)1377 1783 y Fr(is)h(optimally)d(ro)q (oted)228 1833 y(at)i Fq(\036)f Fr(with)h(resp)q(ect)i(to)e Fq(\026)p Fr(,)f(if)445 1926 y(lim)7 b(inf)459 1950 y Fo(n)p Fn(!1)909 1898 y Fp(jh)p Fq( )964 1904 y Fo(n)986 1898 y Fq(;)g(\036)p Fp(ij)1058 1882 y Fm(2)p 572 1916 841 2 v 572 1954 a Fp(k)p Fr(\()p Fq(A)i Fp(\000)h Fq(E)r Fr(\))p Fq( )767 1960 y Fo(n)789 1954 y Fp(k)d Fr(\(1)i(+)h(Im)c Fh(m)1005 1960 y Fo(\026)1028 1954 y Fr(\()p Fq(E)11 b Fr(+)f Fq(i)p Fp(k)p Fr(\()p Fq(A)f Fp(\000)h Fq(E)r Fr(\))p Fq( )1337 1960 y Fo(n)1359 1954 y Fp(k)p Fr(\)\))1429 1926 y Fq(>)i Fr(0)p Fq(:)-1278 b Fr(\(1.5\))228 2043 y Ff(R)n(emark.)18 b Fr(Note)13 b(that)f(an)g(optimally)f(ro)q(oted)h (SO)o(AEV)h(m)o(ust)g(also)e(b)q(e)i(ro)q(oted,)g(and)f(that)h(an)o(y) 228 2093 y(ro)q(oted)e(SO)o(AEV)f(is)g(also)g(optimally)e(ro)q(oted)j (if)e(lim)e(sup)1097 2103 y Fo(\017)p Fn(!)14 b Fm(0)1183 2093 y Fr(Im)7 b Fh(m)1284 2099 y Fo(\026)1306 2093 y Fr(\()p Fq(E)t Fr(+)r Fq(i\017)p Fr(\))12 b Fq(<)g Fp(1)p Fr(.)k(The)11 b(con-)228 2142 y(cept)h(of)e(optimal)f(ro)q(oting)h(is)g (th)o(us)h(of)f(in)o(terest)i(only)e(for)g Fq(E)r Fr('s)h(where)h(lim)7 b(sup)1429 2153 y Fo(\017)p Fn(!)14 b Fm(0)1515 2142 y Fr(Im)7 b Fh(m)1616 2148 y Fo(\026)1638 2142 y Fr(\()p Fq(E)e Fr(+)228 2192 y Fq(i\017)p Fr(\))12 b(=)f Fp(1)p Fr(,)i(namely)m(,)g(on)g(the)i(essen)o(tial)f(supp)q(ort)h(of)e(the)i (singular)e(part)h(of)f Fq(\026)p Fr(.)303 2242 y(Our)g(next)g(theorem) h(is)f(a)f(fairly)g(immediate)g(consequence)j(of)d(Theorem)i(1.1.)i(It) d(relates)228 2292 y(lo)q(cal)f(scaling)h(b)q(eha)o(vior)h(of)f(sp)q (ectral)h(measures)h(to)e(the)i(existence)g(of)e(appropriate)g(t)o(yp)q (es)i(of)228 2342 y(SO)o(AEV's.)303 2417 y Fg(Theorem)h Fr(1.5)p Fg(.)k Ff(L)n(et)15 b Fq(\036)c Fp(2)g(H)p Ff(.)19 b(Then)228 2467 y Fr(\(i\))14 b Ff(The)h(fol)r(lowing)f(thr)n(e)n(e)h (sets)f(c)n(oincide)p Fr(:)694 2502 y Fi(\()728 2572 y Fq(E)772 2500 y Fi(\014)772 2525 y(\014)772 2550 y(\014)772 2575 y(\014)772 2599 y(\014)805 2572 y Fr(lim)7 b(sup)837 2606 y Fo(\017)p Fn(!)p Fm(0)944 2544 y Fq(\026)969 2529 y Fo(\036)991 2544 y Fr(\()p Fq(I)1025 2550 y Fo(\017)1041 2544 y Fr(\()p Fq(E)r Fr(\)\))p 944 2563 179 2 v 1014 2601 a(2)p Fq(\017)1139 2572 y(>)12 b Fr(0)1204 2502 y Fi(\))1244 2572 y Fq(;)p eop %%Page: 4 4 4 3 bop 228 113 a Fm(4)407 b(M.)15 b(CHRIST,)f(A.)h(KISELEV,)g(AND)e (Y.)i(LAST)409 213 y Fp(f)p Fq(E)e Fp(j)e Ff(ther)n(e)k(exists)g(a)g (SO)o(AEV)f(for)h Fq(A)g Ff(and)g Fq(E)i Ff(which)e(is)g(r)n(o)n(ote)n (d)f(at)h Fq(\036)p Fp(g)p Fq(;)228 281 y Ff(and)231 358 y Fp(f)p Fq(E)e Fp(j)e Ff(ther)n(e)k(exists)f(a)i(SO)o(AEV)e(for)h Fq(A)g Ff(and)g Fq(E)i Ff(which)e(is)f(optimal)r(ly)h(r)n(o)n(ote)n(d)f (at)h Fq(\036)g Ff(w.r.t.)e Fq(\026)1665 343 y Fo(\036)1687 358 y Fp(g)p Fq(:)228 436 y Fr(\(ii\))i Ff(F)m(or)g Fq(\026)e Fr(=)g Fq(\026)485 442 y Fo(\036)522 436 y Ff(and)k(every)e Fq(E)g Fp(2)e Fj(R)i Ff(and)h Fq(\013)d Fp(2)f Fr([0)p Fq(;)7 b Fr(1])p Ff(,)14 b Fq(D)1154 421 y Fo(\013)1153 446 y(\026)1178 436 y Fr(\()p Fq(E)r Fr(\))f(=)g Fp(1)i Fr(\()p Ff(r)n(esp)n(e)n(ctively)22 b Fq(>)13 b Fr(0\))j Ff(if)228 487 y(and)f(only)h(if)e(ther)n(e)g(exists)h(a)g(SO)o(AEV,)g Fp(f)p Fq( )912 493 y Fo(n)934 487 y Fp(g)955 472 y Fn(1)955 497 y Fo(n)p Fm(=1)1019 487 y Ff(,)g(for)f Fq(A)h Ff(and)h Fq(E)r Ff(,)f(such)g(that)552 591 y Fr(lim)7 b(inf)566 616 y Fo(n)p Fn(!1)727 563 y Fp(jh)p Fq( )782 569 y Fo(n)804 563 y Fq(;)g(\036)p Fp(ij)876 547 y Fm(2)p 679 581 262 2 v 679 619 a Fp(k)p Fr(\()p Fq(A)i Fp(\000)h Fq(E)r Fr(\))p Fq( )874 625 y Fo(n)896 619 y Fp(k)917 607 y Fo(\013)957 591 y Fr(=)i Fp(1)18 b Fr(\()p Ff(r)n(esp)n(e)n(ctively)26 b Fq(>)12 b Fr(0\))p Fq(:)-1171 b Fr(\(1.6\))228 692 y(\(iii\))14 b Ff(L)n(et)g Fq(\027)j Ff(b)n(e)e(any)h(\014nite)f(p)n (ositive)g(Bor)n(el)f(me)n(asur)n(e)h(on)g Fj(R)p Ff(,)g(and)h(let)e Fq(S)g Fp(\032)e Fj(R)j Ff(b)n(e)g(a)g(Bor)n(el)g(set.)228 742 y(If)g(for)f(e)n(ach)h Fq(E)f Fp(2)e Fq(S)18 b Ff(ther)n(e)c (exists)h(a)g(SO)o(AEV)g(for)g Fq(A)g Ff(and)g Fq(E)i Ff(that)f(is)e(optimal)r(ly)h(r)n(o)n(ote)n(d)f(at)h Fq(\036)228 792 y Ff(w.r.t.)e Fq(\027)s Ff(,)g(then)j Fq(\027)s Fe(\026)o Fq(S)i Ff(is)c(absolutely)h(c)n(ontinuous)h(w.r.t.) d Fq(\026)1146 777 y Fo(\036)1168 792 y Ff(.)303 872 y Fr(F)m(or)g(an)o(y)g(\014nite)i(p)q(ositiv)o(e)e(Borel)h(measure)h Fq(\026)f Fr(on)g Fj(R)p Fr(,)f(the)h(set)640 949 y Fq(S)665 955 y Fo(\026)699 949 y Fp(\021)e(f)p Fq(E)i Fp(j)k Fr(lim)7 b(sup)870 983 y Fo(\017)p Fn(!)p Fm(0)966 949 y Fr(\()p Fq(\026)p Fr(\()p Fq(I)1041 955 y Fo(\017)1057 949 y Fr(\()p Fq(E)r Fr(\)\))p Fq(=)p Fr(2)p Fq(\017)p Fr(\))12 b Fq(>)f Fr(0)p Fp(g)228 1050 y Fr(supp)q(orts)17 b Fq(\026)f Fr(\(namely)m(,)f Fq(\026)p Fr(\()p Fj(R)c Fp(n)f Fq(S)750 1056 y Fo(\026)773 1050 y Fr(\))15 b(=)g(0\))h(and)g(it)f(is)h(minimal) d(in)j(the)h(sense)g(that)f(no)g(set)h(of)228 1100 y(strictly)d (smaller)g(Leb)q(esgue)h(measure)g(supp)q(orts)h Fq(\026)p Fr(.)i Fq(S)1106 1106 y Fo(\026)1143 1100 y Fr(is)c(th)o(us)g(often)g (called)g(the)h(essen)o(tial)228 1150 y(supp)q(ort)i(of)f Fq(\026)p Fr(.)27 b(The)17 b(essence)i(of)d(Theorem)h(1.5)f(can)h(b)q (e)g(understo)q(o)q(d)h(as)f(sa)o(ying)f(that)h(the)228 1199 y(essen)o(tial)g(supp)q(ort)g(of)f Fq(\026)627 1184 y Fo(\036)665 1199 y Fr(coincides)i(with)e(the)h(set)g(of)f(energies)i Fq(E)g Fr(for)f(whic)o(h)f(there)i(exist)228 1249 y(SO)o(AEV's)12 b(for)g Fq(A)f Fr(and)h Fq(E)i Fr(that)d(are)i(optimally)c(ro)q(oted)j (at)g Fq(\036)f Fr(with)h(resp)q(ect)i(to)e Fq(\026)1498 1234 y Fo(\036)1520 1249 y Fr(.)17 b(Our)12 b(more)228 1299 y(elab)q(orate)h(form)o(ulation)d(and)j(the)g(fact)f(that)h(w)o(e) g(de\014ne)g(ro)q(oted)h(SO)o(AEV's)f(separately)g(from)228 1349 y(optimally)h(ro)q(oted)i(ones)h(are)f(motiv)n(ated)f(b)o(y)h(t)o (w)o(o)g(main)f(things.)24 b(First,)17 b(w)o(e)f(w)o(an)o(t)f(to)h(ha)o (v)o(e)228 1399 y(a)f(form)o(ulation)e(that)i(is)g(useful)g(from)f(the) i(p)q(ersp)q(ectiv)o(e)h(of)e(sp)q(ectral)h(analysis.)21 b(That)15 b(is,)g(w)o(e)228 1449 y(w)o(an)o(t)d(a)g(theorem)g(that)h(w) o(ould)e(b)q(e)i(helpful)f(in)f(situations)h(where,)i(a)e(priori,)f(w)o (e)h(do)g(not)h(kno)o(w)228 1498 y(an)o(ything)d(ab)q(out)h(the)h(sp)q (ectral)g(measure)g Fq(\026)920 1504 y Fo(\036)942 1498 y Fr(,)f(and)g(where)h(w)o(e)g(ma)o(y)e(hop)q(e)h(to)g(c)o(haracterize) i(its)228 1548 y(prop)q(erties)k(b)o(y)f(constructing)h(appropriate)f (SO)o(AEV's.)24 b(Second,)17 b(w)o(e)f(will)e(see)j(b)q(elo)o(w)f(that) 228 1598 y(b)q(oth)c(ro)q(oted)g(and)g(optimally)d(ro)q(oted)k(SO)o (AEV's)f(are)g(natural)g(ob)r(jects)h(when)f(one)g(considers)228 1648 y(the)g(issue)h(of)e(con)o(v)o(ergence)i(to)f(generalized)h (eigenfunctions.)k(Roughly)11 b(sp)q(eaking,)g(requiring)228 1698 y(SO)o(AEV's)17 b(to)f(b)q(e)h(ro)q(oted)g(is)f(a)g(su\016cien)o (t)h(condition)f(to)g(ensure)i(that)e(they)h(con)o(v)o(erge)h(in)d(a) 228 1747 y(certain)h(v)o(ery)f(w)o(eak)g(sense)i(to)e(generalized)h (eigenfunctions)g(of)e Fq(A)p Fr(.)23 b(Requiring)14 b(them)h(to)h(b)q(e)228 1797 y(optimally)8 b(ro)q(oted)j(is)g(a)f (su\016cien)o(t)h(condition)f(to)h(ensure)h(that)e(they)i(con)o(v)o (erge)f(to)f(generalized)228 1847 y(eigenfunctions)j(in)g(a)g (stronger,)h(more)f(natural,)g(sense.)19 b(W)m(e)13 b(b)q(eliev)o(e)h (that)f(these)h(conditions)228 1897 y(are)20 b(also)e(necessary)k(\(at) d(least)g(from)g(the)h(practical)f(p)q(ersp)q(ectiv)o(e)j(that)d(one)h (cannot)f(\014nd)228 1947 y(alternativ)o(e)h(\\sensible")h (conditions\))g(in)f(order)i(to)e(ensure)j(con)o(v)o(ergence)f(in)e (the)i(precise)228 1996 y(senses)16 b(that)e(w)o(e)g(consider)h(b)q (elo)o(w.)303 2046 y(Fix)e Fq(\036)e Fp(2)g(H)k Fr(and)e(an)h (orthonormal)f(basis)h Fp(f)p Fq(\016)1019 2052 y Fo(n)1042 2046 y Fp(g)1063 2031 y Fn(1)1063 2057 y Fo(n)p Fm(=1)1141 2046 y Fr(of)f Fp(H)p Fr(.)18 b(Let)723 2128 y Fp(H)758 2134 y Fo(\036)792 2128 y Fp(\021)p 835 2092 392 2 v 11 w(f)p Fq(f)t Fr(\()p Fq(A)p Fr(\))p Fq(\036)13 b Fp(j)e Fq(f)16 b Fp(2)11 b Fq(C)1109 2134 y Fn(1)1144 2128 y Fr(\()p Fj(R)p Fr(\))p Fp(g)228 2205 y Fr(b)q(e)h(the)g(cyclic)f (subspace)i(spanned)f(b)o(y)f Fq(A)g Fr(and)h Fq(\036)p Fr(,)f(and)g(let)g Fq(P)1178 2211 y Fo(\036)1211 2205 y Fr(b)q(e)h(the)g(orthogonal)e(pro)r(jection)228 2255 y(on)f Fp(H)316 2261 y Fo(\036)338 2255 y Fr(.)17 b(By)10 b(the)g(sp)q(ectral)h(theorem,)f Fq(A)p Fe(\026)q Fp(H)900 2261 y Fo(\036)931 2255 y Fr(is)g(unitarily)e(equiv)n(alen)o(t)h(to)h (m)o(ultiplication)d(b)o(y)i Fq(E)228 2306 y Fr(on)k Fq(L)313 2291 y Fm(2)332 2306 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)444 2291 y Fo(\036)465 2306 y Fr(\()p Fq(E)r Fr(\)\),)14 b(and)f(so)h(the)g(v)o(ectors)g Fp(f)p Fq(P)962 2312 y Fo(\036)984 2306 y Fq(\016)1002 2312 y Fo(n)1025 2306 y Fp(g)1046 2291 y Fn(1)1046 2316 y Fo(n)p Fm(=1)1124 2306 y Fr(ha)o(v)o(e)f(represen)o(tations)i(as)f(functions)228 2357 y(in)j Fq(L)308 2342 y Fm(2)327 2357 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)439 2342 y Fo(\036)460 2357 y Fr(\()p Fq(E)r Fr(\)\).)29 b(F)m(or)17 b(eac)o(h)h Fq(n)p Fr(,)g(w)o(e)f(let)h Fq(u)964 2363 y Fo(E)991 2357 y Fr(\()p Fq(n)p Fr(\))g(b)q(e)g(the)g Fq(L)1229 2342 y Fm(2)1248 2357 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)1360 2342 y Fo(\036)1381 2357 y Fr(\()p Fq(E)r Fr(\)\))18 b(function)f(that)228 2407 y(corresp)q(onds)h(to)d Fq(P)537 2413 y Fo(\036)559 2407 y Fq(\016)577 2413 y Fo(n)600 2407 y Fr(.)24 b(Since)17 b Fq(u)771 2413 y Fo(E)799 2407 y Fr(\()p Fq(n)p Fr(\))f(is)f(de\014ned)i(a.e.)f(w.r.t.)e Fq(\026)1276 2392 y Fo(\036)1298 2407 y Fr(,)i(w)o(e)h(also)e(ha)o(v)o (e)h(that)g(the)228 2458 y(sequence)f Fp(f)p Fq(u)444 2464 y Fo(E)472 2458 y Fr(\()p Fq(n)p Fr(\))p Fp(g)550 2443 y Fn(1)550 2469 y Fo(n)p Fm(=1)628 2458 y Fr(is)e(de\014ned)i (a.e.)e(w.r.t.)f Fq(\026)1023 2443 y Fo(\036)1045 2458 y Fr(.)18 b(W)m(e)13 b(call)g Fp(f)p Fq(u)1267 2464 y Fo(E)1294 2458 y Fr(\()p Fq(n)p Fr(\))p Fp(g)1372 2443 y Fn(1)1372 2469 y Fo(n)p Fm(=1)1450 2458 y Fr(the)h(generalized)228 2508 y(eigenfunction)19 b(of)f Fq(A)i Fr(that)f(corresp)q(onds)i(to)e Fq(\036)f Fr(and)h Fq(E)i Fr(in)e(the)h(represen)o(tation)g Fp(f)p Fq(\016)1602 2514 y Fo(n)1625 2508 y Fp(g)1646 2493 y Fn(1)1646 2518 y Fo(n)p Fm(=1)1711 2508 y Fr(.)228 2558 y(Our)g(next)g(theorem)g(relates)h(ro)q(oted)f(and)f(optimally)f (ro)q(oted)i(SO)o(AEV's)g(to)f(generalized)228 2608 y(eigenfunctions.)p eop %%Page: 5 5 5 4 bop 488 113 a Fm(APPR)o(O)o(XIMA)m(TE)15 b(EIGENVECTORS)f(AND)g (SPECTRAL)g(THEOR)m(Y)244 b(5)303 213 y Fg(Theorem)16 b Fr(1.6)p Fg(.)k Ff(L)n(et)15 b Fq(\036)c Fp(2)g(H)p Ff(.)19 b(Then)228 262 y Fr(\(i\))d Ff(Ther)n(e)f(exists)h(an)h (orthonormal)e(b)n(asis,)h Fp(f)p Fq(\016)975 268 y Fo(m)1007 262 y Fp(g)1028 247 y Fn(1)1028 273 y Fo(m)p Fm(=1)1101 262 y Ff(,)h(of)f Fp(H)p Ff(,)g(such)h(that)f(for)f(a.e.)h Fq(E)i Ff(w.r.t.)228 312 y Fq(\026)253 297 y Fo(\036)275 312 y Ff(,)d(every)h(SO)o(AEV,)f Fp(f)p Fq( )634 318 y Fo(n)657 312 y Fp(g)678 297 y Fn(1)678 323 y Fo(n)p Fm(=1)742 312 y Ff(,)h(for)f Fq(A)g Ff(and)i Fq(E)r Ff(,)e(that)h(is)f (r)n(o)n(ote)n(d)h(at)f Fq(\036)p Ff(,)g(ob)n(eys)h(for)f(every)h Fq(m)p Ff(,)228 362 y Fp(h)p Fq( )271 368 y Fo(n)294 362 y Fq(;)7 b(\036)p Fp(i)354 347 y Fn(\000)p Fm(1)397 362 y Fp(h)p Fq( )440 368 y Fo(n)463 362 y Fq(;)g(P)509 368 y Fo(\036)530 362 y Fq(\016)548 368 y Fo(m)580 362 y Fp(i)16 b(!)g Fq(u)694 368 y Fo(E)721 362 y Fr(\()p Fq(m)p Fr(\))i Ff(as)g Fq(n)e Fp(!)f(1)i Fr(\()p Ff(wher)n(e)f Fp(f)p Fq(u)1200 368 y Fo(E)1228 362 y Fr(\()p Fq(m)p Fr(\))p Fp(g)1317 347 y Fn(1)1317 372 y Fo(m)p Fm(=1)1408 362 y Ff(is)h(the)g(gener)n(alize)n(d)228 412 y(eigenfunction)f(of)e Fq(A)h Ff(that)g(c)n(orr)n(esp)n(onds)g(to)g Fq(\036)f Ff(and)i Fq(E)h Ff(in)e(the)g(r)n(epr)n(esentation)f Fp(f)p Fq(\016)1535 418 y Fo(m)1567 412 y Fp(g)1588 397 y Fn(1)1588 422 y Fo(m)p Fm(=1)1661 412 y Fr(\))p Ff(.)228 462 y Fr(\(ii\))j Ff(F)m(or)h(any)h(orthonormal)f(b)n(asis,)h Fp(f)p Fq(\016)862 468 y Fo(m)894 462 y Fp(g)915 447 y Fn(1)915 472 y Fo(m)p Fm(=1)988 462 y Ff(,)g(of)f Fp(H)h Ff(and)g(for)f(any)h(\014nite)g(p)n(ositive)f(Bor)n(el)228 511 y(me)n(asur)n(e)11 b Fq(\027)i Ff(on)e Fj(R)p Ff(,)h(we)e(have)i (that)f(for)g(a.e.)f Fq(E)j Ff(w.r.t.)c Fq(\027)s Ff(,)i(every)g(SO)o (AEV,)g Fp(f)p Fq( )1420 517 y Fo(n)1442 511 y Fp(g)1463 496 y Fn(1)1463 522 y Fo(n)p Fm(=1)1527 511 y Ff(,)h(for)e Fq(A)h Ff(and)228 561 y Fq(E)r Ff(,)i(that)g(is)f(optimal)r(ly)h(r)n(o) n(ote)n(d)g(at)g Fq(\036)f Ff(w.r.t.)f Fq(\027)s Ff(,)h(ob)n(eys)i(for) e(every)h Fq(m)p Ff(,)g Fp(h)p Fq( )1343 567 y Fo(n)1366 561 y Fq(;)7 b(\036)p Fp(i)1426 546 y Fn(\000)p Fm(1)1470 561 y Fp(h)p Fq( )1513 567 y Fo(n)1536 561 y Fq(;)g(P)1582 567 y Fo(\036)1603 561 y Fq(\016)1621 567 y Fo(m)1653 561 y Fp(i)12 b(!)228 611 y Fq(u)252 617 y Fo(E)280 611 y Fr(\()p Fq(m)p Fr(\))17 b Ff(as)g Fq(n)f Fp(!)f(1)h Fr(\()p Ff(wher)n(e)g Fp(f)p Fq(u)756 617 y Fo(E)784 611 y Fr(\()p Fq(m)p Fr(\))p Fp(g)873 596 y Fn(1)873 621 y Fo(m)p Fm(=1)964 611 y Ff(is)h(the)g(gener)n(alize)n(d)f (eigenfunction)i(of)f Fq(A)g Ff(that)228 661 y(c)n(orr)n(esp)n(onds)e (to)f Fq(\036)h Ff(and)h Fq(E)h Ff(in)e(the)f(r)n(epr)n(esentation)h Fp(f)p Fq(\016)1099 667 y Fo(m)1131 661 y Fp(g)1152 646 y Fn(1)1152 671 y Fo(m)p Fm(=1)1225 661 y Fr(\))p Ff(.)228 763 y(R)n(emarks.)26 b Fr(1.)h(The)17 b(existence)h(of)e(an)h (appropriate)f(basis)h(in)f(\(i\))h(comes)g(from)f(c)o(ho)q(osing)g(a) 228 813 y(basis)d(for)g(whic)o(h)g(the)h Fq(u)605 819 y Fo(E)633 813 y Fr(\()p Fq(m)p Fr(\)'s)g(are)f(con)o(tin)o(uous)h (functions)f(of)g Fq(E)r Fr(.)k(The)d(need)g(for)f(optimally)228 863 y(ro)q(oted)e(SO)o(AEV's)h(in)f(\(ii\))f(comes)i(from)e(the)i Fq(u)959 869 y Fo(E)987 863 y Fr(\()p Fq(m)p Fr(\)'s)f(b)q(eing)g (arbitrary)g Fq(L)1405 847 y Fm(2)1435 863 y Fr(functions)g(rather)228 912 y(than)j(con)o(tin)o(uous.)303 962 y(2.)38 b(It)21 b(ma)o(y)e(app)q(ear)j(strange)f(that)g(\(ii\))f(is)h(form)o(ulated)e (without)i(an)o(y)f(reference)j(to)228 1012 y Fq(\026)253 997 y Fo(\036)275 1012 y Fr(.)31 b(Note,)19 b(ho)o(w)o(ev)o(er,)h(that) e(b)o(y)g(\(iii\))f(of)h(Theorem)g(1.5,)g(the)h(restriction)g(of)f Fq(\027)i Fr(to)e(the)h(set)228 1062 y(where)14 b(the)h(optimally)c(ro) q(oted)j(SO)o(AEV's)g(exist)g(is)f(absolutely)g(con)o(tin)o(uous)g (w.r.t.)g Fq(\026)1599 1047 y Fo(\036)1621 1062 y Fr(.)18 b(The)228 1112 y(eigenfunctions)c Fp(f)p Fq(u)544 1118 y Fo(E)572 1112 y Fr(\()p Fq(m)p Fr(\))p Fp(g)661 1097 y Fn(1)661 1122 y Fo(m)p Fm(=1)748 1112 y Fr(are)h(th)o(us)f(w)o(ell)f (de\014ned)i(a.e.)e(w.r.t.)g Fq(\027)j Fr(on)e(this)g(set.)303 1161 y(3.)j(The)12 b(generalized)h(eigenfunctions)f(that)g(w)o(e)g (consider)h(here)g(are)f(connected)i(with)e(the)228 1211 y(cyclic)i(subspace)i Fp(H)551 1217 y Fo(\036)573 1211 y Fr(,)e(and)g(it)g(is)h(sometimes)f(more)g(natural)g(to)g(consider)i (a)e(represen)o(tation)228 1261 y(in)h(a)h(basis)g(of)f Fp(H)503 1267 y Fo(\036)541 1261 y Fr(rather)i(than)f(in)f(a)h(basis)g (of)f Fp(H)p Fr(.)25 b(The)16 b(theorem)h(remains)f(true)h(for)e(suc)o (h)228 1311 y(represen)o(tations.)20 b(Of)13 b(particular)g(in)o (terest)i(is)f(the)g(case)g(where)h(linear)e(com)o(binations)g(of)g (the)228 1361 y(monomials)c Fp(f)p Fq(A)485 1346 y Fo(n)508 1361 y Fq(\036)p Fp(g)554 1346 y Fn(1)554 1371 y Fo(n)p Fm(=1)629 1361 y Fr(happ)q(en)j(to)g(b)q(e)g(dense)h(in)e Fp(H)1068 1367 y Fo(\036)1102 1361 y Fr(\(this)g(is)h(true)g(for)g(an)o (y)f Fq(\036)g Fr(if)g Fq(A)g Fr(happ)q(ens)228 1410 y(to)k(b)q(e)g(b)q(ounded,)g(and)g(it)f(is)h(also)f(true)i(for)f(man)o (y)f(natural)g(c)o(hoices)i(of)e(a)g(v)o(ector)i Fq(\036)f Fr(in)f(man)o(y)228 1460 y(cases)h(of)e(un)o(b)q(ounded)i Fq(A)p Fr('s\).)j(In)c(suc)o(h)g(a)g(case,)g Fp(H)1013 1466 y Fo(\036)1049 1460 y Fr(has)g(a)g(natural)f(basis)h(whic)o(h)g (is)f(obtained)228 1510 y(from)d Fp(f)p Fq(A)375 1495 y Fo(n)397 1510 y Fq(\036)p Fp(g)443 1495 y Fn(1)443 1520 y Fo(n)p Fm(=1)518 1510 y Fr(b)o(y)h(Gram-Sc)o(hmidt)f (orthonormalization)f(\(this)i(basis)g(coincides)g(with)g(the)228 1560 y(orthogonal)k(p)q(olynomials)g(of)i(the)g(sp)q(ectral)h(measure)g Fq(\026)1146 1545 y Fo(\036)1168 1560 y Fr(\),)f(and)g Fq(A)p Fe(\026)p Fp(H)1380 1566 y Fo(\036)1419 1560 y Fr(is)g(w)o(ell)f(kno)o(wn)g(to)228 1610 y(ha)o(v)o(e)h(a)g (tridiagonal)e(matrix)h(represen)o(tation)j(in)e(this)g(basis)g(\(see,) i(e.g.,)e([)p Fs(29)o Fr(]\).)27 b(In)18 b(suc)o(h)f(a)228 1660 y(case,)d(the)h(basis)f(in)f(\(i\))h(can)g(b)q(e)g(c)o(hosen)h(to) f(b)q(e)g(this)g(natural)g(basis.)303 1757 y(Theorem)k(1.6)e(relates)j (SO)o(AEV's)f(to)g(general)g(discrete)h(represen)o(tations)h(of)d (general-)228 1807 y(ized)g(eigenfunctions)f(whic)o(h)h(are)g(w)o(ell)e (de\014ned)j(for)e(arbitrary)g(self-adjoin)o(t)g(op)q(erators.)26 b(In)228 1856 y(man)o(y)9 b(practical)h(cases,)h(ho)o(w)o(ev)o(er,)g (one)f(w)o(an)o(ts)g(to)g(iden)o(tify)f(generalized)h(eigenfunctions)g (with)228 1906 y(solutions)15 b(of)g(a)h(generalized)h(eigen)o(v)n (alue)e(equation)g(in)h(a)f(particular)h(\(natural\))g(space)h(rep-)228 1956 y(resen)o(tation.)24 b(While)15 b(there)i(are)f(cases,)h(lik)o(e)e (discrete)j(Sc)o(hr\177)-21 b(odinger)16 b(op)q(erators,)h(where)g(the) 228 2006 y(natural)11 b(represen)o(tation)i(is)e(a)g(sp)q(ecial)g(case) i(of)d(our)i(more)f(general)h(discrete)h(represen)o(tations,)228 2056 y(there)18 b(are)f(other)h(cases,)g(lik)o(e)e(con)o(tin)o(uous)h (Sc)o(hr\177)-21 b(odinger)17 b(op)q(erators)h(\(or)f(other)g (di\013eren)o(tial)228 2105 y(op)q(erators\),)12 b(where)h(the)f (natural)f(represen)o(tations)j(of)d(generalized)h(eigenfunctions)g (are)g(con-)228 2155 y(tin)o(uous.)18 b(Let)812 2229 y Fq(H)850 2212 y Fm(\012)847 2239 y Fo(V)887 2229 y Fr(=)12 b Fp(\000)p Fr(\001)d(+)h Fq(V)f Fr(\()p Fq(x)p Fr(\))-910 b(\(1.7\))228 2309 y(denote)11 b(the)g(Sc)o(hr\177)-21 b(odinger)11 b(op)q(erator)g(de\014ned)h(on)e(a)g(domain)f(\012)j Fp(\032)g Fj(R)1299 2294 y Fo(d)1329 2309 y Fr(with)e(a)g(smo)q(oth)g (b)q(ound-)228 2359 y(ary)f(and)h(satisfying)f(Diric)o(hlet)g(b)q (oundary)h(conditions)f(on)h Fq(@)r Fr(\012.)17 b(\(Note)10 b(that)g(w)o(e)g(are)g(primarily)228 2409 y(in)o(terested)15 b(in)f(cases)h(where)g(the)g(domain)d(\012)i(is)g(un)o(b)q(ounded,)g (and)g(the)g(case)h(\012)d(=)g Fj(R)1578 2393 y Fo(d)1611 2409 y Fr(is)i(also)228 2458 y(included.\))25 b(Recall)15 b(that)i(for)e(a)h(wide)g(class)h(of)f(Sc)o(hr\177)-21 b(odinger)16 b(op)q(erators)h(of)f(this)g(t)o(yp)q(e,)h(one)228 2508 y(has)c(a)g(generalized)h(eigenfunction)e(expansion)h(theorem)h (with)f(con)o(tin)o(uum)f(eigenfunctions.)228 2558 y(\(see)k([)p Fs(1)o(,)h(3,)g(4,)f(9,)h(22,)g(27)o(,)g(28)p Fr(]\).)i(That)c(is,)f (for)g(eac)o(h)h(v)o(ector)g Fq(\036)e Fp(2)f(H)h Fr(=)g Fq(L)1441 2543 y Fm(2)1459 2558 y Fr(\(\012)p Fq(;)7 b(dx)p Fr(\))14 b(\(where)228 2608 y Fq(dx)e Fr(denotes)i(the)g Fq(d)p Fr(-dimensional)d(Leb)q(esgue)j(measure)g(on)e(\012\),)h(the)h (unitary)e(map)g Fq(U)1570 2614 y Fo(\036)1593 2608 y Fr(,)g(whic)o(h)p eop %%Page: 6 6 6 5 bop 228 113 a Fm(6)407 b(M.)15 b(CHRIST,)f(A.)h(KISELEV,)g(AND)e (Y.)i(LAST)228 213 y Fr(maps)h Fq(H)377 198 y Fm(\012)374 224 y Fo(V)402 213 y Fe(\026)q Fp(H)455 219 y Fo(\036)493 213 y Fr(to)f(m)o(ultiplication)f(b)o(y)i(the)g(v)n(ariable)f(on)h Fq(L)1193 198 y Fm(2)1212 213 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)1324 198 y Fo(\036)1345 213 y Fr(\),)16 b(is)g(kno)o(wn)g(to)g (b)q(e)g(an)228 268 y(in)o(tegral)f(op)q(erator)h(with)f(k)o(ernel)p 770 232 154 2 v 16 w Fq(u)794 274 y Fo(\036)815 268 y Fr(\()p Fq(x;)7 b(E)r Fr(\),)15 b(where)i Fq(u)1096 274 y Fo(\036)1118 268 y Fr(\()p Fq(x;)7 b(E)r Fr(\),)14 b(for)h(\014xed)h Fq(E)r Fr(,)g(is)f(in)g Fq(L)1601 253 y Fm(2)1601 279 y(Lo)q(c)1671 268 y Fr(\(in)228 318 y Fq(x)p Fr(\))f(and)f(solv)o(es)770 397 y(\()p Fq(H)824 380 y Fm(\012)821 407 y Fo(V)859 397 y Fp(\000)d Fq(E)r Fr(\))p Fq(u)974 403 y Fo(\036)996 397 y Fr(\()p Fq(x;)d(E)r Fr(\))k(=)h(0)-952 b(\(1.8\))228 477 y(\(in)14 b(a)g(distributional)f (sense)j(if)e(the)h(p)q(oten)o(tial)f(is)g(not)h(su\016cien)o(tly)f(w)o (ell-b)q(eha)o(v)o(ed\).)20 b(W)m(e)14 b(will)228 527 y(sa)o(y)f(that)h(the)h Fq(u)485 533 y Fo(\036)507 527 y Fr(\()p Fq(x;)7 b(E)r Fr(\)'s)13 b(are)h(the)h(generalized)f (eigenfunctions)g(of)g Fq(H)1368 512 y Fm(\012)1365 539 y Fo(V)1393 527 y Fr(,)g(corresp)q(onding)g(to)228 577 y Fq(\036)p Fr(,)g(if)f(their)i(complex)f(conjugates)h(constitute)g (the)g(k)o(ernel)g(of)f(the)h(unitary)f(map)g Fq(U)1562 583 y Fo(\036)1584 577 y Fr(.)20 b(These)228 628 y(generalized)14 b(eigenfunctions)g(are)g(generally)g(only)f(de\014ned)h(a.e.)f(w.r.t.)g Fq(\026)1404 613 y Fo(\036)1426 628 y Fr(.)18 b(In)13 b(this)h(setting,)228 678 y(\(ii\))f(of)g(Theorem)i(1.6)e(has)h(the)g (follo)o(wing)d(analog.)303 759 y Fg(Theorem)16 b Fr(1.7)p Fg(.)k Ff(L)n(et)d Fq(\036)f Fp(2)f(H)h Fr(=)h Fq(L)878 744 y Fm(2)896 759 y Fr(\(\012)p Fq(;)7 b(dx)p Fr(\))17 b Ff(and)h(let)e Fq(H)1221 744 y Fm(\012)1218 770 y Fo(V)1264 759 y Ff(b)n(e)i(a)f(self-adjoint)g(op)n(er)n(ator)228 809 y(on)e Fq(L)315 793 y Fm(2)333 809 y Fr(\(\012)p Fq(;)7 b(dx)p Fr(\))14 b Ff(for)f(which)h(the)h(gener)n(alize)n(d)f (eigenfunctions)h Fq(u)1231 815 y Fo(\036)1253 809 y Fr(\()p Fq(x;)7 b(E)r Fr(\))13 b Ff(ar)n(e)h(wel)r(l)f(de\014ne)n(d)j Fr(\()p Ff(as)228 858 y(functions)e(in)g Fq(L)485 843 y Fm(2)485 870 y(Lo)q(c)554 858 y Ff(in)f Fq(x)h Ff(for)f(a.e.)h Fq(E)h Ff(w.r.t)d Fq(\026)955 864 y Fo(\036)977 858 y Ff(,)i(whose)g(c)n(omplex)g(c)n(onjugates)g(c)n(onstitute)g(the)228 908 y(kernel)k(of)g(the)h(unitary)f(map)h Fq(U)752 914 y Fo(\036)774 908 y Fr(\))p Ff(.)30 b(L)n(et)18 b Fq(P)935 914 y Fo(\036)975 908 y Ff(b)n(e)g(the)h(ortho)n(gonal)f(pr)n(oje)n (ction)g(on)h Fp(H)1602 914 y Fo(\036)1624 908 y Ff(,)g(and)228 958 y(let)14 b Fq(\027)k Ff(b)n(e)d(any)g(\014nite)h(p)n(ositive)f(Bor) n(el)f(me)n(asur)n(e)h(on)h Fj(R)p Ff(.)j(Then)d(for)e(any)i(c)n(omp)n (actly)f(supp)n(orte)n(d)228 1008 y Fq(\021)d Fp(2)g Fq(L)329 993 y Fm(2)348 1008 y Fr(\(\012)p Fq(;)7 b(dx)p Fr(\))p Ff(,)13 b(for)h(a.e.)h Fq(E)h Ff(w.r.t.)d Fq(\027)s Ff(,)h(every)h(SO)o(AEV,)f Fp(f)p Fq( )1187 1014 y Fo(n)1209 1008 y Fp(g)1230 993 y Fn(1)1230 1018 y Fo(n)p Fm(=1)1295 1008 y Ff(,)g(for)g Fq(H)1426 993 y Fm(\012)1423 1019 y Fo(V)1467 1008 y Ff(and)h Fq(E)r Ff(,)g(that)f(is)228 1058 y(optimal)r(ly)g(r)n(o)n(ote)n(d)h(at)g Fq(\036)f Ff(w.r.t.)f Fq(\027)s Ff(,)h(ob)n(eys)507 1098 y Fi(Z)548 1154 y Fp(h)p Fq( )591 1160 y Fo(n)614 1154 y Fq(;)7 b(\036)p Fp(i)674 1137 y Fn(\000)p Fm(1)718 1154 y Fr(\()p Fq(P)761 1160 y Fo(\036)783 1154 y Fq( )810 1160 y Fo(n)832 1154 y Fr(\)\()p Fq(x)p Fr(\))p 904 1118 79 2 v Fq(\021)q Fr(\()p Fq(x)p Fr(\))h Fq(dx)j Fp(!)1100 1098 y Fi(Z)1148 1154 y Fq(u)1172 1160 y Fo(\036)1194 1154 y Fr(\()p Fq(x;)c(E)r Fr(\))p 1302 1118 V Fq(\021)q Fr(\()p Fq(x)p Fr(\))f Fq(dx:)-1216 b Fr(\(1.9\))228 1287 y Ff(R)n(emarks.)18 b Fr(1.)e(Note)11 b(that)g(\(1.9\))e(ensures)k(that)d Fp(h)p Fq( )1015 1293 y Fo(n)1038 1287 y Fq(;)d(\036)p Fp(i)1098 1271 y Fn(\000)p Fm(1)1142 1287 y Fr(\()p Fq(P)1185 1293 y Fo(\036)1207 1287 y Fq( )1234 1293 y Fo(n)1256 1287 y Fr(\)\()p Fq(x)p Fr(\))k(con)o(v)o(erges)h(to)e Fq(u)1593 1293 y Fo(\036)1615 1287 y Fr(\()p Fq(x;)d(E)r Fr(\))228 1336 y(in)13 b Fq(L)304 1321 y Fm(2)304 1348 y(Lo)q(c)359 1336 y Fr(.)303 1386 y(2.)21 b(While)14 b(w)o(e)h(are)h(primarily)d(in)o(terested)k(in)e(Sc)o(hr\177)-21 b(odinger)16 b(op)q(erators)g(here,)g(note)f(that)228 1436 y(the)j(theorem)h(only)e(relies)i(on)f(prop)q(erties)h(of)f(the)g (generalized)h(eigenfunctions)f Fq(u)1581 1442 y Fo(\036)1603 1436 y Fr(\()p Fq(x;)7 b(E)r Fr(\),)228 1487 y(and)16 b(it)g(do)q(es)h(not)f(require)h(an)o(y)f(explicit)g(assumptions)g(on)h Fq(H)1236 1472 y Fm(\012)1233 1498 y Fo(V)1278 1487 y Fr(itself)e(\(b)q(ey)o(ond)i(it)f(b)q(eing)g(a)228 1537 y(self-adjoin)o(t)c(op)q(erator)j(on)e(the)i(appropriate)f(space\).)303 1614 y(Using)c(sequences)j(of)d(appro)o(ximate)g(eigen)o(v)o(ectors,)i (w)o(e)f(can)g(also)f(study)h(the)g(m)o(ultiplicit)o(y)228 1664 y(of)16 b(the)h(sp)q(ectrum.)28 b(Namely)m(,)15 b(the)i(follo)o(wing)d(natural)i(extension)i(of)e(\(iii\))f(of)h (Theorem)h(1.5)228 1714 y(holds:)303 1795 y Fg(Theorem)f Fr(1.8)p Fg(.)k Ff(L)n(et)14 b Fq(\027)j Ff(b)n(e)d(any)h(\014nite)g(p) n(ositive)f(Bor)n(el)g(me)n(asur)n(e)g(on)h Fj(R)p Fq(;)f Ff(and)h(let)f Fq(S)g Fp(\032)e Fj(R)228 1845 y Ff(b)n(e)h(a)g(Bor)n (el)g(set.)18 b(L)n(et)13 b Fp(f)p Fq(\036)617 1851 y Fo(k)637 1845 y Fp(g)658 1829 y Fo(n)658 1856 y(k)q Fm(=1)734 1845 y Ff(b)n(e)g(an)h(orthonormal)e(set)i(of)f(ve)n(ctors.)18 b(Assume)13 b(that)g(for)g(e)n(ach)228 1896 y Fq(E)g Fp(2)f Fq(S)r Ff(,)i(ther)n(e)g(exist)f Fq(n)h Ff(SO)o(AEV's,)g Fp(f)p Fq( )853 1881 y Fo(k)852 1906 y(j)873 1896 y Fp(g)p Fq(;)f(k)g Fr(=)f(1)p Fq(;)7 b(:)g(:)g(:)t(;)g(n;)13 b Ff(for)g Fq(A)h Ff(and)h Fq(E)r Ff(,)f(such)g(that)g Fp(f)p Fq( )1637 1881 y Fo(k)1636 1906 y(j)1658 1896 y Fp(g)f Ff(is)228 1945 y(optimal)r(ly)i(r)n(o)n(ote)n(d)g(at)h Fq(\036)610 1951 y Fo(k)646 1945 y Ff(w.r.t.)d Fq(\027:)i Ff(Then)h(the)g(multiplicity)e(of)i(the)g(sp)n(e)n(ctrum)f(on)h(the)g (set)f Fq(S)228 1995 y Ff(is)f(at)g(le)n(ast)g Fq(n;)g Ff(and)h Fq(\027)s Fe(\026)o Fq(S)i Ff(is)d(absolutely)h(c)n(ontinuous) g(with)f(r)n(esp)n(e)n(ct)f(to)i(the)f(sp)n(e)n(ctr)n(al)f(me)n(asur)n (es)228 2045 y(in)i(at)g(le)n(ast)f Fq(n)h Ff(ortho)n(gonal)g(cyclic)g (subsp)n(ac)n(es)g(of)g Fq(A)p Ff(.)303 2154 y Fr(The)f(rest)h(of)e (this)h(pap)q(er)g(is)g(organized)g(as)f(follo)o(ws.)k(In)d(Section)g (2,)f(w)o(e)h(pro)o(v)o(e)g(Theorem)228 2203 y(1.1,)c(and)h(in)g (Section)h(3,)f(w)o(e)h(pro)o(v)o(e)f(Theorem)h(1.5.)k(In)c(Section)g (4,)f(w)o(e)g(describ)q(e)i(some)f(general)228 2253 y(consequences)i (of)c(Theorems)i(1.1)e(and)h(1.5,)f(and)h(in)g(Section)g(5,)g(w)o(e)g (describ)q(e)i(the)f(application)228 2303 y(of)18 b(these)i(results)g (to)f(study)g(sp)q(ectral)h(prop)q(erties)g(of)e(a)h(concrete)i(family) 16 b(of)j(Sc)o(hr\177)-21 b(odinger)228 2353 y(op)q(erators.)20 b(Note)15 b(that)f(w)o(e)h(do)f(not)g(pro)o(vide)h(here)g(pro)q(ofs)f (of)g(Theorems)h(1.6,)e(1.7,)g(and)i(1.8,)228 2403 y(nor)g(do)f(w)o(e)h (pro)o(vide)g(pro)q(ofs)g(for)f(the)h(results)h(describ)q(ed)h(in)d (Sections)i(4)e(and)h(5.)20 b(Those)c(will)228 2452 y(b)q(e)e(giv)o(en) g(in)f(a)h(future)g(publication.)303 2502 y(W)m(e)g(w)o(ould)g(lik)o(e) g(to)h(thank)g(F.)f(Gesztesy)m(,)i(S.)f(Jitomirsk)n(a)o(y)o(a,)e(R.)h (Killip,)f(and)h(B.)h(Simon)228 2552 y(for)e(useful)h(discussions.)p eop %%Page: 7 7 7 6 bop 488 113 a Fm(APPR)o(O)o(XIMA)m(TE)15 b(EIGENVECTORS)f(AND)g (SPECTRAL)g(THEOR)m(Y)244 b(7)720 213 y Fs(2.)24 b(Pro)q(of)14 b(of)i(Theorem)e(1.1)303 287 y Fr(It)k(is)h(clearly)f(su\016cien)o(t)h (to)g(consider)g(the)g(case)h(where)g Fp(k)p Fq(\036)p Fp(k)e Fr(=)i(1.)32 b(Since)19 b(the)g(cyclic)228 337 y(subspace)h Fp(H)441 343 y Fo(\036)482 337 y Fr(is)f(in)o(v)n(arian)o (t)e(under)j Fq(A)p Fr(,)f(w)o(e)g(note)h(that)e(it)h(is)g(su\016cien)o (t)g(to)g(consider)g Fq( )j Fp(2)228 388 y(H)263 394 y Fo(\036)285 388 y Fr(,)e(and)g(w)o(e)f(can)h(iden)o(tify)f Fp(H)744 394 y Fo(\036)786 388 y Fr(with)g Fq(L)914 373 y Fm(2)933 388 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)1045 373 y Fo(\036)1066 388 y Fr(\),)20 b(where)h Fq(A)f Fr(acts)g(as)g(the)g (op)q(erator)g(of)228 440 y(m)o(ultiplication)11 b(b)o(y)j(the)h(v)n (ariable.)j(Fix)c Fq(E)i Fr(with)1014 406 y Fi(R)1042 440 y Fr(\()p Fq(x)9 b Fp(\000)h Fq(E)r Fr(\))1182 425 y Fn(\000)p Fm(4)1233 440 y Fq(d\026)1280 425 y Fo(\036)1302 440 y Fr(\()p Fq(x)p Fr(\))i(=)g Fp(1)p Fr(,)i(and)g(for)f(ev)o(ery)228 489 y Fq(\016)g(>)f Fr(0,)h(de\014ne)583 592 y Fq(C)613 598 y Fo(E)r(;\016)678 592 y Fp(\021)722 534 y Fi(\024)744 536 y(Z)785 592 y Fr(\(\()p Fq(x)d Fp(\000)f Fq(E)r Fr(\))941 575 y Fm(2)969 592 y Fr(+)h Fq(\016)1031 575 y Fm(2)1049 592 y Fr(\))1065 575 y Fn(\000)p Fm(2)1117 592 y Fq(d\026)1164 575 y Fo(\036)1186 592 y Fr(\()p Fq(x)p Fr(\))1242 534 y Fi(\025)1264 542 y Fn(\000)p Fm(1)p Fo(=)p Fm(2)1349 592 y Fq(;)-1133 b Fr(\(2.1\))228 690 y(and)14 b(a)f(function)h Fq( )533 696 y Fo(E)r(;\016)598 690 y Fp(2)e Fq(L)666 675 y Fm(2)684 690 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)796 675 y Fo(\036)818 690 y Fr(\))14 b(with)f Fp(k)p Fq( )990 696 y Fo(E)r(;\016)1044 690 y Fp(k)e Fr(=)h(1,)h(b)o(y)669 761 y Fq( )696 767 y Fo(E)r(;\016)750 761 y Fr(\()p Fq(x)p Fr(\))e Fp(\021)h Fq(C)891 767 y Fo(E)r(;\016)945 761 y Fr(\(\()p Fq(x)d Fp(\000)h Fq(E)r Fr(\))1101 743 y Fm(2)1129 761 y Fr(+)f Fq(\016)1190 743 y Fm(2)1209 761 y Fr(\))1225 743 y Fn(\000)p Fm(1)1270 761 y Fq(:)-1054 b Fr(\(2.2\))228 832 y(De\014ne)14 b(also)g(a)f(function)h Fq(\017)d Fr(=)h Fq(\017)p Fr(\()p Fq(E)r(;)7 b(\016)r Fr(\),)13 b(b)o(y)h(requiring)f(the)i(equalit)o(y)429 938 y Fq(\017)d Fr(=)g Fp(k)p Fr(\()p Fq(A)d Fp(\000)g Fq(E)r Fr(\))p Fq( )696 944 y Fo(E)r(;\016)751 938 y Fp(k)i Fr(=)h Fq(C)857 944 y Fo(E)r(;\016)917 880 y Fi(\024)939 882 y(Z)1063 910 y Fr(\()p Fq(x)d Fp(\000)g Fq(E)r Fr(\))1202 895 y Fm(2)p 993 929 299 2 v 993 967 a Fr(\(\()p Fq(x)g Fp(\000)g Fq(E)r Fr(\))1148 955 y Fm(2)1177 967 y Fr(+)g Fq(\016)1238 955 y Fm(2)1257 967 y Fr(\))1273 955 y Fm(2)1303 938 y Fq(d\026)1350 921 y Fo(\036)1372 938 y Fr(\()p Fq(x)p Fr(\))1428 880 y Fi(\025)1450 887 y Fm(1)p Fo(=)p Fm(2)1509 938 y Fq(:)-1293 b Fr(\(2.3\))228 1032 y(One)14 b(can)g(easily)e(v)o(erify)h(that)h Fq(\017)f Fr(is)g(a)g(con)o(tin)o (uous)h(function)f(of)f Fq(\016)r Fr(,)h(that)h Fq(\017)d Fp(!)g Fr(0)i(as)h Fq(\016)f Fp(!)e Fr(0)j(\(b)q(e-)228 1093 y(cause)g(w)o(e)g(assume)542 1060 y Fi(R)570 1093 y Fr(\()p Fq(x)8 b Fp(\000)h Fq(E)r Fr(\))708 1078 y Fn(\000)p Fm(4)760 1093 y Fq(d\026)807 1078 y Fo(\036)828 1093 y Fr(\()p Fq(x)p Fr(\))j(=)g Fp(1)p Fr(\),)h(and)g(that)h Fq(\017)d Fp(!)1274 1060 y Fi(\002R)1319 1093 y Fr(\()p Fq(x)e Fp(\000)h Fq(E)r Fr(\))1459 1078 y Fm(2)1484 1093 y Fq(d\026)1531 1078 y Fo(\036)1553 1093 y Fr(\()p Fq(x)p Fr(\))1609 1060 y Fi(\003)1626 1068 y Fm(1)p Fo(=)p Fm(2)1690 1093 y Fr(=)228 1143 y Fp(k)p Fr(\()p Fq(A)i Fp(\000)h Fq(E)r Fr(\))p Fq(\036)p Fp(k)18 b Fr(as)g Fq(\016)j Fp(!)e(1)p Fr(.)31 b(Th)o(us,)19 b(giv)o(en)f(0)h Fq(<)g(\017)g(<)g Fp(k)p Fr(\()p Fq(A)12 b Fp(\000)h Fq(E)r Fr(\))p Fq(\036)p Fp(k)p Fr(,)18 b(w)o(e)h(can)g(alw)o(a)o(ys)e(\014nd)228 1193 y(a)f(p)q(ositiv)o(e)h Fq(\016)i Fr(for)e(whic)o(h)g(\(2.3\))f(is) h(ob)q(ey)o(ed.)28 b(Recall)16 b(that)1181 1160 y Fi(R)1209 1193 y Fr(\(\()p Fq(x)11 b Fp(\000)h Fq(E)r Fr(\))1369 1178 y Fm(2)1399 1193 y Fr(+)f Fq(\016)1462 1178 y Fm(2)1481 1193 y Fr(\))1497 1178 y Fn(\000)p Fm(1)1549 1193 y Fq(d\026)1596 1178 y Fo(\036)1617 1193 y Fr(\()p Fq(x)p Fr(\))17 b(=)228 1243 y(Im)6 b Fh(m)328 1249 y Fo(\036)350 1243 y Fr(\()p Fq(E)12 b Fr(+)d Fq(i\016)r Fr(\))p Fq(=\016)r Fr(.)19 b(W)m(e)13 b(ha)o(v)o(e)403 1346 y Fp(jh)p Fq( )458 1352 y Fo(E)r(;\016)512 1346 y Fq(;)7 b(\036)p Fp(ij)584 1329 y Fm(2)613 1346 y Fr(=)657 1288 y Fi(\024)679 1290 y(Z)727 1346 y Fq( )754 1352 y Fo(E)r(;\016)808 1346 y Fr(\()p Fq(x)p Fr(\))g Fq(d\026)918 1329 y Fo(\036)940 1346 y Fr(\()p Fq(x)p Fr(\))996 1288 y Fi(\025)1018 1296 y Fm(2)1048 1346 y Fr(=)1092 1288 y Fi(\024)1114 1346 y Fq(C)1144 1352 y Fo(E)r(;\016)1203 1318 y Fr(Im)f Fh(m)1303 1324 y Fo(\036)1325 1318 y Fr(\()p Fq(E)12 b Fr(+)d Fq(i\016)r Fr(\))p 1203 1337 274 2 v 1329 1375 a Fq(\016)1481 1288 y Fi(\025)1503 1296 y Fm(2)1528 1346 y Fq(;)-1312 b Fr(\(2.4\))228 1440 y(and)14 b(from)f(\(2.3\))g(w)o(e)h(obtain)351 1535 y Fq(\017)368 1518 y Fm(2)398 1535 y Fr(=)e(\()p Fq(C)488 1541 y Fo(E)r(;\016)542 1535 y Fr(\))558 1518 y Fm(2)583 1479 y Fi(Z)637 1507 y Fr(\()p Fq(x)d Fp(\000)g Fq(E)r Fr(\))776 1492 y Fm(2)804 1507 y Fr(+)h Fq(\016)866 1492 y Fm(2)894 1507 y Fp(\000)f Fq(\016)955 1492 y Fm(2)p 637 1525 338 2 v 656 1564 a Fr(\(\()p Fq(x)g Fp(\000)h Fq(E)r Fr(\))812 1552 y Fm(2)840 1564 y Fr(+)f Fq(\016)901 1552 y Fm(2)920 1564 y Fr(\))936 1552 y Fm(2)986 1535 y Fq(d\026)1033 1518 y Fo(\036)1054 1535 y Fr(\()p Fq(x)p Fr(\))j(=)g(\()p Fq(C)1212 1541 y Fo(E)r(;\016)1266 1535 y Fr(\))1282 1518 y Fm(2)1305 1507 y Fr(Im)7 b Fh(m)1406 1513 y Fo(\036)1428 1507 y Fr(\()p Fq(E)k Fr(+)f Fq(i\016)r Fr(\))p 1305 1525 274 2 v 1432 1564 a Fq(\016)1593 1535 y Fp(\000)f Fq(\016)1654 1518 y Fm(2)1673 1535 y Fq(:)-1457 b Fr(\(2.5\))228 1628 y(By)14 b(com)o(bining)f(\(2.4\))g(and)h(\(2.5\)) o(,)g(w)o(e)g(obtain)622 1721 y Fp(jh)p Fq( )677 1727 y Fo(E)r(;\016)731 1721 y Fq(;)7 b(\036)p Fp(ij)803 1703 y Fm(2)832 1721 y Fr(=)12 b(\()p Fq(\017)909 1703 y Fm(2)937 1721 y Fr(+)d Fq(\016)998 1703 y Fm(2)1017 1721 y Fr(\))1038 1692 y(Im)e Fh(m)1139 1698 y Fo(\036)1161 1692 y Fr(\()p Fq(E)k Fr(+)f Fq(i\016)r Fr(\))p 1038 1711 V 1165 1749 a Fq(\016)1316 1721 y(:)-1100 b Fr(\(2.6\))228 1803 y(One)17 b(can)g(easily)f(v)o(erify)h(that)g(Im)6 b Fh(m)826 1809 y Fo(\036)848 1803 y Fr(\()p Fq(E)14 b Fr(+)d Fq(i\016)r Fr(\))p Fq(=\016)19 b Fr(is)d(a)h(monotonely)f(decreasing)h(function) 228 1853 y(of)i Fq(\016)r Fr(,)h(while)f Fq(\016)r Fr(Im)7 b Fh(m)568 1859 y Fo(\036)590 1853 y Fr(\()p Fq(E)15 b Fr(+)f Fq(i\016)r Fr(\))20 b(is)f(a)g(monotonely)g(increasing)h (function)f(of)g Fq(\016)r Fr(.)35 b(Th)o(us,)21 b(if)228 1903 y Fq(\017)h(>)h(\016)r Fr(,)e(w)o(e)g(ha)o(v)o(e)f(Im)6 b Fh(m)645 1909 y Fo(\036)667 1903 y Fr(\()p Fq(E)16 b Fr(+)e Fq(i\016)r Fr(\))p Fq(=\016)25 b(>)d Fr(Im)7 b Fh(m)1045 1909 y Fo(\036)1067 1903 y Fr(\()p Fq(E)16 b Fr(+)d Fq(i\017)p Fr(\))p Fq(=\017)p Fr(,)22 b(and)e(if)f Fq(\017)k Fp(\024)f Fq(\016)r Fr(,)g(w)o(e)e(ha)o(v)o(e)228 1953 y Fq(\016)r Fr(Im)6 b Fh(m)348 1959 y Fo(\036)370 1953 y Fr(\()p Fq(E)12 b Fr(+)d Fq(i\016)r Fr(\))j Fp(\025)g Fq(\017)7 b Fr(Im)f Fh(m)701 1959 y Fo(\036)723 1953 y Fr(\()p Fq(E)11 b Fr(+)f Fq(i\017)p Fr(\).)18 b(In)c(either)g(case,)h (w)o(e)f(see)h(that)f(\(2.6\))f(implies)696 2023 y Fp(jh)p Fq( )751 2029 y Fo(E)r(;\016)804 2023 y Fq(;)7 b(\036)p Fp(ij)876 2006 y Fm(2)905 2023 y Fq(>)12 b(\017)7 b Fr(Im)f Fh(m)1074 2029 y Fo(\036)1096 2023 y Fr(\()p Fq(E)11 b Fr(+)f Fq(i\017)p Fr(\))p Fq(:)-1027 b Fr(\(2.7\))228 2094 y(This)12 b(pro)o(v)o(es)h(the)g(leftmost)g(inequalit)o(y)e(in)h (\(1.2\))o(.)18 b(Since)13 b(the)g(middle)f(inequalit)o(y)f(is)h(ob)o (vious,)228 2144 y(it)k(remains)h(to)g(pro)o(v)o(e)h(the)f(righ)o (tmost)g(inequalit)o(y)m(.)25 b(Consider)18 b(an)o(y)e Fq( )j Fp(2)d Fq(L)1456 2129 y Fm(2)1475 2144 y Fr(\()p Fj(R)p Fq(;)7 b(d\026)1587 2129 y Fo(\036)1608 2144 y Fr(\))17 b(with)228 2194 y Fp(k)p Fq( )q Fp(k)11 b Fr(=)h(1)i(and)f Fp(k)p Fr(\()p Fq(A)d Fp(\000)f Fq(E)r Fr(\))p Fq( )q Fp(k)j(\024)g Fq(\017)p Fr(.)18 b(Then)387 2296 y Fp(jh)p Fq( )q(;)7 b(\036)p Fp(ij)515 2279 y Fm(2)544 2296 y Fr(=)588 2237 y Fi(\024)610 2240 y(Z)659 2296 y Fq( )q Fr(\()p Fq(x)p Fr(\))g Fq(d\026)797 2279 y Fo(\036)819 2296 y Fr(\()p Fq(x)p Fr(\))875 2237 y Fi(\025)897 2246 y Fm(2)927 2296 y Fr(=)971 2237 y Fi(\024)992 2240 y(Z)1046 2268 y Fp(j)p Fq(x)h Fp(\000)i Fq(E)h Fp(\000)f Fq(i\017)p Fp(j)p Fq( )q Fr(\()p Fq(x)p Fr(\))p 1046 2286 297 2 v 1088 2325 a Fp(j)p Fq(x)f Fp(\000)g Fq(E)i Fp(\000)f Fq(i\017)p Fp(j)1355 2296 y Fq(d\026)1402 2279 y Fo(\036)1423 2296 y Fr(\()p Fq(x)p Fr(\))1479 2237 y Fi(\025)1501 2246 y Fm(2)240 2296 y Fr(\(2.8\))435 2412 y Fp(\024)479 2354 y Fi(\024)501 2356 y(Z)543 2412 y Fr(\(\()p Fq(x)f Fp(\000)g Fq(E)r Fr(\))698 2395 y Fm(2)726 2412 y Fr(+)h Fq(\017)785 2395 y Fm(2)803 2412 y Fr(\))p Fp(j)p Fq( )q Fr(\()p Fq(x)p Fr(\))p Fp(j)927 2395 y Fm(2)953 2412 y Fq(d\026)1000 2395 y Fo(\036)1021 2412 y Fr(\()p Fq(x)p Fr(\))1077 2354 y Fi(\025)d(\024)1128 2356 y(Z)1170 2412 y Fr(\(\()p Fq(x)i Fp(\000)g Fq(E)r Fr(\))1325 2395 y Fm(2)1353 2412 y Fr(+)h Fq(\017)1412 2395 y Fm(2)1430 2412 y Fr(\))1446 2395 y Fn(\000)p Fm(1)1498 2412 y Fq(d\026)1545 2395 y Fo(\036)1567 2412 y Fr(\()p Fq(x)p Fr(\))1623 2354 y Fi(\025)592 2528 y Fr(=)i(\()p Fp(k)p Fr(\()p Fq(A)d Fp(\000)h Fq(E)r Fr(\))p Fq( )q Fp(k)869 2510 y Fm(2)897 2528 y Fr(+)g Fq(\017)956 2510 y Fm(2)974 2528 y Fr(\))995 2499 y(Im)d Fh(m)1096 2505 y Fo(\036)1118 2499 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)p Fr(\))p 995 2518 270 2 v 1122 2556 a Fq(\017)1282 2528 y Fp(\024)h Fr(2)p Fq(\017)c Fr(Im)g Fh(m)1471 2534 y Fo(\036)1493 2528 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)p Fr(\))p Fq(;)228 2608 y Fr(whic)o(h)k(completes)g(the)h(pro)q(of.)p 1692 2608 2 29 v 1694 2581 25 2 v 1694 2608 V 1719 2608 2 29 v eop %%Page: 8 8 8 7 bop 228 113 a Fm(8)407 b(M.)15 b(CHRIST,)f(A.)h(KISELEV,)g(AND)e (Y.)i(LAST)228 213 y Ff(R)n(emarks.)30 b Fr(1.)h(Note)18 b(that)g(if)g(w)o(e)g(happ)q(en)g(to)g(get)h Fq(\017)f Fr(=)h Fq(\016)h Fr(in)e(\(2.3\))o(,)h(then)f(b)q(oth)h(suprema)228 262 y(in)c(\(1.2\))h(are)g(attained)g(\(with)g(the)g(function)g Fq( )995 268 y Fo(E)r(;\016)1065 262 y Fr(de\014ned)h(ab)q(o)o(v)o(e\)) f(and)g(are)g(equal)g(to)g(the)228 312 y(upp)q(er)g(b)q(ound.)22 b(It)15 b(is)g(not)h(hard)f(to)g(see)h(that)g(there)g(are)g(measures)g (and)f Fq(\017)p Fr('s)g(for)g(whic)o(h)g(this)228 362 y(actually)e(happ)q(ens.)303 412 y(2.)42 b(Note)22 b(that)h(the)f (condition)855 378 y Fi(R)882 412 y Fr(\()p Fq(x)15 b Fp(\000)g Fq(E)r Fr(\))1033 397 y Fn(\000)p Fm(4)1085 412 y Fq(d\026)1132 397 y Fo(\036)1153 412 y Fr(\()p Fq(x)p Fr(\))26 b(=)f Fp(1)d Fr(is)g(used)h(only)e(to)h(en-)228 462 y(sure)h(that)g(the)g(leftmost)f(inequalit)o(y)f(holds)i(for)f (arbitrarily)f(small)g Fq(\017)p Fr('s.)44 b(In)23 b(case)g(that)228 478 y Fi(R)255 511 y Fr(\()p Fq(x)10 b Fp(\000)f Fq(E)r Fr(\))395 496 y Fn(\000)p Fm(4)447 511 y Fq(d\026)494 496 y Fo(\036)516 511 y Fr(\()p Fq(x)p Fr(\))j Fq(<)g Fp(1)p Fr(,)h(this)h(inequalit)o(y)f(w)o(ould)g(still)g(hold)h(for)g Fq(\017)p Fr('s)f(larger)h(than)g([)1613 478 y Fi(R)1641 511 y Fr(\()p Fq(x)9 b Fp(\000)228 565 y Fq(E)r Fr(\))277 550 y Fn(\000)p Fm(2)328 565 y Fq(d\026)375 550 y Fo(\036)397 565 y Fr(\()p Fq(x)p Fr(\))p Fq(=)481 532 y Fi(R)508 565 y Fr(\()p Fq(x)g Fp(\000)g Fq(E)r Fr(\))647 550 y Fn(\000)p Fm(4)699 565 y Fq(d\026)746 550 y Fo(\036)768 565 y Fr(\()p Fq(x)p Fr(\)])836 550 y Fm(1)p Fo(=)p Fm(2)888 565 y Fr(.)17 b(The)e(condition)e Fq(\017)e(<)h Fp(k)p Fr(\()p Fq(A)d Fp(\000)g Fq(E)r Fr(\))p Fq(\036)p Fp(k)k Fr(is)h(required)h(in)228 616 y(order)g(to)f(ha)o(v)o(e)f(an)h (inequalit)o(y)f(for)h(sup)p Fp(fjh)p Fq( )q(;)7 b(\036)p Fp(ij)1009 601 y Fm(2)1039 616 y Fp(j)k(k)p Fq( )q Fp(k)h Fr(=)g(1)p Fq(;)7 b Fp(k)p Fr(\()p Fq(A)i Fp(\000)g Fq(E)r Fr(\))p Fq( )q Fp(k)k Fr(=)f Fq(\017)p Fp(g)p Fr(.)18 b(Without)228 666 y(it,)13 b Fp(k)p Fr(\()p Fq(A)c Fp(\000)h Fq(E)r Fr(\))p Fq( )q Fp(k)i Fr(=)f Fq(\017)j Fr(migh)o(t)f(not)h(ev)o (en)g(b)q(e)h(attainable.)i(The)d(inequalit)o(y)431 736 y Fq(\017)7 b Fr(Im)g Fh(m)556 742 y Fo(\036)578 736 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)p Fr(\))i Fq(<)f Fr(sup)q Fp(fjh)p Fq( )q(;)c(\036)p Fp(ij)992 719 y Fm(2)1021 736 y Fp(j)k(k)p Fq( )q Fp(k)g Fr(=)h(1)p Fq(;)7 b Fp(k)p Fr(\()p Fq(A)i Fp(\000)g Fq(E)r Fr(\))p Fq( )q Fp(k)j(\024)g Fq(\017)p Fp(g)-1291 b Fr(\(2.9\))228 805 y(holds)16 b(without)g(an)o(y)g(restriction)h(from)f(ab)q(o)o(v)o(e)g(on)h Fq(\017)p Fr(.)25 b(The)17 b(righ)o(tmost)f(inequalit)o(y)f(is)h(inde-) 228 855 y(p)q(enden)o(t)f(of)f(those)h(t)o(w)o(o)f(conditions)g(and)h (it)f(holds)g(whenev)o(er)h(the)g(appropriate)g(suprem)o(um)228 905 y(is)g(de\014ned,)g(namely)m(,)f(when)h(the)h(distance)f(of)g Fq(E)i Fr(from)d Fq(\033)q Fr(\()p Fq(A)p Fr(\))h(is)g(less)g(then)h Fq(\017)f Fr(\(alternativ)o(ely)m(,)228 955 y(w)o(e)g(can)g(sa)o(y)g (that)f(it)h(alw)o(a)o(ys)f(holds)g(if)g(w)o(e)h(de\014ne)h(the)g (suprem)o(um)f(of)f(the)h(empt)o(y)g(set)h(to)f(b)q(e)228 1005 y(zero\).)720 1096 y Fs(3.)24 b(Pro)q(of)14 b(of)i(Theorem)e(1.5) 303 1171 y Ff(Pr)n(o)n(of)h(of)h Fr(\(i\).)22 b(The)16 b(\014rst)g(set)g(is)f(w)o(ell)f(kno)o(wn)h(to)g(coincide)h(with)e(the) i(set)g(of)f Fq(E)r Fr('s)g(where)228 1220 y(lim)7 b(sup)355 1230 y Fo(\017)p Fn(!)p Fm(0)427 1220 y Fr(Im)g Fh(m)528 1226 y Fo(\036)550 1220 y Fr(\()p Fq(E)h Fr(+)e Fq(i\017)p Fr(\))12 b Fq(>)g Fr(0.)17 b(Let)12 b(us)h(denote)g(this)f(set)h(b)o(y) f Fq(S)1275 1226 y Fo(\026)1298 1220 y Fr(.)17 b(Let)c(us)f(also)g (denote)h(the)228 1270 y(second)e(set)g(b)o(y)f Fq(S)498 1276 y Fm(r)523 1270 y Fr(and)g(the)g(third)g(set)h(b)o(y)f Fq(S)907 1276 y Fm(or)939 1270 y Fr(.)17 b(W)m(e)9 b(will)g(sho)o(w)h (that)g(these)h(three)h(sets)f(coincide)228 1320 y(b)o(y)i(sho)o(wing)f Fq(S)467 1326 y Fo(\026)501 1320 y Fp(\032)g Fq(S)570 1326 y Fm(or)613 1320 y Fp(\032)g Fq(S)682 1326 y Fm(r)709 1320 y Fp(\032)g Fq(S)778 1326 y Fo(\026)801 1320 y Fr(.)17 b(Let)d Fq(E)g Fp(2)d Fq(S)1013 1326 y Fo(\026)1035 1320 y Fr(,)i(then)h(clearly)1285 1287 y Fi(R)1313 1320 y Fr(\()p Fq(x)7 b Fp(\000)h Fq(E)r Fr(\))1449 1305 y Fn(\000)p Fm(4)1501 1320 y Fq(d\026)1548 1305 y Fo(\036)1569 1320 y Fr(\()p Fq(x)p Fr(\))k(=)g Fp(1)228 1370 y Fr(and)h(there)i(exists)f (a)g(sequence)h Fp(f)p Fq(\017)772 1376 y Fo(n)795 1370 y Fp(g)816 1355 y Fn(1)816 1380 y Fo(n)p Fm(=1)893 1370 y Fr(with)f Fq(\017)1005 1376 y Fo(n)1039 1370 y Fq(<)d Fp(k)p Fr(\()p Fq(A)e Fp(\000)g Fq(E)r Fr(\))p Fq(\036)p Fp(k)23 b(8)p Fq(n)p Fr(,)13 b Fq(\017)1408 1376 y Fo(n)1442 1370 y Fp(!)e Fr(0)i(as)h Fq(n)d Fp(!)g(1)p Fr(,)228 1420 y(so)h(that)h(inf)416 1426 y Fo(n)446 1420 y Fr(Im)6 b Fh(m)547 1426 y Fo(\036)569 1420 y Fr(\()p Fq(E)i Fr(+)e Fq(i\017)693 1426 y Fo(n)716 1420 y Fr(\))12 b Fq(>)g(C)i(>)e Fr(0.)17 b(Theorem)c(1.1)f(th)o(us)h(implies)e(that)h(there)i(exists)f (a)228 1469 y(SO)o(AEV)g Fp(f)p Fq( )433 1475 y Fo(n)455 1469 y Fp(g)476 1454 y Fn(1)476 1480 y Fo(n)p Fm(=1)553 1469 y Fr(with)f Fp(k)p Fr(\()p Fq(A)6 b Fp(\000)g Fq(E)r Fr(\))p Fq( )834 1475 y Fo(n)858 1469 y Fp(k)12 b Fr(=)f Fq(\017)951 1475 y Fo(n)997 1469 y Fp(8)p Fq(n)p Fr(,)h(suc)o(h)h(that) g Fp(jh)p Fq( )1305 1475 y Fo(n)1327 1469 y Fq(;)7 b(\036)p Fp(ij)1399 1454 y Fm(2)1428 1469 y Fp(\025)12 b Fq(\017)1489 1475 y Fo(n)1512 1469 y Fr(Im)6 b Fh(m)1612 1475 y Fo(\036)1634 1469 y Fr(\()p Fq(E)j Fr(+)228 1519 y Fq(i\017)259 1525 y Fo(n)282 1519 y Fr(\))34 b Fp(8)p Fq(n)p Fr(.)28 b(Since)18 b(Im)6 b Fh(m)632 1525 y Fo(\036)654 1519 y Fr(\()p Fq(E)14 b Fr(+)e Fq(i\017)790 1525 y Fo(n)813 1519 y Fr(\))17 b Fq(>)g(C)s Fr(,)g(w)o(e)h(see)g(that)g(there)g(m)o(ust)f(b)q(e)h (some)g(constan)o(t)237 1563 y(~)228 1573 y Fq(C)c(>)e Fr(0,)h(so)h(that)g Fp(jh)p Fq( )558 1579 y Fo(n)580 1573 y Fq(;)7 b(\036)p Fp(ij)652 1558 y Fm(2)681 1573 y Fp(\025)734 1563 y Fr(~)725 1573 y Fq(C)r(\017)774 1579 y Fo(n)797 1573 y Fr(\(1)i(+)g(Im)e Fh(m)985 1579 y Fo(\036)1007 1573 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)1138 1579 y Fo(n)1160 1573 y Fr(\)\).)19 b(Th)o(us,)13 b Fp(f)p Fq( )1387 1579 y Fo(n)1410 1573 y Fp(g)1431 1558 y Fn(1)1431 1584 y Fo(n)p Fm(=1)1509 1573 y Fr(is)g(optimally)228 1625 y(ro)q(oted)i(at)f Fq(\036)h Fr(w.r.t.)e Fq(\026)586 1610 y Fo(\036)608 1625 y Fr(,)h(and)h(it)f(follo)o(ws)f(that)i Fq(E)f Fp(2)f Fq(S)1097 1631 y Fm(or)1129 1625 y Fr(.)20 b(W)m(e)14 b(ha)o(v)o(e)h(th)o(us)g(sho)o(wn)f Fq(S)1573 1631 y Fo(\026)1609 1625 y Fp(\032)f Fq(S)1679 1631 y Fm(or)1711 1625 y Fr(.)228 1675 y(Since)h(the)h(inclusion)e Fq(S)606 1681 y Fm(or)649 1675 y Fp(\032)f Fq(S)718 1681 y Fm(r)748 1675 y Fr(is)h(ob)o(vious,)g(it)h(remains)f(to)h(sho)o(w)g Fq(S)1326 1681 y Fm(r)1353 1675 y Fp(\032)e Fq(S)1422 1681 y Fo(\026)1445 1675 y Fr(.)303 1724 y(Let)k Fq(E)g Fp(2)e Fq(S)493 1730 y Fm(r)508 1724 y Fr(,)i(then)g(there)h(exists)f (a)g(SO)o(AEV,)f Fp(f)p Fq( )1112 1730 y Fo(n)1135 1724 y Fp(g)1156 1709 y Fn(1)1156 1735 y Fo(n)p Fm(=1)1220 1724 y Fr(,)h(with)f(inf)1394 1730 y Fo(n)1424 1724 y Fp(jh)p Fq( )1479 1730 y Fo(n)1501 1724 y Fq(;)7 b(\036)p Fp(ij)1573 1709 y Fm(2)1591 1724 y Fq(=)p Fp(k)p Fr(\()p Fq(A)j Fp(\000)228 1774 y Fq(E)r Fr(\))p Fq( )304 1780 y Fo(n)327 1774 y Fp(k)h Fq(>)h(C)i(>)e Fr(0.)k(Let)10 b Fq(\017)627 1780 y Fo(n)661 1774 y Fr(=)h Fp(k)p Fr(\()p Fq(A)p Fp(\000)p Fq(E)r Fr(\))p Fq( )880 1780 y Fo(n)904 1774 y Fp(k)p Fr(,)e(then)h(b)o(y)f(Theorem)h(1.1)e(w)o(e)i(ha)o(v)o(e) f(2)p Fq(\017)1508 1780 y Fo(n)1530 1774 y Fr(Im)d Fh(m)1630 1780 y Fo(\036)1653 1774 y Fr(\()p Fq(E)r Fr(+)228 1824 y Fq(i\017)259 1830 y Fo(n)282 1824 y Fr(\))23 b Fp(\025)h(jh)p Fq( )432 1830 y Fo(n)455 1824 y Fq(;)7 b(\036)p Fp(ij)527 1809 y Fm(2)544 1824 y Fr(.)40 b(Th)o(us,)23 b(w)o(e)e(obtain)g(inf)977 1830 y Fo(n)1007 1824 y Fr(Im)6 b Fh(m)1108 1830 y Fo(\036)1130 1824 y Fr(\()p Fq(E)16 b Fr(+)e Fq(i\017)1270 1830 y Fo(n)1293 1824 y Fr(\))24 b Fq(>)g(C)r(=)p Fr(2)f Fq(>)h Fr(0,)e(and)f(so)228 1874 y(lim)7 b(sup)355 1884 y Fo(\017)p Fn(!)p Fm(0)427 1874 y Fr(Im)g Fh(m)528 1880 y Fo(\036)550 1874 y Fr(\()p Fq(E)13 b Fr(+)e Fq(i\017)p Fr(\))16 b Fq(>)g Fr(0.)26 b(This)16 b(implies)g Fq(E)h Fp(2)f Fq(S)1181 1880 y Fo(\026)1204 1874 y Fr(,)g(and)g(w)o(e)h(th)o(us)g(conclude)g (that)228 1924 y Fq(S)253 1930 y Fm(r)280 1924 y Fp(\032)12 b Fq(S)349 1930 y Fo(\026)371 1924 y Fr(.)p 1692 1924 2 29 v 1694 1897 25 2 v 1694 1924 V 1719 1924 2 29 v 303 2013 a Ff(Pr)n(o)n(of)19 b(of)h Fr(\(ii\).)33 b(Supp)q(ose)20 b(that)g(there)g(exists)g(a)f(SO)o(AEV,)g Fp(f)p Fq( )1332 2019 y Fo(n)1355 2013 y Fp(g)1376 1998 y Fn(1)1376 2023 y Fo(n)p Fm(=1)1440 2013 y Fr(,)h(for)f Fq(A)h Fr(and)f Fq(E)r Fr(,)228 2063 y(suc)o(h)i(that)f(\(1.6\))g(holds.)37 b(Let)21 b Fq(\017)772 2069 y Fo(n)817 2063 y Fr(=)i Fp(k)p Fr(\()p Fq(A)13 b Fp(\000)h Fq(E)r Fr(\))p Fq( )1075 2069 y Fo(n)1098 2063 y Fp(k)p Fr(,)21 b(then)g(b)o(y)f(Theorem)h(1.1)f (w)o(e)g(ha)o(v)o(e)228 2112 y Fq(\017)245 2097 y Fm(1)p Fn(\000)p Fo(\013)245 2123 y(n)311 2112 y Fr(Im)6 b Fh(m)411 2118 y Fo(\036)433 2112 y Fr(\()p Fq(E)12 b Fr(+)d Fq(i\017)564 2118 y Fo(n)587 2112 y Fr(\))j Fp(\025)g(jh)p Fq( )714 2118 y Fo(n)736 2112 y Fq(;)7 b(\036)p Fp(ij)808 2097 y Fm(2)825 2112 y Fq(=)p Fr(2)p Fp(k)p Fr(\()p Fq(A)i Fp(\000)h Fq(E)r Fr(\))p Fq( )1062 2118 y Fo(n)1085 2112 y Fp(k)1106 2097 y Fo(\013)1129 2112 y Fr(,)j(and)h(so)g(clearly)614 2184 y(lim)7 b(inf)629 2209 y Fo(n)p Fn(!1)736 2184 y Fq(\017)753 2167 y Fm(1)p Fn(\000)p Fo(\013)753 2194 y(n)819 2184 y Fr(Im)g Fh(m)920 2190 y Fo(\036)942 2184 y Fr(\()p Fq(E)k Fr(+)f Fq(i\017)1073 2190 y Fo(n)1096 2184 y Fr(\))h(=)h Fp(1)f Fr(\()c Fq(>)12 b Fr(0\))p Fq(:)228 2268 y Fr(Similarly)m(,)19 b(if)h(there)i(exists)g(a)e (sequence,)k Fp(f)p Fq(\017)972 2274 y Fo(n)995 2268 y Fp(g)1016 2253 y Fn(1)1016 2278 y Fo(n)p Fm(=1)1080 2268 y Fr(,)e(where)g Fq(\017)1258 2274 y Fo(n)1303 2268 y Fp(!)h Fr(0)d(as)h Fq(n)i Fp(!)g(1)p Fr(,)e(and)228 2318 y(lim)7 b(inf)343 2324 y Fo(n)p Fn(!1)439 2318 y Fq(\017)456 2303 y Fm(1)p Fn(\000)p Fo(\013)456 2328 y(n)522 2318 y Fr(Im)f Fh(m)622 2324 y Fo(\036)644 2318 y Fr(\()p Fq(E)14 b Fr(+)d Fq(i\017)779 2324 y Fo(n)802 2318 y Fr(\))16 b(=)g Fp(1)g Fr(\()7 b Fq(>)16 b Fr(0\),)h(then)g(w)o (e)g(can)g(c)o(ho)q(ose)h(an)e(appropriate)228 2368 y(SO)o(AEV,)c Fp(f)p Fq( )444 2374 y Fo(n)466 2368 y Fp(g)487 2353 y Fn(1)487 2378 y Fo(n)p Fm(=1)552 2368 y Fr(,)f(with)h Fp(k)p Fr(\()p Fq(A)5 b Fp(\000)g Fq(E)r Fr(\))p Fq( )854 2374 y Fo(n)877 2368 y Fp(k)11 b Fr(=)h Fq(\017)970 2374 y Fo(n)993 2368 y Fr(,)f(so)h(that)g(\(1.6\))f(holds.)17 b(Since)c(w)o(e)f(kno)o(w)f(that)228 2418 y Fq(D)263 2402 y Fo(\013)262 2428 y(\026)287 2418 y Fr(\()p Fq(E)r Fr(\))k(=)g Fp(1)f Fr(\()7 b Fq(>)15 b Fr(0\))h(if)e(and)i(only)f(if)g (lim)7 b(sup)975 2428 y Fo(\017)p Fn(!)p Fm(0)1047 2418 y Fq(\017)1064 2402 y Fm(1)p Fn(\000)p Fo(\013)1130 2418 y Fr(Im)g Fh(m)1231 2424 y Fo(\036)1253 2418 y Fr(\()p Fq(E)13 b Fr(+)e Fq(i\017)p Fr(\))j(=)h Fp(1)g Fr(\()7 b Fq(>)14 b Fr(0\),)i(this)228 2469 y(pro)o(v)o(es)e(\(ii\).)p 1692 2469 V 1694 2442 25 2 v 1694 2469 V 1719 2469 2 29 v 303 2558 a Ff(Pr)n(o)n(of)h(of)h Fr(\(iii\).)k(Let)c Fq(E)h Fr(b)q(e)e(suc)o(h)h(that)f(there)h(exists)g(a)f(SO)o(AEV,)g Fp(f)p Fq( )1410 2564 y Fo(n)1432 2558 y Fp(g)1453 2543 y Fn(1)1453 2568 y Fo(n)p Fm(=1)1518 2558 y Fr(,)g(for)f Fq(A)h Fr(and)228 2608 y Fq(E)j Fr(that)d(is)h(optimally)d(ro)q(oted)k (at)e Fq(\036)h Fr(w.r.t.)e Fq(\027)s Fr(.)23 b(Let)16 b Fq(\017)1089 2614 y Fo(n)1127 2608 y Fr(=)e Fp(k)p Fr(\()p Fq(A)d Fp(\000)g Fq(E)r Fr(\))p Fq( )1371 2614 y Fo(n)1394 2608 y Fp(k)p Fr(,)k(then)h Fq(\017)1555 2614 y Fo(n)1592 2608 y Fp(!)f Fr(0)g(as)p eop %%Page: 9 9 9 8 bop 488 113 a Fm(APPR)o(O)o(XIMA)m(TE)15 b(EIGENVECTORS)f(AND)g (SPECTRAL)g(THEOR)m(Y)244 b(9)228 213 y Fq(n)15 b Fp(!)g(1)p Fr(,)h(and)g(there)i(m)o(ust)e(b)q(e)h(a)f(p)q(ositiv)o(e)g(constan)o (t)h Fq(C)i Fr(suc)o(h)e(that)f(for)g(su\016cien)o(tly)g(small)228 262 y Fq(\017)245 268 y Fo(n)267 262 y Fr(,)d Fp(jh)p Fq( )347 268 y Fo(n)369 262 y Fq(;)7 b(\036)p Fp(ij)441 247 y Fm(2)470 262 y Fq(>)12 b(C)s(\017)564 268 y Fo(n)586 262 y Fr(\(1)c(+)f(Im)g Fh(m)771 268 y Fo(\027)792 262 y Fr(\()p Fq(E)i Fr(+)f Fq(i\017)919 268 y Fo(n)942 262 y Fr(\)\))k Fq(>)f(C)s(\017)1079 268 y Fo(n)1102 262 y Fr(Im)6 b Fh(m)1202 268 y Fo(\027)1223 262 y Fr(\()p Fq(E)j Fr(+)f Fq(i\017)1350 268 y Fo(n)1373 262 y Fr(\).)18 b(By)13 b(Theorem)h(1.1,)228 312 y(w)o(e)g(also)f(ha)o(v)o(e)g(2)p Fq(\017)505 318 y Fo(n)527 312 y Fr(Im)7 b Fh(m)628 318 y Fo(\036)650 312 y Fr(\()p Fq(E)j Fr(+)f Fq(i\017)779 318 y Fo(n)802 312 y Fr(\))j Fp(\025)g(jh)p Fq( )929 318 y Fo(n)951 312 y Fq(;)7 b(\036)p Fp(ij)1023 297 y Fm(2)1040 312 y Fr(,)13 b(and)h(th)o(us)g(w)o(e)g(get)f(2Im)7 b Fh(m)1488 318 y Fo(\036)1510 312 y Fr(\()p Fq(E)k Fr(+)d Fq(i\017)1639 318 y Fo(n)1662 312 y Fr(\))k Fq(>)228 362 y(C)s Fr(Im)6 b Fh(m)361 368 y Fo(\027)382 362 y Fr(\()p Fq(E)11 b Fr(+)f Fq(i\017)513 368 y Fo(n)535 362 y Fr(\))k(for)g(small)f Fq(\017)755 368 y Fo(n)777 362 y Fr(.)18 b(This)c(implies)f(that)720 459 y(lim)7 b(inf)746 484 y Fo(\017)p Fn(!)p Fm(0)848 431 y Fr(Im)f Fh(m)948 437 y Fo(\027)969 431 y Fr(\()p Fq(E)11 b Fr(+)f Fq(i\017)p Fr(\))p 847 449 270 2 v 847 487 a(Im)c Fh(m)948 493 y Fo(\036)970 487 y Fr(\()p Fq(E)11 b Fr(+)e Fq(i\017)p Fr(\))1133 459 y Fq(<)j Fp(1)p Fq(:)-1003 b Fr(\(3.1\))228 555 y(It)15 b(is)f(w)o(ell)g(kno)o(wn)h(that)f(the)i(part)f(of)f Fq(\027)j Fr(whic)o(h)d(is)h(singular)f(w.r.t.)g Fq(\026)1331 561 y Fo(\036)1367 555 y Fr(is)h(supp)q(orted)h(on)f(the)228 605 y(set)d(of)f Fq(E)r Fr('s)g(where)i(lim)582 611 y Fo(\017)p Fn(!)p Fm(0)648 605 y Fr(\(Im)7 b Fh(m)765 611 y Fo(\027)785 605 y Fr(\()p Fq(E)g Fr(+)t Fq(i\017)p Fr(\))p Fq(=)p Fr(Im)g Fh(m)1044 611 y Fo(\036)1066 605 y Fr(\()p Fq(E)g Fr(+)t Fq(i\017)p Fr(\)\))12 b(=)g Fp(1)p Fr(,)f(and)g(th)o(us)h(w)o(e)f(conclude)228 654 y(that)j(the)g (restriction)h(of)e(this)h(part)g(to)g Fq(S)j Fr(m)o(ust)d(v)n(anish.)p 1692 654 2 29 v 1694 628 25 2 v 1694 654 V 1719 654 2 29 v 660 754 a Fs(4.)24 b(Some)15 b(general)g(consequences)303 829 y Fr(Let)20 b Fq(H)421 814 y Fm(\012)418 841 y Fo(V)466 829 y Fr(b)q(e)g(the)g(Sc)o(hr\177)-21 b(odinger)20 b(op)q(erator)h(on) e(\012)i Fp(\032)g Fj(R)1209 814 y Fo(d)1248 829 y Fr(de\014ned)g(b)o (y)f(\(1.7\))o(.)36 b(Denote)228 880 y(b)o(y)17 b Fq(B)320 886 y Fo(R)365 880 y Fr(the)h(ball)e(of)g(radius)i Fq(R)f Fr(cen)o(tered)i(at)e(the)h(origin,)e(and)h(denote)i(b)o(y)e Fq(W)1510 865 y Fo(l)1504 890 y(m)1553 880 y Fr(the)g(usual)228 930 y(Sob)q(olev)d(spaces)i(of)e(functions)h Fq(f)20 b Fr(suc)o(h)15 b(that)g Fq(D)999 915 y Fo(l)1012 930 y Fq(f)20 b Fr(exists)15 b(in)f(the)h(distributional)f(sense)i(and)228 946 y Fi(R)255 980 y Fr(\()p Fp(j)p Fq(u)p Fp(j)319 965 y Fo(m)358 980 y Fr(+)8 b Fp(j)p Fq(D)445 965 y Fo(l)458 980 y Fq(u)p Fp(j)494 965 y Fo(m)524 980 y Fr(\))f Fq(dx)k(<)h Fp(1)p Fr(.)17 b(One)d(general)g(consequence)h(of)e(Theorem)g(1.5)f(is) h(that)h(w)o(e)f(can)228 1029 y(reco)o(v)o(er)i(\(and)f(pro)o(vide)f(a) h(simpler)g(pro)q(of)f(of)s(\))h(Theorem)g(1)g(of)f([)p Fs(20)o Fr(].)18 b(That)c(is)303 1107 y Fg(Theorem)i Fr(4.1)p Fg(.)k Ff(Assume)15 b(that)g(the)g(p)n(otential)g Fq(V)9 b Fr(\()p Fq(x)p Fr(\))15 b Ff(b)n(elongs)g(to)g Fq(L)1395 1092 y Fn(1)1395 1119 y Fm(lo)q(c)1454 1107 y Ff(and)g(is)g(b)n(ounde)n(d)228 1157 y(fr)n(om)20 b(b)n(elow,)h(and)h Fr(\012)f Ff(is)f(a)h(domain)h(with)e(pie)n(c)n(ewise)g(smo)n(oth)h(b)n (oundary.)38 b(Supp)n(ose)22 b(that)228 1207 y(ther)n(e)12 b(exists)h(a)g Fq(W)520 1192 y Fm(2)514 1218 y(2)p Fo(;)p Fm(Lo)q(c)608 1207 y Ff(distributional)f(solution)g Fq(u)p Fr(\()p Fq(x;)7 b(E)r Fr(\))12 b Ff(of)h(the)g(gener)n(alize)n(d)f (eigenfunction)228 1257 y(e)n(quation)781 1329 y Fr(\()p Fq(H)835 1312 y Fm(\012)832 1340 y Fo(V)870 1329 y Fp(\000)e Fq(E)r Fr(\))p Fq(u)p Fr(\()p Fq(x;)d(E)r Fr(\))k(=)h(0)-941 b(\(4.1\))228 1404 y Ff(satisfying)13 b(the)g Fr(\()p Ff(Dirichlet)p Fr(\))f Ff(b)n(oundary)i(c)n(onditions)g(and)f(such)h (that)f(for)f(some)i Fq(\013)p Ff(,)f Fr(0)e Fp(\024)h Fq(\013)f Fp(\024)h Fr(1)p Ff(,)228 1454 y(we)i(have)595 1538 y Fr(lim)7 b(inf)606 1564 y Fo(R)p Fn(!1)717 1538 y Fq(R)749 1521 y Fn(\000)p Fo(\013)832 1481 y Fi(Z)796 1600 y Fo(B)821 1604 y Fd(R)844 1600 y Fn(\\)p Fm(\012)907 1538 y Fp(j)p Fq(u)p Fr(\()p Fq(x;)g(E)r Fr(\))p Fp(j)1063 1521 y Fm(2)1086 1538 y Fq(dx)k(<)h Fp(1)f Fr(\()c(=)12 b(0\))p Fq(:)-1128 b Fr(\(4.2\))228 1669 y Ff(Fix)15 b(some)g(c)n(omp)n(actly)g(supp)n(orte)n(d)g Fq(\036)p Fr(\()p Fq(x)p Fr(\))c Fp(2)g Fq(L)943 1654 y Fm(2)962 1669 y Fr(\(\012\))p Ff(,)k(such)g(that)775 1707 y Fi(Z)775 1826 y Fm(\012)823 1763 y Fq(\036)p Fr(\()p Fq(x)p Fr(\))p Fq(u)p Fr(\()p Fq(x;)7 b(E)r Fr(\))g Fq(dx)j Fp(6)p Fr(=)i(0)p Fq(:)228 1886 y Ff(Then)j(we)g(have)792 1958 y Fq(D)827 1941 y Fo(\013)826 1970 y(\026)846 1962 y Fd(\036)869 1958 y Fr(\()p Fq(E)r Fr(\))c Fq(>)h Fr(0)g(\()7 b(=)k Fp(1)p Fr(\))p Fq(:)303 2037 y Ff(R)n(emark.)17 b Fr(Our)10 b(new)f(pro)q(of)g(of)g(this)g(theorem)h(is)f(based)h(on)f(using)g (appropriately)g(smo)q(othened)228 2087 y(cuto\013s)15 b(of)e(the)h(solution)f(in)h(order)h(to)e(construct)j(an)d (appropriately)h(ro)q(oted)g(SO)o(AEV.)303 2159 y(Theorem)g(4.1)g(is)g (related)h(to)f(the)h(w)o(ell-kno)o(wn)e(theorem)i(of)f(Sc)o(h'nol)g (\(see)h([)p Fs(3,)h(26,)h(27)p Fr(]\))228 2209 y(whic)o(h)e(sa)o(ys)g (that)g(the)h(existence)h(of)e(a)g(p)q(olynomially)d(b)q(ounded)k (\(but)f(not)g Fq(L)1480 2194 y Fm(2)1499 2209 y Fr(\))g(solution)g(of) 228 2259 y(\(4.1\))e(implies)f(that)h(the)h(energy)h Fq(E)g Fr(b)q(elongs)f(to)f(the)h(essen)o(tial)g(sp)q(ectrum)h(of)e Fq(H)1507 2244 y Fm(\012)1504 2271 y Fo(V)1532 2259 y Fq(:)g Fr(Ho)o(w)o(ev)o(er,)228 2309 y(Sc)o(h'nol's)18 b(theorem)i(giv)o(es)f(no)g(further)h(information)e(on)h(the)h (structure)h(of)e(the)h(essen)o(tial)228 2359 y(sp)q(ectrum)e(or)e(on)h (sp)q(ectral)h(m)o(ultiplicit)o(y)m(.)23 b(A)17 b(ma)r(jor)e(adv)n(an)o (tage)h(of)g(our)h(new)g(approac)o(h)f(is)228 2409 y(that)e(it)g(allo)o (ws)e(us)j(to)f(go)f(b)q(ey)o(ond)i(merely)f(c)o(haracterizing)g(the)h (b)q(eha)o(vior)f(of)f(sp)q(ectral)j(mea-)228 2458 y(sures,)f(and)g(to) f(also)g(prob)q(e)h(sp)q(ectral)g(m)o(ultiplicit)o(y)e(through)h(the)h (existence)h(of)e(appropriate)228 2508 y(solutions.)26 b(Roughly)16 b(sp)q(eaking,)h(the)h(existence)g(of)f(a)f(n)o(um)o(b)q (er,)i(sa)o(y)e Fq(m)p Fr(,)i(of)e(linearly)g(inde-)228 2558 y(p)q(enden)o(t)e(solutions)f(ha)o(ving)f(appropriate)i(deca)o(y)f (prop)q(erties)i(on)e(a)g(su\016cien)o(tly)h(large)f(set)h(of)228 2608 y(energies)g(can)g(b)q(e)g(used)g(to)f(deduce)i(that)f(the)g(sp)q (ectrum)h(has)e(m)o(ultiplicit)o(y)e(of)i(at)g(least)h Fq(m)f Fr(on)p eop %%Page: 10 10 10 9 bop 228 113 a Fm(10)390 b(M.)15 b(CHRIST,)f(A.)h(KISELEV,)g(AND)e (Y.)i(LAST)228 213 y Fr(this)d(set.)18 b(There)13 b(are)g(v)n(arious)e (precise)i(form)o(ulations)e(that)h(one)g(can)g(giv)o(e)g(to)f(this)h (result.)19 b(W)m(e)228 262 y(will)12 b(presen)o(t)k(a)d(form)o (ulation)f(with)i(resp)q(ect)i(to)e(Hausdor\013)g(measures.)303 312 y(Let)h Fq(h)402 297 y Fo(\013)441 312 y Fr(b)q(e)g(the)h Fq(\013)p Fr(-dimensional)d(Hausdor\013)j(measure)g(on)e Fj(R)p Fr(.)22 b(Recall)14 b(that)h(for)g(a)g(mea-)228 362 y(surable)f(set)h Fq(S)r Fr(,)710 441 y Fq(h)734 424 y Fo(\013)758 441 y Fr(\()p Fq(S)r Fr(\))e(=)i(lim)873 468 y Fo(\016)q Fn(!)p Fm(0)978 441 y Fr(inf)946 468 y Fo(\016)q Fm(-co)o(v)o(ers)1068 402 y Fi(X)1088 489 y Fo(\015)1135 441 y Fp(j)p Fq(I)1165 447 y Fo(\015)1186 441 y Fp(j)1198 424 y Fo(\013)1221 441 y Fq(;)-1005 b Fr(\(4.3\))228 555 y(where)14 b(the)g(in\014m)o(um)f(is)g(tak)o(en)g(o) o(v)o(er)h(all)e(co)o(v)o(erings)i(of)f Fq(S)j Fr(b)o(y)d(coun)o(table) h(unions)f(of)g(in)o(terv)n(als)228 605 y Fq(I)246 611 y Fo(\015)278 605 y Fr(with)e(size)h(not)f(exceeding)h Fq(\016)r Fr(.)17 b(When)11 b Fq(\013)g Fr(=)h(1,)f Fq(h)1019 590 y Fo(\013)1053 605 y Fr(coincides)h(with)e(the)i(Leb)q(esgue)h (measure.)303 685 y Fg(Theorem)j Fr(4.2)p Fg(.)k Ff(L)n(et)h Fq(H)708 670 y Fm(\012)705 696 y Fo(V)755 685 y Ff(b)n(e)g(the)g (Schr\177)-21 b(odinger)21 b(op)n(er)n(ator)g(on)g Fr(\012)i Fp(\032)g Fj(R)1496 670 y Fo(d)1537 685 y Ff(de\014ne)n(d)f(by)228 735 y Fr(\(1.7\))o Ff(,)12 b(with)g Fq(V)21 b Fp(2)11 b Fq(L)538 719 y Fn(1)538 746 y Fm(lo)q(c)582 735 y Ff(.)18 b(Assume)12 b(that)g(for)g(every)g(ener)n(gy)g Fq(E)i Ff(in)e(a)g(Bor)n(el)g(set)g Fq(S)j Ff(of)d(p)n(ositive)g Fq(h)1699 719 y Fo(\013)228 784 y Ff(me)n(asur)n(e,)g(we)f(have)h Fq(n)f Ff(line)n(arly)g(indep)n(endent)i Fq(W)1002 769 y Fm(2)996 796 y(2)p Fo(;)p Fm(Lo)q(c)1088 784 y Ff(solutions)f Fq(u)1283 790 y Fo(i)1297 784 y Fr(\()p Fq(x;)7 b(E)r Fr(\))k Ff(of)g(the)g(gener)n(alize)n(d)228 834 y(eigenfunction)16 b(e)n(quation)775 908 y Fr(\()p Fq(H)829 891 y Fm(\012)826 918 y Fo(V)864 908 y Fp(\000)9 b Fq(E)r Fr(\))p Fq(u)978 914 y Fo(i)992 908 y Fr(\()p Fq(x;)e(E)r Fr(\))k(=)h(0)-948 b(\(4.4\))228 983 y Ff(satisfying)13 b(the)g Fr(\()p Ff(Dirichlet)p Fr(\))g Ff(b)n(oundary)h(c)n(onditions)g(and)g(such)g (that)f(for)g(some)g Fq(\013;)g Fr(0)e Fp(\024)h Fq(\013)f Fp(\024)h Fr(1)p Fq(;)228 1033 y Ff(we)i(have)754 1106 y Fr(lim)7 b(inf)766 1133 y Fo(R)p Fn(!1)876 1106 y Fq(R)908 1089 y Fn(\000)p Fo(\013)958 1106 y Fp(k)p Fq(u)1003 1112 y Fo(i)1016 1106 y Fp(k)1037 1089 y Fm(2)1037 1117 y Fo(B)1062 1121 y Fd(R)1099 1106 y Fq(<)12 b Fp(1)p Fq(:)-969 b Fr(\(4.5\))228 1194 y Ff(Then)15 b(the)g(sp)n(e)n(ctrum)f (has)i(multiplicity)d(at)i(le)n(ast)g Fq(n)f Ff(on)i(the)f(set)f Fq(S)r Ff(.)303 1272 y(R)n(emarks.)k Fr(1.)g(P)o(oten)o(tially)13 b(the)h(most)g(useful)g(is)f(the)i(case)g Fq(\013)c Fr(=)h(1,)h(where)i (Theorem)f(1.5)228 1322 y(implies)d(the)i(existence)i(of)c(absolutely)h (con)o(tin)o(uous)h(sp)q(ectrum)h(of)e(m)o(ultiplicit)o(y)e Fq(n)i Fr(on)g(the)h(set)228 1372 y Fq(S)r Fr(.)303 1421 y(2.)k(Results)d(parallel)f(to)h(Theorems)g(4.1)f(and)h(4.2)f(hold)g (in)g(the)i(discrete)g(setting,)f(where)228 1471 y(the)g(Sc)o(hr\177) -21 b(odinger)15 b(op)q(erator)f Fq(h)718 1456 y Fm(\012)718 1482 y Fo(v)758 1471 y Fr(is)f(de\014ned)i(b)o(y)607 1553 y Fq(h)631 1536 y Fm(\012)631 1563 y Fo(v)657 1553 y Fq(u)p Fr(\()p Fq(n)p Fr(\))c(=)877 1513 y Fi(X)793 1604 y Fo(m)p Fn(2)p Fm(\012)p Fo(;)6 b Fn(j)p Fo(m)p Fn(\000)p Fo(n)p Fn(j)p Fm(=1)1029 1553 y Fq(u)p Fr(\()p Fq(m)p Fr(\))j(+)h Fq(v)q Fr(\()p Fq(n)p Fr(\))p Fq(u)p Fr(\()p Fq(n)p Fr(\))p Fq(:)-1115 b Fr(\(4.6\))228 1672 y(No)14 b(restrictions)h(on)e(the)i(p)q(oten)o(tial)e(are)h(needed)i (in)d(this)h(case.)708 1774 y Fs(5.)24 b(A)16 b(concrete)f(application) 303 1849 y Fr(In)d(this)g(section)h(w)o(e)f(sk)o(etc)o(h)h(a)f (concrete)i(application)c(of)i(our)g(results)h(to)f(sp)q(ectral)h (analy-)228 1899 y(sis)f(of)g(Sc)o(hr\177)-21 b(odinger)13 b(op)q(erators.)18 b(Complete)13 b(pro)q(ofs)f(will)f(app)q(ear)i (elsewhere.)19 b(The)13 b(result)g(w)o(e)228 1948 y(presen)o(t)f(here)h (can)e(b)q(e)h(view)o(ed)f(as)g(a)g(generalization)f(of)h(recen)o(t)i (results)f(concerning)g(the)f(abso-)228 1998 y(lutely)f(con)o(tin)o (uous)h(sp)q(ectrum)h(of)e(one-dimensional)f(Sc)o(hr\177)-21 b(odinger)11 b(op)q(erators)h(with)e(deca)o(ying)228 2048 y(p)q(oten)o(tials)j([)p Fs(5,)j(7,)g(25)p Fr(])d(to)h(Sc)o (hr\177)-21 b(odinger)14 b(op)q(erators)h(on)f(strips.)303 2126 y Fg(Theorem)i Fr(5.1)p Fg(.)k Ff(L)n(et)d Fr(\012)h Ff(b)n(e)g(a)f(cylinder)h(in)f Fj(R)1051 2111 y Fo(d)1071 2126 y Fq(;)g Fr(\012)f(=)h Fq(D)c Fp(\002)e Fr(\()p Fp(\0001)p Fq(;)c Fp(1)p Fr(\))p Fq(;)17 b Ff(wher)n(e)g Fq(D)i Ff(is)e(a)228 2176 y(c)n(omp)n(act)e(domain)g(in)g Fj(R)617 2161 y Fo(d)p Fn(\000)p Fm(1)693 2176 y Ff(with)f(a)h(smo)n (oth)f(b)n(oundary.)20 b(L)n(et)14 b Fp(f)p Fq(E)1278 2182 y Fo(n)1300 2176 y Fp(g)1321 2161 y Fn(1)1321 2186 y Fo(n)p Fm(=1)1400 2176 y Ff(b)n(e)h(the)g(eigenvalues)228 2226 y(of)h(the)h(Dirichlet)f(L)n(aplacian)h Fp(\000)p Fr(\001)782 2211 y Fo(D)829 2226 y Ff(on)g Fq(D)i Ff(in)d(incr)n(e)n (asing)h(or)n(der)f Fr(\()p Ff(c)n(ounting)i(multiplicity)p Fr(\))p Ff(.)228 2280 y(If)f(the)h(p)n(otential)g Fq(V)9 b Fr(\()p Fq(x)p Fr(\))18 b Ff(satis\014es)g Fp(j)p Fq(V)9 b Fr(\()p Fq(x)p Fr(\))p Fp(j)16 b(\024)h Fq(C)s Fr(\(1)11 b(+)h Fp(j)p Fq(x)p Fp(j)p Fr(\))1152 2265 y Fn(\000)1181 2254 y Fc(1)p 1181 2259 15 2 v 1181 2275 a(2)1201 2265 y Fn(\000)p Fo(\017)1260 2280 y Ff(for)17 b(some)h Fq(\017)f(>)g Fr(0)p Ff(,)h(then)g(the)228 2330 y(absolutely)f(c)n(ontinuous)i(sp)n (e)n(ctrum)e(of)g(the)g(op)n(er)n(ator)g Fq(H)1135 2315 y Fm(\012)1132 2341 y Fo(V)1178 2330 y Ff(has)h(the)f(same)h(structur)n (e)f(as)g(the)228 2380 y(absolutely)d(c)n(ontinuous)h(sp)n(e)n(ctrum)f (of)g(the)h(unp)n(erturb)n(e)n(d)f(op)n(er)n(ator)f Fq(H)1340 2365 y Fm(\012)1337 2390 y(0)1366 2380 y Ff(.)19 b(Namely,)14 b(the)g(abso-)228 2430 y(lutely)g(c)n(ontinuous)i(sp)n(e)n(ctrum)f(c)n (overs)f Fr([)p Fq(E)891 2436 y Fm(1)909 2430 y Fq(;)7 b Fp(1)p Fr(\))14 b Ff(and)i(has)f(multiplicity)f Fr(2)p Fq(n)g Ff(on)h Fr([)p Fq(E)1536 2436 y Fo(n)1558 2430 y Fq(;)7 b(E)1608 2436 y Fo(n)p Fm(+1)1672 2430 y Fr(])p Ff(.)303 2508 y(R)n(emarks.)18 b Fr(1.)g(W)m(e)c(use)g(metho)q(ds)h(of) e([)p Fs(5)o Fr(])h(to)f(pro)o(v)o(e)h(this)g(result.)303 2558 y(2.)j(Our)d(pro)q(of)g(also)f(\014nds)h(the)g(asymptotic)f(b)q (eha)o(vior)h(of)f(solutions)g(of)g(the)h(generalized)228 2608 y(eigen)o(v)n(alue)f(equation)h(for)f(a.e.)18 b Fq(E)r Fr(.)p eop %%Page: 11 11 11 10 bop 488 113 a Fm(APPR)o(O)o(XIMA)m(TE)15 b(EIGENVECTORS)f(AND)g (SPECTRAL)g(THEOR)m(Y)227 b(11)303 213 y Fr(The)12 b(outline)g(of)f (the)i(pro)q(of)f(is)g(the)h(follo)o(wing:)i(Giv)o(en)c Fq(E)j Fp(6)p Fr(=)d Fq(E)1283 219 y Fo(n)1318 213 y Fr(in)h(the)g(sp)q(ectrum,)i(let)e Fq(N)228 262 y Fr(b)q(e)i(suc)o(h)h (that)f Fq(E)499 268 y Fo(N)542 262 y Fq(<)e(E)i(<)e(E)706 268 y Fo(N)s Fm(+1)780 262 y Fq(:)h Fr(W)m(e)h(call)f(all)g(mo)q(des)h (\(eigenfunctions\))h Fq(\036)1468 268 y Fm(1)1486 262 y Fr(\()p Fq(y)q Fr(\))p Fq(;)7 b(:::;)g(\036)1638 268 y Fo(N)1668 262 y Fr(\()p Fq(y)q Fr(\))228 312 y(of)13 b Fp(\000)p Fr(\001)342 297 y Fo(D)385 312 y Fr(corresp)q(onding)i(to)e Fq(E)733 318 y Fm(1)752 312 y Fq(;)7 b(:::;)g(E)857 318 y Fo(N)899 312 y Fr(op)q(en,)13 b(while)h(the)g(rest)h(of)e(the)h(mo)q (des)g Fq(\036)1542 318 y Fo(N)s Fm(+1)1615 312 y Fr(\()p Fq(y)q Fr(\))p Fq(;)7 b(:::)228 362 y Fr(are)18 b(closed.)29 b(W)m(e)17 b(consider)h(the)g(decomp)q(osition)f(of)g(the)h(Hilb)q(ert) g(space)g Fp(H)g Fr(=)g Fp(H)1567 368 y Fm(O)1606 362 y Fr(+)12 b Fp(H)1685 368 y Fm(C)1711 362 y Fr(,)228 412 y(where)h Fp(H)381 418 y Fm(O)421 412 y Fr(is)f(generated)i(b)o(y)e (functions)g(of)g(the)h(t)o(yp)q(e)g Fq(f)t Fr(\()p Fq(x)p Fr(\))p Fq(\036)1195 418 y Fo(i)1209 412 y Fr(\()p Fq(y)q Fr(\))p Fq(;)g(i)f Fr(=)f(1)p Fq(;)c(:::;)g(N)e Fr(,)10 b Fq(f)16 b Fp(2)11 b Fq(L)1614 397 y Fm(2)1633 412 y Fr(\()p Fq(dx)p Fr(\),)228 462 y(while)j Fp(H)372 468 y Fm(C)413 462 y Fr(is)g(generated)i(b)o(y)f(functions)g(of)f(the)h(t)o (yp)q(e)g Fq(f)t Fr(\()p Fq(x)p Fr(\))p Fq(\036)1203 468 y Fo(i)1218 462 y Fr(\()p Fq(y)q Fr(\),)g Fq(i)e Fr(=)h Fq(N)g Fr(+)c(1)p Fq(;)d(:)g(:)g(:)e(:)14 b Fr(Let)h Fq(P)1683 468 y Fm(O)1711 462 y Fr(,)228 511 y Fq(P)255 517 y Fm(C)294 511 y Fr(b)q(e)g(the)f(orthogonal)f(pro)r(jections)i(on) e(these)j(subspaces.)k(One)15 b(can)f(use)h(the)f(metho)q(ds)h(of)228 561 y([)p Fs(5)o Fr(])f(to)f(study)i(the)f(\014nite)g(dimensional)f (system)706 637 y(\()p Fp(\000)p Fr(\001)p Fq(P)816 643 y Fm(O)853 637 y Fr(+)c Fq(P)921 643 y Fm(O)948 637 y Fq(V)h(P)1009 643 y Fm(O)1036 637 y Fr(\))p Fq(u)1076 643 y Fm(o)1106 637 y Fr(=)i Fq(E)r(u)1207 643 y Fm(o)1225 637 y Fq(;)-1009 b Fr(\(5.1\))228 715 y(and)14 b(to)h(sho)o(w)f(that)h (for)f(a.e.)20 b Fq(E)14 b Fp(2)f Fr([)p Fq(E)831 721 y Fo(n)853 715 y Fq(;)7 b(E)903 721 y Fo(n)p Fm(+1)966 715 y Fr(],)14 b(it)g(has)h(2)p Fq(n)f Fr(linearly)g(indep)q(enden)o(t) i(b)q(ounded)228 764 y(solutions.)26 b(Using)16 b(a)h(certain)g (compactness)h(argumen)o(t)e(to)h(con)o(trol)f(mo)q(des)h(of)f Fp(\000)p Fr(\001)1599 749 y Fo(D)1646 764 y Fr(that)228 814 y(are)i(closed)g(for)g(a)f(giv)o(en)h(energy)g(region,)g(w)o(e)g (also)g(sho)o(w)f(that)h(for)g(a.e.)f Fq(E)j Fp(2)e Fr([)p Fq(E)1564 820 y Fo(n)1585 814 y Fq(;)7 b(E)1635 820 y Fo(n)p Fm(+1)1699 814 y Fr(],)228 864 y(there)14 b(exist)e(2)p Fq(n)g Fr(linearly)g(indep)q(enden)o(t)i(b)q(ounded)f(solutions)f(of)g (the)h(generalized)g(eigen)o(v)n(alue)228 914 y(equation)i(\()p Fq(H)453 899 y Fm(\012)450 925 y Fo(V)489 914 y Fp(\000)c Fq(E)r Fr(\))p Fq(u)j Fr(=)h(0.)22 b(By)16 b(Theorem)g(4.2,)f(it)g (follo)o(ws)f(that)h(the)h(m)o(ultiplicit)o(y)e(of)h(the)228 964 y(absolutely)g(con)o(tin)o(uous)g(sp)q(ectrum)h(in)f(this)h(region) f(is)g(at)g(least)g(2)p Fq(n)p Fr(.)22 b(W)m(e)15 b(then)h(use)g(b)q (ounds)228 1014 y(on)h(generalized)h(eigenfunctions)f(pro)o(v)o(ed)h (in)f([)p Fs(22)o Fr(])g(to)g(sho)o(w)g(that)g(the)h(m)o(ultiplicit)o (y)d(of)i(the)228 1063 y(absolutely)c(con)o(tin)o(uous)h(sp)q(ectrum)h (cannot)f(b)q(e)h(higher.)863 1172 y Fs(References)245 1238 y Fu([1])i(N.)f(Aronsza)r(jn,)f Fl(On)i(a)g(pr)n(oblem)h(of)f (Weyl)g(in)g(the)g(the)n(ory)h(of)e(Sturm-Liouvil)r(le)i(e)n(quations,) h Fu(Am.)d(J.)300 1279 y(Math.)11 b Fb(79)i Fu(\(1957\),)c(597{610)245 1321 y([2])17 b(J.)12 b(Avron)f(and)f(B.)i(Simon,)f Fl(T)m(r)n(ansient) h(and)i(r)n(e)n(curr)n(ent)g(sp)n(e)n(ctrum,)g Fu(J.)e(F)m(unct.)e (Anal.)h Fb(43)h Fu(\(1981\),)e(1{31)245 1362 y([3])17 b(J.)12 b(Berezanskii,)c(Expansions)h(in)i(Eigenfunction)o(s)e(of)i (Selfadjoin)o(t)d(Op)q(erators,)h(T)m(ransl.)i(Math.)f(Mono.,)300 1404 y(V)m(ol.)i Fb(17)p Fu(,)g(Amer.)g(Math.)f(So)q(c.,)f(Pro)o (vidence,)f(R.I.,)h(1968)245 1445 y([4])17 b(F.A.)f(Berezin)e(and)h (M.A.)h(Sh)o(ubin,)e Fl(The)j(Schr\177)-18 b(odinger)18 b(Equation,)g Fu(Klu)o(w)o(er)e(Academic)f(Publishers,)300 1487 y(Dordrec)o(h)o(t,)9 b(Boston,)h(1991)245 1528 y([5])17 b(M.)c(Christ)e(and)g(A.)h(Kiselev,)f Fl(A)o(bsolutely)k(c)n(ontinuous) f(sp)n(e)n(ctrum)g(for)f(one-dimensional)h(Schr\177)-18 b(odinger)300 1570 y(op)n(er)n(ators)18 b(with)e(slow)r(ly)g(de)n(c)n (aying)j(p)n(otentials:)24 b(Some)18 b(optimal)f(r)n(esults,)g Fu(J.)e(Amer.)h(Math.)f(So)q(c.)g Fb(11)300 1612 y Fu(\(1998\),)10 b(771{797)245 1653 y([6])17 b(D.)j(Damanik,)f Ft(\013)p Fl(-c)n(ontinuity)i(pr)n(op)n(erties)g(of)f(one-dimensional)h (quasicrystals,)h Fu(Comm)o(un.)d(Math.)300 1695 y(Ph)o(ys.)11 b Fb(192)i Fu(\(1998\),)c(169{182)245 1736 y([7])17 b(P)m(.)9 b(Deift)e(and)g(R.)h(Killip,)g Fl(On)i(the)h(absolutely)g(c)n (ontinuous)f(sp)n(e)n(ctrum)h(of)e(one-dimensional)i(Schr\177)-18 b(odinger)300 1778 y(op)n(er)n(ators)14 b(with)f(squar)n(e)g(summable)h (p)n(otentials,)f Fu(Comm)o(un.)f(Math.)f(Ph)o(ys.)g Fb(203)i Fu(\(1999\),)c(341{347)245 1819 y([8])17 b(R.)g(del)f(Rio,)h (S.)g(Jitomirsk)n(a)o(y)o(a,)f(Y.)h(Last,)g(and)f(B.)g(Simon,)i Fl(Op)n(er)n(ators)g(with)g(singular)f(c)n(ontinuous)300 1861 y(sp)n(e)n(ctrum,)12 b(IV.)e(Hausdor\013)g(dimensions,)g(r)n(ank)h (one)g(p)n(erturb)n(ations,)g(and)f(lo)n(c)n(alization,)i Fu(J.)c(Anal.)g(Math.)300 1902 y Fb(69)13 b Fu(\(1996\),)c(153{200)245 1944 y([9])17 b(W.)12 b(Donogh)o(ue,)d Fl(On)k(p)n(erturb)n(ation)i(of) d(sp)n(e)n(ctr)n(a,)i Fu(Comm)o(un.)e(Pure)f(Appl.)g(Math.)g Fb(18)h Fu(\(1965\),)e(559{579)228 1985 y([10])16 b(D.)11 b(Gilb)q(ert)d(and)i(D.)g(P)o(earson,)f Fl(On)j(sub)n(or)n(dinacy)i (and)e(analysis)f(of)h(the)g(sp)n(e)n(ctrum)h(of)f(one-dimensional)300 2027 y(Schr\177)-18 b(odinger)15 b(op)n(er)n(ators,)f Fu(J.)d(Math.)g(Anal.)g(Appl.)g Fb(128)i Fu(\(1987\),)d(30{56)228 2068 y([11])16 b(D.)c(Gilb)q(ert,)e Fl(On)k(sub)n(or)n(dinacy)h(and)f (analysis)f(of)g(the)h(sp)n(e)n(ctrum)g(of)f(Schr\177)-18 b(odinger)15 b(op)n(er)n(ators)g(with)d(two)300 2110 y(singular)i(endp)n(oints,)e Fu(Pro)q(c.)e(Ro)o(y)m(.)h(So)q(c.)g(Edin) o(burgh)e(Sect.)h(A)i Fb(112)h Fu(\(1989\),)d(213{229)228 2151 y([12])16 b(I.)c(Guarneri)d Fl(Sp)n(e)n(ctr)n(al)15 b(pr)n(op)n(erties)f(of)f(quantum)g(di\013usion)g(on)g(discr)n(ete)h (lattic)n(es)p Fu(,)e(Europh)o(ys.)e(Lett.)h Fb(10)300 2193 y Fu(\(1989\),)f(95{100)228 2234 y([13])16 b(I.)c(Guarneri)e Fl(On)k(an)g(estimate)f(c)n(onc)n(erning)j(quantum)e(di\013usion)g(in)f (the)h(pr)n(esenc)n(e)h(of)f(a)f(fr)n(actal)i(sp)n(e)n(c-)300 2276 y(trum)p Fu(,)d(Europh)o(ys.)d(Lett.)i Fb(21)i Fu(\(1993\),)c (729{733)228 2317 y([14])16 b(J.)d(Janas)f(and)f(S.)h(Nab)q(ok)o(o,)g Fl(Jac)n(obi)j(matric)n(es)f(with)f(absolutely)i(c)n(ontinuous)g(sp)n (e)n(ctrum,)g Fu(Pro)q(c.)c(Amer.)300 2359 y(Math.)g(So)q(c.)g Fb(127)h Fu(\(1999\),)e(791{800)228 2400 y([15])16 b(S.)8 b(Jitomirsk)n(a)o(y)o(a)d(and)i(Y.)h(Last,)g Fl(Dimensional)h (Hausdor\013)h(pr)n(op)n(erties)g(of)g(singular)f(c)n(ontinuous)h(sp)n (e)n(ctr)n(a,)300 2442 y Fu(Ph)o(ys.)h(Rev.)g(Lett.)g Fb(76)h Fu(\(1996\),)e(1765{1769)228 2483 y([16])16 b(S.)c(Jitomirsk)n (a)o(y)o(a)e(and)h(Y.)h(Last,)f Fl(Power)j(law)f(sub)n(or)n(dinacy)i (and)e(singular)h(sp)n(e)n(ctr)n(a.)g(I.)e(Half-line)i(op)n(er-)300 2525 y(ators,)f Fu(Acta)e(Math.,)g(to)g(app)q(ear)228 2566 y([17])16 b(S.)10 b(Jitomirsk)n(a)o(y)o(a)e(and)h(Y.)h(Last,)g Fl(Power)h(law)h(sub)n(or)n(dinacy)h(and)f(singular)f(sp)n(e)n(ctr)n (a.)i(II.)d(Line)i(op)n(er)n(ators,)300 2608 y Fu(preprin)o(t)p eop %%Page: 12 12 12 11 bop 228 113 a Fm(12)390 b(M.)15 b(CHRIST,)f(A.)h(KISELEV,)g(AND)e (Y.)i(LAST)228 213 y Fu([18])h(A.)f(Kiselev,)f Fl(A)o(bsolutely)j(c)n (ontinuous)f(sp)n(e)n(ctrum)h(of)e(one-dimensional)i(Schr\177)-18 b(odinger)17 b(op)n(er)n(ators)g(and)300 254 y(Jac)n(obi)11 b(matric)n(es)f(with)f(slow)r(ly)h(de)n(cr)n(e)n(asing)h(p)n (otentials,)g Fu(Comm)o(un.)d(Math.)f(Ph)o(ys.)h Fb(179)h Fu(\(1996\),)e(377{400)228 296 y([19])16 b(A.)11 b(Kiselev,)f Fl(A)o(bsolutely)j(c)n(ontinuous)g(sp)n(e)n(ctrum)f(of)g(p)n(erturb)n (e)n(d)i(Stark)e(op)n(er)n(ators,)h Fu(to)d(app)q(ear)f(in)h(T)m(rans.) 300 337 y(Amer.)i(Math.)f(So)q(c.)228 379 y([20])16 b(A.)i(Kiselev)f (and)f(Y.)i(Last,)g Fl(Solutions,)i(sp)n(e)n(ctrum)f(and)g(dynamics)g (of)f(Schr\177)-18 b(odinger)21 b(op)n(er)n(ators)e(on)300 420 y(in\014nite)13 b(domains,)g Fu(preprin)o(t)228 462 y([21])j(A.)d(Kiselev,)f(Y.)h(Last,)f(and)g(B.)g(Simon,)g Fl(Mo)n(di\014e)n(d)j(Pr\177)-19 b(ufer)15 b(and)f(EF)o(GP)g(tr)n (ansforms)g(and)g(the)g(sp)n(e)n(ctr)n(al)300 503 y(analysis)e(of)g (one-dimensional)h(Schr\177)-18 b(odinger)13 b(op)n(er)n(ators,)g Fu(Comm)o(un.)d(Math.)g(Ph)o(ys.)f Fb(194)j Fu(\(1998\),)d(1{45)228 545 y([22])16 b(Y.)10 b(Last)e(and)g(B.)g(Simon,)h Fl(Eigenfunctions,)i (tr)n(ansfer)g(matric)n(es,)g(and)g(absolutely)g(c)n(ontinuous)g(sp)n (e)n(ctrum)300 586 y(of)i(one-dimensional)h(Schr\177)-18 b(odinger)15 b(op)n(er)n(ators,)f Fu(In)o(v)o(en)o(t.)9 b(math.)i Fb(135)i Fu(\(1999\),)d(329{367)228 628 y([23])16 b(Y.)f(Last,)g Fl(Quantum)h(dynamics)h(and)f(de)n(c)n(omp)n(ositions)h (of)e(singular)h(c)n(ontinuous)h(sp)n(e)n(ctr)n(a,)g Fu(J.)e(F)m(unct.)300 669 y(Anal.)c Fb(142)i Fu(\(1996\),)d(406{445)228 711 y([24])16 b(M.)e(Reed)e(and)g(B.)h(Simon,)f Fl(Metho)n(ds)j(of)f (Mo)n(dern)h(Mathematic)n(al)h(Physics,)f(I.)f(F)m(unctional)h(A)o (nalysis)p Fu(,)300 752 y(Academic)c(Press,)g(London,)f(San)g(Diego,)g (1980)228 794 y([25])16 b(C.)e(Remling,)e Fl(The)i(absolutely)i(c)n (ontinuous)e(sp)n(e)n(ctrum)i(of)e(one-dimensional)h(Schr\177)-18 b(odinger)16 b(op)n(er)n(ators)300 835 y(with)d(de)n(c)n(aying)i(p)n (otentials,)e Fu(Comm)o(un.)f(Math.)f(Ph)o(ys.)f Fb(193)j Fu(\(1998\),)d(151{170)228 877 y([26])16 b(I.)c(Sc)o(h'nol,)d Fl(On)k(the)g(b)n(ehavior)i(of)e(the)g(Schr\177)-18 b(odinger)15 b(e)n(quation,)f Fu(Math.)d(Sb.)f Fb(42)j Fu(\(1957\),)c(273{286)228 918 y([27])16 b(B.)c(Simon,)e Fl(Schr\177)-18 b(odinger)15 b(semigr)n(oups,)f Fu(Bull.)c(Amer.)i(Math.)f(So)q(c.)g Fb(7)p Fu(\(1982\),)e(447{525)228 960 y([28])16 b(B.)c(Simon,)e Fl(Sp)n(e)n(ctr)n(al)k(analysis)f(of)g(r)n(ank)g(one)g(p)n(erturb)n (ations)h(and)f(applic)n(ations,)h Fu(CRM)e(Lecture)e(Notes)300 1001 y(V)m(ol.)j(8)f(\(J.)g(F)m(eldman,)f(R.)i(F)m(ro)q(ese,)e(L.)h (Rosen,)g(eds.\),)f(pp.)h(109{149,)e(Amer.)i(Math.)g(So)q(c.,)f(Pro)o (vidence,)300 1043 y(RI,)h(1995)228 1084 y([29])k(B.)9 b(Simon,)f Fl(The)i(classic)n(al)g(moment)h(pr)n(oblem)g(as)f(a)f (self-adjoint)h(\014nite)g(di\013er)n(enc)n(e)i(op)n(er)n(ator,)g Fu(Adv)n(ances)300 1126 y(in)f(Math.)g Fb(137)i Fu(\(1998\),)d(82{203) 228 1167 y([30])16 b(G.)c(Stolz,)g Fl(Bounde)n(d)j(solutions)e(and)h (absolute)h(c)n(ontinuity)f(of)g(Sturm-Liouvil)r(le)h(op)n(er)n(ators,) g Fu(J.)d(Math.)300 1209 y(Anal.)f(Appl.)g Fb(169)i Fu(\(1992\),)d (210{228)303 1287 y Fv(Dep)n(ar)n(tment)g(of)i(Ma)n(thema)n(tics,)f (University)i(of)g(Calif)o(ornia,)g(Berkeley)e(CA)i(94720)303 1328 y Fl(E-mail)g(addr)n(ess)s Fu(:)i Fa(mchrist@mat)o(h.b)o(er)o(kel) o(ey.)o(edu)303 1399 y Fv(Dep)n(ar)n(tment)6 b(of)i(Ma)n(thema)n(tics,) g(University)h(of)g(Chica)o(go,)h(5734)g(S.)f(University)g(A)l(ve,)g (Chica)o(go,)228 1440 y(IL)k(60637)303 1482 y Fl(E-mail)g(addr)n(ess)s Fu(:)i Fa(kiselev@mat)o(h.u)o(ch)o(ica)o(go.)o(edu)303 1552 y Fv(Insitute)e(of)f(Ma)n(thema)n(tics,)g(The)g(Hebrew)g (University,)h(91904)i(Jer)o(usalem,)d(Israel)h(and)h(De-)228 1594 y(p)n(ar)n(tment)d(of)h(Ma)n(thema)n(tics,)f(Calif)o(ornia)j (Institute)f(of)f(Technology,)h(P)m(asadena,)g(CA)g(91125)303 1635 y Fl(E-mail)g(addr)n(ess)s Fu(:)i Fa(ylast@its.c)o(alt)o(ec)o(h.e) o(du)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910241410845--