%!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: DDEV.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMSL10 CMR9 CMCSC10 CMSL9 CMBX10 CMR10 CMTI10 CMEX10 %%+ MSBM7 CMMI6 CMMI10 CMSY10 CMR8 CMMI8 CMR6 MSBM10 CMSY8 CMTI9 MSAM10 %%+ CMSY6 CMSS9 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: %+ /usr/local/teTeX/bin/powerpc-apple-darwin-current/dvips -R -Poutline %+ -o /tmp/altpdflatex.716-1163109146/DDEV.ps DDEV.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.11.09:2252 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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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 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 13 /circlecopyrt put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 60 /Rfractur put dup 65 /A put dup 68 /D put dup 75 /K put dup 77 /M put dup 82 /R put dup 83 /S put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 6 /Sigma put dup 12 /fi put dup 14 /ffi put dup 35 /numbersign put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 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 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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 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 67 /C put dup 75 /K put dup 76 /L 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 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 119 /w put dup 121 /y put readonly def /FontBBox{-301 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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 49 /infinity put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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 65 /A put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 77 /M put dup 78 /N put dup 79 /O put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 4E47B39E6712EB8908A3265E5FAB40567E866C244814449F1E993AAB422C3F1D DFA8C7118584F2E5197FD4BFA3A8AE9E953C6CD4672C0FF51E41C3A919749C1A F06650DF4C5E17492164BDBCDF22609A74BFA7F69960A64B9F949FFC2A807458 8579366C4F41BDE1FDFBCC4845FA19BBB6963D65EE8532549274BAEBDFF24FA6 03235D1BE37C06B1938AF369DA75BF38DDBC87A1FF445EAA16E1895ABE9506B9 211955753E447865D33CEF007391D2666A046277A30A49804FFCED3FEA5EB2C3 E52EE14A9F75241EA10C91974CDA6236EB840FD44D6DDE4D9B3266C3B99BD38B D835BCA8CB819C073480FB972CC028D218F6A1D344CE1B63F4FBF2C826F412E1 6E0B05A26125865A14FD7B7030B478BB8BC6BC395335C3BA940E1C348267F4F9 0AF97BBEE253511940F1048E175D3569F7D05A28851B6F50765FEB6C9654FEDC 1BF52F535DB5BB90C1BD5D2EBF75E0AEBE82B20507F3C28A03746781018D4EB2 298E4F2C27ACF73FA73EBE43F014BB575AAD516C0407B29E1653375135ECB74D C91372F06FA8EF37C31AF3FA48AE65318EAA6C34830A5377ABB2DFA5DA53A574 433484BA1466709A4B186761655C8E482833B697673E847C691079E7F1DCB8D6 1AD91101D757B83E2090337D525AEECB028FB3C9F6A6E6AD2F322CFDC5A833E6 1CE4EDBF41FD34FD61630581D222F854A76C2EA9FD72796A7C9CC1F6C2FCCD16 E95CA05826A4ECFADA6A5FB83C41A7131E52BA6585DD6DD78515D8F7327DFC6F 9404F893B80205FD2A53E2CC072B11E080696CEC743CD9382971DC7729B1DCAF 6F52CB9570F9A03E8C4D5B77D20A0B30FACA5F8A07D745285B1E8BE3D4EBD5F9 0450B8EFCD741E22EC7079DE4960915DBFCF2021C0D7776150530155265BF072 81B490811929E87A700E19FA187B6DB40010FDA4747742EF0D1EC98588BAB758 2E0430EB4E8087763FB88A443277DC39D831B30FE4BAA4AA7CFC50292E4694C6 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 73 /I put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: 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 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /nu put dup 27 /sigma put dup 30 /phi put dup 31 /chi put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 76 /L put dup 83 /S put dup 86 /V put dup 87 /W put dup 97 /a put dup 99 /c 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 114 /r put dup 116 /t put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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%%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 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 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 88 /X put dup 89 /Y put dup 97 /a put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 110 /n put dup 114 /r put dup 115 /s put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /beta put dup 14 /delta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon 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 70 /F put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 80 /P put dup 82 /R put dup 86 /V put dup 87 /W put dup 88 /X put dup 96 /lscript 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 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 87 /W put dup 100 /d put dup 107 /k put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 73 /I put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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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 16 /dotlessi put dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight 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 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 120 /x put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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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 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 58 /colon put dup 65 /A put dup 67 /C put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T 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 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 readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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 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 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 88 /X 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 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 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % 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 (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I 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 89 /Y put dup 97 /a put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (DDEV.dvi) @start /Fa 135[35 1[35 40 28 29 26 1[40 38 40 61 18 38 20 18 40 38 23 34 40 34 40 37 32[51 17[21 1[21 44[{}25 74.7198 /CMSS9 rf /Fb 242[83 13[{}1 83.022 /CMSY10 rf /Fc 134[44 1[60 1[46 32 33 33 1[46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 11[62 60 46 61 65 57 1[62 76 52 3[62 1[54 57 63 60 1[62 7[42 42 42 42 42 42 42 42 1[42 1[23 28 23 2[32 32 1[65 2[69 35[{}53 83.022 /CMR10 rf /Fd 136[69 2[37 38 3[48 53 80 27 50 3[48 1[44 1[42 33[72 18[27 46[{}13 83.022 /CMBX10 rf /Fe 206[61 49[{}1 49.8132 /CMSY6 rf /Ff 252[71 3[{}1 90.9091 /MSAM10 rf /Fg 133[31 37 35 51 35 41 25 31 32 35 39 39 43 63 20 35 1[24 39 35 24 35 39 35 35 39 12[55 43 3[59 57 69 4[57 59 1[52 58 2[57 18[24 27 24 44[{}38 74.7198 /CMTI9 rf /Fh 143[59 5[20 51[0 3[47 71 19 14[71 11[55 14[55 2[35 2[55{}11 66.4176 /CMSY8 rf 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1[21 4[21 19[38 38 3[64 14[{}32 74.7198 /CMSL9 rf /Fw 135[46 1[46 46 44 34 45 1[42 47 46 56 38 47 1[23 46 48 1[42 47 44 43 46 10[61 61 59 46 60 1[55 2[74 51 1[42 3[53 55 62 59 58 61 8[41 41 41 41 41 2[41 41 1[24 28 24 24[41 2[23 16[{}47 74.7198 /CMCSC10 rf /Fx 132[38 1[41 41 55 41 43 30 30 30 41 43 38 43 64 21 2[21 43 38 23 34 43 34 43 38 3[21 1[21 2[58 79 58 58 55 43 57 1[52 60 58 70 48 60 39 28 58 60 50 52 59 55 54 58 6[21 38 38 38 38 38 38 38 38 38 38 1[21 26 21 2[30 30 40[{}65 74.7198 /CMR9 rf /Fy 166[62 3[62 60 46 61 1[57 65 62 76 52 2[30 62 2[57 63 60 1[62 45[42 19[{}17 83.022 /CMSL10 rf /Fz 134[48 2[48 51 35 36 36 2[45 51 1[25 2[25 1[45 1[40 51 2[45 7[68 4[66 51 67 1[62 71 1[83 3[33 68 1[59 62 69 66 1[68 18[25 30 25 44[{}31 90.9091 /CMSL10 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 685 361 a Fz(HARD)m(Y-TYPE)47 b(ESTIMA)-8 b(TES)44 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y Fl(4)p 101 5597 499 4 v 101 5688 a Fg(Date)6 b Fx(:)27 b(No)n(v)n(em)n(b)r(er)e(9,)h(2006.)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 101 -103 a Fl(2)233 b(JEAN)23 b(DOLBEA)n(UL)-6 b(T,)23 b(JA)-8 b(VIER)24 b(DUO)n(ANDIK)n(OETXEA,)e(MARIA)h(J.)h (ESTEBAN,)f(AND)g(LUIS)h(VEGA)101 96 y Ft(where)957 213 y Fn(\013)h Ft(:=)1161 112 y Fq(\020)1253 172 y Fl(0)86 b Fk(\033)1414 184 y Fo(k)1231 240 y Fk(\033)1271 252 y Fo(k)1395 240 y Fl(0)1467 112 y Fq(\021)1562 213 y Fn(;)106 b(k)29 b Ft(=)c(1)p Fn(;)31 b Ft(2)p Fn(;)g Ft(3)p Fn(;)107 b(\014)30 b Ft(:=)2446 112 y Fq(\020)2516 174 y Fp(I)2542 183 y Fj(2)2681 174 y Fl(0)2528 243 y(0)78 b Fh(\000)p Fp(I)2722 252 y Fj(2)2771 112 y Fq(\021)101 383 y Ft(and)38 b Fi(I)321 398 y Fk(d)399 383 y Ft(is)g(the)g(iden)m (tit)m(y)i(op)s(erator)e(on)g Fi(C)1580 350 y Fk(d)1620 383 y Ft(.)63 b(Here)39 b Fn(\033)i Ft(=)c(\()p Fn(\033)2219 398 y Fk(k)2262 383 y Ft(\))2297 398 y Fk(k)r Fl(=1)p Fk(;)p Fl(2)p Fk(;)p Fl(3)2578 383 y 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y(energy)d(lev)m(el.)42 b Ft(One)27 b(can)g(indeed)h(pro)m(v)m(e)g(that)g(in)f(the)h (non-relativistic)i(limit)e Fn(\025)2851 3143 y Fl(1)2890 3129 y Ft([)p Fn(V)21 b Ft(])28 b(con)m(v)m(erges)h(to)f(the)101 3237 y(lo)m(w)m(est)33 b(eigen)m(v)-5 b(alue)32 b(of)e(the)h(limiting)g (Sc)m(hr\177)-45 b(odinger)30 b(op)s(erator)h(with)f(the)h(same)f(p)s (oten)m(tial)i Fn(V)20 b Ft(.)201 3345 y(Notations)37 b(in)e(\()p Fm(\003)p Ft(\))h(deserv)m(e)g(some)g(precisions.)55 b(W)-8 b(e)37 b(refer)e(to)h([1])g(for)f(more)g(details.)57 b(The)34 b(spinor)101 3462 y Fn(\036)g Ft(=)292 3389 y Fq(\000)334 3422 y Fk(\036)376 3431 y Fj(1)334 3494 y Fk(\036)376 3503 y Fj(2)411 3389 y Fq(\001)487 3462 y Ft(tak)m(es)j(its)f(v)-5 b(alues)35 b(in)g Fi(C)1309 3429 y Fl(2)1383 3462 y Ft(and)f(b)m(y)h Fm(j)p Fn(\036)p Fm(j)1799 3429 y Fl(2)1840 3462 y Ft(,)h Fm(jr)p Fn(\036)p Fm(j)2081 3429 y Fl(2)2156 3462 y Ft(and)f Fm(j)p Fn(\033)26 b Fm(\001)e(r)15 b Fn(\036)p Fm(j)2660 3429 y Fl(2)2735 3462 y Ft(w)m(e)35 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3717 y Fl(1)349 3703 y Fm(\000)20 b Fn(@)488 3717 y Fl(3)528 3703 y Fn(\036)582 3717 y Fl(2)621 3703 y Fm(j)646 3670 y Fl(2)686 3703 y Ft(.)201 3810 y(The)28 b(pro)s(of)h(of)g(Inequalit)m(y)h(\()p Fm(\003)p Ft(\))g(relies)f(on)g(the)g(follo)m(wing)i(observ)-5 b(ations.)41 b(The)28 b(eigenfunction)i(asso-)101 3918 y(ciated)i(to)f Fn(\025)533 3932 y Fl(1)573 3918 y Ft([)p Fn(V)20 b Ft(])31 b(can)f(b)s(e)g(c)m(haracterized)j(as)d(a)h(critical) h(p)s(oin)m(t)f(of)f(the)h(Ra)m(yleigh)h(quotien)m(t)1451 4121 y Fn(R)q Ft([)p Fn( )s Ft(])26 b(:=)1836 4060 y Fm(h)q Fn( )s(;)15 b(H)2050 4074 y Fk(V)2126 4060 y Fn( )t Fm(i)p 1790 4100 480 4 v 1790 4185 a(k)p Fn( )s Fm(k)1942 4154 y Fl(2)1942 4220 y Fk(L)1990 4201 y Fj(2)2026 4220 y Fl(\()p Fp(R)2101 4201 y Fj(3)2136 4220 y Fk(;)p Fp(C)2204 4201 y Fj(4)2239 4220 y Fl(\))2305 4121 y Fn(;)101 4373 y Ft(whic)m(h)33 b(is)g(obtained)h(b)m(y)e(decomp)s(osing)h Fn( )h Ft(=)1697 4300 y Fq(\000)1740 4333 y Fk(\036)1739 4405 y(\037)1783 4300 y Fq(\001)1857 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5599 y Fl(2)1949 5636 y Fn(dx)26 b Fm(\025)f(\000)2256 5512 y Fq(Z)2306 5719 y Fp(R)2354 5700 y Fj(3)2408 5636 y Fn(V)35 b Fm(j)p Fn(\036)p Fm(j)2600 5599 y Fl(2)2666 5636 y Fn(dx)91 b Fm(8)25 b Fn(\036)g Fm(2)g Fn(L)3159 5599 y Fl(2)3198 5636 y Ft(\()p Fi(R)3299 5599 y Fl(3)3339 5636 y Fn(;)15 b Fi(C)3445 5599 y Fl(2)3484 5636 y Ft(\))p eop end %%Page: 4 4 TeXDict begin 4 3 bop 101 -103 a Fl(4)233 b(JEAN)23 b(DOLBEA)n(UL)-6 b(T,)23 b(JA)-8 b(VIER)24 b(DUO)n(ANDIK)n(OETXEA,)e(MARIA)h(J.)h (ESTEBAN,)f(AND)g(LUIS)h(VEGA)101 96 y Ft(and,)40 b(under)d(some)i(tec) m(hnical)h(assumptions,)g(w)m(e)f(also)g(sho)m(w)f(that)h(the)g (equalit)m(y)h(case)f(is)g(ac)m(hiev)m(ed)101 204 y(only)31 b(if)f Fn(V)51 b Ft(is)30 b(the)h(Coulom)m(b)f(p)s(oten)m(tial.)42 b(In)30 b(suc)m(h)g(a)h(case)g(\003[)p Fn(V)21 b Ft(])k(=)g Fn(\025)2489 218 y Fl(1)2529 204 y Ft([)p Fn(V)20 b Ft(])31 b(:)40 b(See)31 b(Theorem)f(1)h(b)s(elo)m(w.)201 312 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Ft(\).)42 b(Let)1018 1915 y(\003[)p Fn(V)20 b Ft(])26 b(:=)f(min)1399 1970 y Fh(\006)1518 1915 y Fn(\025)1571 1929 y Fh(\006)1645 1915 y Fn(;)197 b(\025)1920 1929 y Fh(\006)2004 1915 y Ft(:=)2125 1832 y Fq(p)p 2216 1832 508 4 v 83 x Ft(1)21 b Fm(\000)f Ft(4)15 b Fn(A)2501 1929 y Fh(\006)2561 1915 y Ft([)p Fn(V)20 b Ft(])2684 1888 y Fl(2)2739 1915 y Fn(:)201 2108 y Ft(The)30 b(fundamen)m(tal)g(example)h(is)f(the)h (Coulom)m(b)f(p)s(oten)m(tial)i Fn(V)20 b Ft(\()p Fn(x)p Ft(\))26 b(=)2634 2072 y Fk(\027)p 2614 2087 79 4 v 2614 2141 a Fh(j)p Fk(x)p Fh(j)2734 2108 y Ft(for)k(whic)m(h)1219 2312 y Fn(A)1287 2326 y Fh(\006)1372 2312 y Ft(=)1478 2251 y Fn(\027)p 1478 2291 51 4 v 1481 2375 a Ft(2)1629 2312 y(and)91 b(\003[)p Fn(V)20 b Ft(])26 b(=)2175 2222 y Fq(p)p 2266 2222 247 4 v 90 x Ft(1)20 b Fm(\000)g Fn(\027)2473 2286 y Fl(2)2537 2312 y Fn(:)101 2510 y Fu(Remark.)36 b Ft(\(a\))28 b Fs(Note)e(that)h(for)g(any)f Fn(V)46 b Fm(2)25 b(A)1626 2525 y Fl(rad)1731 2510 y Fs(,)j Ft(\003[)p Fn(V)20 b Ft(])26 b Fm(2)f Ft([0)p Fn(;)15 b Ft(1])p Fs(.)41 b(Mor)-5 b(e)g(over,)28 b Fn(V)46 b Fm(6\021)25 b Ft(0)h Fs(implies)h Ft(\003[)p Fn(V)21 b Ft(])k Fn(<)g Ft(1)101 2618 y Fs(and)34 b Ft(\003[)p Fn(V)20 b Ft(])26 b(=)f(0)33 b Fs(only)g(if)g Fn(A)1018 2632 y Fh(\006)1077 2618 y Ft([)p Fn(V)20 b Ft(])26 b(=)f(1)p Fn(=)p Ft(2)p Fs(.)101 2737 y Ft(\(b\))37 b Fs(If)g Fn(V)54 b Fm(2)33 b(A)633 2752 y Fl(rad)739 2737 y Fs(,)38 b(then)1023 2714 y Fq(e)1011 2737 y Fn(V)21 b Ft(\()p Fn(x)p Ft(\))34 b(=)f Fm(j)p Fn(x)p Fm(j)1447 2704 y Fh(\000)p Fl(2)1541 2737 y Fn(V)21 b Ft(\()p Fm(j)p Fn(x)p Fm(j)1752 2704 y Fh(\000)p Fl(1)1847 2737 y Ft(\))37 b Fs(is)g(also)i(in)d Fm(A)2400 2752 y Fl(rad)2543 2737 y Fs(and)i Ft(\003[)p Fn(V)21 b Ft(])33 b(=)h(\003[)3148 2714 y Fq(e)3137 2737 y Fn(V)20 b Ft(])p Fs(.)56 b(This)37 b(is)g(a)101 2857 y(simple)j(c)-5 b(onse)g(quenc)g(e)39 b(of)f Fn(A)1081 2871 y Fl(+)1141 2857 y Ft([)p Fn(V)20 b Ft(])36 b(=)g Fn(A)1475 2871 y Fh(\000)1534 2857 y Ft([)1570 2834 y Fq(e)1559 2857 y Fn(V)20 b Ft(])39 b Fs(and)h Fn(A)1947 2871 y Fh(\000)2006 2857 y Ft([)p Fn(V)20 b Ft(])36 b(=)g Fn(A)2340 2871 y Fl(+)2399 2857 y Ft([)2435 2834 y Fq(e)2424 2857 y Fn(V)21 b Ft(])p Fs(.)59 b(Note)39 b(that)g(for)h(the)e(Coulomb) 101 2977 y(p)-5 b(otential)484 2954 y Fq(e)473 2977 y Fn(V)45 b Ft(=)25 b Fn(V)c Fs(.)201 3143 y Ft(With)31 b(the)f(ab)s(o)m(v)m(e)i(notations,)g(w)m(e)e(can)h(state)h(our)e(main) g(results.)101 3310 y Fu(Theorem)35 b(1.)42 b Fs(Assume)32 b(that)i Fn(V)46 b Fm(2)25 b(A)1451 3325 y Fl(rad)1556 3310 y Fs(.)42 b(F)-7 b(or)34 b(any)f Fn(\036)25 b Fm(2)g Fn(L)2197 3277 y Fl(2)2237 3310 y Ft(\()p Fi(R)2338 3277 y Fl(3)2377 3310 y Fn(;)15 b Fi(C)2483 3277 y Fl(2)2523 3310 y Ft(\))32 b Fs(and)i(any)f Fn(\025)25 b Fm(2)3105 3236 y Fq(\000)3167 3310 y Fm(\000)20 b Ft(1)p Fn(;)15 b Ft(\003[)p Fn(V)22 b Ft(])3531 3236 y Fq(\003)3569 3310 y Fs(,)101 3533 y Ft(\(1\))791 3409 y Fq(Z)842 3615 y Fp(R)890 3596 y Fj(3)944 3533 y Fn(V)35 b Fm(j)p Fn(\036)p Fm(j)1136 3495 y Fl(2)1201 3533 y Fn(dx)26 b Fm(\024)1422 3409 y Fq(Z)1472 3615 y Fp(R)1520 3596 y Fj(3)1603 3471 y Fm(j)p Fn(\033)e Fm(\001)c(r)15 b Fn(\036)p Fm(j)1919 3438 y Fl(2)p 1584 3512 394 4 v 1584 3595 a Ft(1)21 b(+)f Fn(\025)g Ft(+)g Fn(V)2013 3533 y(dx)h Ft(+)f(\(1)h Fm(\000)f Fn(\025)p Ft(\))2519 3409 y Fq(Z)2570 3615 y Fp(R)2618 3596 y Fj(3)2672 3533 y Fm(j)p Fn(\036)p Fm(j)2776 3495 y Fl(2)2841 3533 y Fn(dx)25 b(:)101 3748 y Fs(Mor)-5 b(e)g(over,)33 b(if)e Fn(A)683 3762 y Fh(\006)767 3748 y Fn(<)873 3712 y Fl(1)p 873 3727 36 4 v 873 3779 a(2)949 3748 y Fs(and)i Ft(\003[)p Fn(V)20 b Ft(])26 b Fn(<)f Ft(1)p Fs(,)31 b(then)h(ther)-5 b(e)32 b(exists)g(a)f(non-trivial)h (function)g Fn(\036)25 b Fm(2)g Fn(L)3320 3715 y Fl(2)3359 3748 y Ft(\()p Fi(R)3460 3715 y Fl(3)3500 3748 y Fn(;)15 b Fi(C)3606 3715 y Fl(2)3645 3748 y Ft(\))101 3856 y Fs(such)33 b(that)i(\(1\))e(holds)i(as)f(an)f(e)-5 b(quality)34 b(with)g Fn(\025)26 b Ft(=)g(\003[)p Fn(V)20 b Ft(])33 b Fs(if)g(and)h(only)g(if)f Fn(V)53 b Fs(is)33 b(the)g(Coulomb)h(p)-5 b(otential)101 3972 y Fn(V)21 b Ft(\()p Fn(x)p Ft(\))42 b(=)482 3937 y Fk(\027)p 462 3952 79 4 v 462 4005 a Fh(j)p Fk(x)p Fh(j)593 3972 y Fs(with)h Fn(\027)48 b Fm(2)42 b Ft(\(0)p Fn(;)15 b Ft(1\))p Fs(,)46 b Fn(\027)i Ft(=)1476 3894 y Fq(p)p 1567 3894 383 4 v 78 x Ft(1)20 b Fm(\000)g Ft(\003[)p Fn(V)h Ft(])1910 3946 y Fl(2)1992 3972 y Fs(and)42 b Fn(\036)h Fs(is)e(a)i(r)-5 b(adial)44 b(function)e(such)g(that)h Fn(\036)g Ft(=)101 4124 y Fn(B)20 b(r)234 4036 y Fh(p)p 292 4036 164 3 v 292 4092 a Fl(1)p Fh(\000)p Fk(\027)421 4073 y Fj(2)456 4092 y Fh(\000)p Fl(1)566 4124 y Fn(e)608 4092 y Fh(\000)p Fk(\027)c(r)784 4124 y Fs(for)33 b(some)h(c)-5 b(onstant)34 b Fn(B)c Fm(2)25 b Fi(C)1775 4092 y Fl(2)1814 4124 y Fs(.)201 4291 y Ft(Without)g(further)f(restrictions)i(on)e Fn(\036)p Ft(,)j(b)s(oth)d(sides)g(of)h(the)g(inequalit)m(y)h(can)f(b)s (e)g(in\014nite.)38 b(As)25 b(already)101 4399 y(explained)j(in)f(the)g (in)m(tro)s(duction,)h(the)g(motiv)-5 b(ation)29 b(for)e(Hardy-t)m(yp)s (e)g(estimates)i(lik)m(e)g(\(1\))f(comes)g(from)101 4507 y(the)h(Dirac)i(op)s(erator.)40 b(The)29 b(goal)h(is)f(to)h(giv)m(e)h (a)e(lo)m(w)m(er)h(estimate)h(for)e(the)g Fz(ground)f(state)j(lev)m(el) j Ft(and)28 b(one)101 4615 y(can)j(pro)m(v)m(e)g(the)g(follo)m(wing)h (result.)101 4781 y Fu(Theorem)j(2.)42 b Fs(L)-5 b(et)33 b Fn(V)45 b Fm(2)25 b(A)1082 4796 y Fl(rad)1188 4781 y Fs(.)42 b(If)32 b Fn(\026)25 b Fm(2)g Ft(\()p Fm(\000)p Ft(1)p Fn(;)15 b Ft(1\))34 b Fs(is)f(an)g(eigenvalue)f(of)h Fn(H)2674 4795 y Fk(V)2734 4781 y Fs(,)g(then)1684 4935 y Fn(\026)25 b Fm(\025)g Ft(\003[)p Fn(V)c Ft(])k Fn(:)101 5089 y Fs(As)35 b(a)g(c)-5 b(onse)g(quenc)g(e,)35 b(if)f Fn(V)49 b Fm(2)29 b(A)1211 5104 y Fl(rad)1317 5089 y Fs(,)35 b Fn(H)1456 5103 y Fk(V)1551 5089 y Fs(has)g(only)h(nonne)-5 b(gative)35 b(eigenvalues)f(in)h Ft(\()p Fm(\000)p Ft(1)p Fn(;)15 b Ft(1\))36 b Fs(and)g Ft(\003[)p Fn(V)20 b Ft(])101 5197 y Fs(is)33 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Fs(If)f Fn(V)46 b Fm(2)25 b(A)1032 111 y Fl(rad)1168 96 y Fs(and)33 b Ft(0)25 b Fm(\024)g Fn(W)13 b Ft(\()p Fn(x)p Ft(\))26 b Fm(\024)f Fn(V)20 b Ft(\()p Fn(x)p Ft(\))32 b Fs(a.e.,)f(then)39 b Ft(\(1\))32 b Fs(holds)h(with)f Fn(W)43 b Fs(inste)-5 b(ad)33 b(of)e Fn(V)101 204 y Fs(and)j Fn(\025)25 b Fm(2)442 130 y Fq(\000)503 204 y Fm(\000)20 b Ft(1)p Fn(;)15 b Ft(\003[)p Fn(V)22 b Ft(])867 130 y Fq(\003)905 204 y Fs(.)41 b(Note)33 b(that)h(the)f(left-hand)g(side)g(of)52 b Ft(\(1\))33 b Fs(de)-5 b(cr)g(e)g(ases)35 b(and)e(the)g(right-hand)h (side)101 312 y(incr)-5 b(e)g(ases)34 b(when)f Fn(V)53 b Fs(is)33 b(r)-5 b(eplac)g(e)g(d)35 b(by)d Fn(W)13 b Fs(.)41 b(Note)33 b(also)h(that)g Fn(W)45 b Fs(do)-5 b(es)34 b(not)f(ne)-5 b(e)g(d)33 b(to)g(b)-5 b(e)33 b(r)-5 b(adial.)101 420 y Ft(\(b\))33 b Fs(If)f Fn(V)403 434 y Fk(a)445 420 y Ft(\()p Fn(x)p Ft(\))26 b(=)f Fn(V)20 b Ft(\()p Fn(a)h Ft(+)f Fn(x)p Ft(\))33 b Fs(is)f(in)h Fm(A)1359 435 y Fl(rad)1497 420 y Fs(for)g(some)g Fn(a)26 b Fm(2)f 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Fi(R)1359 1126 y Fl(3)1398 1159 y Ft(,)31 b(de\014ne)882 1329 y(\003)945 1291 y Fh(\003)985 1329 y Ft([)p Fn(W)13 b Ft(])25 b(:=)g(sup)1432 1255 y Fq(\010)1485 1329 y Ft(\003[)p Fn(W)1672 1291 y Fh(\003)1659 1351 y Fk(a)1711 1329 y Ft(])51 b(:)g Fn(a)25 b Fm(2)g Fi(R)2088 1291 y Fl(3)2142 1329 y Fn(;)41 b(A)2276 1343 y Fh(\006)2335 1329 y Ft([)p Fn(W)2459 1291 y Fh(\003)2446 1351 y Fk(a)2498 1329 y Ft(])26 b Fm(\024)f Ft(1)p Fn(=)p Ft(2)2780 1255 y Fq(\011)2875 1329 y Fn(:)101 1495 y Ft(The)30 b(follo)m(wing)i (corollary)g(is)e(a)h(simple)f(consequence)h(of)g(Theorem)f(1)h(and)f (the)g(preceding)g(remarks.)101 1669 y Fu(Corollary)40 b(3.)j Fs(Assume)35 b(that)h Fn(W)1324 1636 y Fh(\003)1311 1692 y Fk(a)1398 1669 y Fs(is)f(in)f Fm(A)1684 1684 y Fl(rad)1825 1669 y Fs(for)i(some)f Fn(a)30 b Fm(2)f Fi(R)2441 1636 y Fl(3)2480 1669 y Fs(.)49 b(F)-7 b(or)36 b(any)f Fn(\036)30 b Fm(2)f Fn(L)3141 1636 y Fl(2)3180 1669 y Ft(\()p Fi(R)3281 1636 y Fl(3)3321 1669 y Fn(;)15 b Fi(C)3427 1636 y Fl(2)3466 1669 y Ft(\))36 b Fs(and)101 1777 y(any)e Fn(\025)25 b Fm(2)440 1704 y Fq(\000)502 1777 y Fm(\000)19 b Ft(1)p Fn(;)c Ft(\003)740 1744 y Fh(\003)781 1777 y Ft([)p Fn(W)e Ft(])930 1704 y Fq(\003)968 1777 y Fs(,)766 1890 y Fq(Z)816 2096 y Fp(R)864 2077 y Fj(3)918 2013 y Fn(W)28 b Fm(j)p Fn(\036)p Fm(j)1136 1976 y Fl(2)1201 2013 y Fn(dx)e Fm(\024)1422 1890 y Fq(Z)1472 2096 y Fp(R)1520 2077 y Fj(3)1616 1952 y Fm(j)p Fn(\033)d Fm(\001)e(r)15 b Fn(\036)p Fm(j)1932 1919 y Fl(2)p 1584 1993 420 4 v 1584 2076 a Ft(1)21 b(+)f Fn(\025)g Ft(+)g Fn(W)2039 2013 y(dx)g Ft(+)g(\(1)h Fm(\000)f Fn(\025)p Ft(\))2544 1890 y Fq(Z)2595 2096 y Fp(R)2643 2077 y Fj(3)2697 2013 y Fm(j)p Fn(\036)p Fm(j)2801 1976 y Fl(2)2866 2013 y Fn(dx)26 b(;)101 2231 y Fs(and)34 b Ft(\003)341 2198 y Fh(\003)380 2231 y Ft([)p Fn(W)13 b Ft(])33 b Fs(is)f(a)h(lower)h(b) -5 b(ound)34 b(for)f(al)5 b(l)33 b(eigenvalues)f(of)h Fn(H)2178 2245 y Fk(W)2291 2231 y Fs(in)f(the)h(gap)h Ft(\()p Fm(\000)p Ft(1)p Fn(;)15 b Ft(1\))p Fs(.)201 2405 y Ft(Note)32 b(that)f(as)f(a)h(sp)s(ecial)g(case)g(corresp)s (onding)f(to)h Fn(\025)25 b Ft(=)g(\003[)p Fn(V)c Ft(],)280 2508 y Fq(Z)331 2714 y Fp(R)379 2695 y Fj(3)432 2632 y Fn(V)36 b Fm(j)p Fn(\036)p Fm(j)625 2594 y Fl(2)690 2632 y Fn(dx)26 b Fm(\024)911 2508 y Fq(Z)961 2714 y Fp(R)1009 2695 y Fj(3)1159 2570 y Fm(j)p Fn(\033)e Fm(\001)c(r)15 b Fn(\036)p Fm(j)1475 2537 y Fl(2)p 1073 2611 528 4 v 1073 2694 a Ft(1)21 b(+)f(\003[)p Fn(V)g Ft(])h(+)f Fn(V)1636 2632 y(dx)h Ft(+)e(\(1)i Fm(\000)f Ft(\003[)p Fn(V)h Ft(]\))2275 2508 y Fq(Z)2326 2714 y Fp(R)2374 2695 y Fj(3)2428 2632 y Fm(j)p Fn(\036)p Fm(j)2532 2594 y Fl(2)2597 2632 y Fn(dx)92 b Fm(8)24 b Fn(\036)i Fm(2)f Fn(L)3091 2594 y Fl(2)3130 2632 y Ft(\()p Fi(R)3231 2594 y Fl(3)3270 2632 y Fn(;)15 b Fi(C)3376 2594 y Fl(2)3416 2632 y Ft(\))26 b Fn(:)101 2847 y Ft(Suc)m(h)42 b(an)g(inequalit)m(y)h(is)g(the)f (generalization)j(to)e(Dirac)g(op)s(erators)g(of)f(the)g(Hardy)g (inequalit)m(y)i(for)101 2955 y(Sc)m(hr\177)-45 b(odinger)30 b(op)s(erators)h(established)f(in)g([4)q(].)1192 3198 y(3.)46 b Fr(Pr)n(oof)34 b(of)g(Theorems)f(1)h(and)g(2)101 3359 y Ft(3.1.)47 b Fu(Preliminary)h(results.)e Ft(The)40 b(P)m(auli)i(matrices)g(are)f(Hermitian)g(and)g(satisfy)g(the)g(follo)m (wing)101 3467 y(prop)s(erties)30 b(:)1051 3596 y Fn(\033)1103 3610 y Fk(j)1155 3596 y Fn(\033)1207 3611 y Fk(k)1270 3596 y Ft(+)20 b Fn(\033)1413 3611 y Fk(k)1470 3596 y Fn(\033)1522 3610 y Fk(j)1584 3596 y Ft(=)25 b(2)15 b Fn(\016)1780 3611 y Fk(j)t(k)1871 3596 y Fi(I)1906 3610 y Fl(2)1961 3596 y Fn(;)106 b Fm(8)25 b Fn(j;)31 b(k)e Ft(=)c(1)p Fn(;)31 b Ft(2)p Fn(;)g Ft(3)25 b Fn(:)101 3744 y Ft(With)33 b(a)f(standard)f(abuse)h(of)g(notations,)i(eac)m(h)f (time)g(a)f(scalar)h Fn(\016)j Ft(app)s(ears)31 b(in)h(an)g(iden)m(tit) m(y)h(in)m(v)m(olving)101 3852 y(op)s(erators)j(acting)i(on)d(t)m(w)m (o-spinors,)k(it)d(has)g(to)g(b)s(e)g(understo)s(o)s(d)e(as)i Fn(\016)19 b Fi(I)2635 3866 y Fl(2)2675 3852 y Ft(,)37 b(where)e Fi(I)3040 3866 y Fl(2)3116 3852 y Ft(is)h(the)g(iden)m(tit)m (y)101 3960 y(op)s(erator)31 b(on)f Fi(C)660 3927 y Fl(2)699 3960 y Ft(.)41 b(F)-8 b(or)31 b(an)m(y)g Fn(a;)15 b(b)26 b Fm(2)e Fi(C)1404 3927 y Fl(3)1443 3960 y Ft(,)31 b(w)m(e)g(ha)m(v)m (e)1220 4126 y(\()p Fn(\033)24 b Fm(\001)c Fn(a)p Ft(\)\()p Fn(\033)k Fm(\001)d Fn(b)p Ft(\))k(=)g Fn(a)c Fm(\001)f Fn(b)g Ft(+)g Fn(i)15 b(\033)24 b Fm(\001)d Ft(\()p Fn(a)f Fm(\002)g Fn(b)p Ft(\))26 b Fn(:)101 4293 y Ft(Applying)k(this)g(form)m (ula)h(to)g Fn(a)25 b Ft(=)g Fn(x)31 b Ft(and)e Fn(b)d Ft(=)f Fm(r)p Ft(,)30 b(w)m(e)h(obtain)g(the)f(follo)m(wing)i(result.) 101 4466 y Fu(Lemma)j(4.)42 b Fs(The)33 b(fol)5 b(lowing)33 b(e)-5 b(quality)34 b(holds:)899 4609 y Fn(x)p 874 4650 103 4 v 874 4733 a Fm(j)p Fn(x)p Fm(j)1006 4671 y(\001)21 b(r)k Ft(=)1249 4570 y Fq(\020)1303 4671 y Fn(\033)f Fm(\001)1459 4609 y Fn(x)p 1434 4650 V 1434 4733 a Fm(j)p Fn(x)p Fm(j)1546 4570 y Fq(\021)q(\020)1655 4671 y Fn(\033)f Fm(\001)e(r)1852 4570 y Fq(\021)1926 4671 y Ft(+)2055 4609 y(1)p 2027 4650 V 2027 4733 a Fm(j)p Fn(x)p Fm(j)2139 4570 y Fq(\020)2194 4671 y Fn(\033)i Fm(\001)e Fn(L)2377 4570 y Fq(\021)2524 4671 y Fm(8)j Fn(x)i Fm(2)e Fi(R)2828 4633 y Fl(3)2893 4671 y Fn(;)101 4886 y Fs(wher)-5 b(e)34 b Fn(L)25 b Ft(:=)g Fm(\000)p Fn(i)15 b(x)21 b Fm(\002)f(r)32 b Fs(is)g(the)h(orbital)h(angular)g(momentum)g(op)-5 b(er)g(ator.)201 5060 y Ft(The)33 b(next)g(result)g(is)h(concerned)f (with)g(the)h(sp)s(ectrum)e(of)i(the)f(op)s(erator)h Fn(\033)25 b Fm(\001)d Fn(L)p Ft(.)50 b(W)-8 b(e)34 b(shall)g(denote) 101 5168 y(b)m(y)24 b Fn(X)296 5183 y Fk(k)362 5168 y Ft(the)f(sp)s(ectral)h(space)g(of)g Fn(\033)9 b Fm(\001)d Fn(L)24 b Ft(asso)s(ciated)g(to)h(the)e(eigen)m(v)-5 b(alue)25 b Fn(k)s Ft(,)h(and)c(b)m(y)i Fn(P)2911 5183 y Fk(k)2977 5168 y Ft(the)g(corresp)s(onding)101 5276 y(pro)5 b(jector)31 b(in)f Fn(H)682 5243 y Fl(1)722 5276 y Ft(\()p Fi(R)823 5243 y Fl(3)862 5276 y Fn(;)15 b Fi(C)968 5243 y Fl(2)1008 5276 y Ft(\).)101 5449 y Fu(Lemma)43 b(5.)i Fs(The)39 b(sp)-5 b(e)g(ctrum)40 b(of)f Fn(\033)28 b Fm(\001)d Fn(L)38 b Fs(is)h(the)g(discr)-5 b(ete)40 b(set)f Fm(f)p Fn(k)h Fm(2)35 b Fi(Z)51 b Ft(:)h Fn(k)40 b Fm(6)p Ft(=)c Fm(\000)p Ft(1)p Fm(g)j Fs(and)h Ft(Ker)o(\()p Fn(\033)29 b Fm(\001)24 b Fn(L)p Ft(\))101 5557 y Fs(c)-5 b(ontains)34 b(al)5 b(l)32 b(r)-5 b(adial)34 b(functions.)42 b(Mor)-5 b(e)g(over,)33 b(if)f Fn(\036)g Fs(is)g(a)g(c)-5 b(ontinuous)33 b(function,)g(then)f Fn(P)3156 5572 y Fk(k)3199 5557 y Fn(\036)p Ft(\(0\))27 b(=)e(0)32 b Fs(for)101 5665 y(any)i Fn(k)28 b Fm(2)d Fi(Z)20 b Fm(n)g(f)p Ft(0)p Fn(;)15 b Fm(\000)p Ft(1)p Fm(g)p Fs(.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 101 -103 a Fl(6)233 b(JEAN)23 b(DOLBEA)n(UL)-6 b(T,)23 b(JA)-8 b(VIER)24 b(DUO)n(ANDIK)n(OETXEA,)e(MARIA)h(J.)h (ESTEBAN,)f(AND)g(LUIS)h(VEGA)101 96 y Fs(Pr)-5 b(o)g(of.)43 b Ft(The)26 b(sp)s(ectrum)g(of)h(the)g(op)s(erator)g(1)13 b(+)g Fn(\033)j Fm(\001)d Fn(L)27 b Ft(is)g(the)g(discrete)g(set)h Fm(f\006)p Ft(1)p Fn(;)15 b Fm(\006)p Ft(2)p Fn(;)g Fm(\001)g(\001)g (\001)i(g)p Ft(,)29 b(see)e([7)q(].)39 b(This)101 204 y(can)31 b(b)s(e)f(seen)g(b)m(y)g(noticing)i(that)1092 429 y(1)20 b(+)g Fn(\033)k Fm(\001)c Fn(L)25 b Ft(=)g Fn(J)1611 391 y Fl(2)1671 429 y Fm(\000)20 b Fn(L)1824 391 y Fl(2)1883 429 y Ft(+)1984 367 y(1)p 1984 408 46 4 v 1984 491 a(4)2055 429 y Fn(;)106 b(J)34 b Ft(:=)26 b Fn(L)20 b Ft(+)2575 367 y Fn(\033)p 2575 408 56 4 v 2580 491 a Ft(2)2665 429 y Fn(:)101 657 y Ft(Then,)27 b(the)g(fact)h(that)f(the)g(sp)s(ectrum)f(of)h Fn(J)1589 624 y Fl(2)1655 657 y Ft(\(resp.)39 b Fn(L)1979 624 y Fl(2)2018 657 y Ft(\))27 b(is)g(the)g(set)h Fm(f)p Fn(j)5 b Ft(\()p Fn(j)20 b Ft(+)13 b(1\))41 b(:)f Fn(j)31 b Ft(=)3083 621 y Fl(1)p 3083 636 36 4 v 3083 689 a(2)3128 657 y Fn(;)3179 621 y Fl(3)p 3179 636 V 3179 689 a(2)3224 657 y Fn(;)15 b(:)g(:)g(:)h Fm(g)28 b Ft(\(resp.)101 775 y Fm(f)p Fn(`)p Ft(\()p Fn(`)21 b Ft(+)f(1\))41 b(:)g Fn(`)25 b Ft(=)g Fn(j)h Fm(\006)879 740 y Fl(1)p 879 755 V 879 807 a(2)924 775 y Fn(;)41 b(j)30 b Ft(=)1163 740 y Fl(1)p 1163 755 V 1163 807 a(2)1209 775 y Fn(;)1274 740 y Fl(3)p 1274 755 V 1274 807 a(2)1319 775 y Fn(;)h(:)15 b(:)g(:)h Fm(g)31 b Ft(pro)m(v)m(es)g(the)g(statemen)m(t)h(on)e(the)h (sp)s(ectrum)e(of)h(1)21 b(+)f Fn(\033)j Fm(\001)e Fn(L)p Ft(.)201 894 y(F)-8 b(or)39 b(all)h Fn(k)i Fm(6)p Ft(=)d(0,)j Fm(\000)p Ft(1,)f Fn(P)1058 909 y Fk(k)1101 894 y Fn(\036)e Ft(is)g(a)g(linear)g(com)m(bination)h(of)f(functions)f Fn( )2739 861 y Fk(k)2736 917 y(m)2803 894 y Ft(,)j Fn(m)e Ft(=)g(1,.)16 b(.)f(.)h Fn(m)3370 909 y Fk(k)3451 894 y Ft(whic)m(h)101 1002 y(dep)s(end)29 b(only)h(on)h(the)f(angular)h(v) -5 b(ariable)31 b Fn(!)i Ft(and)d(not)g(on)h Fm(j)p Fn(x)p Fm(j)p Ft(,)g(with)f(co)s(e\016cien)m(ts)1268 1236 y Fn(f)1323 1198 y Fk(k)1313 1258 y(m)1379 1236 y Ft(\()p Fn(r)s Ft(\))c(=)1615 1112 y Fq(Z)1665 1318 y Fk(S)1712 1299 y Fj(2)1766 1236 y Fn(P)1824 1251 y Fk(k)1867 1236 y Fn(\036)p Ft(\()p Fn(r)18 b(!)s Ft(\))d Fn( )2187 1198 y Fk(k)2184 1258 y(m)2251 1236 y Ft(\()p Fn(!)s Ft(\))26 b Fn(d!)101 1474 y Ft(whic)m(h)k(are)h(con)m(tin)m(uous)g(functions)f (of)h Fn(r)c Ft(=)e Fm(j)p Fn(x)p Fm(j)p Ft(.)42 b(According)31 b(to)g([7],)g(Section)g(4.6.4,)1265 1708 y Fn(f)1320 1670 y Fk(k)1310 1730 y(m)1376 1708 y Ft(\(0\))c(=)e Fn(P)1672 1723 y Fk(k)1715 1708 y Fn(\036)p Ft(\(0\))1899 1584 y Fq(Z)1951 1790 y Fk(S)1998 1771 y Fj(2)2051 1708 y Fn( )2113 1670 y Fk(k)2110 1730 y(m)2192 1708 y Fn(d!)k Ft(=)c(0)g Fn(;)101 1946 y Ft(whic)m(h)30 b(pro)m(v)m(es)h(that)g Fn(P)897 1961 y Fk(k)940 1946 y Fn(\036)p Ft(\(0\))c(=)e(0)31 b(for)f Fn(k)e Fm(6)p Ft(=)d(0,)31 b Fm(\000)p Ft(1.)1750 b Ff(\003)201 2158 y Ft(The)30 b(main)h(p)s(oin)m(ts)f(are)i(that)f Fn(L)f Ft(comm)m(utes)i(with)e(all)i(radial)f(functions)g(and)f(that)h (0)g(is)g(not)g(in)f(the)101 2266 y(sp)s(ectrum)f(of)i(1)21 b(+)f Fn(\033)j Fm(\001)d Fn(L)p Ft(.)41 b(The)30 b(follo)m(wing)i (result)e(is)g(adapted)h(from)f([1].)101 2468 y Fu(Lemma)e(6.)37 b Fs(F)-7 b(or)28 b(any)g Fn(k)s Fs(,)g Fn(l)12 b Fm(2)e Ft(Sp)s(ec)o(\()p Fn(\033)h Fm(\001)d Fn(L)p Ft(\))p Fs(,)29 b Fn(k)13 b Fm(6)p Ft(=)d Fn(l)r Fs(,)28 b Fn(P)1885 2483 y Fk(k)1928 2395 y Fq(\000)1970 2468 y Fn(\033)11 b Fm(\001)d(r)2142 2395 y Fq(\001)2184 2417 y Fl(2)2223 2468 y Fn(P)2281 2483 y Fk(l)2333 2468 y Fm(\021)25 b Fn(P)2487 2483 y Fk(l)2513 2395 y Fq(\000)2555 2468 y Fn(\033)11 b Fm(\001)d(r)2727 2395 y Fq(\001)2769 2417 y Fl(2)2808 2468 y Fn(P)2866 2483 y Fk(k)2935 2468 y Fm(\021)25 b Ft(0)i Fs(in)g Fn(H)3292 2435 y Fl(1)3331 2468 y Ft(\()p Fi(R)3432 2435 y Fl(3)3472 2468 y Fn(;)15 b Fi(C)3578 2435 y Fl(2)3617 2468 y Ft(\))p Fs(.)101 2654 y(Pr)-5 b(o)g(of.)43 b Ft(A)35 b(direct)g(computation)h(sho)m(ws)f (that)g(the)g(an)m(ti-comm)m(utator)j Fm(f)p Fn(\033)27 b Fm(\001)c(r)p Fn(;)15 b Ft(1)24 b(+)f Fn(\033)j Fm(\001)d Fn(L)p Fm(g)33 b Ft(=)g(0,)j Fz(i.e.)p Ft(,)101 2762 y Fn(\033)24 b Fm(\001)c(r)30 b Ft(an)m(ticomm)m(utes)i(with)e(1)21 b(+)f Fn(\033)j Fm(\001)d Fn(L)p Ft(.)41 b(Hence)31 b(\()p Fn(\033)24 b Fm(\001)c(r)p Ft(\))2058 2729 y Fl(2)2128 2762 y Ft(comm)m(utes)31 b(with)f(1)21 b(+)e Fn(\033)24 b Fm(\001)c Fn(L)p Ft(.)41 b(Let)30 b Fn(\036)3383 2777 y Fk(k)3452 2762 y Fm(2)24 b Fn(X)3612 2777 y Fk(k)3655 2762 y Ft(,)101 2870 y Fn(\036)155 2885 y Fk(l)207 2870 y Fm(2)h Fn(X)368 2885 y Fk(l)394 2870 y Ft(.)41 b(Then)775 3095 y(\()p Fn(\033)24 b Fm(\001)c(r)15 b Fn(\036)1076 3110 y Fk(k)1119 3095 y Fn(;)g(\033)24 b Fm(\001)c(r)15 b Fn(\036)1425 3110 y Fk(l)1451 3095 y Ft(\))84 b(=)1803 3034 y(1)p 1733 3074 186 4 v 1733 3157 a Fn(l)22 b Ft(+)e(1)1944 3021 y Fq(\000)1986 3095 y Fn(\036)2040 3110 y Fk(k)2082 3095 y Fn(;)15 b Ft(\()p Fn(\033)25 b Fm(\001)20 b(r)p Ft(\))2390 3058 y Fl(2)2445 3095 y Ft(\(1)h(+)f Fn(\033)j Fm(\001)e Fn(L)p Ft(\))15 b Fn(\036)2924 3110 y Fk(l)2950 3021 y Fq(\001)1570 3320 y Ft(=)1803 3259 y(1)p 1733 3300 V 1733 3383 a Fn(l)22 b Ft(+)e(1)1944 3247 y Fq(\000)1986 3320 y Fn(\036)2040 3335 y Fk(k)2082 3320 y Fn(;)15 b Ft(\(1)22 b(+)e Fn(\033)j Fm(\001)e Fn(L)p Ft(\)\()p Fn(\033)j Fm(\001)c(r)p Ft(\))2800 3283 y Fl(2)2855 3320 y Fn(\036)2909 3335 y Fk(l)2935 3247 y Fq(\001)1570 3546 y Ft(=)1803 3484 y(1)p 1733 3525 V 1733 3608 a Fn(l)i Ft(+)e(1)1944 3472 y Fq(\000)1986 3546 y Ft(\(1)h(+)f Fn(\033)j Fm(\001)d Fn(L)p Ft(\))15 b Fn(\036)2464 3561 y Fk(k)2508 3546 y Fn(;)g Ft(\()p Fn(\033)24 b Fm(\001)c(r)p Ft(\))2815 3508 y Fl(2)2870 3546 y Fn(\036)2924 3561 y Fk(l)2950 3472 y Fq(\001)1570 3776 y Ft(=)1733 3714 y Fn(k)k Ft(+)c(1)p 1733 3755 207 4 v 1744 3838 a Fn(l)i Ft(+)e(1)1965 3702 y Fq(\000)2007 3776 y Fn(\036)2061 3791 y Fk(k)2104 3776 y Fn(;)15 b Ft(\()p Fn(\033)24 b Fm(\001)c(r)p Ft(\))2411 3738 y Fl(2)2466 3776 y Fn(\036)2520 3791 y Fk(l)2546 3702 y Fq(\001)2628 3776 y Fn(;)101 4001 y Ft(whic)m(h)30 b(is)h(imp)s(ossible)f(except)h(if)f(\()p Fn(\033)24 b Fm(\001)d(r)15 b Fn(\036)1566 4016 y Fk(k)1608 4001 y Fn(;)g(\033)24 b Fm(\001)d(r)15 b Fn(\036)1915 4016 y Fk(l)1941 4001 y Ft(\))25 b(=)g(0.)1443 b Ff(\003)201 4180 y Ft(The)30 b(radial)h(symmetry)f(of)g(the)h(p)s(oten)m(tial)h (can)e(b)s(e)g(tak)m(en)h(in)m(to)h(accoun)m(t)g(as)e(follo)m(ws.)101 4365 y Fu(Corollary)f(7.)37 b Fs(A)n(ny)26 b(function)h Fn(\036)e Fm(2)g Fn(L)1444 4332 y Fl(2)1483 4365 y Ft(\()p Fi(R)1584 4332 y Fl(3)1624 4365 y Fn(;)15 b Fi(C)1730 4332 y Fl(2)1769 4365 y Ft(\))27 b Fs(c)-5 b(an)27 b(b)-5 b(e)27 b(written)g Fn(\036)f Ft(=)2580 4297 y Fq(P)2676 4392 y Fk(k)r Fh(2)p Fp(Z)p Fk(;)11 b(k)r Fh(6)p Fl(=)p Fh(\000)p Fl(1)3040 4365 y Fn(\036)3094 4380 y Fk(k)3163 4365 y Fs(with)28 b Fn(\036)3409 4380 y Fk(k)3477 4365 y Fm(2)d Fn(X)3638 4380 y Fk(k)101 4477 y Fs(and)34 b(mor)-5 b(e)g(over,)34 b(if)f Fn(W)45 b Fs(is)32 b(a)h(r)-5 b(adial)35 b(function,)940 4582 y Fq(Z)991 4788 y Fp(R)1039 4769 y Fj(3)1093 4705 y Fn(W)27 b Fm(j)p Fn(\036)p Fm(j)1310 4668 y Fl(2)1376 4705 y Fn(dx)e Ft(=)1703 4619 y Fq(X)1596 4817 y Fk(k)r Fh(2)p Fp(Z)p Fk(;)11 b(k)r Fh(6)p Fl(=)p Fh(\000)p Fl(1)1956 4582 y Fq(Z)2007 4788 y Fp(R)2055 4769 y Fj(3)2109 4705 y Fn(W)27 b Fm(j)p Fn(\036)2301 4720 y Fk(k)2344 4705 y Fm(j)2369 4668 y Fl(2)2434 4705 y Fn(dx)f(;)940 4874 y Fq(Z)991 5080 y Fp(R)1039 5061 y Fj(3)1093 4998 y Fn(W)h Fm(j)p Fn(\033)d Fm(\001)d(r)15 b Fn(\036)p Fm(j)1523 4960 y Fl(2)1588 4998 y Fn(dx)25 b Ft(=)1915 4911 y Fq(X)1808 5109 y Fk(k)r Fh(2)p Fp(Z)p Fk(;)11 b(k)r Fh(6)p Fl(=)p Fh(\000)p Fl(1)2168 4874 y Fq(Z)2218 5080 y Fp(R)2266 5061 y Fj(3)2320 4998 y Fn(W)28 b Fm(j)p Fn(\033)c Fm(\001)c(r)15 b Fn(\036)2725 5013 y Fk(k)2768 4998 y Fm(j)2793 4960 y Fl(2)2858 4998 y Fn(dx)25 b(:)101 5285 y Ft(3.2.)47 b Fu(Tw)m(o)f(useful)h (inequalities.)f Ft(Here)41 b(w)m(e)g(follo)m(w)h(the)f(approac)m(h)g (of)f([4)q(])h(for)f(the)h(Sc)m(hr\177)-45 b(odinger)101 5393 y(op)s(erator.)41 b(Let)31 b Fn(\036)26 b Fm(2)e(D)s Ft(\()p Fi(R)1005 5360 y Fl(3)1045 5393 y Fn(;)15 b Fi(C)1151 5360 y Fl(2)1190 5393 y Ft(\))31 b(and)e(write)1005 5635 y Fm(j)p Fn(\036)p Ft(\()p Fn(x)p Ft(\))p Fm(j)1231 5597 y Fl(2)1297 5635 y Ft(=)c Fm(<)1474 5507 y Fq(\022)1540 5511 y(Z)1631 5537 y Fh(1)1591 5717 y Fk(r)1721 5635 y Fm(\000)p Ft(2)p 1852 5556 233 4 v 15 w Fn(\036)p Ft(\()p Fn(t)15 b(!)s Ft(\))h(\()p Fn(!)24 b Fm(\001)c(r)p Fn(\036)p Ft(\()p Fn(t)15 b(!)s Ft(\)\))26 b Fn(dt)2710 5507 y Fq(\023)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 965 -103 a Fl(HARD)n(Y-TYPE)29 b(ESTIMA)-6 b(TES)31 b(F)n(OR)g(DIRA)n(C)f(OPERA)-6 b(TORS)828 b(7)101 96 y Ft(for)30 b Fn(r)e Ft(=)d Fm(j)p Fn(x)p Fm(j)p Ft(,)31 b Fn(!)d Ft(=)d Fn(x=r)s Ft(.)41 b(Assume)30 b(that)h Fn(W)42 b Ft(is)31 b(a)g(radially)g(symmetric)f(function.)325 185 y Fq(Z)375 391 y Fp(R)423 372 y Fj(3)477 309 y Fn(W)e Fm(j)p Fn(\036)p Fm(j)695 271 y Fl(2)760 309 y Fn(dx)83 b Ft(=)1096 185 y Fq(Z)1147 391 y Fk(S)1194 372 y Fj(2)1247 309 y Fn(d!)1370 185 y Fq(Z)1460 211 y Fh(1)1420 391 y Fl(0)1550 309 y Fn(W)13 b Ft(\()p Fn(r)18 b(!)s Ft(\))d Fm(j)p Fn(\036)p Ft(\()p Fn(r)k(!)s Ft(\))p Fm(j)2147 271 y Fl(2)2202 309 y Fn(r)2246 271 y Fl(2)2300 309 y Fn(dr)942 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4 v 1978 1932 a Fn(\036)2032 1832 y Fq(\021)2102 1777 y( )2351 1871 y Fn(\033)h Fm(\001)c(r)15 b Fn(\036)p 2184 1912 602 4 v 2184 1930 a Fq(p)p 2275 1930 511 4 v 77 x Fn(\016)s Ft(\(1)22 b(+)d Fn(\026)h Ft(+)g Fn(V)h Ft(\))2795 1777 y Fq(!)2897 1932 y Fn(g)2940 1946 y Fk(W)3021 1932 y Ft(\()p Fm(j)p Fn(x)p Fm(j)p Ft(\))26 b Fn(dx)3318 1777 y Fq(!)795 2219 y Fm(\024)f(k)p Fn(g)979 2233 y Fk(W)1060 2219 y Fm(k)1105 2237 y Fk(L)1153 2218 y Fe(1)1218 2237 y Fl(\(0)p Fk(;)p Fh(1)p Fl(\))1418 2090 y Fq(\024)1476 2157 y Ft(1)p 1476 2198 46 4 v 1477 2281 a Fn(\016)1547 2095 y Fq(Z)1597 2301 y Fp(R)1645 2282 y Fj(3)1729 2157 y Fm(j)p Fn(\033)e Fm(\001)e(r)15 b Fn(\036)p Fm(j)2045 2124 y Fl(2)p 1709 2198 396 4 v 1709 2281 a Ft(1)21 b(+)e Fn(\026)h Ft(+)g Fn(V)2140 2219 y(dx)g Ft(+)g Fn(\016)2409 2095 y Fq(Z)2460 2301 y Fp(R)2508 2282 y Fj(3)2546 2219 y Ft(\(1)h(+)f Fn(\026)g Ft(+)g Fn(V)g Ft(\))15 b Fm(j)p Fn(\036)p Fm(j)3131 2181 y Fl(2)3197 2219 y Fn(dx)3296 2090 y Fq(\025)3360 2219 y Fn(:)101 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3156 y Ft(for)32 b Fn(r)e Ft(=)d Fm(j)p Fn(x)p Fm(j)p Ft(,)33 b Fn(!)d Ft(=)d Fn(x=r)s Ft(.)45 b(Assume)31 b(that)i Fn(W)44 b Ft(is)31 b(a)i(radially)f(symmetric)g(function)f(and)h(that)g Fn(\036)p Ft(\(0\))d(=)e(0.)101 3264 y(Then)379 3331 y Fq(Z)429 3537 y Fp(R)477 3518 y Fj(3)531 3455 y Fn(W)h Fm(j)p Fn(\036)p Fm(j)749 3417 y Fl(2)814 3455 y Fn(dx)83 b Ft(=)1150 3331 y Fq(Z)1200 3537 y Fk(S)1247 3518 y Fj(2)1301 3455 y Fn(d!)1423 3331 y Fq(Z)1514 3357 y Fh(1)1474 3537 y Fl(0)1604 3455 y Fn(W)13 b Ft(\()p Fn(r)18 b(!)s Ft(\))d Fm(j)p Fn(\036)p Ft(\()p Fn(r)j(!)s Ft(\))p Fm(j)2200 3417 y Fl(2)2255 3455 y Fn(r)2299 3417 y Fl(2)2354 3455 y Fn(dr)996 3704 y Ft(=)83 b(2)15 b Fm(<)1291 3576 y Fq(\022)1358 3580 y(Z)1409 3787 y Fk(S)1456 3768 y Fj(2)1509 3704 y Fn(d!)1632 3580 y Fq(Z)1722 3607 y Fh(1)1682 3787 y Fl(0)1812 3704 y Fn(W)e Ft(\()p Fn(r)18 b(!)s Ft(\))d Fn(r)2159 3667 y Fl(2)2213 3704 y Fn(dr)2319 3580 y Fq(Z)2410 3607 y Fk(r)2370 3787 y Fl(0)p 2463 3625 V 2463 3704 a Fn(\036)p 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Fq(!)801 5352 y Fm(\024)e(k)p Fn(h)994 5366 y Fk(W)1075 5352 y Fm(k)1120 5371 y Fk(L)1168 5352 y Fe(1)1233 5371 y Fl(\(0)p Fk(;)p Fh(1)p Fl(\))1433 5224 y Fq(\024)1491 5291 y Ft(1)p 1491 5331 46 4 v 1492 5415 a Fn(\016)1562 5228 y Fq(Z)1612 5435 y Fp(R)1660 5416 y Fj(3)1744 5291 y Fm(j)p Fn(\033)e Fm(\001)e(r)15 b Fn(\036)p Fm(j)2060 5258 y Fl(2)p 1724 5331 396 4 v 1724 5415 a Ft(1)21 b(+)e Fn(\026)h Ft(+)g Fn(V)2155 5352 y(dx)g Ft(+)g Fn(\016)2424 5228 y Fq(Z)2475 5435 y Fp(R)2523 5416 y Fj(3)2561 5352 y Ft(\(1)h(+)f Fn(\026)g Ft(+)g Fn(V)g Ft(\))15 b Fm(j)p Fn(\036)p Fm(j)3146 5315 y Fl(2)3212 5352 y Fn(dx)3311 5224 y Fq(\025)3375 5352 y Fn(:)101 5580 y Ft(Again)31 b(w)m(e)g(use)f(the)h(fact)g(that)1190 5506 y Fq(\012)1233 5580 y Fm(j)p Fn(x)p Fm(j)1335 5547 y Fh(\000)p Fl(1)1445 5580 y Fn(h)1497 5594 y Fk(W)1593 5580 y Fn(\033)24 b Fm(\001)c Fn(L)15 b(\036;)g(\036)1939 5506 y Fq(\013)2013 5580 y Ft(is)30 b(real.)201 5688 y(Summarizing,)g(w)m(e)h(ha)m(v)m(e)g (the)g(follo)m(wing)h(result.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 101 -103 a Fl(8)233 b(JEAN)23 b(DOLBEA)n(UL)-6 b(T,)23 b(JA)-8 b(VIER)24 b(DUO)n(ANDIK)n(OETXEA,)e(MARIA)h(J.)h (ESTEBAN,)f(AND)g(LUIS)h(VEGA)101 96 y Fu(Lemma)35 b(8.)42 b Fs(With)34 b(the)f(ab)-5 b(ove)33 b(notations,)i(for)f(any)f Fn(\036)26 b Fm(2)g(D)s Ft(\()p Fi(R)2310 63 y Fl(3)2349 96 y Fn(;)15 b Fi(C)2455 63 y Fl(2)2494 96 y Ft(\))p Fs(,)33 b(for)h(any)f Fn(\016)d(>)25 b Ft(0)p Fs(,)34 b Fn(\026)25 b(>)h Fm(\000)p Ft(1)p Fs(,)33 b(and)101 204 y(any)h(nonne)-5 b(gative)33 b Fn(V)20 b Fs(,)101 467 y Ft(\(4\))408 272 y Fq(D\020)518 373 y Fn(W)33 b Ft(+)759 337 y Fl(2)p 738 352 79 4 v 738 405 a Fh(j)p Fk(x)p Fh(j)842 373 y Fn(g)885 387 y Fk(W)981 373 y Fn(\033)23 b Fm(\001)e Fn(L)1164 272 y Fq(\021)1218 373 y Fn(\036;)15 b(\036)1366 272 y Fq(E)p 408 446 1014 4 v 659 530 a Fm(k)p Fn(g)747 544 y Fk(W)828 530 y Fm(k)873 548 y Fk(L)921 529 y Fe(1)987 548 y Fl(\(0)p Fk(;)p Fh(1)p Fl(\))1457 467 y Fm(\024)1563 406 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1031 y Fp(R)1711 1012 y Fj(3)1795 887 y Fm(j)p Fn(\033)24 b Fm(\001)c(r)15 b Fn(\036)p Fm(j)2111 854 y Fl(2)p 1775 927 396 4 v 1775 1010 a Ft(1)21 b(+)f Fn(\026)g Ft(+)g Fn(V)2206 948 y(dx)h Ft(+)f Fn(\016)2476 824 y Fq(Z)2526 1031 y Fp(R)2574 1012 y Fj(3)2613 948 y Ft(\(1)h(+)f Fn(\026)g Ft(+)g Fn(V)g Ft(\))15 b Fm(j)p Fn(\036)p Fm(j)3198 911 y Fl(2)3264 948 y Fn(dx)25 b(:)101 1171 y Ft(3.3.)47 b Fu(Sp)s(ectral)35 b(decomp)s(osition.)47 b Ft(F)-8 b(or)31 b(a)g(giv)m(en)g Fn(V)46 b Fm(2)25 b(A)2164 1186 y Fl(rad)2269 1171 y Ft(,)31 b(de\014ne)524 1536 y Fn(A)592 1551 y Fk(k)660 1536 y Ft(:=)781 1241 y Fq(8)781 1322 y(>)781 1350 y(>)781 1377 y(>)781 1404 y(>)781 1432 y(<)781 1595 y(>)781 1622 y(>)781 1650 y(>)781 1677 y(>)781 1704 y(:)903 1368 y Ft(sup)910 1443 y Fk(r)r(>)p Fl(0)1055 1239 y Fq(\022)1243 1306 y Ft(1)p 1132 1347 267 4 v 1132 1434 a Fn(r)1176 1407 y Fl(2\()p Fk(k)r Fl(+1\))1439 1244 y Fq(Z)1530 1270 y Fk(r)1489 1450 y Fl(0)1583 1368 y Fn(V)20 b Ft(\()p Fn(s)p Ft(\))15 b Fn(s)1827 1330 y Fl(2\()p Fk(k)r Fl(+1\))2065 1368 y Fn(ds)2155 1239 y Fq(\023)2488 1368 y Ft(if)25 b Fn(k)k Fm(2)24 b Fi(Z)15 b Fn(;)41 b(k)28 b Fm(\025)d Ft(0)h Fn(;)903 1702 y Ft(sup)910 1778 y Fk(r)r(>)p Fl(0)1055 1574 y Fq(\022)1243 1641 y Ft(1)p 1132 1681 V 1132 1768 a Fn(r)1176 1742 y Fl(2\()p Fk(k)r Fl(+1\))1439 1578 y Fq(Z)1530 1605 y Fl(+)p Fh(1)1489 1784 y Fk(r)1675 1702 y Fn(V)20 b Ft(\()p Fn(s)p Ft(\))15 b Fn(s)1919 1664 y Fl(2\()p Fk(k)r Fl(+1\))2157 1702 y Fn(ds)2247 1574 y Fq(\023)2488 1702 y Ft(if)25 b Fn(k)k Fm(2)24 b Fi(Z)15 b Fn(;)41 b(k)28 b Fm(\024)d(\000)p Ft(2)h Fn(:)101 1901 y Ft(Since)32 b Fn(V)51 b Ft(is)32 b(nonnegativ)m(e,)h (observ)m(e)f(that)h Fn(A)1660 1916 y Fk(k)1729 1901 y Fm(\024)27 b Fn(A)1895 1915 y Fl(0)1966 1901 y Ft(for)k(all)i Fn(k)d Fm(2)c Fi(Z)p Fn(;)31 b(k)f Fm(\025)d Ft(0)k(and)g(that)h Fn(A)3210 1916 y Fk(k)3280 1901 y Fm(\024)27 b Fn(A)3446 1915 y Fh(\000)p Fl(2)3572 1901 y Ft(for)101 2009 y(all)32 b Fn(k)c Fm(2)d Fi(Z)p Fn(;)30 b(k)e Fm(\024)d(\000)p Ft(2.)41 b(So,)31 b(if)f Fn(V)51 b Ft(b)s(elongs)30 b(to)h Fm(A)1707 2024 y Fl(rad)1813 2009 y Ft(,)f Fn(A)1936 2024 y Fk(k)2004 2009 y Fm(\024)25 b Ft(1)p Fn(=)p Ft(2)32 b(for)e(all)h Fn(k)e Fm(2)c Fi(Z)p Fn(;)30 b(k)e Fm(6)p Ft(=)d Fm(\000)p Ft(1.)101 2172 y(3.3.1.)48 b Fs(Nonne)-5 b(gative)34 b(sp)-5 b(e)g(ctrum)34 b(of)g Fn(\033)24 b Fm(\001)d Fn(L)p Fs(.)45 b Ft(Assume)31 b(\014rst)f(that)i Fn(k)e Fm(2)c Fi(Z)p Ft(,)31 b Fn(k)f Fm(\025)c Ft(0,)32 b(and)e(let)i(us)f(solv)m(e)h(the)101 2280 y(equation)101 2469 y(\(6\))1065 b Fn(W)1367 2484 y Fk(k)1430 2469 y Ft(+)1531 2407 y(2)p Fn(k)p 1531 2448 96 4 v 1557 2531 a(r)1651 2469 y(g)1694 2484 y Fk(k)1763 2469 y Ft(=)25 b Fn(V)111 b Fm(8)25 b Fn(r)i Fm(2)e Ft(\(0)p Fn(;)15 b Fm(1)p Ft(\))101 2646 y(where)30 b Fn(g)407 2661 y Fk(k)475 2646 y Ft(:=)c Fn(g)640 2671 y Fo(W)697 2688 y(k)774 2646 y Ft(with)k(the)h(notation)g(\(2\),)h Fz(i.e.,)1339 2861 y Fn(g)1382 2876 y Fk(k)1425 2861 y Ft(\()p Fn(r)s Ft(\))26 b(:=)1715 2800 y(1)p 1696 2840 83 4 v 1696 2924 a Fn(r)1740 2897 y Fl(2)1804 2737 y Fq(Z)1895 2764 y Fk(r)1855 2944 y Fl(0)1948 2861 y Fn(W)2034 2876 y Fk(k)2077 2861 y Ft(\()p Fn(s)p Ft(\))15 b Fn(s)2248 2824 y Fl(2)2302 2861 y Fn(ds)25 b(:)101 3052 y Ft(Since)385 3113 y Fq(\022)452 3242 y Fn(r)496 3204 y Fl(2)p Fk(k)588 3118 y Fq(Z)679 3144 y Fk(r)639 3324 y Fl(0)732 3242 y Fn(s)775 3204 y Fl(2)830 3242 y Fn(W)916 3257 y Fk(k)958 3242 y Ft(\()p Fn(s)p Ft(\))h Fn(ds)1187 3113 y Fq(\023)1254 3136 y Fh(0)1302 3242 y Ft(=)f Fn(r)1442 3204 y Fl(2\()p Fk(k)r Fl(+1\))1680 3242 y Fn(W)1766 3257 y Fk(k)1828 3242 y Ft(+)20 b(2)p Fn(k)f(r)2074 3204 y Fl(2)p Fk(k)r Fh(\000)p Fl(1)2272 3118 y Fq(Z)2363 3144 y Fk(r)2323 3324 y Fl(0)2416 3242 y Fn(s)2459 3204 y Fl(2)2513 3242 y Fn(W)2599 3257 y Fk(k)2642 3242 y Ft(\()p Fn(s)p Ft(\))25 b Fn(ds)h Ft(=)f Fn(r)3036 3204 y Fl(2\()p Fk(k)r Fl(+1\))3273 3242 y Fn(V)46 b(;)101 3443 y Ft(Eq.)41 b(\(6\))31 b(can)g(b)s(e)f (solv)m(ed)h(b)m(y)f(taking)1221 3638 y Fn(g)1264 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Fk(L)1762 4349 y Fe(1)1828 4368 y Fl(\(0)p Fk(;)p Fh(1)p Fl(\))2037 4350 y Ft(=)25 b Fn(A)2201 4365 y Fk(k)101 4477 y Ft(b)m(y)40 b(de\014nition)f(of)h Fn(A)831 4492 y Fk(k)874 4477 y Ft(.)68 b(Collecting)42 b(the)e(ab)s(o)m(v)m(e)h(estimates)g(and)e(using)g(Lemma)h(8,)j(for)c Fn(\036)i Ft(=)f Fn(\036)3536 4492 y Fk(k)3620 4477 y Fm(2)101 4585 y(D)s Ft(\()p Fi(R)275 4552 y Fl(3)314 4585 y Fn(;)15 b Fi(C)420 4552 y Fl(2)460 4585 y Ft(\))31 b(suc)m(h)f(that)1239 4694 y Fn(\033)23 b Fm(\001)d Fn(L)15 b(\036)1490 4709 y Fk(k)1559 4694 y Ft(=)25 b Fn(k)18 b(\036)1774 4709 y Fk(k)1842 4694 y Fn(;)106 b(k)29 b Fm(2)c Fi(Z)15 b Fn(;)40 b(k)28 b Fm(\025)d Ft(0)h Fn(;)101 4821 y Ft(w)m(e)31 b(obtain)448 4876 y Fq(Z)498 5082 y Fp(R)546 5063 y Fj(3)600 5000 y Fn(V)k Fm(j)p Fn(\036)767 5015 y Fk(k)810 5000 y Fm(j)835 4962 y Fl(2)900 5000 y Fn(dx)84 b Ft(=)1236 4871 y Fq(\034)1305 4899 y(\020)1359 5000 y Fn(W)1445 5015 y Fk(k)1508 5000 y Ft(+)1637 4938 y(2)p 1608 4979 103 4 v 1608 5062 a Fm(j)p 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Fl(2)p 855 5618 V 856 5701 a Ft(1)21 b(+)f Fn(\026)g Ft(+)g Fn(V)1288 5639 y(dx)h Ft(+)f(\(1)h Fm(\000)f Fn(\026)p Ft(\))1796 5515 y Fq(Z)1847 5721 y Fp(R)1895 5702 y Fj(3)1948 5639 y Fm(j)p Fn(\036)2027 5654 y Fk(k)2070 5639 y Fm(j)2095 5601 y Fl(2)2160 5639 y Fn(dx)h Fm(\000)2371 5515 y Fq(Z)2421 5721 y Fp(R)2469 5702 y Fj(3)2523 5639 y Fn(V)36 b Fm(j)p Fn(\036)2691 5654 y Fk(k)2734 5639 y Fm(j)2759 5601 y Fl(2)2824 5639 y Fn(dx)25 b Fm(\025)g Ft(0)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 965 -103 a Fl(HARD)n(Y-TYPE)29 b(ESTIMA)-6 b(TES)31 b(F)n(OR)g(DIRA)n(C)f(OPERA)-6 b(TORS)828 b(9)101 96 y Ft(if)31 b Fn(\026)f Ft(and)f Fn(\016)35 b Ft(satisfy)30 b(the)h(equations)1572 251 y(1)21 b Fm(\000)f Fn(\026)25 b Ft(=)g Fn(\016)1948 214 y Fl(2)2003 251 y Ft(\(1)c(+)f Fn(\026)p Ft(\))26 b Fn(;)1572 390 y(A)1640 405 y Fk(k)1698 390 y Fn(\016)1741 352 y Fl(2)1802 390 y Fm(\000)20 b Fn(\016)k Ft(+)c Fn(A)2116 405 y Fk(k)2184 390 y Ft(=)25 b(0)g Fn(:)101 545 y Ft(Solutions)31 b(to)g(the)f(ab)s(o) m(v)m(e)i(system)e(of)h(scalar)g(equations)g(are)g(giv)m(en)h(b)m(y) 1253 751 y Fn(\016)d Ft(=)c Fn(\016)1461 713 y Fh(\006)1458 780 y Fk(k)1546 751 y Ft(=)1708 689 y(1)p 1652 730 157 4 v 1652 813 a(2)p Fn(A)1765 828 y Fk(k)1849 623 y Fq(\022)1916 751 y Ft(1)20 b Fm(\006)2072 642 y Fq(q)p 2163 642 329 4 v 109 x Ft(1)h Fm(\000)f Ft(4)15 b Fn(A)2448 719 y Fl(2)2448 780 y Fk(k)2491 623 y Fq(\023)2599 751 y Fn(;)1253 1016 y(\026)25 b Ft(=)g Fn(\026)1484 977 y Fh(\006)1484 1045 y Fk(k)1568 1016 y Ft(=)1674 951 y(1)c Fm(\000)f(j)p Fn(\016)1899 913 y Fh(\006)1896 980 y Fk(k)1959 951 y Fm(j)1984 918 y Fl(2)p 1674 995 350 4 v 1674 1084 a Ft(1)h(+)f Fm(j)p Fn(\016)1899 1046 y Fh(\006)1896 1113 y Fk(k)1959 1084 y Fm(j)1984 1058 y Fl(2)2059 1016 y Ft(=)25 b Fm(\007)2226 907 y Fq(q)p 2316 907 329 4 v 2316 1016 a Ft(1)c Fm(\000)f Ft(4)15 b Fn(A)2601 984 y Fl(2)2601 1045 y Fk(k)2670 1016 y Fn(:)101 1281 y Ft(W)-8 b(e)33 b(observ)m(e)e(that)h Fn(\026)836 1243 y Fl(+)836 1311 y Fk(k)921 1281 y Fn(<)27 b Ft(0)g Fn(<)f(\026)1243 1243 y Fh(\000)1243 1311 y Fk(k)1328 1281 y Ft(=)1426 1177 y Fq(q)p 1517 1177 V 104 x Ft(1)20 b Fm(\000)g Ft(4)15 b Fn(A)1801 1250 y Fl(2)1801 1311 y Fk(k)1845 1281 y Ft(.)43 b(W)-8 b(e)32 b(can)f(therefore)h(c)m(ho)s(ose)g Fn(\016)e Ft(=)d Fn(\016)3120 1243 y Fh(\000)3117 1311 y Fk(k)3211 1281 y Ft(and)j Fn(\026)d Ft(=)f Fn(\026)3622 1243 y Fh(\000)3622 1311 y Fk(k)101 1408 y Ft(in)k(order)g(to)h(maximize)h Fn(\026)p Ft(,)e(and)g(get)635 1495 y Fq(Z)685 1701 y Fp(R)733 1683 y Fj(3)825 1557 y Fm(j)p Fn(\033)24 b Fm(\001)c(r)15 b Fn(\036)1116 1572 y Fk(k)1159 1557 y Fm(j)1184 1524 y Fl(2)p 797 1598 455 4 v 797 1687 a Ft(1)21 b(+)f Fn(\026)1009 1649 y Fh(\000)1009 1717 y Fk(k)1088 1687 y Ft(+)g Fn(V)1287 1619 y(dx)g Ft(+)g(\(1)h Fm(\000)f Fn(\026)1744 1581 y Fh(\000)1744 1648 y Fk(k)1803 1619 y Ft(\))1853 1495 y Fq(Z)1904 1701 y Fp(R)1952 1683 y Fj(3)2006 1619 y Fm(j)p Fn(\036)2085 1634 y Fk(k)2128 1619 y Fm(j)2153 1581 y Fl(2)2218 1619 y 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3728 y Fn(;)15 b Fi(C)1644 3695 y Fl(2)1684 3728 y Ft(\),)30 b(then)g(equalit)m(y)h(o)s(ccurs)f (in)f(\(4\))i(for)f Fn(W)38 b Ft(=)25 b Fn(W)3303 3743 y Fk(k)3345 3728 y Ft(,)31 b(for)e(an)m(y)101 3836 y Fn(k)g Fm(\025)c Ft(0,)31 b(and)e(equalit)m(y)j(also)f(o)s(ccurs)f(in)g (\(5\))i(for)e Fn(W)37 b Ft(=)25 b Fn(W)2056 3851 y Fk(k)2099 3836 y Ft(,)30 b(for)g(an)m(y)h Fn(k)e Fm(\024)24 b(\000)p Ft(2.)41 b(If)30 b(w)m(e)h(decomp)s(ose)f Fn(\036)h Ft(on)101 3944 y(the)e(eigenspaces)i(of)e Fn(\033)21 b Fm(\001)c Fn(L)29 b Ft(as)1154 3875 y Fq(P)1250 3970 y Fk(k)r Fh(6)p Fl(=)p Fh(\000)p Fl(1)1453 3944 y Fn(P)1511 3959 y Fk(k)1554 3944 y Fn(\036)p Ft(,)h(this)e(means)h(that)h(for)f(an)m(y)g Fn(k)2666 3958 y Fl(0)2731 3944 y Fm(2)c Fi(Z)p Ft(,)k Fn(k)2979 3958 y Fl(0)3044 3944 y Fm(6)p Ft(=)c Fm(\000)p Ft(1,)k(suc)m(h)g(that)101 4055 y Fn(\036)155 4070 y Fk(k)192 4079 y Fj(0)256 4055 y Ft(:=)c Fn(P)435 4070 y Fk(k)472 4079 y Fj(0)511 4055 y Fn(\036)h Fm(6)p Ft(=)f(0,)246 4188 y(\(1\))43 b(if)30 b Fn(k)534 4202 y Fl(0)599 4188 y Fm(\025)25 b Ft(0,)31 b(then)f Fn(g)1046 4203 y Fk(k)1083 4212 y Fj(0)1148 4188 y Fm(\021)24 b Fn(A)1311 4203 y Fk(k)1348 4212 y Fj(0)1417 4188 y Ft(is)31 b(constan)m(t)h(a.e.)246 4296 y(\(2\))43 b(if)30 b Fn(k)534 4310 y Fl(0)599 4296 y Fm(\024)25 b(\000)p Ft(2,)31 b(then)f Fn(h)1126 4311 y Fk(k)1163 4320 y Fj(0)1227 4296 y Fm(\021)25 b Fn(A)1391 4311 y Fk(k)1428 4320 y Fj(0)1497 4296 y Ft(is)31 b(constan)m(t)g(a.e.) 101 4436 y(Hence)j(a)f(deriv)-5 b(ation)33 b(with)g(resp)s(ect)g(to)g Fn(r)i Ft(of)e Fn(A)1768 4451 y Fk(k)1805 4460 y Fj(0)1873 4436 y Fm(\021)2096 4400 y Fl(1)p 1983 4415 261 4 v 1983 4478 a Fk(r)2017 4456 y Fj(2\()p Fo(k)2104 4471 y Fj(0)2139 4456 y(+1\))2284 4362 y Fq(R)2345 4389 y Fk(r)2327 4467 y Fl(0)2398 4436 y Fn(V)20 b Ft(\()p Fn(s)p Ft(\))15 b Fn(s)2642 4403 y Fl(2\()p Fk(k)2741 4412 y Fj(0)2776 4403 y Fl(+1\))2913 4436 y Fn(ds)33 b Ft(in)f(the)h(\014rst)f(case,)101 4573 y(and)e(of)h Fn(A)450 4588 y Fk(k)487 4597 y Fj(0)550 4573 y Fm(\021)769 4537 y Fl(1)p 656 4552 V 656 4615 a Fk(r)690 4593 y Fj(2\()p Fo(k)777 4608 y Fj(0)812 4593 y(+1\))958 4500 y Fq(R)1018 4526 y Fl(+)p Fh(1)1001 4605 y Fk(r)1163 4573 y Fn(V)20 b Ft(\()p Fn(s)p Ft(\))15 b Fn(s)1407 4540 y Fl(2\()p Fk(k)1506 4549 y Fj(0)1541 4540 y Fl(+1\))1678 4573 y Fn(ds)30 b Ft(in)h(the)f(second)h(case,)g (sho)m(ws)f(that)1392 4774 y Fn(V)20 b Ft(\()p Fn(r)s Ft(\))25 b(=)1710 4712 y Fn(\027)p 1710 4753 51 4 v 1714 4836 a(r)1862 4774 y Fm(8)g Fn(r)i Fm(2)e Ft(\(0)p Fn(;)15 b Fm(1)p Ft(\))27 b Fn(;)101 4963 y Ft(with)53 b Fn(\027)43 b Ft(=)37 b(2)15 b Fm(j)p Fn(k)659 4977 y Fl(0)725 4963 y Ft(+)25 b(1)p Fm(j)15 b Fn(A)974 4978 y Fk(k)1011 4987 y Fj(0)1050 4963 y Ft(.)63 b(F)-8 b(or)39 b(this)e(p)s(oten)m(tial)i(w) m(e)g(compute)f(all)g Fn(A)2603 4978 y Fk(k)2646 4963 y Ft('s)g(and)f(observ)m(e)h(that)h Fn(A)3533 4977 y Fl(0)3610 4963 y Ft(=)101 5071 y Fn(A)169 5085 y Fh(\000)p Fl(2)289 5071 y Ft(=)25 b Fn(\027)6 b(=)p Ft(2)29 b(and)f(that)h Fn(A)993 5086 y Fk(k)1061 5071 y Fn(<)c(\027)6 b(=)p Ft(2)29 b(for)f(all)h Fn(k)g Fm(6)p Ft(=)c Fm(\000)p Ft(2,)k(0.)41 b(So,)28 b Fn(P)2248 5086 y Fk(k)2292 5071 y Fn(\036)d Ft(=)g(0)k(for)f(all)h Fn(k)g Fm(6)p Ft(=)c Fm(\000)p Ft(2)p Fn(;)15 b Ft(0.)41 b(F)-8 b(or)29 b Fn(k)3449 5085 y Fl(0)3514 5071 y Ft(=)c(0,)101 5178 y Fm(\000)p Ft(2,)41 b Fn(\036)337 5193 y Fk(k)374 5202 y Fj(0)452 5178 y Ft(=)e Fn(P)620 5193 y Fk(k)657 5202 y Fj(0)696 5178 y Fn(\036)g Ft(=)f Fn(f)943 5193 y Fk(k)980 5202 y Fj(0)1018 5178 y Ft(\()p Fn(r)s Ft(\))15 b Fn(F)1205 5193 y Fk(k)1242 5202 y Fj(0)1282 5178 y Ft(,)41 b(where)d Fn(F)1677 5193 y Fk(k)1714 5202 y Fj(0)1791 5178 y Ft(is)h(an)f (eigenfunction)i(of)f Fn(\033)29 b Fm(\001)c Fn(L)39 b Ft(with)f(eigen)m(v)-5 b(alue)41 b Fn(k)3641 5192 y Fl(0)101 5286 y Ft(dep)s(ending)29 b(only)h(on)g(the)g(angular)g(v)-5 b(ariables)30 b(and)g(not)g(on)g Fm(j)p Fn(x)p Fm(j)p Ft(.)40 b(Equalit)m(y)31 b(in)f(\(4\))h(and)e Fn(k)3196 5300 y Fl(0)3261 5286 y Ft(=)c(0)31 b(means)101 5396 y(that)300 5322 y Fq(\000)341 5396 y Fn(\033)25 b Fm(\001)494 5360 y Fk(x)p 474 5375 79 4 v 474 5428 a Fh(j)p Fk(x)p Fh(j)563 5322 y Fq(\001)620 5396 y Ft(\()p Fn(\033)e Fm(\001)e(r)p Ft(\))30 b Fn(\036)971 5410 y Fl(0)1038 5396 y Ft(=)e Fm(\000)p Fn(\016)1251 5358 y Fh(\000)1248 5422 y Fl(0)1310 5396 y Ft(\(1)22 b(+)f Fn(\026)1559 5358 y Fh(\000)1559 5422 y Fl(0)1639 5396 y Ft(+)g Fn(V)f Ft(\))15 b Fn(\036)1908 5410 y Fl(0)1948 5396 y Ft(,)32 b(that)h(is,)f(from)g(Lemma)f(4,)i Fn(f)3012 5410 y Fl(0)3051 5396 y Ft(\()p Fn(r)s Ft(\))f(is)g(a)g(solution)101 5516 y(to)f(the)g(equation)1329 5659 y Fn(f)1384 5621 y Fh(0)1374 5681 y Fl(0)1438 5659 y Ft(=)25 b Fm(\000)15 b Fn(\016)1663 5621 y Fh(\000)1660 5685 y Fl(0)1738 5558 y Fq(\020)1792 5659 y Ft(1)21 b(+)f Fn(\026)2004 5621 y Fh(\000)2004 5685 y Fl(0)2083 5659 y Ft(+)2184 5597 y Fn(\027)p 2184 5638 51 4 v 2188 5721 a(r)2244 5558 y Fq(\021)2329 5659 y Fn(f)2374 5673 y Fl(0)2428 5659 y Fn(;)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 965 -103 a Fl(HARD)n(Y-TYPE)29 b(ESTIMA)-6 b(TES)31 b(F)n(OR)g(DIRA)n(C)f(OPERA)-6 b(TORS)793 b(11)101 97 y Ft(where)30 b Fn(\016)407 58 y Fh(\000)404 123 y Fl(0)467 97 y Ft(\(1)21 b(+)f Fn(\026)714 58 y Fh(\000)714 123 y Fl(0)773 97 y Ft(\))25 b(=)g Fn(\027)36 b Ft(is)31 b(solv)m(ed)g(b)m(y)f Fn(\027)21 b(\016)1610 58 y Fh(\000)1607 123 y Fl(0)1695 97 y Ft(=)k(1)c Fm(\000)1948 20 y(p)p 2024 20 247 4 v 77 x Ft(1)f Fm(\000)g Fn(\027)2231 70 y Fl(2)2270 97 y Ft(.)41 b(Solving)31 b(the)f(equation)h(yields)1381 268 y Fn(f)1426 282 y Fl(0)1465 268 y Ft(\()p Fn(r)s Ft(\))26 b(=)f Fn(A)15 b(r)1828 231 y Fh(\000)p Fl(1+)1973 175 y Fh(p)p 2031 175 164 3 v 2031 231 a Fl(1)p Fh(\000)p Fk(\027)2160 212 y Fj(2)2215 268 y Fn(e)2257 231 y Fh(\000)p Fk(\027)h(r)101 414 y Ft(for)36 b(some)g(constan)m(t)i Fn(A)d Fm(2)f Fi(C)p Ft(.)57 b(On)35 b(the)h(other)h(hand,)f(equalit)m (y)i(for)e(\(5\))h(and)e Fn(k)2898 428 y Fl(0)2973 414 y Ft(=)f Fm(\000)p Ft(2)i(means)g(that)101 450 y Fq(\000)143 524 y Fn(\033)25 b Fm(\001)297 488 y Fk(x)p 278 503 79 4 v 278 556 a Fh(j)p Fk(x)p Fh(j)366 450 y Fq(\001)423 524 y Ft(\()q Fn(\033)e Fm(\001)e(r)p Ft(\))30 b Fn(\036)775 538 y Fh(\000)p Fl(2)899 524 y Ft(=)g Fn(\016)1043 485 y Fh(\000)1040 550 y(\000)p Fl(2)1135 524 y Ft(\(1)23 b(+)f Fn(\026)1386 485 y Fh(\000)1386 550 y(\000)p Fl(2)1502 524 y Ft(+)g Fn(V)e Ft(\))15 b Fn(\036)1772 538 y Fh(\000)p Fl(2)1867 524 y Ft(,)34 b(that)g(is,)g(from)f(Lemma)g(4,)i Fn(f)2941 538 y Fh(\000)p Fl(2)3047 524 y Ft(\()p Fn(r)s Ft(\))e(is)g(a)h(solution)101 644 y(to)d(the)g(equation)1138 790 y Fn(f)1193 753 y Fh(0)1183 813 y(\000)p Fl(2)1302 790 y Ft(=)25 b Fn(\016)1441 752 y Fh(\000)1438 817 y(\000)p Fl(2)1548 690 y Fq(\020)1602 790 y Ft(1)c(+)f Fn(\026)1814 752 y Fh(\000)1814 817 y(\000)p Fl(2)1928 790 y Ft(+)2029 729 y Fn(\027)p 2029 770 51 4 v 2033 853 a(r)2090 690 y Fq(\021)2174 790 y Fn(f)2219 804 y Fh(\000)p Fl(2)2333 790 y Fm(\000)2434 729 y Ft(2)p 2434 770 46 4 v 2435 853 a Fn(r)2505 790 y(f)2550 804 y Fh(\000)p Fl(2)101 964 y Ft(where)30 b Fn(\016)407 926 y Fh(\000)404 990 y(\000)p Fl(2)499 964 y Ft(\(1)21 b(+)f Fn(\026)746 926 y Fh(\000)746 990 y(\000)p Fl(2)840 964 y Ft(\))26 b(=)e Fn(\027)36 b Ft(is)31 b(solv)m(ed)g(b)m(y)f Fn(\027)21 b(\016)1677 926 y Fh(\000)1674 990 y(\000)p Fl(2)1794 964 y Ft(=)k(1)c Fm(\000)2047 888 y(p)p 2122 888 247 4 v 2122 964 a Ft(1)g Fm(\000)f Fn(\027)2330 938 y Fl(2)2369 964 y Ft(.)41 b(Solving)31 b(the)f(equation)h(yields)1375 1143 y Fn(f)1420 1157 y Fh(\000)p Fl(2)1526 1143 y Ft(\()p Fn(r)s Ft(\))26 b(=)f Fn(A)15 b(r)1889 1105 y Fh(\000)p Fl(1)p Fh(\000)2034 1050 y(p)p 2092 1050 164 3 v 2092 1105 a Fl(1)p Fh(\000)p Fk(\027)2221 1086 y Fj(2)2275 1143 y Fn(e)2317 1105 y Fk(\027)h(r)101 1289 y Ft(for)26 b(some)g(constan)m(t)i Fn(A)d Fm(2)g Fi(C)p Ft(.)39 b(Hence)27 b Fn(f)1442 1303 y Fh(\000)p Fl(2)1561 1289 y Ft(is)f(not)h(in)e Fn(L)1969 1256 y Fl(2)2009 1289 y Ft(\()p Fi(R)2110 1256 y Fl(3)2149 1289 y Ft(\))h(unless)g 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Ft(+)g(\(1)h Fm(\000)f Fn(V)g Ft(\))15 b Fn(\036)26 b Ft(=)f Fn(\026)15 b(\036)25 b(:)101 2721 y Ft(Multiplying)31 b(the)g(last)g(equation)g(b) m(y)f Fn(\036)h Ft(and)f(in)m(tegrating)i(o)m(v)m(er)g Fi(R)2407 2688 y Fl(3)2446 2721 y Ft(,)e(w)m(e)h(get)530 2922 y Fn(f)575 2937 y Fk(\036)621 2922 y Ft(\()p Fn(\026)p Ft(\))25 b(:=)892 2798 y Fq(Z)943 3004 y Fp(R)991 2986 y Fj(3)1075 2861 y Fm(j)p Fn(\033)e Fm(\001)e(r)15 b Fn(\036)p Fm(j)1391 2828 y Fl(2)p 1055 2901 V 1055 2984 a Ft(1)20 b(+)g Fn(\026)g Ft(+)g Fn(V)1486 2922 y(dx)g Ft(+)g(\(1)h Fm(\000)f Fn(\026)p Ft(\))1993 2798 y Fq(Z)2044 3004 y Fp(R)2092 2986 y Fj(3)2146 2922 y Fm(j)p Fn(\036)p Fm(j)2250 2884 y Fl(2)2315 2922 y Fn(dx)h Fm(\000)2525 2798 y Fq(Z)2576 3004 y Fp(R)2624 2986 y Fj(3)2678 2922 y Fn(V)35 b Fm(j)p Fn(\036)p Fm(j)2870 2884 y Fl(2)2936 2922 y Fn(dx)25 b Ft(=)g(0)h Fn(:)101 3119 y Ft(By)31 b(monotonicit)m(y)h(of)f Fn(f)943 3134 y Fk(\036)989 3119 y Ft(,)f Fn(f)1089 3134 y Fk(\036)1135 3119 y Ft(\()p Fn(\025)p 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