Content-Type: multipart/mixed; boundary="-------------0303281907417" This is a multi-part message in MIME format. ---------------0303281907417 Content-Type: text/plain; name="03-141.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-141.keywords" Quantum Ornstein-Uhlenbeck Semigroups, noncommutative elliptic operators, standard forms, Dirichlet forms, CCR algebras, quasi-free states. ---------------0303281907417 Content-Type: application/postscript; name="QOUS.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="QOUS.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: QOUS.dvi %%CreationDate: Sat Mar 29 10:03:12 2003 %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR8 CMR10 CMBX10 CMBX12 CMSY10 CMMI12 CMR7 %%+ CMSY8 MSBM10 CMEX10 CMMI8 CMTI12 CMMI6 CMR6 CMSY6 MSAM10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 QOUS %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.03.29:1002 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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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 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e67a35269b877f161 ead3b9314f0ebb23d1a803608fb8bed47522b2a7814eba409f33973ca88ebb86 148d3d0d91d8056e573e23f17b04c8dfeb3d040cab25f1da3caad77b904b64f1 98dc18b1abbbeb700e3bcad39189c15c31bdc0009f94094d223f303408ab2800 533c624b77a8b576fcb13a492a5aac1035d7766bf5e2ba738c1de82bcb395cb4 344d6f452463521d7a6d8d979625e6900ac2b993de3d6a15cf4473176be41862 e61b3f9e109d500405e6f01936bd4cd77ca687b490df2a2e501e141fbecc7082 86f85fa428a8121af362ef3bee57af 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 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dfa618d367b001af bfc9543651fadf6389c92ef4a113933bfb4ce45b9ec1da149da8266d7ae816ac 591763c1ce0975ca2c6fad4dbf00da2d5ea9005dfe188787b84606f607dcb299 dbd62d54c0d9a3f63151b4605487b20b6612bacb429274d9c41584d6b0364b8e 3d9974f806134b12b859a7d761a1cf69b88c3b8fcd7df349ecea60c16041cf86 b3759ca18300e771a59922287a619d7d1e85a6c05fd807d62fb128a85222d66a 5ffdfc23a68bf4b49d6cb703a4b61c83a49b8e965c46262f03314b3b201179ba 4523b7f14bc486d8c369a8681b7a2837f5f82810d48ab9c8063598298c56340e 20e031c2603386b7564b3ff929be77050ce216cf7ccca14fd4916821b5e568bc 46c3ae7c6c63e4bb5f9cff87417005f6403af598ac4fbf73ff557e9124485569 1327cc0942e6061e12c0f107926d130a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 9391472f46a58f4d47b46f95c41dbec5e4518446480e8974913480c0da9f92bc 300aa0015bb8f9b3c2a25d5120886f62b2df404d1b579f01d97195616f552934 e1f979b7ceffab3ce61dee1e4f585e219043cd509c73738a8640bc56f140f093 3a2a0a5aacf4ab20455d4fe0ba59a261a47c000b10d468e73ed6d0b7b9297625 d712e1b77522db962a08c15b1229def94f7102455440c79fb38d12693c4d30ae 9da40e05613e3de42234d7636cc35a394dc57496a409d703d1c0a3a909c278c5 64d1e878e9e78365efa363cce7790b4b5490a8e281528a9b009d4cfc917042d7 8d164ecf2f1a8a9134a81ecf3f940661adb050e4d8db07ee36aa95e8ae818677 c78ae35f7f96df9dd822f2c977157e92b8cac5d14b0e6edead6f3d9554b600d4 7bc69037a4cb8dfa05a66eb9e938d92a1cd50ba41fc486a415cf3b1eec386c0b 1b7471f4bcb055db1419c1c9fe9abc383116d4efa92cee4ef8b8efd52879 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 57 /nine put dup 58 /colon put dup 65 /A 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 75 /K put dup 76 /L put dup 77 /M 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 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 7731f6a36e5ea695f4efc3bf5669b7f60ab606df5819149c0c8a4817f06dc172 0543467645daeffe4457293e0c0407c4ec7857558d0452de2ac773f141783edf 81b30215ae8d2d062aeb3cb69eb0d62bf04780997c9c027a22896f456b784378 7f2ccdd4804375321e7946d3d254521249a4d28a360f4d3163912a688c16e807 15d0440bd27bb78fcbc89bc28028aab2f594f8b641e45b9c599180b69663eef0 03465a2425aa4f8b29dade493002fb335b23eb145108ef3ab828755d3753ea8d f0683265497b1cb5cb484c13f7a4e003fa6ab3ed5e24fcc6125cee4e33936122 3cc907af3c64e5aaad882c356e673e6ca2509a71920e99549d8797e9af709e51 4ebefe61aa0478a60f3a2b7b3d47542fbd126303e31718e6b9c711d202dff7af eb7c19098c9e2a52aa57844eb07e1bfe9067dbe7d0bcd87c86f4ea099deb2373 edf149ab7955979e26ad415a5c29c1dfd02b5f0678dc863d3d6e537ad3b40da7 6529df395597aad7f22d53ed7e16573ce620e179f10fc8e66d1eef855b5427e2 9a7aee650c79f4ee340b1bcc4aef263bba31c829effbd9df68ce46b8fd90d280 0c6578962434e58b2a1c8ae85c4654718c99105befa5d22b215ed4526a3ec28d 85dc2b5e6083260207507833764a3366739e5394386cb88392e7a2595355cbf7 8e2a10f10cc16c4b9da2a57bd7de1967f14099f165439cc419d0fc6e7e4ca766 352fe48a1a08eb047830921078987a678ac77bde7ea81c6ef5c7aed2955f80c3 3421e89357b4513866c70707876d920f5c3624234f176c521a1665dbf5c6b0e8 c8cf86571ae47a7230f9f55a86d97a01e541f9b85f89c5e5494899e6a3c3a8f5 9c0defd988bfc518223c59d54acfc8e7c8927f52f33d20318b0b8941b113b0ec 9d21b459df89dc4937512af344122a970bb9018a4d2772a5f6fb4cd0ff7662f5 51616ba4f2e5b747f50555fca7afe1db1aeb2a20bfed7117f889a865df70eae0 01a7d6bc6d839dddefbc46456be52219c089cc53eb9b918bf2cd74fc898bc7d6 3e070e33258d22db31caab7987b40a952580740b335f8c367b94f29cd9ae77e9 67f4a43ef380d7c0f4c93eef1460f388914fe36f9ee450c26fcbd1856446cfa3 4bc69f1b8c37995c00d55217dfb5cd6e0369a123c450935689d636ec4b0e7c86 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76e1bc7d72c83b7f6b7af882a2229de7ffe8cee7f67a8cfa6f3d8b0f9ff4566f 170cb801967c6de0a46fd0ab7d51dc366f79729e92bbbb12f2d596c91d466499 43626cab02705c8d4c48fdcc9f3dcc3f990d3fe9dede11004c05888dcf88f434 1769f34035a7a8334ded3bedac42 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 21 /lambda put dup 28 /tau put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 72 /H put dup 74 /J put dup 93 /sharp put dup 98 /b put dup 102 /f put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 116 /t put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 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c19ef83c7371c9dd1d841253e2c5f9f620fb10433c08bad7350fc58f219ec49d e0912904d296274279aaf57f122a399ab8665d696a1036bcc9bb95 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 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 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7af0421249bf3142ddccb4b7fa81fc5cc51e540fb4922cc06be13731474c0a1f 2277f51a0c4b4cf330ef97343829d409f6fc994ae5a7667ec13ba110c9ecf68d 303492d8c4557e18a8ff42773ad7ddfe9cefb66d2e0d052ae7e17ced55033659 975ed42dfd1a98e5bfa1ac1d1e90237e3a16bacba571cd69e6a83ccb61171b9f 4e47613a264cb8d7ca3c8087303677ab8a99bd0c357d1d35afacd6de4fdf043e d50b70a0937fb00586f8c6759183651359c4d58d61bc367ad8a6516d682fea39 16543cfa52f3c973bb8ad7488f71f4b5a9de0557f7013723d23d6d17ba4b59ff 536456b4468b2fee2faf016a98d119fe5a956a7b305c6f27b1f67b91bc10d1c6 42f90000653b722b294bf89e35fe6fb13b6b65d33636213f55ac244ce1894a72 a7fd546c68793b292f914631b84f3547a445edf11fc76d3fb3cac8ce266d40eb bcd73952194ba1b88f1031f4a5c3cca1bc84556fb63f4dec3e68b5df6a634d9c 00fabddb46ca95e3b540a46fb8d5925d139c484f0e06e802ed954ebde20663b3 65a1ada2365d2e9fd52f4874c80c4cc8205b5cb16f683214a4a4028968227e39 bccdca40e26c7980f1bece1a0c6ecf6880420270c918a52a4e3bd433b1a07ada 0a2efc8bd401ea8ae25a3bb44f3df89577aeedb61e7dbce9ac6688be223dee78 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03ad78e7fe3ede25ebd7b90e10c8ffdb9b6b4b460b11e84d5d1ed2c307 f30f1dd56f09206083aeb1e5c4efa99c7550086966a11485846744c64511501b 824efac74c4f7f82a5899bd4661ec1e8799ed0d5c8127cc83617f26d23e551b1 ce866a85f935a218c6629183510c5b437e16e972bddf37b94ebeb85a00f5e8ab 4268bd40960af4c45c83d7cb1a3cb5cf3abd14135731f347a030662b439c09cc 9be4024b789884240244b14cfe3e91d02c33e64e62adac7f76187ecdb3ca3ba3 9be671caf8b9834d53892d7b7b479d3b8bd7ff6e41be0ba274add8e0e8cc28c9 6398caad5abac95bc9e715e284f4c2bcf821351085a9e2f74ea924ef915b04bf 5f5d34e5edd95b02ab660c828fbf4a5264cd2db3d4b61603f9b3c67f59dbdcba 6dcf85f4148c9f537dbc17b5291c0f60e4e5d3ec7bd18d68bac76a3069d39cbe 056718ec25dbc5f53a3cec7a9c6d17b7c17a5cd6a7ba919575f92c2f55bc5a5b 356c79deaa597fbc3da512906cb9a5adeb6ae2bc8a00a3fc9d12173a3531a0b3 485716e26edfa038ebeedd835ab03eec796d7ff85f6dcc11d6576f86c5f1ada9 f818f0329cc1bd98a456cce5ef897ac428bcda126b201b4c358bc1b8ac4d3630 b9917105f05622325cd2d267d833c406f1dc3d9d27f1cea251213077f11a461b 0be1c4980a3b6a3e21666635a7051d53b5eb35ba10baa41f6aa50588cfbb6038 f51ecd1a6a199054fe48bc97a8ed28b844c1188dde62d4629681d0edea16b0fa 1d78dc49a7fe446bd56fddc59e8fbfbc488ef0814cc4c692867bb1939753cd9a 81e5e84aa6ca20e5d5316b319d0f9eb6c07a63d8e9d0c67856832eb66ae4308f d3b53b6f62206d662b38ddc766819ca4d7cab8103678083603aef80d428a5c4b 92db49c05d278888f37c57d73d0b240c137df49f62a87ec6e2df3310f6a81e41 1610e5c8673c42ffceee9269f46ee287ae79c17b7de72bc69d3147bdb03c2d90 5f04f17975c4afa85f8afd936e97a47170bff5f1ca025a9f5801be5ecc7f9802 6139ada11283781814fefe0aa5205e5026d21db0f84b8a3e7db819355d4f8098 ba15811bd3346d2a4e772b5bcd91ff59461980656387a588c589ffe4f8402abf 465402f67918836449fcb86f4a916e4f9566013482bd81c2ea85f0dd65b85f9c cd5075941ed19806055f83e1d2dd134950acb50dd5db3e32bf32587b961af8d9 321cb46e8d641b0d89acd655362e4c40c3ebd9a5e2d174a47e71f435faf8b934 7d331cd611461037ae798f1b52f917d6bb5762b25b359837922da8fc6a19a4d3 8735e91d1e51edbbb0d4a4fdb7eded2a5a26f91c635ebb980ee810e93b639aad 8f68b36e4c6394b9f4d08a9957a8800e5c23b254e88b26382d4c9f6bd28ffdaf d0fca23dcd0b8a38dc7cf94744fa9760ada67fdd655ce958e1acd15f55e13147 239d5690533df800183246c1e64a76b35003d5270b67697b0201fcccefb8ef23 2b3af23afd3c1a5ed04b42c518974d90fcfcc47aab66a938cb52f8581b2c6f2a 538bd6cc75c2853427d0dd8d31e6d7e77c47f0416cdc79e0105658445924e2c5 490ad228e2132b8ac9b9cdbd20dc1425a4a4ab64191c702968 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 52 /four put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 64 /at put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y 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{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 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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 49 /one put dup 50 /two put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784dd32fbf3ff701f6b07 72c3661e0561ac614e9d741b232a0a42532bf9f91c1b6e4f7fb5df6a7d57a413 6087c7366279bea1da2e5b45cb2cb412ed06370ab7acfffde951760e5780f1a2 a5b0d3a36404e2cc8fe0d4974abb14ad10292e5dc1052bedb764929bbcd286d1 c08d9f7ccdccaa67dbc6be59a8ceda22dab94428ae070410145447d9aec13d02 017fb1f46e77ceedee642474b15cfcc31626213f04ab512e32081ba880b0f341 58eda42b8e5c337362cdc37858b7ce30fd20ff6407d3d2cc2fdf219725fe5cb9 cc52abff5ab3d8ffcda3bdac17a3a73171d598ff8f4ca694c7c42785193a055c d6041f1727a1decb1bc19834aa6afba9ced9ebedd351095c46f9635ea198b83f c2d35601180b315bf665497dcbc23afe3ed42c803a9d0e8f00021831ddc48526 30b565a2b60211b54517b353c56b506c1f8cd337af6ada68340a07236d8d47b2 4d8257058b300efd0373bab5c4ba73c24195b8c34f99b1357b47d5810e13b4e2 55102ce9702253480412830f50c112def6874aef7121f3802b0334983a62ee04 6736f791aa683776ea768fd0a090a28c4b8459 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 33 /omega put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 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b2262e6cdce2c0c677539ea2767fc38a52ce72539d5ce9d63949184de711e125 246f6535b93101ca0a900b9131bda52720ab58777214cc014a7f6e7f9659e7ae 9226b3d0d316ce21914ff1fe1c7ca1ce2f3d7d502d7e6eedddd24ba90af719de 1de4885aef5233cbea77682326edb54bf1bf1ae33c3a348b7f0f5ef0dea9f80c 579b605d68b6b05d6203dfc67064e167a45d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright 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 63 /perpendicular put dup 65 /A put dup 69 /E put dup 72 /H put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor 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 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd 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4579bdb1ec790a97280eb586b95f0900dfba1d8089e374d371433226e4818798 62391132551119faa99764e6c24a6291b7e5935baabe0e5cb8ef519394ee6f51 02609a26c930884d32da8c89646385c1 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 68 /D put dup 69 /E put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 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 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{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c 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e36aacb564c81ea68da60838cf10153faf91c17f3bc6083e9e833cf15c1a99d5 ff81eb74812b475bd2679666880fed90815a22d54c3f5a166e8b94ca5b199b20 ba993ea39760ddf81888a66e370f8dd0990cf39abe0a6b614e8d6a843b08a5f9 580b6b2dcc7b47997a357c840811e4b73a9fca346b481183e6a22f2331a86e2c f2cd245b004a38be04cd2605692df4dfd0ff019b25c7b4f7a6a553d6db878c87 30a5f83532ba57810aa8554b67d9234eb727b1be2e7b568d7742edcf1623307e 7c0d6aa3d17836131be3699da683b956b090224fda0af85dc6ac10b962a8228d 262384efb735446f682a0da7f4112cdabe4f922eacfcb7b8c88bf550b60957e7 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 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 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 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 73 /I put dup 109 /m put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e50ce572fcc9fc546f deba1eeaeb3eeb7b26c3392319eb3a0b6b0bf883e84d85906b18c1debe42efb1 78ceaa2d89346e5b9ad4eb8347bec3280a4fb9fb98eddbc39dbed37d3777182a 7ac4d6e1bb2be120817d28d7689d4fd1919a6cfe47e96361c75d5dd51ac53c06 382a9f1d0939cb89385d387a295e2e407cb3b56786a12b977ca6967bc974f9b5 4f2be44aaa24764e348be7c280974931f326d0274da6a9b2d3f4a6dd2b3db254 23fc7e6307a83a38ec0e9499368adc5bfd5c67d733a1696426d3b121bb953692 df1f98ed0cd79054698f5386971031d322163563ed7424f8262be5a7f1a4bd02 7e68886c0c0b874be93aaf0d4a3ec3c6d57088f3ebcc939833a5fa0da6a7c82d aa81e1bef6b61246406c2cc925c716ba3d193e9e93c6db1746f326bf920add75 fcb11fb81b2c464d7acbd40f9c5bcf4cdf1f891e2c169ae735249df73a402125 7653dc68bbdf398960412671c6219536885062fea3ebb3c9f8d447df0c447635 86b29a83d884c9bd4007bb5051c1da1e3e5b5160921f5931d0a6af0d7b930d6b c2672dad214756577ebb8866f357619ac43ba5f7cfc734833500ed65e520e3aa e8656caa5d611e8fbf338b67212580787836169903ae3e1aeeaa5342e5c973c5 5b8261d991732acb15d41c6aed22ca074388be3e192638612bf5c1c2dcb2a166 9ca6b0c62d1c37598a105dbc01c8af18ca19c465a8b2b0ef4f5f8557a81868bb 33a81b8c3aa4883aceae7ce25a1e24ff7903c066b58e0871d86d35f3741ab432 63b5160950bb807acb56e68a3ebc952fa0569b65e25010290e5ea70789977617 4cedf8b5500809f307804b03defc49cb94a6b5becb3675b62c4fd0f4e8021715 f33e0a9d714eacde13bb5fca67b4b9d3ebd038857846435beef8a5a2b8bc5e2d eed193e735ec3a9ab55d9fd13f9ee48bea1480fbc9b2fa048acfbb3d9958e2bb 6955df6becbcf7c310d3c84764544cffb4007abd592878405b0fe8ea22f8f4d8 b874c92656823deb2fdd5a564dc12d78894ee4a96989bbcce6b6106186c02390 c496181ccaff5c4e7a235d154545e37575af503688727de723ef61e5b8bce5f1 548984c4528824cbeb1668ff2c49992317b1aa00dcf911a5c5b62a8b1d7ca4d1 f4a4af2c335af411d7db27a3470c8ae189d5c5275e37b05b558672b0593e497d e82c9967bc67ab869c7721c2f4dcffe9a465d8e4898b5ac0a46b6118565ab07a bfd0432ba20ae1ac141cf990e2542c84589dc80741bbe9ba22e220279b8a7a74 037f7266746d65a26bcfa1a4ab960e94f6742e7226f3009c18ecdbe1011e5516 47feff2a03614df8d0ab657280557d9703404b933772c1c9349648aa5bbeecee 717ad56f35b2f53c1c106dd13d142674b082d143709826859d609794abd8333a 3f7063959ad9bc967a8497e1d6c1fe845c01327dedb0ee184adbc9804adaec01 72f5aa02c2aceaf025d099343537557c1b81e1e4a5424790b2e6b8fb17e57e62 f088e8a4e8dd182dd1242df6fad115856484585f9c1f802b5b14113db51331af ed4db19cb7e6f18a21d68440cd03b4e5b7de0463061cc971272820e36dd7af2f 1035e7a29a2fbffd0b23cf65eeb6d0f427a54b060722750de0509df47e99979f 5dd7d009001c25f8a49a7ad9f3944be4604fd97ac9d0bf14d9ca6c13164b5567 9ad319fa16a9a713c17d9fbbc1132a4cda0c007949db01fbdc1500f3591f3ade c6bd86600fba4c9b706034f32a362936d4587fdd406e5953e8cfa83cbec0e701 40d8379151cb4ddeded42618d96549f69d8cd68f2d800e4a37eb3a14faeb7dd9 919de6a8bfc1c186992c286180353fc68a57157bdd620a03a3298ff8c7289bb9 c31121d3684a7a6a11f1f3a7bf2b42b6be9246c120cbd8cffe67ae377498b7e5 d1647a405a8512453651dfe2244bb51795206f0b4ed99612c14532e856ce90b7 e9fd24b3ed9583db4aca6801f31847b6c218261a3502a046bb0cb61e4af7d525 a28e042fe3b855a3fa4ffd063734b221111ec7b646440b2b8391f3beb67c18ce 3c3350eea5643d80c26b35ea7ec28cc356cb255e1648f0b6b6a88f07b8413341 5202523447dfef31225ae4a3e0be6f3161e5788f7f423f1d1e8b1918b5271d1c bea8ed5f7913b8cd87062563cba1fff670401368dd632b074e19987c345cf0fa 35f248260a5530e17f6215211a8af4a9abb21930b7a2b238d2053acd3a86d981 8d0639012ac1ea202df4eb18ecffb8feab7815dc6dc8ba89452c07ca55615ebb a590d9dbe57523ad3f521c2a78d19ecf98b3b891f9d3bcebf870ec7db97a97ef 32140b3c5b1999662d05de208220b8a48c25385f3b8b725aa33c22afc83e8a5b f1136abf81b700a39c61de425ee068944ad8f1767532b5581f9e3fc606f18b09 333ee68465b8a3bbe1b2617589503e4635146b51472e306ab87d02f133f14d02 f6a0bbd8469b9898f7d39c2edaa5f94f8d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 67 /C put dup 70 /F put dup 79 /O put dup 81 /Q put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c 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 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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@start /Fa 252[1033 3[{}1 1328.35 /MSAM10 rf /Fb 207[243 48[{}1 664.176 /CMSY6 rf /Fc 205[406 50[{}1 664.176 /CMR6 rf /Fd 150[296 105[{}1 664.176 /CMMI6 rf /Fe 133[531 631 598 863 598 697 432 531 548 598 664 664 731 1063 332 598 399 399 664 598 399 598 664 598 598 664 3[399 1[399 797 4[966 930 731 948 996 881 996 1[1165 815 999 682 501 2[848 881 981 930 1[966 6[399 664 4[664 664 664 664 664 1[399 465 399 2[531 531 27[731 12[{}59 1328.35 /CMTI12 rf /Ff 133[438 5[340 2[420 474 1[571 832 281 1[381 320 2[452 3[403 4[366 18[514 1[773 10[470 1[261 25[585 4[418 6[549 8[528 12[{}20 885.568 /CMMI8 rf /Fg 144[886 886 8[738 10[738 1697 1919 7[1402 62[793 793 4[627 627 775 775 4[553 553 609 609{}17 1328.35 /CMEX10 rf /Fh 165[886 7[959 3[959 10[959 67[{}4 1328.35 /MSBM10 rf /Fi 206[941 255 14[941 11[732 14[732 2[470 2[732{}7 885.568 /CMSY8 rf /Fj 134[584 3[615 430 437 434 1[615 553 615 922 307 584 1[307 615 2[492 1[492 1[553 27[753 4[861 5[307 11[307 369 45[{}21 1106.96 /CMR10 rf /Fk 205[441 441 49[{}2 774.872 /CMR7 rf /Fl 133[604 5[470 613 1[577 653 1[776 1138 387 676 538 444 2[641 603 676 1[553 683 9[1228 1[883 762 800 1[1030 838 1[1042 1258 2[719 574 1078 1026 2[1080 931 988 975 2[1012 650 1012 361 361 18[852 5[810 846 3[565 739 1[738 568 1[783 759 749 2[642 2[576 675 734 827 11[{}52 1328.35 /CMMI12 rf /Fm 143[1107 4[664 369 517 517 664 664 6[886 886 1[886 886 3[1312 5[1085 924 1[1090 1595 4[1122 2[701 3[1061 1[1033 6[738 0 0 3[886 1328 7[1328 7[1328 6[1033 4[1033 1033 2[1033 2[664 3[1033 1[1033 1033 1033 2[664 1033 369 1033{}40 1328.35 /CMSY10 rf /Fn 134[1136 1[1554 1136 1196 837 849 879 1[1196 1076 1196 1793 598 1136 1[598 1196 1[658 982 1196 956 1[1046 12[1497 1196 1605 1[1470 2[2041 3[801 4[1649 9[598 4[1076 1076 1076 1076 1076 2[598 46[{}34 1912.83 /CMBX12 rf /Fo 134[789 789 1079 1[830 581 589 610 1[830 747 830 1245 415 789 457 415 830 747 457 682 830 664 830 726 12[1039 1[1115 1[1021 2[1418 898 1170 5[981 1145 2[1128 6[415 747 747 747 1[747 747 747 747 747 747 1[415 4[581 581 27[830 12[{}46 1328.35 /CMBX12 rf /Fp 139[542 550 574 14[620 774 678 31[1054 65[{}7 1212.12 /CMBX10 rf /Fq 133[539 640 640 875 640 673 471 478 475 640 673 606 673 1010 337 640 1[337 673 606 370 539 673 539 673 606 7[909 1[1246 1[909 875 673 892 2[943 909 1111 1[943 2[909 1[791 1[926 875 9[606 1[606 2[606 1[606 606 606 1[337 404 337 2[471 471 40[{}50 1212.12 /CMR10 rf /Fr 146[784 35[339 11[732 8[470 470 470 470 470 4[732 1[366 366 17[470 22[{}12 885.568 /CMR8 rf /Fs 128[650 4[578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 3[361 1[361 795 975 975 1336 975 975 939 723 957 1012 885 1012 975 1192 813 1011 668 469 975 1021 849 885 993 939 921 975 3[1012 1[361 361 650 650 650 650 650 650 650 650 650 650 1[361 434 361 1012 650 506 506 17[650 7[1084 723 723 759 939 1012 939 6[1084 1[{}84 1328.35 /CMR12 rf /Ft 134[1113 3[1173 813 825 813 1[1173 1053 1173 1773 573 1113 1[573 1173 1053 633 933 1[933 1173 1053 11[1592 1[1173 1[1653 1[1653 8[1382 2[1533 21[693 45[{}26 2295.84 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 8087 12675 a Ft(Construction)692 b(of)h(a)g(F)-180 b(amily)693 b(of)g(Quan)-60 b(tum)10805 15442 y(Ornstein-Uhlen)g(b)60 b(ec)-60 b(k)692 b(Semigroups)15568 25083 y Fs(Ch)-36 b(ul)434 b(Ki)g(Ko)22179 24601 y Fr(1)23139 25083 y Fs(and)f(Y)-108 b(ong)433 b(Mo)36 b(on)434 b(P)-36 b(ark)35367 24601 y Fr(2)7107 27848 y Fq(Departmen)i(t)405 b(of)f(Mathematics,)h(Y)-101 b(onsei)404 b(Univ)-34 b(ersit)g(y)-101 b(,)404 b(Seoul)g(120-749,)g (Korea)22847 31559 y Fp(Abstract)5387 33806 y Fq(F)-101 b(or)484 b(a)g(giv)-34 b(en)484 b(quasi-free)f(state)i(on)f(the)g(CCR)h (algebra)f(o)-34 b(v)g(er)484 b(one)g(dimensional)g(Hilb)34 b(ert)3569 35312 y(space,)337 b(a)321 b(family)g(of)h(Mark)-34 b(o)g(vian)321 b(semigroups)h(whic)-34 b(h)322 b(lea)-34 b(v)g(e)321 b(the)g(quasi-free)g(state)h(in)-34 b(v)-67 b(arian)-34 b(t)321 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b(forms,)f(CCR)g(algebras,)h(quasi-free)f (states.)1757 50145 y Fo(Mathematical)500 b(sub)83 b(ject)499 b(classi\014cation)i(\(2000\))435 b Fs(:)578 b(46L55,)435 b(46L57.)317 54582 y Fn(1.)2152 b(In)-60 b(tro)60 b(duction)1618 57890 y Fs(The)379 b(purp)36 b(ose)378 b(of)i(this)f(pap)36 b(er)379 b(is)g(to)h(\014nd)e(the)g(explicit)i(expressions)g(of)g(the)e (generators)i(of)g(Mark)-36 b(o-)317 59882 y(vian)457 b(semigroups)e(on)h(the)g(CCR)g(algebra)h Fm(A)e Fs(o)-36 b(v)g(er)457 b(one)e(dimensional)i(Hilb)36 b(ert)455 b(space,)462 b(whic)-36 b(h)456 b(lea)-36 b(v)g(e)317 61875 y(a)498 b(giv)-36 b(en)498 b(\(\014xed\))f(quasi-free)i(state)f Fl(!)545 b Fs(on)497 b Fm(A)h Fs(in)-36 b(v)-72 b(arian)-36 b(t.)770 b(By)499 b(means)e(of)i(noncomm)-36 b(utativ)g(e)497 b(elliptic)317 63868 y(op)36 b(erators)495 b([BP])h(and)e(Diric)-36 b(hlet)495 b(forms)g(on)g(standard)f(forms)h(of)h(v)-36 b(on)495 b(Neumann)f(algebras)h([Cip1],)317 65860 y(w)-36 b(e)436 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b(alue.)587 b(Th)-36 b(us)435 b(the)h(semigroups)g(are)317 11291 y(ergo)36 b(dic)434 b(and)f(tend)g(to)h(the)f(equilibrium)h(exp) 36 b(onen)-36 b(tially)434 b(fast.)2268 13284 y(The)540 b(need)f(to)h(construct)e(Mark)-36 b(o)g(vian)541 b(semigroups)f(on)f (v)-36 b(on)540 b(Neumann)f(algebras,)567 b(whic)-36 b(h)540 b(are)317 15276 y(symmetric)353 b(with)g(resp)36 b(ect)353 b(to)g(a)g(non-tracial)g(state,)370 b(is)353 b(clear)g(for)h(v)-72 b(arious)354 b(applications)f(to)g(op)36 b(en)353 b(sys-)317 17269 y(tems)516 b([Da)-36 b(v],)540 b(quan)-36 b(tum)515 b(statistical)j(mec)-36 b(hanics)516 b([BR],)i(and)e(quan)-36 b(tum)516 b(probabilit)-36 b(y)516 b(theory)h([Acc,)317 19261 y(AFL,)425 b(P)-36 b(art].)576 b(Although)425 b(on)g(the)f(abstract)h(lev)-36 b(el)427 b(w)-36 b(e)425 b(ha)-36 b(v)g(e)426 b(quite)f(a)h(w)-36 b(ell-dev)g(elop)36 b(ed)425 b(theory)h([Cip1,)317 21254 y(GL1,)380 b(GL2],)h(the)366 b(progress)h(in)g(concrete)g(applications) h(is)f(v)-36 b(ery)368 b(slo)-36 b(w.)557 b(W)-108 b(e)367 b(w)-36 b(ould)367 b(lik)-36 b(e)368 b(to)f(men)-36 b(tion)367 b(a)317 23246 y(few)396 b(w)-36 b(orks)396 b(in)f(this)g(direction.)566 b(The)395 b(completely)h(p)36 b(ositiv)-36 b(e)396 b(Hamiltonian)g (semigroup)f(for)h(quan)-36 b(tum)317 25239 y(spin)538 b(c)-36 b(hains)539 b(in)f(the)h(ground)e(state)i(represen)-36 b(tation)538 b(has)h(b)36 b(een)538 b(considered)g(in)g([Mat].)895 b(In)538 b([MZ1,)317 27231 y(MZ2],)434 b(the)e(generalized)i (conditional)f(exp)36 b(ectation)434 b(is)f(used)f(to)h(construct)f (generators)i(of)f(spin-\015ip)317 29224 y(t)-36 b(yp)36 b(e)337 b(dynamics)g(for)h(quan)-36 b(tum)336 b(spin)h(systems.)546 b(In)337 b([P)-36 b(ar],)358 b(one)337 b(of)g(authors)g(ga)-36 b(v)g(e)338 b(a)f(general)h(construc-)317 31216 y(tion)449 b(metho)36 b(d)448 b(of)i(Diric)-36 b(hlet)449 b(forms)h(on)f(standard) f(forms)h(of)h(v)-36 b(on)449 b(Neumann)f(algebras)i(and)e(applied)317 33209 y(the)541 b(metho)36 b(d)541 b(to)g(construct)f(translation)i(in) -36 b(v)-72 b(arian)-36 b(t)541 b(Mark)-36 b(o)g(vian)543 b(semigroups)e(for)h(quan)-36 b(tum)540 b(spin)317 35201 y(systems.)550 b(The)347 b(metho)36 b(d)347 b(of)h([P)-36 b(ar])348 b(has)f(b)36 b(een)347 b(extended)f(to)i(construct)e (symmetric)i(Mark)-36 b(o)g(vian)348 b(semi-)317 37194 y(groups)479 b(on)g(the)f(CCR)i(and)f(CAR)h(algebras)f(with)g(resp)36 b(ect)479 b(to)g(quasi-free)h(states)f([BKP1,)492 b(BKP2],)317 39186 y(and)383 b(on)h(quan)-36 b(tum)382 b(mec)-36 b(hanical)384 b(systems)g([BK].)g(In)g([CFL],)g(quan)-36 b(tum)383 b(O-U)f(semigroups)i(w)-36 b(ere)383 b(con-)317 41179 y(structed)434 b(b)-36 b(y)435 b(means)h(of)g(noncomm)-36 b(utativ)g(e)435 b(Diric)-36 b(hlet)436 b(forms.)584 b(It)435 b(ma)-36 b(y)436 b(b)36 b(e)435 b(w)-36 b(orth)435 b(to)h(men)-36 b(tion)434 b(that)317 43171 y(for)464 b(a)f(giv)-36 b(en)464 b(quasi-free)f(state)g Fl(!)511 b Fs(on)463 b(the)f(CCR)i(algebra)g Fm(A)p Fs(,)470 b(the)463 b(O-U)f(semigroup)h(constructed)f(in)317 45164 y([CFL])386 b(b)36 b(elongs)385 b(to)h(the)e(family)j(of)f(semigroups)f(w)-36 b(e)386 b(constructed)e(in)h(this)g(pap)36 b(er.)562 b(See)385 b(Section)g(3)g(for)317 47156 y(the)433 b(details.)2268 49149 y(Let)471 b(us)h(describ)36 b(e)471 b(the)g(con)-36 b(ten)g(t)471 b(of)i(this)e(pap)36 b(er)471 b(brie\015y)-108 b(.)693 b(Let)471 b Fm(A)h Fs(b)36 b(e)471 b(the)g Fl(C)39669 48667 y Fi(\003)40195 49149 y Fs(-algebra)h(generated)317 51142 y(b)-36 b(y)360 b(the)f(W)-108 b(eyl)360 b(op)36 b(erators)360 b Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\),)374 b Fl(z)428 b Fm(2)369 b Fh(C)p Fl(:)360 b Fs(See)f(Section)g(5.2)i(of)f ([BR])g(for)h(the)e(details.)554 b(The)359 b(quasi-free)317 53134 y(state)434 b Fl(!)481 b Fs(on)433 b Fm(A)h Fs(is)f(giv)-36 b(en)435 b(b)-36 b(y)9756 56586 y Fl(!)48 b Fs(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))368 b(=)g(exp)18667 55510 y Fg(\010)19737 56586 y Fm(\000)21198 55687 y Fs(1)p 21198 56281 651 54 v 21198 57497 a(4)21981 56586 y Fm(j)p Fl(z)59 b Fm(j)23382 56038 y Fr(2)23909 56586 y Fs(\(1)295 b(+)g Fl(e)27270 56038 y Fi(\000)p Ff(\014)28630 56586 y Fs(\)\(1)g Fm(\000)h Fl(e)32519 56038 y Fi(\000)p Ff(\014)33879 56586 y Fs(\))34385 56038 y Fi(\000)p Fr(1)35643 55510 y Fg(\011)36418 56586 y Fl(;)885 b(z)429 b Fm(2)368 b Fh(C)p Fl(;)6766 b Fs(\(1.1\))317 59907 y(where)434 b Fl(\014)443 b Fm(2)368 b Fs(\(0)p Fl(;)221 b Fm(1)p Fs(\))435 b(is)f(the)f(in)-36 b(v)g(erse)433 b(temp)36 b(erature,)433 b(whic)-36 b(h)433 b(is)h(\014xed)g(throughout)e(in)h(this)h(pap)36 b(er.)2268 61900 y(Let)427 b(\()p Fm(H)6220 62099 y Ff(!)6892 61900 y Fl(;)221 b(\031)8212 62099 y Ff(!)8884 61900 y Fl(;)g Fs(\012)10405 62099 y Ff(!)11079 61900 y Fs(\))427 b(b)36 b(e)427 b(the)f(GNS)h(represen)-36 b(tation)426 b([BR])i(of)g(\()p Fm(A)p Fl(;)221 b(!)48 b Fs(\))427 b(and)g(let)g Fm(M)369 b Fs(=)g Fl(\031)44567 62099 y Ff(!)45239 61900 y Fs(\()p Fm(A)p Fs(\))47312 61417 y Fi(00)47877 61900 y Fs(.)576 b(F)-108 b(or)317 63892 y(eac)-36 b(h)535 b Fl(z)602 b Fm(2)542 b Fh(C)p Fl(;)536 b Fs(w)-36 b(e)535 b(denote)g(b)-36 b(y)535 b(\010)16991 64091 y Ff(!)17664 63892 y Fs(\()p Fl(z)59 b Fs(\))535 b(the)g(in\014nitesimal)g(generator)h(of)g(the)f(strongly)h(con)-36 b(tin)g(uous)317 65885 y(unitary)434 b(group)f Fl(\031)9329 66084 y Ff(!)10001 65885 y Fs(\()p Fl(W)181 b Fs(\()p Fl(tz)59 b Fs(\)\))p Fl(;)221 b(t)368 b Fm(2)h Fh(R)g Fs(:)17658 69205 y Fl(\031)18396 69404 y Ff(!)19068 69205 y Fs(\()p Fl(W)181 b Fs(\()p Fl(tz)59 b Fs(\)\))368 b(=)h(exp)27591 68129 y Fg(\000)28200 69205 y Fl(it)p Fs(\010)30053 69404 y Ff(!)30725 69205 y Fs(\()p Fl(z)59 b Fs(\))32400 68129 y Fg(\001)33009 69205 y Fl(:)14667 b Fs(\(1.2\))317 72526 y(Let)433 b Fl(Q)h Fs(and)f Fl(P)615 b Fs(b)36 b(e)433 b(the)g(p)36 b(osition)434 b(and)f(momen)-36 b(tum)433 b(op)36 b(erators)433 b(giv)-36 b(en)435 b(b)-36 b(y)18447 75847 y Fl(Q)369 b Fs(=)g(\010)22166 76046 y Ff(!)22839 75847 y Fs(\(1\))p Fl(;)1522 b(P)550 b Fs(=)368 b(\010)30091 76046 y Ff(!)30764 75847 y Fs(\()p Fl(i)p Fs(\))p Fl(:)15456 b Fs(\(1.3\))25189 79850 y(2)p eop %%Page: 3 3 3 2 bop 2268 7306 a Fs(F)-108 b(rom)465 b(no)-36 b(w)466 b(on)f(w)-36 b(e)466 b(write)g Fm(H)436 b Fs(=)424 b Fm(H)19713 7505 y Ff(!)20385 7306 y Fs(,)473 b Fl(\030)21787 7505 y Fr(0)22737 7306 y Fs(=)423 b(\012)25111 7505 y Ff(!)26249 7306 y Fs(and)465 b Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\))423 b(=)g Fl(\031)34490 7505 y Ff(!)35162 7306 y Fs(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))p Fl(;)664 b Fm(8)p Fl(z)483 b Fm(2)423 b Fh(C)p Fs(.)675 b(Let)465 b Fl(\033)48513 7505 y Ff(t)49374 7306 y Fs(b)36 b(e)317 9298 y(the)433 b(group)g(of)i(automorphisms)e(on)g Fm(M)h Fs(de\014ned)e(b)-36 b(y)16485 12573 y Fl(\033)17224 12772 y Ff(t)17620 12573 y Fs(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))368 b(=)g Fl(W)181 b Fs(\()p Fl(e)25982 12024 y Fi(\000)p Ff(i\014)46 b(t)28002 12573 y Fl(z)59 b Fs(\))p Fl(;)665 b Fm(8)p Fl(z)429 b Fm(2)368 b Fh(C)p Fl(:)13495 b Fs(\(1.4\))317 15847 y(The)498 b(state)h Fl(!)546 b Fs(satis\014es)498 b(the)g Fl(\033)48 b Fs(-KMS)497 b(condition)i([BR])g(and)f(th)-36 b(us)497 b Fl(\033)34599 16046 y Ff(t)34994 15847 y Fl(;)443 b(t)479 b Fm(2)g Fh(R)p Fl(;)499 b Fs(is)g(the)e(group)h(of)h(the)317 17840 y(mo)36 b(dular)365 b(automorphisms.)556 b(Let)365 b(\001)g(and)g Fl(J)488 b Fs(b)36 b(e)365 b(the)g(mo)36 b(dular)365 b(op)36 b(erator)366 b(and)f(mo)36 b(dular)365 b(conjugation)317 19832 y(resp)36 b(ectiv)-36 b(ely)586 b(asso)36 b(ciated)585 b(to)g(\()p Fm(M)p Fl(;)221 b(\030)18892 20031 y Fr(0)19419 19832 y Fs(\).)1031 b(Then)584 b Fl(\033)25603 20031 y Ff(t)25998 19832 y Fs(\()p Fl(A)p Fs(\))626 b(=)f(\001)31332 19350 y Ff(it)32048 19832 y Fl(A)p Fs(\001)34107 19350 y Fi(\000)p Ff(it)35554 19832 y Fl(;)443 b(A)626 b Fm(2)g(M)p Fl(:)585 b Fs(Let)f Fm(M)46088 19350 y Fi(0)46983 19832 y Fs(b)36 b(e)584 b(the)317 21825 y(comm)-36 b(utan)g(t)506 b(of)h Fm(M)p Fs(.)797 b(The)506 b(map)g Fl(j)790 b Fs(:)714 b Fm(M)493 b(!)h(M)25179 21343 y Fi(0)25996 21825 y Fs(is)507 b(the)f(an)-36 b(ti-linear)506 b Fm(\003)p Fs(-isomorphism)g(de\014ned) f(b)-36 b(y)317 23817 y Fl(j)75 b Fs(\()p Fl(A)p Fs(\))369 b(=)f Fl(J)123 b(AJ)-22 b(;)444 b(A)369 b Fm(2)f(M)p Fl(:)429 b Fs(The)f(natural)g(p)36 b(ositiv)-36 b(e)429 b(cone)f Fm(P)538 b Fs(asso)36 b(ciated)429 b(with)f(\()p Fm(M)p Fl(;)221 b(\030)41804 24016 y Fr(0)42331 23817 y Fs(\))428 b(is)g(the)g(closure)317 25810 y(of)324 b(the)f(set)g Fm(f)p Fl(Aj)75 b Fs(\()p Fl(A)p Fs(\))p Fl(\030)10545 26009 y Fr(0)11440 25810 y Fs(:)369 b Fl(A)g Fm(2)g(Mg)p Fs(.)542 b(Then)323 b(\()p Fm(M)p Fl(;)221 b Fm(H)13 b Fl(;)221 b Fm(P)109 b Fl(;)221 b(J)123 b Fs(\))325 b(is)f(the)e(natural)i(standard)e(form)i(asso)36 b(ciated)317 27802 y(to)434 b(\()p Fm(M)p Fl(;)221 b(\030)5158 28001 y Fr(0)5684 27802 y Fs(\).)2268 29795 y(Next)450 b(let)f(us)g(describ) 36 b(e)449 b(the)g(explicit)h(forms)f(of)h(the)f(generators)g(of)h (Mark)-36 b(o)g(vian)451 b(semigroups)e(on)317 31787 y Fm(M)483 b Fs(whic)-36 b(h)483 b(lea)-36 b(v)g(e)484 b Fl(!)531 b Fs(in)-36 b(v)-72 b(arian)-36 b(t.)726 b(In)483 b([BP],)h(the)e(authors)h(constructed)f(Mark)-36 b(o)g(vian)484 b(semigroups)f(on)317 33780 y(v)-36 b(on)358 b(Neumann)f(algebras)h(b) -36 b(y)358 b(the)f(quan)-36 b(tum)356 b(F)-108 b(eynman-Kac)358 b(form)-36 b(ula.)553 b(Let)357 b Fm(N)554 b Fs(b)36 b(e)357 b(a)h(v)-36 b(on)358 b(Neumann)317 35772 y(algebra)386 b(and)e Fl(b)369 b Fs(=)f Fl(b)10156 35290 y Fi(\003)11051 35772 y Fm(2)h(N)196 b Fl(:)384 b Fs(Let)h Fl(\016)434 b Fs(b)36 b(e)385 b(a)g(densely)f(de\014ned)g(*-deriv)-72 b(ation)385 b(on)f Fm(N)580 b Fs(suc)-36 b(h)384 b(that)h Fo(1)369 b Fm(2)f Fl(D)36 b Fs(\()p Fl(\016)50 b Fs(\))317 37765 y(and)433 b Fl(\016)50 b Fs(\()p Fo(1)p Fs(\))369 b(=)g(0.)578 b(Let)433 b Fl(G)11926 37283 y Ff(b)12818 37765 y Fs(b)36 b(e)433 b(an)h(elliptic)g(op)36 b(erator)434 b(on)f Fm(N)629 b Fs(de\014ned)432 b(b)-36 b(y)7359 41039 y Fl(D)36 b Fs(\()p Fl(G)10007 40491 y Ff(b)10465 41039 y Fs(\))1107 b(=)f Fl(D)36 b Fs(\()p Fl(\016)16444 40491 y Fr(2)16970 41039 y Fs(\))p Fl(;)30200 b Fs(\(1.5\))7500 43569 y Fl(G)8526 43020 y Ff(b)8984 43569 y Fs(\()p Fl(A)p Fs(\))1107 b(=)f Fm(\000)15362 42670 y Fs(1)p 15362 43263 651 54 v 15362 44480 a(2)16146 43569 y Fl(\016)16772 43020 y Fr(2)17297 43569 y Fs(\()p Fl(A)p Fs(\))295 b Fm(\000)h Fl(b\016)50 b Fs(\()p Fl(A)p Fs(\))294 b Fm(\000)i Fl(\016)50 b Fs(\()p Fl(A)p Fs(\))p Fl(b)294 b Fs(+)30597 42670 y(1)p 30597 43263 V 30597 44480 a(2)31380 43569 y([)p Fl(b;)221 b Fs([)p Fl(b;)g(A)p Fs(]])p Fl(;)667 b(A)369 b Fm(2)g Fl(D)36 b Fs(\()p Fl(G)42344 43020 y Ff(b)42801 43569 y Fs(\))p Fl(;)317 46969 y Fs(where)578 b([)p Fl(A;)221 b(B)67 b Fs(])616 b(=)f Fl(AB)461 b Fm(\000)393 b Fl(B)67 b(A;)665 b(A;)221 b(B)683 b Fm(2)614 b(N)196 b Fs(.)1012 b(It)578 b(w)-36 b(as)579 b(sho)-36 b(wn)577 b(that)h(the)f(closed)i(extension)f(of)h Fl(G)50253 46487 y Ff(b)317 48962 y Fs(generates)434 b(a)g(Mark)-36 b(o)g(vian)434 b(semigroup)g(on)f Fm(N)630 b Fs([BP].)2268 50955 y(W)-108 b(e)555 b(are)h(searc)-36 b(hing)555 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74374 y(3)31169 74046 y Fs(\)[)p Fl(Q;)221 b Fs([)p Fl(Q;)g(A)p Fs(]])12750 76529 y Fm(\000)p Fl(i\015)72 b(\013)15801 76728 y Fr(1)16327 75453 y Fg(\000)16936 76529 y Fl(Q)p Fs([)p Fl(P)36 b(;)221 b(A)p Fs(])297 b(+)e([)p Fl(P)36 b(;)221 b(A)p Fs(])p Fl(Q)26906 75453 y Fg(\001)27812 76529 y Fm(\000)295 b Fl(i\015)72 b(\013)31158 76728 y Fr(2)31684 76529 y Fl(\013)32511 76728 y Fr(3)33037 75453 y Fg(\000)33646 76529 y Fl(P)181 b Fs([)p Fl(Q;)221 b(A)p Fs(])297 b(+)d([)p Fl(Q;)221 b(A)p Fs(])p Fl(P)43905 75453 y Fg(\001)44516 76529 y Fl(;)25189 79850 y Fs(3)p eop %%Page: 4 4 4 3 bop 317 7306 a Fs(where)436 b Fl(\015)507 b Fs(is)436 b(the)f(normalized)h(constan)-36 b(t)435 b(whic)-36 b(h)436 b(will)h(b)36 b(e)435 b(c)-36 b(hosen)435 b(as)h Fl(\015)445 b Fs(=)372 b(\(1)297 b(+)f Fl(\013)40527 6824 y Fr(2)40519 7634 y(2)41053 7306 y Fs(\))41559 6824 y Fi(\000)p Fr(1)42817 7306 y Fs(.)584 b(Since)436 b Fl(P)616 b Fs(and)317 9298 y Fl(Q)434 b Fs(are)g(un)-36 b(b)36 b(ounded)431 b(op)36 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75516 y Ff(J)50350 75998 y Fl(:)25189 79850 y Fs(5)p eop %%Page: 6 6 6 5 bop 317 7306 a Fs(F)-108 b(or)605 b(an)-36 b(y)605 b(t)-36 b(w)g(o)605 b(elemen)-36 b(ts)604 b Fl(\030)64 b(;)444 b(\021)708 b Fm(2)660 b(H)19032 6824 y Ff(J)19684 7306 y Fs(,)648 b Fl(\030)725 b Fm(\024)661 b Fl(\021)652 b Fs(means)605 b Fl(\021)459 b Fm(\000)412 b Fl(\030)725 b Fm(2)660 b(P)109 b Fs(,)648 b(th)-36 b(us)604 b Fm(H)40722 6824 y Ff(J)41979 7306 y Fs(has)h(an)g(ordered)317 9298 y(structure.)1194 b(An)639 b(an)-36 b(ti-unitary)638 b(in)-36 b(v)g(olution)640 b Fl(J)762 b Fs(on)639 b Fm(H)651 b Fs(b)-36 b(y)639 b Fl(J)123 b Fs(\()p Fl(\030)500 b Fs(+)434 b Fl(i\021)48 b Fs(\))718 b(:=)h Fl(\030)499 b Fm(\000)436 b Fl(i\021)48 b(;)663 b Fm(8)221 b Fl(\030)64 b(;)221 b(\021)768 b Fm(2)718 b(H)49697 8816 y Ff(J)50350 9298 y Fs(,)317 11291 y(preserv)-36 b(es)466 b Fm(P)574 b Fs(and)465 b Fm(H)11122 10809 y Ff(J)11774 11291 y Fs(.)675 b(An)-36 b(y)465 b Fl(J)123 b Fs(-real)466 b(elemen)-36 b(t)465 b 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Fl(\030)359 b Fs(+)295 b Fl(i\021)416 b Fm(2)369 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))18492 10809 y Ff(J)19439 11291 y Fs(+)295 b Fl(iD)36 b Fs(\()p Fm(E)119 b Fs(\))24138 10809 y Ff(J)25159 11291 y Fs(=)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))433 b(where)h Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))36626 10809 y Ff(J)37647 11291 y Fs(:=)369 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))295 b Fm(\\)f Fl(H)44993 10809 y Ff(J)45645 11291 y Fs(.)2268 13284 y(Let)323 b(us)g(denote)g(b)-36 b(y)323 b(Pro)72 b(j\()p Fl(\030)64 b(;)221 b Fm(Q)p Fs(\))324 b(the)f(pro)72 b(jection)324 b(of)g(the)e(v)-36 b(ector)324 b Fl(\030)433 b Fm(2)369 b(H)36850 12801 y Ff(J)37825 13284 y Fs(on)-36 b(to)323 b(the)g(closed,)346 b(con)-36 b(v)g(ex)317 15276 y(cone)434 b Fm(Q)369 b(\032)g(H)7271 14794 y Ff(J)7923 15276 y Fs(.)578 b(F)-108 b(or)433 b Fl(\030)64 b(;)221 b(\021)418 b Fm(2)368 b(H)15855 14794 y Ff(J)16507 15276 y Fs(,)434 b(de\014ne)17999 18597 y Fl(\030)360 b Fm(_)295 b Fl(\021)1154 b Fs(:=)1107 b(Pro)72 b(j)q(\()p Fl(\030)64 b(;)221 b(\021)343 b Fs(+)295 b Fm(P)109 b Fs(\))p Fl(;)17999 20922 y(\030)360 b Fm(^)295 b Fl(\021)1154 b Fs(:=)1107 b(Pro)72 b(j)q(\()p Fl(\030)64 b(;)221 b(\021)343 b Fm(\000)295 b(P)109 b Fs(\))p Fl(:)317 24242 y Fo(De\014nition)500 b(2.1)21 b(.)596 b Fe(A)461 b Fl(J)123 b Fe(-r)-66 b(e)g(al,)460 b(r)-66 b(e)g(al-value)g(d,)460 b(densely)h(de\014ne)-66 b(d)460 b(quadr)-66 b(atic)460 b(form)h Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))460 b Fe(is)h(c)-66 b(al)66 b(le)-66 b(d)317 26235 y(Markovian)464 b(with)h(r)-66 b(esp)g(e)g(ct)463 b(to)i Fl(\030)15998 26434 y Fr(0)16893 26235 y Fm(2)368 b(P)574 b Fe(if)10222 29556 y Fl(\030)433 b Fm(2)369 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))15426 29007 y Ff(J)16542 29556 y Fe(implies)465 b Fl(\030)359 b Fm(^)295 b Fl(\030)23669 29755 y Fr(0)24564 29556 y Fm(2)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))465 b Fe(and)g Fm(E)119 b Fs([)p Fl(\030)359 b Fm(^)295 b Fl(\030)35612 29755 y Fr(0)36138 29556 y Fs(])369 b Fm(\024)g(E)119 b Fs([)p Fl(\030)64 b Fs(])p Fl(:)317 32877 y Fe(A)465 b(close)-66 b(d)464 b(Markovian)g(form)h(is)f(c)-66 b(al)66 b(le)-66 b(d)465 b(a)g(Dirichlet)e(form.)2268 35976 y Fs(Next,)f(w)-36 b(e)456 b(state)f(main)h(results)f(whic)-36 b(h)456 b(will)h(b)36 b(e)455 b(used)g(in)g(the)g(sequel.)645 b(F)-108 b(or)455 b(the)h(pro)36 b(ofs)456 b(w)-36 b(e)456 b(refer)317 37969 y(to)434 b([Cip1].)317 41068 y Fo(Prop)42 b(osition)500 b(2.1)21 b(.)995 b Fe(L)-66 b(et)597 b Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))597 b Fe(b)-66 b(e)597 b(a)g Fl(J)123 b Fe(-r)-66 b(e)g(al,)629 b(r)-66 b(e)g(al)597 b(value)-66 b(d,)630 b(densely)597 b(de\014ne)-66 b(d)597 b(close)-66 b(d)597 b(form.)317 43061 y(Assume)465 b(that)g(the)f(fol)66 b(lowing)465 b(pr)-66 b(op)g(erties)464 b(hold:)1192 46160 y(\(a\))651 b Fl(\030)4137 46359 y Fr(0)5032 46160 y Fm(2)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))p Fe(,)1258 49260 y(\(b\))651 b Fm(E)119 b Fs(\()p Fl(\030)64 b(;)221 b(\030)64 b Fs(\))370 b Fm(\025)f Fs(0)465 b Fe(for)g Fl(\030)433 b Fm(2)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))p Fe(,)1258 52359 y(\(c\))651 b Fl(\030)433 b Fm(2)369 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))8773 51877 y Ff(J)9890 52359 y Fe(implies)464 b Fl(\030)14908 52558 y Fi(\006)16064 52359 y Fm(2)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))465 b Fe(and)g Fm(E)119 b Fs(\()p Fl(\030)25149 52558 y Fr(+)25936 52359 y Fl(;)221 b(\030)27086 52558 y Fi(\000)27874 52359 y Fs(\))369 b Fm(\024)g Fs(0)p Fl(:)317 55459 y Fe(Then)464 b Fm(E)584 b Fe(is)465 b(a)g(Dirichlet)e(form)i(if)f(and)h(only)f(if)g Fm(E)119 b Fs(\()p Fl(\030)64 b(;)221 b(\030)27044 55658 y Fr(0)27571 55459 y Fs(\))369 b Fm(\025)g Fs(0)465 b Fe(for)g(al)66 b(l)465 b Fl(\030)434 b Fm(2)368 b Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))295 b Fm(\\)g(P)109 b Fe(.)317 58558 y Fs(The)506 b(ab)36 b(o)-36 b(v)g(e)507 b(result)e(follo)-36 b(ws)508 b(from)e(Prop)36 b(osition)507 b(4.5)f(\(b\))f(of)i([Cip1])g (and)e(Prop)36 b(osition)507 b(4.10)g(\(ii\))f(of)317 60551 y([Cip1].)317 63650 y Fo(Theorem)499 b(2.1)21 b(.)p Fe(\(The)-66 b(or)g(em)651 b(4.11)i(of)e([Cip1]\))h(L)-66 b(et)651 b Fm(f)p Fl(T)28363 63849 y Ff(t)28758 63650 y Fm(g)29422 63849 y Ff(t)p Fi(\025)p Fr(0)31672 63650 y Fe(b)-66 b(e)651 b(a)h Fl(J)123 b Fe(-r)-66 b(e)g(al,)698 b(str)-66 b(ongly)651 b(c)-66 b(ontinuous,)317 65643 y(symmetric)410 b(semigr)-66 b(oup)411 b(on)h Fm(H)424 b Fe(and)412 b(let)f Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))412 b Fe(b)-66 b(e)411 b(the)g(asso)-66 b(ciate)g(d)411 b(densely)g(de\014ne)-66 b(d)411 b Fl(J)123 b Fe(-r)-66 b(e)g(al,)421 b(r)-66 b(e)g(al)317 67635 y(value)g(d)465 b(quadr)-66 b(atic)464 b(form.)597 b(Then)464 b(the)h(fol)66 b(lowings)465 b(ar)-66 b(e)464 b(e)-66 b(quivalent:)1192 70735 y(\(a\))651 b Fm(f)p Fl(T)4995 70934 y Ff(t)5390 70735 y Fm(g)6054 70934 y Ff(t)p Fi(\025)p Fr(0)8117 70735 y Fe(is)465 b(sub-Markovian.)1258 73834 y(\(b\))651 b Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))465 b Fe(is)f(a)h(Dirichlet)f(form)g(.)25189 79850 y Fs(7)p eop %%Page: 8 8 8 7 bop 317 7306 a Fn(3.)2152 b(Diric)-60 b(hlet)717 b(forms)f(:)957 b(Preliminaries)715 b(and)i(main)g(results)1618 10614 y Fs(Let)433 b Fm(A)g Fs(b)36 b(e)433 b(the)h Fl(C)10479 10132 y Fi(\003)11004 10614 y Fs(-algebra)g(generated)f(b)-36 b(y)434 b(the)f(W)-108 b(eyl)434 b(op)36 b(erators)434 b Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\),)434 b Fl(z)428 b Fm(2)368 b Fh(C)p Fs(,)435 b(satisfying)14655 13869 y Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\))17739 13320 y Fi(\003)19371 13869 y Fs(=)1107 b Fl(W)181 b Fs(\()p Fm(\000)p Fl(z)59 b Fs(\))p Fl(;)11786 16193 y(W)181 b Fs(\()p Fl(z)59 b Fs(\))p Fl(W)181 b Fs(\()p Fl(z)17448 15645 y Fi(0)17758 16193 y Fs(\))1107 b(=)g Fl(e)22093 15645 y Fi(\000)23012 15283 y Fd(i)p 22957 15444 406 40 v 22957 15992 a Fc(2)23496 15645 y Fr(Im\()55 b(\026)-525 b Ff(z)37 b(z)25935 15332 y Fb(0)26233 15645 y Fr(\))26654 16193 y Fl(W)181 b Fs(\()p Fl(z)354 b Fs(+)295 b Fl(z)31497 15645 y Fi(0)31808 16193 y Fs(\))p Fl(;)664 b Fm(8)p Fl(z)59 b(;)221 b(z)35985 15645 y Fi(0)36666 16193 y Fm(2)369 b Fh(C)p Fl(;)8796 b Fs(\(3.1\))317 19448 y(where)502 b(\026)-729 b Fl(z)483 b Fs(is)424 b(a)f(complex)h(conjugate)g(of)g Fl(z)59 b Fs(.)576 b Fm(A)422 b Fs(is)i(the)f(CCR)h(algebra)g(o)-36 b(v)g(er)424 b(one)f(dimensional)h(Hilb)36 b(ert)317 21441 y(space.)579 b(Let)433 b Fl(!)481 b Fs(b)36 b(e)433 b(a)h(quasi-free)g(state)g(on)f Fm(A)h Fs(de\014ned)e(b)-36 b(y)9498 24827 y Fl(!)48 b Fs(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))368 b(=)g(exp)18409 23751 y Fg(\010)19479 24827 y Fm(\000)20940 23928 y Fs(1)p 20940 24521 651 54 v 20940 25738 a(4)21723 24827 y Fm(j)p Fl(z)59 b Fm(j)23124 24278 y Fr(2)23650 24827 y Fs(\(1)296 b(+)e Fl(e)27011 24278 y Fi(\000)p Ff(\014)28372 24827 y Fs(\)\(1)h Fm(\000)g Fl(e)32260 24278 y Fi(\000)p Ff(\014)33621 24827 y Fs(\))34127 24278 y Fi(\000)p Fr(1)35384 23751 y Fg(\011)36159 24827 y Fl(;)665 b Fm(8)p Fl(z)428 b Fm(2)369 b Fh(C)p Fl(;)6507 b Fs(\(3.2\))317 28081 y(where)434 b Fl(\014)443 b(>)368 b Fs(0)434 b(is)g(the)f(in)-36 b(v)g(erse)434 b(temp)36 b(erature.)2268 30074 y(Let)573 b(\()p Fm(H)6366 30273 y Ff(!)7037 30074 y Fl(;)221 b(\031)8357 30273 y Ff(!)9030 30074 y Fl(;)g Fs(\012)10551 30273 y Ff(!)11224 30074 y Fs(\))573 b(b)36 b(e)573 b(the)f(GNS)g(represen)-36 b(tation)572 b([BR])i(of)f(\()p Fm(A)p Fl(;)221 b(!)48 b Fs(\),)608 b(and)572 b(let)h Fm(M)607 b Fs(=)e Fl(\031)47039 30273 y Ff(!)47711 30074 y Fs(\()p Fm(A)p Fs(\))49784 29592 y Fi(00)50350 30074 y Fs(.)317 32066 y(Denote)i(b)-36 b(y)607 b(\010)7871 32265 y Ff(!)8543 32066 y Fs(\()p Fl(z)59 b Fs(\))607 b(the)g(in\014nitesimal)f(generator)h(of)h(the)e (strongly)i(con)-36 b(tin)g(uous)605 b(unitary)i(group)317 34059 y Fl(\031)1055 34258 y Ff(!)1727 34059 y Fs(\()p Fl(W)181 b Fs(\()p Fl(tz)59 b Fs(\)\))433 b(for)h(all)g Fl(z)429 b Fm(2)368 b Fh(C)i Fs(:)17872 36051 y Fl(\031)18610 36250 y Ff(!)19282 36051 y Fs(\()p Fl(W)181 b Fs(\()p Fl(tz)59 b Fs(\)\))368 b(=)g(exp\()p Fl(it)p Fs(\010)29942 36250 y Ff(!)30614 36051 y Fs(\()p Fl(z)59 b Fs(\)\))p Fl(:)317 38735 y Fs(The)434 b(p)36 b(osition)434 b(and)f(momen)-36 b(tum)432 b(op)36 b(erators)434 b(are)g(de\014ned)e(b)-36 b(y)18086 41990 y Fl(Q)369 b Fs(:=)g(\010)22166 42189 y Ff(!)22839 41990 y Fs(\(1\))p Fl(;)1522 b(P)550 b Fs(:=)369 b(\010)30453 42189 y Ff(!)31125 41990 y Fs(\()p Fl(i)p Fs(\))p Fl(;)15095 b Fs(\(3.3\))317 45245 y(resp)36 b(ectiv)-36 b(ely)-108 b(.)672 b(Since)465 b Fl(!)512 b Fs(is)465 b(an)f(en)-36 b(tire)464 b(analytic)i(state)f([BR])g(b)-36 b(y)465 b(\(3.2\),)473 b(the)464 b(cyclic)i(v)-36 b(ector)465 b(\012)46862 45444 y Ff(!)47999 45245 y Fs(is)g(an)317 47237 y(en)-36 b(tire)433 b(analytic)i(v)-36 b(ector)434 b(for)g Fl(P)615 b Fs(and)433 b Fl(Q)p Fs(.)578 b(The)434 b(annihilation)g(and)f(creation)h(op)36 b(erators)434 b(giv)-36 b(en)434 b(b)-36 b(y)16668 50492 y Fl(D)36 b Fs(\()p Fl(a)18973 50691 y Ff(!)19645 50492 y Fs(\))369 b(=)f Fl(D)36 b Fs(\()p Fl(P)181 b Fs(\))295 b Fm(\\)g Fl(D)36 b Fs(\()p Fl(Q)p Fs(\))369 b(=)f Fl(D)36 b Fs(\()p Fl(a)33735 49944 y Fi(\003)33735 50820 y Ff(!)34407 50492 y Fs(\))p Fl(;)12763 b Fs(\(3.4\))14067 52817 y Fl(a)14750 53016 y Ff(!)15791 52817 y Fs(:=)368 b(2)18182 52268 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)20381 52817 y Fs(\()p Fl(Q)296 b Fs(+)e Fl(iP)181 b Fs(\))p Fl(;)1522 b(a)28054 52268 y Fi(\003)28054 53145 y Ff(!)29095 52817 y Fs(:=)369 b(2)31487 52268 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)33686 52817 y Fs(\()p Fl(Q)295 b Fm(\000)g Fl(iP)181 b Fs(\))p Fl(;)317 56071 y Fs(are)478 b(densely)f(de\014ned,)487 b(closed)478 b(and)e(\()p Fl(a)19706 56270 y Ff(!)20378 56071 y Fs(\))20884 55589 y Fi(\003)21853 56071 y Fs(=)443 b Fl(a)23991 55589 y Fi(\003)23991 56400 y Ff(!)24663 56071 y Fs(,)488 b(and)477 b(satisfy)i(the)e(canonical)h(comm)-36 b(utation)477 b(rela-)317 58064 y(tion\(CCR\):)435 b([)p Fl(a)8244 58263 y Ff(!)8916 58064 y Fl(;)221 b(a)10181 57582 y Fi(\003)10181 58392 y Ff(!)10853 58064 y Fs(])370 b(=)e Fo(1)p Fs(.)579 b(See)433 b(Section)h(5.2.3)h(of)f([BR])g(for)h (the)e(details.)2268 60057 y(In)600 b(the)f(rest)g(of)i(this)e(pap)36 b(er,)641 b(w)-36 b(e)600 b(write)g(that)f Fm(H)664 b Fs(=)652 b Fm(H)30863 60256 y Ff(!)31535 60057 y Fs(,)863 b Fl(\030)33327 60256 y Fr(0)34504 60057 y Fs(=)651 b(\012)37106 60256 y Ff(!)38000 60057 y Fs(,)642 b Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\))651 b(=)h Fl(\031)45140 60256 y Ff(!)45812 60057 y Fs(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))p Fl(;)317 62049 y Fs(\010\()p Fl(z)g Fs(\))667 b(=)f(\010)6215 62248 y Ff(!)6888 62049 y Fs(\()p Fl(z)59 b Fs(\))p Fl(;)665 b Fm(8)p Fl(z)726 b Fm(2)666 b Fh(C)p Fl(;)609 b(a)666 b Fs(=)h Fl(a)18849 62248 y Ff(!)20129 62049 y Fs(and)608 b Fl(a)23516 61567 y Fi(\003)24708 62049 y Fs(=)666 b Fl(a)27069 61567 y Fi(\003)27069 62377 y Ff(!)27741 62049 y Fs(.)1103 b(Let)608 b Fl(\033)32449 62248 y Ff(t)33733 62049 y Fs(:)888 b Fm(M)667 b(!)g(M)609 b Fs(b)36 b(e)608 b(the)g(group)g(of)317 64042 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Ff(t)46996 74536 y Fs(\()p Fl(A)p Fs(\))716 b(=)317 76529 y(\001)1401 76047 y Ff(it)2117 76529 y Fl(A)p Fs(\001)4176 76047 y Fi(\000)p Ff(it)5623 76529 y Fl(;)443 b(A)573 b Fm(2)f(M)p Fl(:)554 b Fs(Let)f Fm(M)15988 76047 y Fi(0)16852 76529 y Fs(b)36 b(e)553 b(the)g(comm)-36 b(utan)g(t)553 b(of)h Fm(M)p Fs(.)938 b(The)553 b(map)g Fl(j)870 b Fs(:)794 b Fm(M)573 b(!)h(M)46613 76047 y Fi(0)47477 76529 y Fs(is)553 b(the)25189 79850 y(8)p eop %%Page: 9 9 9 8 bop 317 7306 a Fs(an)-36 b(ti-linear)463 b Fm(\003)p Fs(-isomorphism)g(de\014ned)f(b)-36 b(y)463 b Fl(j)75 b Fs(\()p Fl(A)p Fs(\))418 b(=)h Fl(J)123 b(AJ)-22 b(;)444 b(A)419 b Fm(2)g(M)p Fl(:)464 b Fs(The)f(natural)g(p)36 b(ositiv)-36 b(e)464 b(cone)f Fm(P)317 9298 y Fs(asso)36 b(ciated)381 b(with)f(\()p Fm(M)p Fl(;)221 b(\030)12616 9497 y Fr(0)13143 9298 y Fs(\))380 b(is)g(the)f(closure)h(of)h(the)f (set)g Fm(f)p Fl(Aj)75 b Fs(\()p Fl(A)p Fs(\))p Fl(\030)32157 9497 y Fr(0)33273 9298 y Fs(:)590 b Fl(A)369 b Fm(2)g(Mg)p Fs(.)561 b(Then)379 b(\()p Fm(M)p Fl(;)221 b Fm(H)13 b Fl(;)221 b Fm(P)109 b Fl(;)221 b(J)123 b Fs(\))317 11291 y(is)434 b(the)f(natural)g(standard)g(form)h(asso)36 b(ciated)435 b(to)e(\()p Fm(M)p Fl(;)221 b(\030)27948 11490 y Fr(0)28475 11291 y Fs(\).)2268 13284 y(Next,)434 b(w)-36 b(e)434 b(in)-36 b(tro)36 b(duce)433 b(t)-36 b(w)g(o)434 b(dense)f(manifolds)h(of)g Fm(H)13 b Fs(.)578 b(F)-108 b(or)434 b(an)-36 b(y)433 b Fl(A)369 b Fm(2)g(M)p Fs(,)434 b(de\014ne)14584 16620 y Fl(A)15559 16819 y Ff(n)16555 16620 y Fs(=)17935 15146 y Fg(\020)18866 15722 y Fl(n)p 18862 16315 786 54 v 18862 17531 a(\031)19780 15146 y Fg(\021)20573 15443 y Fr(1)p Ff(=)p Fr(2)22261 14812 y Fg(Z)23811 16620 y Fl(\033)24550 16819 y Ff(t)24945 16620 y Fs(\()p Fl(A)p Fs(\))p Fl(e)27535 16072 y Fi(\000)p Ff(nt)29178 15759 y Fc(2)29694 16620 y Fl(dt;)1522 b(n)369 b Fm(2)g Fh(N)p Fl(:)11594 b Fs(\(3.5\))317 19998 y(Then)423 b Fl(A)4678 20197 y Ff(n)5727 19998 y Fs(is)g(an)g(en)-36 b(tire)423 b(analytic)h(elemen)-36 b(t)422 b(for)i Fl(\033)24908 20197 y Ff(t)25303 19998 y Fs(,)i Fm(k)p Fl(A)27729 20197 y Ff(n)28355 19998 y Fm(k)369 b(\024)g(k)p Fl(A)p Fm(k)665 b(8)p Fl(n)369 b Fm(2)g Fh(N)423 b Fs(and)g Fl(A)41772 20197 y Ff(n)42767 19998 y Fm(!)369 b Fl(A)423 b Fs(strongly)-108 b(.)317 21990 y(See)433 b(the)g(pro)36 b(of)435 b(of)f(Prop)36 b(osition)434 b(2.5.22)i(in)d([BR].)i(Put)10136 25109 y Fm(M)11731 25308 y Fr(0)12626 25109 y Fs(:=)802 b(the)433 b(algebra)i(generated)e(b)-36 b(y)433 b Fl(A)30300 25308 y Ff(n)30927 25109 y Fl(;)664 b(A)369 b Fm(2)g(M)p Fl(;)664 b(n)370 b Fm(2)e Fh(N)p Fl(;)7146 b Fs(\(3.6\))317 28228 y(and)389 b(denote)f(b)-36 b(y)389 b Fm(H)9832 28427 y Ff(f)98 b(in)11718 28228 y Fs(the)388 b(subset)g(of)i(\014nite)f (linear)g(com)-36 b(binations)389 b(of)h(the)f(v)-36 b(ectors)389 b(of)h(the)f(follo)-36 b(wing)317 30221 y(t)g(yp)36 b(e:)15819 32213 y Fl( )16713 31665 y Ff(n)16665 32542 y(m)17921 32213 y Fs(=)368 b Fl(P)20320 31665 y Ff(m)21208 32213 y Fl(Q)22238 31665 y Ff(n)22865 32213 y Fl(\030)23433 32412 y Fr(0)23958 32213 y Fl(;)886 b(m;)221 b(n)370 b Fs(=)f(0)p Fl(;)221 b Fs(1)p Fl(;)g Fs(2)p Fl(;)g Fm(\001)298 b(\001)d(\001)p Fl(:)317 34839 y Fs(Clearly)451 b Fm(M)6444 35038 y Fr(0)6970 34839 y Fl(\030)7538 35038 y Fr(0)8513 34839 y Fs(and)d Fm(H)12179 35038 y Ff(f)98 b(in)14124 34839 y Fs(are)449 b(dense)f(in)h Fm(H)13 b Fs(,)453 b(and)448 b Fl(P)36 b(;)444 b(Q)449 b Fs(and)f Fl(j)75 b Fs(\()p Fl(P)181 b Fs(\))p Fl(;)886 b(j)75 b Fs(\()p Fl(Q)p Fs(\))448 b(are)h(a\016liated)h(with)f Fm(M)317 36831 y Fs(and)433 b Fm(M)4441 36349 y Fi(0)4752 36831 y Fs(,)h(resp)36 b(ectiv)-36 b(ely)434 b(\(see)g(Lemma)f(4.1\).) 2268 38824 y(F)-108 b(or)433 b(an)-36 b(y)434 b Fl(z)428 b Fm(2)369 b Fh(C)p Fs(,)434 b(w)-36 b(e)434 b(write)15715 41943 y Fl(\033)16454 42142 y Ff(z)16985 41943 y Fs(\()p Fl(a)p Fs(\))368 b(:=)h Fl(e)21393 41394 y Ff(i\014)46 b(z)22817 41943 y Fl(a;)1522 b(\033)26122 42142 y Ff(z)26653 41943 y Fs(\()p Fl(a)27842 41394 y Fi(\003)28367 41943 y Fs(\))369 b(:=)g Fl(e)31587 41394 y Fi(\000)p Ff(i\014)46 b(z)33743 41943 y Fl(a)34426 41394 y Fi(\003)34952 41943 y Fl(;)12724 b Fs(\(3.7\))317 45062 y(and)5880 47054 y Fl(\033)6619 47253 y Ff(z)7149 47054 y Fs(\()p Fl(P)181 b Fs(\))369 b(:=)g Fm(\000)p Fl(i)p Fs(2)13418 46506 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)15617 45978 y Fg(\000)16226 47054 y Fl(e)16829 46506 y Ff(i\014)46 b(z)18253 47054 y Fl(a)295 b Fm(\000)g Fl(e)21162 46506 y Fi(\000)p Ff(i\014)46 b(z)23318 47054 y Fl(a)24001 46506 y Fi(\003)24526 45978 y Fg(\001)25135 47054 y Fl(;)1523 b(\033)27758 47253 y Ff(z)28288 47054 y Fs(\()p Fl(Q)p Fs(\))369 b(:=)g(2)33091 46506 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)35290 45978 y Fg(\000)35899 47054 y Fl(e)36502 46506 y Ff(i\014)46 b(z)37926 47054 y Fl(a)295 b Fs(+)g Fl(e)40814 46506 y Fi(\000)p Ff(i\014)46 b(z)42970 47054 y Fl(a)43653 46506 y Fi(\003)44178 45978 y Fg(\001)44787 47054 y Fl(:)2889 b Fs(\(3.8\))317 49680 y(Denote)538 b(b)-36 b(y)537 b(\010)h(either)f Fl(P)718 b Fs(or)538 b Fl(Q)p Fs(.)890 b(In)537 b(fact,)564 b(one)538 b(ma)-36 b(y)538 b(b)36 b(e)537 b(able)h(to)f(sho)-36 b(w)538 b(that)f(for)h Fl(\030)610 b Fm(2)545 b(M)46747 49879 y Fr(0)47273 49680 y Fl(\030)47841 49879 y Fr(0)48904 49680 y Fs(the)317 51672 y(function)536 b Fl(t)543 b Fm(7!)g Fl(\033)9137 51871 y Ff(t)9533 51672 y Fs(\(\010\))p Fl(\030)600 b Fs(has)536 b(an)g(analytic)h(extension)f(on)g Fh(C)p Fs(,)562 b(whic)-36 b(h)536 b(is)h(denoted)e(b)-36 b(y)535 b Fl(\033)44578 51871 y Ff(z)45109 51672 y Fs(\(\010\))p Fl(\030)64 b Fs(,)562 b(and)317 53665 y(that)433 b Fl(\033)3874 53864 y Ff(z)4405 53665 y Fs(\(\010\))g(is)h(equal)h(to)e(that)h(of)g (\(3.8\))g(on)f Fm(M)23927 53864 y Fr(0)24453 53665 y Fl(\030)25021 53864 y Fr(0)25547 53665 y Fs(.)579 b(See)433 b(Lemma)g(4.1)i(for)f Fl(D)36 b Fs(\(\010\).)2268 55657 y(As)292 b(discussed)f(in)g(In)-36 b(tro)36 b(duction,)320 b(for)292 b(an)-36 b(y)292 b(parameters)f Fl(\013)30310 55856 y Fr(1)30836 55657 y Fl(;)443 b(\013)32467 55856 y Fr(2)32993 55657 y Fl(;)g(\013)34624 55856 y Fr(3)35150 55657 y Fl(;)292 b Fs(w)-36 b(e)292 b(in)-36 b(tro)36 b(duce)291 b(a)h(sesquilinear)317 57650 y(form)434 b(corresp)36 b(onding)433 b(to)h(the)f(op)36 b(erator)434 b Fl(H)536 b Fs(giv)-36 b(en)434 b(in)f(\(1.9\))h(b)-36 b(y)2957 60769 y Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))1107 b(=)f Fm(M)10725 60968 y Fr(0)11251 60769 y Fl(\030)11819 60968 y Fr(0)12345 60769 y Fl(;)35331 b Fs(\(3.9\))2169 63723 y Fm(E)119 b Fs(\()p Fl(\021)48 b(;)221 b(\030)64 b Fs(\))1107 b(=)9263 62825 y Fl(\015)p 9263 63418 747 54 v 9311 64635 a Fs(2)10143 63723 y(\(1)295 b(+)g Fl(\013)13736 63175 y Fr(2)13728 64052 y(2)14263 63723 y Fs(\))p Fm(h)15286 62648 y Fg(\000)15894 63723 y Fl(\033)16633 63930 y Fi(\000)p Ff(i=)p Fr(4)18682 63723 y Fs(\()p Fl(P)181 b Fs(\))295 b Fm(\000)g Fl(j)75 b Fs(\()p Fl(\033)24194 63930 y Fi(\000)p Ff(i=)p Fr(4)26242 63723 y Fs(\()p Fl(P)181 b Fs(\)\))28779 62648 y Fg(\001)29388 62945 y Fi(\003)29913 63723 y Fl(\021)48 b(;)31185 62648 y Fg(\000)31794 63723 y Fl(\033)32533 63930 y Fi(\000)p Ff(i=)p Fr(4)34582 63723 y Fs(\()p Fl(P)181 b Fs(\))295 b Fm(\000)g Fl(j)h Fs(\()p Fl(\033)40315 63930 y Fi(\000)p Ff(i=)p Fr(4)42364 63723 y Fs(\()p Fl(P)181 b Fs(\)\))44901 62648 y Fg(\001)45509 63723 y Fl(\030)498 b Fm(i)9130 66548 y Fs(+)10275 65649 y Fl(\015)p 10275 66242 V 10323 67459 a Fs(2)11155 66548 y(\()p Fl(\013)12496 66000 y Fr(2)12488 66876 y(1)13317 66548 y Fs(+)295 b Fl(\013)15459 66000 y Fr(2)15451 66876 y(3)15986 66548 y Fs(\))p Fm(h)17009 65472 y Fg(\000)17617 66548 y Fl(\033)18356 66755 y Fi(\000)p Ff(i=)p Fr(4)20405 66548 y Fs(\()p Fl(Q)p Fs(\))f Fm(\000)i Fl(j)75 b Fs(\()p Fl(\033)25928 66755 y Fi(\000)p Ff(i=)p Fr(4)27976 66548 y Fs(\()p Fl(Q)p Fs(\)\))30524 65472 y Fg(\001)31133 65769 y Fi(\003)31658 66548 y Fl(\021)48 b(;)32930 65472 y Fg(\000)33539 66548 y Fl(\033)34278 66755 y Fi(\000)p Ff(i=)p Fr(4)36327 66548 y Fs(\()p Fl(Q)p Fs(\))295 b Fm(\000)g Fl(j)i Fs(\()p Fl(\033)42072 66755 y Fi(\000)p Ff(i=)p Fr(4)44120 66548 y Fs(\()p Fl(Q)p Fs(\)\))46668 65472 y Fg(\001)47277 66548 y Fl(\030)497 b Fm(i)9130 69093 y(\000)p Fl(i\015)72 b(\013)12181 69292 y Fr(1)12707 69093 y Fm(h)13224 68017 y Fg(\000)13833 69093 y Fl(\033)14572 69299 y Fi(\000)p Ff(i=)p Fr(4)16620 69093 y Fs(\()p Fl(Q)p Fs(\))295 b(+)g Fl(j)75 b Fs(\()p Fl(\033)22122 69299 y Fi(\000)p Ff(i=)p Fr(4)24171 69093 y Fs(\()p Fl(Q)p Fs(\)\))26719 68017 y Fg(\001)27327 68314 y Fi(\003)27853 69093 y Fl(\021)48 b(;)29125 68017 y Fg(\000)29734 69093 y Fl(\033)30473 69299 y Fi(\000)p Ff(i=)p Fr(4)32521 69093 y Fs(\()p Fl(P)181 b Fs(\))295 b Fm(\000)h Fl(j)g Fs(\()p Fl(\033)38255 69299 y Fi(\000)p Ff(i=)p Fr(4)40303 69093 y Fs(\()p Fl(P)181 b Fs(\)\))42840 68017 y Fg(\001)43449 69093 y Fl(\030)498 b Fm(i)9130 71417 y(\000)p Fl(i\015)72 b(\013)12181 71616 y Fr(2)12707 71417 y Fl(\013)13534 71616 y Fr(3)14060 71417 y Fm(h)14577 70341 y Fg(\000)15186 71417 y Fl(\033)15925 71624 y Fi(\000)p Ff(i=)p Fr(4)17974 71417 y Fs(\()p Fl(P)181 b Fs(\))294 b(+)h Fl(j)75 b Fs(\()p Fl(\033)23464 71624 y Fi(\000)p Ff(i=)p Fr(4)25513 71417 y Fs(\()p Fl(P)181 b Fs(\)\))28050 70341 y Fg(\001)28658 70639 y Fi(\003)29184 71417 y Fl(\021)48 b(;)30456 70341 y Fg(\000)31065 71417 y Fl(\033)31804 71624 y Fi(\000)p Ff(i=)p Fr(4)33852 71417 y Fs(\()p Fl(Q)p Fs(\))295 b Fm(\000)h Fl(j)g Fs(\()p Fl(\033)39597 71624 y Fi(\000)p Ff(i=)p Fr(4)41645 71417 y Fs(\()p Fl(Q)p Fs(\)\))44193 70341 y Fg(\001)44802 71417 y Fl(\030)64 b Fm(i)317 74536 y Fs(for)405 b Fl(\021)48 b(;)442 b(\030)433 b Fm(2)369 b(M)7619 74735 y Fr(0)8145 74536 y Fl(\030)8713 74735 y Fr(0)9239 74536 y Fs(,)410 b(where)404 b Fl(\015)441 b Fs(=)369 b(\(1)235 b(+)f Fl(\013)19707 74054 y Fr(2)19699 74865 y(2)20234 74536 y Fs(\))20740 74054 y Fi(\000)p Fr(1)21997 74536 y Fs(.)569 b(Notice)405 b(that)e Fm(E)523 b Fs(is)405 b(w)-36 b(ell)404 b(de\014ned)f(on)h Fm(M)42701 74735 y Fr(0)43227 74536 y Fl(\030)43795 74735 y Fr(0)44725 74536 y Fs(b)-36 b(y)404 b(Lemma)317 76529 y(4.1.)25189 79850 y(9)p eop %%Page: 10 10 10 9 bop 317 7306 a Fo(Prop)42 b(osition)500 b(3.1)21 b(.)598 b Fe(L)-66 b(et)464 b(the)g(p)-66 b(ar)g(ameters)464 b Fl(\013)23204 7505 y Fr(1)23730 7306 y Fl(;)443 b(\013)25361 7505 y Fr(2)25887 7306 y Fl(;)g(\013)27518 7505 y Fr(3)28413 7306 y Fm(2)369 b Fh(R)465 b Fe(satisfy)g(the)f(fol)66 b(lowing)465 b(r)-66 b(elations:)15485 9483 y Fs(1)p 15485 10076 651 54 v 15485 11293 a(2)16269 10382 y(\(1)295 b(+)g Fl(\013)19862 9833 y Fr(2)19854 10710 y(2)20388 10382 y Fs(\))221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\))369 b(=)g Fm(\000)p Fl(\013)30165 10581 y Fr(1)30913 10382 y Fs(cosh\()p Fl(\014)74 b(=)p Fs(2\))p Fl(;)10529 b Fs(\(3.10\))15485 12592 y(1)p 15485 13185 V 15485 14401 a(2)16269 13490 y(\()p Fl(\013)17610 12942 y Fr(2)17602 13819 y(1)18431 13490 y Fs(+)295 b Fl(\013)20573 12942 y Fr(2)20565 13819 y(3)21100 13490 y Fs(\))221 b(sinh)o(\()p Fl(\014)74 b(=)p Fs(2\))370 b(=)e Fl(\013)29843 13689 y Fr(2)30369 13490 y Fl(\013)31196 13689 y Fr(3)31944 13490 y Fs(cosh\()p Fl(\014)74 b(=)p Fs(2\))p Fl(:)317 16435 y Fe(Then)464 b(the)h(sesquiline)-66 b(ar)464 b(form)g(given)f (by)i(\(3.9\))h(e)-66 b(quals)465 b(to)6011 19379 y Fl(D)36 b Fs(\()p Fm(E)119 b Fs(\))1107 b(=)g Fm(M)13780 19578 y Fr(0)14306 19379 y Fl(\030)14874 19578 y Fr(0)15399 19379 y Fl(;)31627 b Fs(\(3.11\))5223 21704 y Fm(E)119 b Fs(\()p Fl(\021)48 b(;)221 b(\030)64 b Fs(\))1107 b(=)g Fl(\015)12860 21903 y Fr(1)13385 21704 y Fm(h)13902 20628 y Fg(\000)14510 21704 y Fl(\033)15249 21911 y Fi(\000)p Ff(i=)p Fr(4)17298 21704 y Fs(\()p Fl(P)181 b Fs(\))295 b Fm(\000)g Fl(j)75 b Fs(\()p Fl(\033)22810 21911 y Fi(\000)p Ff(i=)p Fr(4)24859 21704 y Fs(\()p Fl(P)181 b Fs(\)\))27396 20628 y Fg(\001)28004 21704 y Fl(\021)48 b(;)29276 20628 y Fg(\000)29885 21704 y Fl(\033)30624 21911 y Fi(\000)p Ff(i=)p Fr(4)32673 21704 y Fs(\()p Fl(P)181 b Fs(\))294 b Fm(\000)i Fl(j)g Fs(\()p Fl(\033)38406 21911 y Fi(\000)p Ff(i=)p Fr(4)40455 21704 y Fs(\()p Fl(P)181 b Fs(\)\))42992 20628 y Fg(\001)43600 21704 y Fl(\030)529 b Fm(i)12185 24029 y Fs(+)p Fl(\015)13872 24228 y Fr(2)14397 24029 y Fm(h)14914 22953 y Fg(\000)15522 24029 y Fl(\033)16261 24235 y Fi(\000)p Ff(i=)p Fr(4)18310 24029 y Fs(\()p Fl(Q)p Fs(\))295 b Fm(\000)g Fl(j)75 b Fs(\()p Fl(\033)23833 24235 y Fi(\000)p Ff(i=)p Fr(4)25881 24029 y Fs(\()p Fl(Q)p Fs(\)\))28429 22953 y Fg(\001)29038 24029 y Fl(\021)48 b(;)30310 22953 y Fg(\000)30919 24029 y Fl(\033)31658 24235 y Fi(\000)p Ff(i=)p Fr(4)33707 24029 y Fs(\()p Fl(Q)p Fs(\))294 b Fm(\000)i Fl(j)g Fs(\()p Fl(\033)39451 24235 y Fi(\000)p Ff(i=)p Fr(4)41500 24029 y Fs(\()p Fl(Q)p Fs(\)\))44048 22953 y Fg(\001)44656 24029 y Fl(\030)64 b Fm(i)317 26973 y Fe(wher)-66 b(e)11823 28966 y Fl(\015)12498 29165 y Fr(1)13393 28966 y Fs(=)368 b Fm(\000)p Fl(\015)72 b(\013)17380 29165 y Fr(1)17906 28966 y Fl(=)221 b Fs(sinh)q(\()p Fl(\014)74 b(=)p Fs(2\))p Fl(;)1550 b(\015)26804 29165 y Fr(2)27698 28966 y Fs(=)369 b Fl(\015)72 b(\013)30653 29165 y Fr(2)31179 28966 y Fl(\013)32006 29165 y Fr(3)32532 28966 y Fl(=)221 b Fs(sinh\()p Fl(\014)74 b(=)p Fs(2\))p Fl(:)8183 b Fs(\(3.12\))317 31516 y Fe(Thus)466 b(by)f(\(3.11\))g Fm(E)584 b Fe(is)465 b(a)g(p)-66 b(ositive)463 b(form.)2268 34518 y Fs(In)434 b(the)g(rest)h(of)g(the)f(pap)36 b(er,)435 b Fe(we)466 b(assume)g(that)g(the)f(r)-66 b(elations)465 b(\(3.10\))i(hold.)581 b Fs(W)-108 b(e)434 b(state)h(the)f(main)317 36510 y(results.)317 39287 y Fo(Theorem)499 b(3.1)21 b(.)612 b Fe(The)469 b(form)g Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))469 b Fe(de\014ne)-66 b(d)468 b(as)i(in)e(\(3.9\))i(\(and)g(so)g(as)g(in)e(\(3.11\)\))j(is)e (closable.)317 41280 y(Denote)497 b(by)h Fs(\()p 7051 40213 820 54 v Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p 10075 40213 V Fm(E)119 b Fs(\)\))498 b Fe(the)g(closur)-66 b(e)498 b(of)f Fs(\()p Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p Fm(E)119 b Fs(\)\))498 b Fe(and)g(by)g Fl(H)601 b Fe(the)497 b(p)-66 b(ositive)497 b(self-adjoint)g(op)-66 b(er)g(ator)317 43272 y(asso)g(ciate)g(d)464 b(to)h Fs(\()p 8461 42205 V Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p 11485 42205 V Fm(E)119 b Fs(\)\))p Fe(.)597 b(Then)465 b(the)f(fol)66 b(lowing)465 b(pr)-66 b(op)g(erties)464 b(hold:)1192 46049 y(\(a\))651 b Fl(\030)4137 46248 y Fr(0)5032 46049 y Fm(2)368 b Fl(D)36 b Fs(\()p Fl(H)103 b Fs(\))464 b Fe(and)h Fl(H)103 b(\030)14332 46248 y Fr(0)15226 46049 y Fs(=)369 b(0)p Fe(,)1258 49041 y(\(b\))p 3569 47974 V 651 w Fm(E)584 b Fe(is)464 b Fl(J)123 b Fe(-r)-66 b(e)g(al,)464 b(that)h(is,)p 14878 47974 V 464 w Fm(E)119 b Fs([)p Fl(J)k(\030)64 b Fs(])369 b(=)p 19644 47974 V 369 w Fm(E)119 b Fs([)p Fl(\030)64 b Fs(])466 b Fe(for)e Fl(\030)434 b Fm(2)368 b Fl(D)36 b Fs(\()p 28237 47974 V Fm(E)119 b Fs(\))p Fl(;)1258 52033 y Fe(\(c\))p 3569 50966 V 651 w Fm(E)g Fs(\()p Fl(\030)5463 52232 y Fr(+)6250 52033 y Fl(;)221 b(\030)7400 52232 y Fi(\000)8188 52033 y Fs(\))369 b Fm(\024)g Fs(0)465 b Fe(for)f(any)h Fl(\030)433 b Fm(2)369 b Fl(D)36 b Fs(\()p 20024 50966 V Fm(E)119 b Fs(\))295 b Fm(\\)f(H)23960 51551 y Ff(J)24613 52033 y Fe(.)317 54810 y(F)-100 b(urthermor)-66 b(e)465 b(the)f(form)g Fs(\()p 13626 53743 V Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p 16650 53743 V Fm(E)120 b Fs(\)\))464 b Fe(is)h(a)g(Dirichlet)e(form.) 2268 57587 y Fs(Recall)504 b(that)e(a)g(symmetric,)521 b(strongly)503 b(con)-36 b(tin)g(uous,)519 b(p)36 b(ositiv)-36 b(e)503 b(preserving)f(semigroup)h Fm(f)p Fl(T)48054 57786 y Ff(t)48449 57587 y Fm(g)49113 57786 y Ff(t)p Fi(\025)p Fr(0)317 59580 y Fs(on)434 b Fm(H)447 b Fs(is)434 b(called)h Fe(er)-66 b(go)g(dic)433 b Fs(if)i(for)f(an)-36 b(y)435 b Fl(\021)48 b(;)442 b(\030)434 b Fm(2)370 b(P)405 b(\000)295 b(f)p Fs(0)p Fm(g)p Fs(,)436 b(there)d(exists)i Fl(t)370 b(>)f Fs(0)435 b(suc)-36 b(h)433 b(that)h Fm(h)p Fl(\021)48 b(;)221 b(T)46404 59779 y Ff(t)46799 59580 y Fl(\030)64 b Fm(i)370 b Fl(>)f Fs(0.)317 61572 y(Let)640 b Fl(H)743 b Fs(b)36 b(e)640 b(the)g(self-adjoin)-36 b(t)641 b(generator)f(of)i Fl(T)24435 61771 y Ff(t)24830 61572 y Fs(:)1209 b Fl(T)27162 61771 y Ff(t)28278 61572 y Fs(=)721 b Fl(e)30614 61090 y Fi(\000)p Ff(tH)32582 61572 y Fl(;)443 b(t)721 b Fm(\025)g Fs(0.)1199 b(An)640 b(eigen)-36 b(v)g(ector)641 b(of)h Fl(H)317 63565 y Fs(corresp)36 b(onding)562 b(to)g(the)g(lo)-36 b(w)g(est)563 b(eigen)-36 b(v)-72 b(alue)563 b(inf)315 b Fl(\033)48 b Fs(\()p Fl(H)103 b Fs(\))561 b(will)i(b)36 b(e)562 b(called)h(a)g Fe(gr)-66 b(ound)583 b(state)562 b Fs(for)h Fl(H)103 b Fs(\(or)317 65557 y Fm(f)p Fl(T)1743 65756 y Ff(t)2139 65557 y Fm(g)2803 65756 y Ff(t)p Fi(\025)p Fr(0)4400 65557 y Fs(\).)572 b(The)412 b(ergo)36 b(dicit)-36 b(y)414 b(of)g(its)f(semigroup)f(is)h (equiv)-72 b(alen)-36 b(t)414 b(to)f(that)f(the)g(lo)-36 b(w)g(est)414 b(eigen)-36 b(v)-72 b(alue)413 b(of)h Fl(H)317 67550 y Fs(has)439 b(m)-36 b(ultiplicit)g(y)439 b(one)f(and)g(there)g (exists)h(a)g(strictly)h(p)36 b(ositiv)-36 b(e)439 b(ground)f(state)g (for)i Fl(H)540 b Fs(\(Theorem)439 b(4.3)317 69542 y(of)434 b([Cip2]\).)317 72544 y Fo(Theorem)499 b(3.2)21 b(.)1163 b Fe(L)-66 b(et)652 b Fm(f)p Fl(T)14016 72743 y Ff(t)14412 72544 y Fm(g)15076 72743 y Ff(t)p Fi(\025)p Fr(0)17327 72544 y Fe(b)-66 b(e)652 b(the)h(symmetric)f(Markovian)g(semigr)-66 b(oup)652 b(asso)-66 b(ciate)g(d)653 b(to)g(the)317 74536 y(Dirichlet)522 b(form)h Fs(\()p 9493 73469 V Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p 12517 73469 V Fm(E)119 b Fs(\)\))524 b Fe(and)f(let)g Fl(H)626 b Fe(b)-66 b(e)523 b(the)g(Dirichlet)f(op)-66 b(er)g(ator,)537 b(i.e.,)f Fl(T)39465 74735 y Ff(t)40339 74536 y Fs(=)477 b Fl(e)42431 74054 y Fi(\000)p Ff(tH)44399 74536 y Fe(.)773 b(Then)523 b(the)317 76529 y(fol)66 b(lowing)465 b(r)-66 b(esults)465 b(hold:)24864 79850 y Fs(10)p eop %%Page: 11 11 11 10 bop 1192 7306 a Fe(\(a\))651 b Fl(H)567 b Fe(is)465 b(essential)66 b(ly)465 b(self-adjoint)e(on)i Fm(H)22642 7505 y Ff(f)98 b(in)24138 7306 y Fe(.)1258 10405 y(\(b\))651 b(The)d(sp)-66 b(e)g(ctrum)648 b(of)g(gener)-66 b(ator)647 b Fl(H)751 b Fe(ar)-66 b(e)648 b(discr)-66 b(ete,)692 b(i.e.,)g Fl(\033)48 b Fs(\()p Fl(H)103 b Fs(\))708 b(=)h Fm(f)p Fl(m\034)40005 10604 y Fr(1)40962 10405 y Fs(+)430 b Fl(n\034)43745 10604 y Fr(2)45202 10405 y Fs(:)930 b Fl(m;)221 b(n)710 b Fs(=)3569 12398 y(0)p Fl(;)221 b Fs(1)p Fl(;)g Fs(2)p Fl(;)g(:::)p Fm(g)p Fl(;)469 b Fe(wher)-66 b(e)465 b Fl(\034)14077 12597 y Fr(1)14971 12398 y Fs(=)369 b(2)p Fl(\015)17677 12597 y Fr(1)18424 12398 y Fs(sinh\()p Fl(\014)74 b(=)p Fs(2\))p Fl(;)664 b(\034)25454 12597 y Fr(2)26349 12398 y Fs(=)369 b(2)p Fl(\015)29055 12597 y Fr(2)29802 12398 y Fs(sinh)o(\()p Fl(\014)74 b(=)p Fs(2\))p Fl(:)1258 15497 y Fe(\(c\))651 b(The)465 b(zer)-66 b(o)464 b(is)h(a)g(simple)g(eigenvalue)e(of)i Fl(H)567 b Fe(with)465 b(eigenve)-66 b(ctor)462 b Fl(\030)35139 15696 y Fr(0)35665 15497 y Fe(.)597 b(Thus)466 b Fm(f)p Fl(T)41375 15696 y Ff(t)41770 15497 y Fm(g)42434 15696 y Ff(t)p Fi(\025)p Fr(0)44497 15497 y Fe(is)f(er)-66 b(go)g(dic.)2268 18929 y Fs(The)574 b(eigenspaces)g(of)g Fl(H)676 b Fs(will)575 b(b)36 b(e)574 b(giv)-36 b(en)574 b(in)g(Theorem)f(5.2)i(\(b\).)998 b(By)574 b(the)f(sp)36 b(ectral)574 b(theorem,)317 20922 y(Theorem)434 b(3.2)g(\(b\))f (implies)h(that)f(for)h(an)-36 b(y)434 b Fl(\030)f Fm(2)369 b(H)446 b Fs(and)433 b Fl(t)369 b(>)g Fs(0)14724 24242 y Fm(k)p Fl(T)16150 24441 y Ff(t)16546 24242 y Fl(\030)359 b Fm(\000)296 b(h)p Fl(\030)19887 24441 y Fr(0)20412 24242 y Fl(;)221 b(\030)64 b Fm(i)p Fl(\030)22711 24441 y Fr(0)23238 24242 y Fm(k)369 b(\024)g Fl(e)26276 23694 y Fi(\000)p Ff(\034)102 b(t)27922 24242 y Fm(k)p Fl(\030)360 b Fm(\000)295 b(h)p Fl(\030)31927 24441 y Fr(0)32453 24242 y Fl(;)221 b(\030)64 b Fm(i)p Fl(\030)34752 24441 y Fr(0)35278 24242 y Fm(k)p Fl(;)317 27563 y Fs(where)434 b Fl(\034)517 b Fs(=)368 b(min)p Fm(f)p Fl(\034)9934 27762 y Fr(1)10460 27563 y Fl(;)443 b(\034)11829 27762 y Fr(2)12355 27563 y Fm(g)p Fl(:)434 b Fs(Th)-36 b(us)433 b Fm(f)p Fl(T)18535 27762 y Ff(t)18930 27563 y Fm(g)19594 27762 y Ff(t)p Fi(\025)p Fr(0)21626 27563 y Fs(con)-36 b(v)g(erges)434 b(to)f(the)g(equilibrium)h(exp)36 b(onen)-36 b(tially)435 b(fast.)317 30663 y Fo(Remark)499 b(3.1)20 b(.)620 b Fe(L)-66 b(et)471 b Fm(f)p Fl(T)12623 30862 y Ff(t)13019 30663 y Fm(g)13683 30862 y Ff(t)p Fi(\025)p Fr(0)15753 30663 y Fe(b)-66 b(e)471 b(a)i(str)-66 b(ongly)471 b(c)-66 b(ontinuous,)473 b(symmetric)d(Markovian)h(semigr)-66 b(oup)472 b(on)317 32655 y Fm(H)13 b Fe(.)597 b(Consider)464 b(the)h(symmetric)e(emb)-66 b(e)g(dding:)21113 35976 y Fl(i)21557 36175 y Fr(0)22673 35976 y Fs(:)591 b Fm(M)812 b(\000)-221 b(!)812 b(H)22442 38301 y Fl(i)22886 38500 y Fr(0)23411 38301 y Fs(\()p Fl(A)p Fs(\))369 b(=)g(\001)28232 37752 y Fr(1)p Ff(=)p Fr(4)29698 38301 y Fl(A\030)31241 38500 y Fr(0)31767 38301 y Fl(:)317 41622 y Fe(De\014ne)464 b(the)g(maps)i Fl(S)10766 41821 y Ff(t)11626 41622 y Fe(on)e Fm(M)h Fe(by)16327 44943 y Fl(S)17127 45142 y Ff(t)17891 44943 y Fs(:)369 b Fm(M)h(!)f(M)p Fl(;)1550 b(i)26233 45142 y Fr(0)27054 44943 y Fm(\016)295 b Fl(S)28813 45142 y Ff(t)29577 44943 y Fs(=)369 b Fl(T)31720 45142 y Ff(t)32411 44943 y Fm(\016)295 b Fl(i)33814 45142 y Fr(0)34340 44943 y Fl(:)12686 b Fs(\(3.13\))317 48263 y Fe(It)579 b(fol)66 b(lows)580 b(fr)-66 b(om)578 b(The)-66 b(or)g(em)578 b(2.12)h(of)g([Cip1])f(that)h Fm(f)p Fl(S)27837 48462 y Ff(t)28232 48263 y Fm(g)28896 48462 y Ff(t)p Fi(\025)p Fr(0)31073 48263 y Fe(is)g(a)g(we)-66 b(akly)579 b(c)-66 b(ontinuous)579 b(Markovian)317 50256 y(semigr)-66 b(oup)464 b(on)h Fm(M)p Fe(.)317 53688 y Fo(Remark)499 b(3.2)20 b(.)567 b Fe(L)-66 b(et)368 b Fm(N)564 b Fe(b)-66 b(e)369 b(the)g(maximal)f(ab)-66 b(elian)368 b(sub)-66 b(algebr)g(a)368 b(of)h Fm(M)h Fe(gener)-66 b(ate)g(d)367 b(by)i Fs(exp)p Fm(f)p Fl(ik)45 b(Q)p Fm(g)p Fl(;)443 b(k)414 b Fm(2)317 55680 y Fh(R)p Fe(.)1189 b(L)-66 b(et)661 b Fl(S)6104 55879 y Ff(t)7161 55680 y Fe(b)-66 b(e)662 b(the)f(Markovian)g(semigr)-66 b(oup)662 b(on)g Fm(M)g Fe(obtaine)-66 b(d)661 b(by)h(\(3.13\).)1189 b(As)662 b(mentione)-66 b(d)660 b(in)317 57673 y(Intr)-66 b(o)g(duction,)399 b(it)385 b(c)-66 b(an)385 b(b)-66 b(e)385 b(che)-66 b(cke)g(d)385 b(that)h(for)f(any)h Fl(\013)25471 57872 y Fr(1)25997 57673 y Fl(;)443 b(\013)27628 57872 y Fr(2)28154 57673 y Fl(;)g(\013)29785 57872 y Fr(3)30680 57673 y Fm(2)369 b Fh(R)386 b Fe(satisfying)f(\(3.10\))i(the)e(r)-66 b(estriction)317 59665 y(of)367 b Fl(S)2547 59864 y Ff(t)3309 59665 y Fe(to)f Fm(N)563 b Fe(gives)366 b(a)h(unique)f(O-U)h(semigr)-66 b(oup)366 b(whose)h(gener)-66 b(ation)365 b(satis\014es)i(the)f(e)-66 b(quation)366 b(in)g(\(1.12\).)2268 63097 y Fs(Before)347 b(closing)f(this)g(section,)364 b(w)-36 b(e)346 b(sho)-36 b(w)346 b(that)f(distinct)g(v)-72 b(alues)347 b(of)f(the)f(parameters)h Fl(\013)44616 63296 y Fr(1)45142 63097 y Fl(;)443 b(\013)46773 63296 y Fr(2)47299 63097 y Fl(;)g(\013)48930 63296 y Fr(3)49825 63097 y Fm(2)317 65089 y Fh(R)469 b Fs(satisfying)i (\(3.10\))f(giv)-36 b(e)470 b(distinct)e(semigroups)h(on)g Fm(H)13 b Fs(.)684 b(It)469 b(follo)-36 b(ws)471 b(from)e(Theorem)g (3.2)h(\(b\))e(that)317 67082 y(the)433 b(sp)36 b(ectrum)433 b(of)h(eac)-36 b(h)434 b(generator)f Fl(H)536 b Fs(is)434 b(c)-36 b(haracterized)433 b(b)-36 b(y)19556 70403 y Fl(\015)20231 70602 y Fr(1)20756 70403 y Fl(=\015)22081 70602 y Fr(2)22976 70403 y Fs(=)369 b Fm(\000)p Fl(\013)26217 70602 y Fr(1)26743 70403 y Fl(=)p Fs(\()p Fl(\013)28726 70602 y Fr(2)29252 70403 y Fl(\013)30079 70602 y Fr(3)30605 70403 y Fs(\))p Fl(:)317 73724 y Fs(W)-108 b(e)434 b(\014rst)f (consider)g(t)-36 b(w)g(o)434 b(sp)36 b(ecial)434 b(examples)g(and)f (then)g(describ)36 b(e)433 b(general)h(case.)24864 79850 y(11)p eop %%Page: 12 12 12 11 bop 317 7306 a Fo(Example)520 b(3.1.)635 b Fe(the)482 b(O-U)g(semigr)-66 b(oup)482 b(studie)-66 b(d)482 b(in)f([CFL].)1299 b Fs(Cho)36 b(ose)453 b Fl(\013)37197 7505 y Fr(3)38124 7306 y Fs(=)401 b(1)452 b(and)g Fm(\000)p Fl(\013)45047 7505 y Fr(1)45974 7306 y Fs(=)401 b Fl(\013)48214 7505 y Fr(2)48740 7306 y Fs(.)635 b(It)317 9298 y(follo)-36 b(ws)436 b(from)e(\(3.10\))g(and)f(\(3.12\))i(that)15776 12290 y Fl(\015)16451 12489 y Fr(1)18094 12290 y Fs(=)1117 b Fl(\015)20898 12489 y Fr(2)21424 12290 y Fl(;)15623 14615 y(\013)16450 14814 y Fr(2)18094 14615 y Fs(=)g(\(cosh\()p Fl(\014)74 b(=)p Fs(2\))296 b Fm(\006)f Fs(1\))p Fl(=)221 b Fs(sinh\()p Fl(\014)74 b(=)p Fs(2\))11983 b(\(3.14\))18083 16940 y Fm(\021)1107 b Fl(\013)21050 17139 y Fr(2)p Ff(;)p Fi(\006)22570 16940 y Fs(\()p Fl(\014)74 b Fs(\))p Fl(:)317 19932 y Fs(Using)434 b(\(3.8\),)g(w)-36 b(e)434 b(obtain)g(from)g (\(3.11\))g(that)6830 22924 y Fm(E)119 b Fs(\()p Fl(\030)64 b(;)221 b(\030)64 b Fs(\))370 b(=)f Fl(\015)13006 21449 y Fg(\020)13799 22924 y Fm(k)14463 21848 y Fg(\000)15072 22924 y Fl(\026a)294 b Fm(\000)i Fl(\025j)75 b Fs(\()p Fl(a)20722 22375 y Fi(\003)21247 22924 y Fs(\))21753 21848 y Fg(\001)22361 22924 y Fl(\030)64 b Fm(k)23657 22375 y Fr(2)24479 22924 y Fs(+)295 b Fm(k)26450 21848 y Fg(\000)27059 22924 y Fl(\025a)28501 22375 y Fi(\003)29321 22924 y Fm(\000)g Fl(\026j)75 b Fs(\()p Fl(a)p Fs(\))33740 21848 y Fg(\001)34348 22924 y Fl(\030)64 b Fm(k)35644 22375 y Fr(2)36170 21449 y Fg(\021)36964 22924 y Fl(;)664 b Fm(8)p Fl(\030)433 b Fm(2)369 b(M)42578 23123 y Fr(0)43104 22924 y Fl(\030)43672 23123 y Fr(0)317 26044 y Fs(where)434 b Fl(\026)368 b Fs(=)h Fl(\025)7366 25562 y Fi(\000)p Fr(1)8992 26044 y Fs(=)g Fl(e)10976 25562 y Ff(\014)46 b(=)p Fr(4)12979 26044 y Fs(and)433 b Fl(\015)441 b Fs(=)368 b Fl(\015)18679 26243 y Fr(1)19574 26044 y Fs(=)g Fl(\015)21629 26243 y Fr(2)22155 26044 y Fs(.)578 b(The)434 b(ab)36 b(o)-36 b(v)g(e)434 b(is)g(the)f(form)h(studied)f(in)g([CFL].)317 28666 y Fo(Example)499 b(3.2.)1301 b Fs(Consider)434 b Fl(\013)16363 28865 y Fr(2)17258 28666 y Fs(=)368 b(1.)579 b(It)434 b(follo)-36 b(ws)435 b(from)f(\(3.10\))h(and)e(\(3.12\))h (that)7200 31658 y Fl(\013)8027 31857 y Fr(1)9660 31658 y Fs(=)1107 b Fm(\000)221 b Fs(tanh\()p Fl(\014)74 b(=)p Fs(2\))p Fl(;)7200 33983 y(\013)8027 34182 y Fr(3)9660 33983 y Fs(=)1107 b(coth\()p Fl(\014)74 b(=)p Fs(2\))295 b Fm(\006)18980 32907 y Fg(\000)19810 33983 y Fs(coth)22267 33411 y Fr(2)22792 33983 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))296 b Fm(\000)f Fs(tanh)30138 33411 y Fr(2)30663 33983 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))33783 32907 y Fg(\001)34392 33204 y Fr(1)p Ff(=)p Fr(2)9660 36307 y Fs(=)1107 b(tanh)14380 35735 y Fr(2)14906 36307 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))p Fl(=)18676 35231 y Fg(\002)19452 36307 y Fs(coth)o(\()p Fl(\014)g(=)p Fs(2\))296 b Fm(\007)26652 35231 y Fg(\000)27482 36307 y Fs(coth)29939 35735 y Fr(2)30465 36307 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))295 b Fm(\000)h Fs(tanh)37810 35735 y Fr(2)38336 36307 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))41456 35231 y Fg(\001)42065 35529 y Fr(1)p Ff(=)p Fr(2)43532 35231 y Fg(\003)44085 36307 y Fl(;)5502 38632 y(\015)6177 38831 y Fr(1)6703 38632 y Fl(=\015)8028 38831 y Fr(2)9660 38632 y Fs(=)1107 b(coth\()p Fl(\014)74 b(=)p Fs(2\))17356 37556 y Fg(\002)18131 38632 y Fs(coth\()p Fl(\014)g(=)p Fs(2\))295 b 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Fs(cosh\()p Fl(\014)74 b(=)p Fs(2\))295 b Fm(\006)19083 48534 y Fg(\000)19692 49610 y Fl(\013)20527 49062 y Fr(2)20519 49939 y(2)21275 49610 y Fs(cosh)23739 49038 y Fr(2)24265 49610 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))295 b Fm(\000)h Fl(\013)29844 49062 y Fr(2)29836 49939 y(1)30592 49610 y Fs(sinh)32911 49038 y Fr(2)33437 49610 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))36557 48534 y Fg(\001)37166 48832 y Fr(1)p Ff(=)p Fr(2)38633 48534 y Fg(\003)39186 49610 y Fl(=)221 b Fs(sinh)q(\()p Fl(\014)74 b(=)p Fs(2\))p Fl(:)317 52602 y Fs(F)-108 b(or)553 b(real)g Fl(\013)6242 52801 y Fr(3)6768 52602 y Fs(,)584 b(one)553 b(requires)g(that)g Fl(\013)18994 52120 y Fr(2)18986 52931 y(2)19742 52602 y Fs(cosh)22206 52030 y Fr(2)22731 52602 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))573 b Fm(\025)g Fl(\013)28865 52120 y Fr(2)28857 52931 y(1)29613 52602 y Fs(sinh)31932 52030 y Fr(2)32458 52602 y Fs(\()p Fl(\014)74 b(=)p Fs(2\).)937 b(By)554 b(the)e(\014rst)g(equation)i(in)317 54595 y(\(3.16\),)435 b(p)36 b(ositiv)-36 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Fs(cosh)15739 73632 y Fr(2)16265 74204 y Fs(\()p Fl(\014)g(=)p Fs(2\))295 b(+)g(cosh\()p Fl(\014)74 b(=)p Fs(2\))26571 73128 y Fg(\003)27125 74204 y Fl(=)221 b Fs(sinh)30316 73632 y Fr(2)30842 74204 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))p Fl(;)10371 76529 y Fs(=)1117 b(coth\()p Fl(\014)74 b(=)p Fs(2\))p Fl(\013)18904 76728 y Fr(2)p Ff(;)p Fr(+)20424 76529 y Fs(\()p Fl(\014)g Fs(\))p Fl(;)24864 79850 y Fs(12)p eop %%Page: 13 13 13 12 bop 317 7306 a Fs(where)493 b Fl(\013)4961 7505 y Fr(2)p Ff(;)p Fr(+)6480 7306 y Fs(\()p Fl(\014)74 b Fs(\))492 b(has)h(b)36 b(een)492 b(de\014ned)g(in)g(\(3.14\).)757 b(The)493 b(ab)36 b(o)-36 b(v)g(e)493 b(implies)g(that)g Fl(\024)39436 7505 y Fr(+)40223 7306 y Fs(\()p Fl(\014)74 b Fs(\))469 b Fl(>)g(\013)44820 7505 y Fr(2)p Ff(;)p Fr(+)46340 7306 y Fs(\()p Fl(\014)74 b Fs(\))p Fl(:)492 b Fs(An)317 9298 y(argumen)-36 b(t)413 b(similar)i(to)f(that)f(used)g (ab)36 b(o)-36 b(v)g(e)414 b(yields)h(that)e Fl(\024)28532 9497 y Fi(\000)29319 9298 y Fs(\()p Fl(\014)74 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Fg(\002)12341 18929 y Fl(\013)13168 19128 y Fr(2)13916 18929 y Fs(cosh\()p Fl(\014)74 b(=)p Fs(2\))295 b Fm(\006)21124 17853 y Fg(\000)21732 18929 y Fl(\013)22567 18381 y Fr(2)22559 19257 y(2)23315 18929 y Fs(cosh)25779 18357 y Fr(2)26305 18929 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))296 b Fm(\000)f Fl(\013)31884 18381 y Fr(2)31876 19257 y(1)32632 18929 y Fs(sinh)34951 18357 y Fr(2)35477 18929 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))38597 17853 y Fg(\001)39206 18150 y Fr(1)p Ff(=)p Fr(2)40673 17853 y Fg(\003)41227 18929 y Fl(=)221 b Fs(sinh\()p Fl(\014)74 b(=)p Fs(2\))9489 21591 y(=)1287 b Fl(\013)12615 21790 y Fr(2)13362 21591 y Fs(coth\()p Fl(\014)74 b(=)p Fs(2\))296 b Fm(\006)20563 20515 y Fg(\000)21172 21591 y Fl(\013)22007 21043 y Fr(2)21999 21920 y(2)22755 21591 y Fs(coth)25212 21019 y Fr(2)25737 21591 y Fs(\()p Fl(\014)74 b(=)p Fs(2\))296 b Fm(\000)30614 20693 y Fs(1)p 30614 21286 651 54 v 30614 22503 a(4)31397 21591 y(\(1)f(+)g Fl(\013)34990 21043 y Fr(2)34982 21920 y(2)35517 21591 y 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Fl(f)35851 27104 y Fr(+)36638 26905 y Fs(\()p Fl(I)37718 27104 y Ff(\014)38346 26905 y Fs(\))375 b Fm([)g Fl(f)41129 27104 y Fi(\000)41916 26905 y Fs(\()p Fl(I)42996 27104 y Ff(\014)43624 26905 y Fs(\))551 b(is)g(a)g(closed)317 28897 y(in)-36 b(terv)-72 b(al)564 b(con)-36 b(taining)563 b(1)g(with)g(non-zero)g(length)f(\(b) -36 b(y)563 b(\(3.15\)\).)968 b(Therefore)564 b(there)e(exist)i (in\014nitely)317 30890 y(man)-36 b(y)434 b(di\013eren)-36 b(t)432 b(semigroups,)i(whic)-36 b(h)433 b(con)-36 b(taining)434 b(the)f(semigroups)h(constructed)e(in)h([CFL].)317 35327 y Fn(4.)2152 b(Mark)-60 b(o)g(v)662 b(prop)60 b(ert)-60 b(y)660 b(of)h(forms)g(:)929 b(Pro)60 b(ofs)661 b(of)g(Prop)60 b(osition)4143 37762 y(3.1)717 b(and)f(Theorem)h(3.1)1618 41070 y Fs(In)556 b(this)g(section)g(w)-36 b(e)557 b(pro)36 b(duce)555 b(the)h(pro)36 b(ofs)557 b(of)g(Prop)36 b(osition)556 b(3.1)i(and)d(Theorem)i(3.1.)947 b(Since)556 b Fl(P)317 43063 y Fs(and)385 b Fl(Q)g Fs(are)h(un)-36 b(b)36 b(ounded)383 b(op)36 b(erators,)395 b(w)-36 b(e)386 b(need)e(to)i(emplo)-36 b(y)385 b(the)g(appro)-36 b(ximation)386 b(pro)36 b(cedure)384 b(used)h(in)317 45055 y([BKP1].)578 b(In)427 b(the)h(rest)f(of)i(this)f (pap)36 b(er)427 b(w)-36 b(e)428 b(denote)g(b)-36 b(y)428 b(\010)g(either)f Fl(P)609 b Fs(or)428 b Fl(Q)g Fs(and)g(b)-36 b(y)428 b Fl(a)41918 44573 y Ff(])42767 45055 y Fs(either)f Fl(a)h Fs(or)g Fl(a)49824 44573 y Fi(\003)50350 45055 y Fs(.)317 48487 y Fo(Lemma)499 b(4.1)21 b(.)1926 b Fe(\(a\))466 b Fm(M)13670 48686 y Fr(0)14196 48487 y Fl(\030)14764 48686 y Fr(0)15755 48487 y Fe(and)f Fm(H)19401 48686 y Ff(f)98 b(in)21362 48487 y Fe(ar)-66 b(e)464 b(dense)h(in)f Fm(H)13 b Fe(.)2268 50479 y(\(b\))466 b(The)e(inclusions)16066 52472 y Fm(M)17661 52671 y Fr(0)18187 52472 y Fl(\030)18755 52671 y Fr(0)19650 52472 y Fm(\032)369 b Fl(D)36 b Fs(\(\010\))664 b Fe(and)g Fm(H)28628 52671 y Ff(f)98 b(in)30493 52472 y Fm(\032)369 b Fl(D)36 b Fs(\(\010\))317 55184 y Fe(hold,)465 b(mor)-66 b(e)g(over)464 b(the)g(r)-66 b(elation)18656 57176 y Fs(\010)p Fl(A\030)21138 57375 y Fr(0)22033 57176 y Fs(=)369 b Fl(j)75 b Fs(\()p Fl(\033)25272 57383 y Fi(\000)p Ff(i=)p Fr(2)27320 57176 y Fs(\()p Fl(A)28801 56628 y Fi(\003)29327 57176 y Fs(\)\)\010)p Fl(\030)31846 57375 y Fr(0)317 59889 y Fe(holds)653 b(for)e(any)g Fl(A)715 b Fm(2)f(M)13649 60088 y Fr(0)14175 59889 y Fe(.)1158 b(A)-33 b(lso)651 b(we)h(have)f Fm(M)25726 60088 y Fr(0)26252 59889 y Fl(\030)26820 60088 y Fr(0)28061 59889 y Fm(\032)715 b Fl(D)36 b Fs(\()p Fl(a)32114 59406 y Ff(])32535 59889 y Fs(\))p Fe(,)697 b Fm(H)35259 60088 y Ff(f)98 b(in)37470 59889 y Fm(\032)715 b Fl(D)36 b Fs(\()p Fl(a)41523 59406 y Ff(])41944 59889 y Fs(\))651 b Fe(and)h Fl(a)46495 59406 y Ff(])46916 59889 y Fl(A\030)48459 60088 y Fr(0)49699 59889 y Fs(=)317 61881 y Fl(j)75 b Fs(\()p Fl(\033)2175 62088 y Fi(\000)p Ff(i=)p Fr(2)4224 61881 y Fs(\()p Fl(A)5705 61399 y Fi(\003)6230 61881 y Fs(\)\))p Fl(a)7925 61399 y Ff(])8346 61881 y Fl(\030)8914 62080 y Fr(0)9440 61881 y Fl(;)664 b(A)369 b Fm(2)g(M)14659 62080 y Fr(0)15185 61881 y Fe(.)317 66641 y(Pr)-66 b(o)g(of:)745 b Fs(\(a\))517 b(Since)g(for)h Fl(A)511 b Fm(2)g(M)p Fs(,)539 b Fl(A)18568 66840 y Ff(n)19712 66641 y Fs(in)517 b(\(3.5\))h(con)-36 b(v)g(erges)518 b(strongly)g(to)f Fl(A)h Fs(as)f Fl(n)512 b Fm(!)g(1)518 b Fs(and)f(since)317 68634 y Fm(M)p Fl(\030)2480 68833 y Fr(0)3419 68634 y Fs(is)413 b(dense)g(in)g Fm(H)13 b Fs(,)417 b Fm(M)13239 68833 y Fr(0)13765 68634 y Fl(\030)14333 68833 y Fr(0)15272 68634 y Fs(is)c(dense)f(in)h Fm(H)13 b Fs(.)571 b(It)413 b(follo)-36 b(ws)415 b(from)e(\(3.2\))h(that)e Fl(!)461 b Fs(is)413 b(an)g(en)-36 b(tire)413 b(analytic)317 70626 y(state,)566 b(and)539 b(so)h Fl(\030)8903 70825 y Fr(0)9969 70626 y Fs(is)f(an)h(en)-36 b(tire)539 b(analytic)h(v)-36 b(ector)540 b(for)g(\010\()p Fl(z)59 b Fs(\))p Fl(;)444 b(z)608 b Fm(2)549 b Fh(C)p Fs(.)897 b(Th)-36 b(us)538 b Fm(H)41011 70825 y Ff(f)98 b(in)43047 70626 y Fs(is)540 b(dense.)895 b(See)317 72619 y(Section)433 b(5.2.3)j(of)e([BR])g(for)g (the)g(details.)24864 79850 y(13)p eop %%Page: 14 14 14 13 bop 2268 7306 a Fs(\(b\))571 b(Since)g Fl(\030)8676 7505 y Fr(0)9773 7306 y Fs(is)h(an)f(en)-36 b(tire)571 b(analytic)h(v)-36 b(ector)572 b(for)g(\010,)607 b Fm(H)31231 7505 y Ff(f)98 b(in)33330 7306 y Fm(\032)604 b Fl(D)36 b Fs(\(\010\).)992 b(Notice)572 b(that)f(for)h(an)-36 b(y)317 9298 y Fl(A)369 b Fm(2)g(M)4511 9497 y Fr(0)5037 9298 y Fs(,)434 b Fl(j)75 b Fs(\()p Fl(\033)7690 9505 y Fi(\000)p Ff(i=)p Fr(2)9738 9298 y Fs(\()p Fl(A)11219 8816 y Fi(\003)11745 9298 y Fs(\)\))p Fl(\030)13325 9497 y Fr(0)14219 9298 y Fs(=)369 b Fl(A\030)17143 9497 y Fr(0)17669 9298 y Fs(.)578 b(F)-108 b(or)433 b Fl(z)c Fs(=)368 b(1)p Fl(;)221 b(i)434 b Fs(and)g Fl(n)369 b Fm(2)f Fh(N)p Fs(,)435 b(w)-36 b(e)434 b(write)f(that)17249 12920 y(\010\()p Fl(z)59 b(;)221 b(n)p Fs(\))371 b(=)d Fm(\000)p Fl(in)25225 11446 y Fg(\020)26019 12920 y Fl(W)27428 11844 y Fg(\000)28232 12022 y Fs(1)p 28169 12615 777 54 v 28169 13831 a Fl(n)29078 12920 y(z)29741 11844 y Fg(\001)30646 12920 y Fm(\000)295 b Fs(1)32624 11446 y Fg(\021)33418 12920 y Fl(:)14258 b Fs(\(4.1\))317 16241 y(Notice)434 b(that)14703 19562 y(\010\()p Fl(z)59 b(;)221 b(n)p Fs(\))p Fl(A\030)20218 19761 y Fr(0)21853 19562 y Fs(=)1106 b(\010\()p Fl(z)59 b(;)221 b(n)p Fs(\))p Fl(j)75 b Fs(\()p Fl(\033)29801 19769 y Fi(\000)p Ff(i=)p Fr(2)31851 19562 y Fs(\()p Fl(A)33332 19014 y Fi(\003)33858 19562 y Fs(\)\))p Fl(\030)35438 19761 y Fr(0)21853 21887 y Fs(=)1106 b Fl(j)75 b Fs(\()p Fl(\033)25829 22093 y Fi(\000)p Ff(i=)p Fr(2)27878 21887 y Fs(\()p Fl(A)29359 21338 y Fi(\003)29884 21887 y Fs(\)\)\010\()p Fl(z)59 b(;)221 b(n)p Fs(\))p Fl(\030)35436 22086 y Fr(0)35963 21887 y Fl(:)317 25207 y Fs(Since)585 b(the)g(sequence)h Fm(f)p Fs(\010\()p Fl(z)59 b(;)221 b(n)p Fs(\))p Fl(\030)17004 25406 y Fr(0)17531 25207 y Fm(g)586 b Fs(con)-36 b(v)g(erges)586 b(to)f(\010)p Fl(\030)28005 25406 y Fr(0)28532 25207 y Fs(,)623 b(it)586 b(follo)-36 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46317 a(n)29820 45406 y Fs(\010)30759 44330 y Fg(\001)31663 45406 y Fm(\000)296 b Fo(1)33739 43931 y Fg(\021o)35418 45406 y Fl(e)36021 44857 y Fi(\000)p Ff(nt)37664 44545 y Fc(2)38401 45406 y Fl(dt:)8129 b Fs(\(4.2\))317 48985 y(Notice)431 b(that)e(for)h(an)-36 b(y)430 b Fl(n)370 b Fm(2)e Fh(N)p Fs(,)431 b(\010)16736 49184 y Ff(n)17793 48985 y Fs(is)f(an)f(en)-36 b(tire)430 b(analytic)h(elemen)-36 b(t)429 b(for)h Fl(\033)37014 49184 y Ff(t)37839 48985 y Fs(and)g Fl(\033)41104 49184 y Ff(z)41634 48985 y Fs(\(\010)43079 49184 y Ff(n)43706 48985 y Fs(\))f(is)i(giv)-36 b(en)430 b(b)-36 b(y)9191 52597 y Fl(\033)9930 52796 y Ff(z)10461 52597 y Fs(\(\010)11906 52796 y Ff(n)12532 52597 y Fs(\))369 b(:=)g Fm(\000)16182 51123 y Fg(\020)17113 51699 y Fl(n)p 17108 52292 786 54 v 17108 53509 a(\031)18026 51123 y Fg(\021)18820 51420 y Fr(1)p Ff(=)p Fr(2)20508 50789 y Fg(Z)22058 52597 y Fl(in)23278 51123 y Fg(n)24163 52597 y Fl(\033)24902 52796 y Ff(t)25298 51123 y Fg(\020)26312 52597 y Fs(exp)28521 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b(ell)434 b(de\014ned)e(on)i Fm(M)18267 24407 y Fr(0)18793 24208 y Fl(\030)19361 24407 y Fr(0)19887 24208 y Fs(.)578 b(W)-108 b(e)434 b(ha)-36 b(v)g(e)434 b(that)5375 27529 y Fl(P)1288 b Fs(=)1107 b Fl(\033)10359 27736 y Fi(\000)p Ff(i=)p Fr(4)12407 27529 y Fs(\()p Fl(\033)13652 27736 y Ff(i=)p Fr(4)14969 27529 y Fs(\()p Fl(P)181 b Fs(\)\))368 b(=)h Fl(i\033)20438 27736 y Fi(\000)p Ff(i=)p Fr(4)22487 27529 y Fs(\()p Fl(e)23596 26981 y Ff(\014)46 b(=)p Fr(4)25165 27529 y Fl(a)25848 26981 y Fi(\003)26669 27529 y Fm(\000)295 b Fl(e)28600 26981 y Fi(\000)p Ff(\014)46 b(=)p Fr(4)30901 27529 y Fl(a)p Fs(\))p Fl(=)32740 26365 y Fm(p)p 33847 26365 651 54 v 1164 x Fs(2)13540 b(\(4.5\))7501 29854 y(=)1107 b Fl(ie)10667 29305 y Ff(\014)46 b(=)p Fr(4)12236 28778 y Fg(\000)12845 29854 y Fl(\033)13584 30060 y Fi(\000)p Ff(i=)p Fr(4)15632 29854 y Fs(\()p Fl(Q)p Fs(\))295 b Fm(\000)h Fl(i\033)20481 30060 y Fi(\000)p Ff(i=)p Fr(4)22529 29854 y Fs(\()p Fl(P)181 b Fs(\))24560 28778 y Fg(\001)25169 29854 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Fl(s)22018 19460 y Fi(\006)22805 19261 y Fl(A)p Fs(\))p Fl(\033)25025 19468 y Ff(i=)p Fr(4)26341 19261 y Fs(\(\010\))p Fl(\030)28860 19460 y Fr(0)29682 19261 y Fm(\000)295 b Fs(\()p Fl(s)32129 19460 y Fi(\006)32916 19261 y Fl(A)p Fs(\))p Fl(j)75 b Fs(\()p Fl(\033)36255 19468 y Ff(i=)p Fr(4)37571 19261 y Fs(\(\010\)\))p Fl(\030)40596 19460 y Fr(0)41122 19261 y Fl(;)5904 b Fs(\(4.13\))9324 21586 y Fl(\016)9900 21785 y Fr(+)10687 21586 y Fs(\()p Fl(\033)11932 21792 y Ff(i=)p Fr(4)13249 21586 y Fs(\(\010\)\))p Fl(\030)16274 21785 y Fi(\006)18168 21586 y Fs(=)1106 b Fl(j)75 b Fs(\()p Fl(s)22018 21785 y Fi(\006)22805 21586 y Fl(A)p Fs(\))p Fl(\033)25025 21792 y Ff(i=)p Fr(4)26341 21586 y Fs(\(\010\))p Fl(\030)28860 21785 y Fr(0)29682 21586 y Fs(+)294 b(\()p Fl(s)32107 21785 y Fi(\006)32894 21586 y Fl(A)p Fs(\))p Fl(j)75 b Fs(\()p Fl(\033)36233 21792 y Ff(i=)p Fr(4)37549 21586 y Fs(\(\010\)\))p Fl(\030)40574 21785 y Fr(0)41100 21586 y Fl(:)317 24907 y Fs(The)428 b(relations)g(analogous)h(to)e (\(4.12\))i(and)e(\(4.13\))i(for)f(\010)28790 25106 y Ff(n)29844 24907 y Fs(hold.)576 b(Th)-36 b(us,)429 b(it)f(follo)-36 b(ws)429 b(from)f(\(3.9\))g(and)317 26899 y(Lemma)453 b(4.2)i(that)p 9933 25832 820 54 v 453 w Fm(E)118 b Fs([)p Fl(\030)11681 27098 y Ff(m;)p Fi(\006)13871 26899 y Fm(\000)309 b Fl(\030)15781 27098 y Ff(n;)p Fi(\006)17401 26899 y Fs(])403 b Fm(!)g Fs(0)453 b(as)h Fl(m;)221 b(n)404 b Fm(!)f(1)p Fs(.)638 b(Since)453 b(\()p 33496 25832 V Fm(E)119 b Fl(;)221 b(D)36 b Fs(\()p 36520 25832 V Fm(E)119 b Fs(\)\))453 b(is)h(closed,)459 b Fl(\030)44924 27098 y Fi(\006)46114 26899 y Fm(2)402 b Fl(D)36 b Fs(\()p 49024 25832 V Fm(E)119 b Fs(\).)317 28892 y(Next)414 b(w)-36 b(e)415 b(will)g(pro)-36 b(v)g(e)414 b(that)f Fm(E)14793 29091 y Ff(n)15420 28892 y Fs([)p Fl(\030)16349 29091 y Fi(\006)17136 28892 y Fs(])h(con)-36 b(v)g(erges)415 b(to)p 25286 27825 V 414 w Fm(E)119 b Fs([)p Fl(\030)27035 29091 y Fi(\006)27822 28892 y Fs(])415 b(as)f Fl(n)g Fs(tends)f(to)h(in\014nit)-36 b(y)-108 b(.)571 b(By)415 b(\(4.12\),)k(\(4.13\))317 30884 y(and)401 b(Lemma)g(4.2,)408 b(w)-36 b(e)402 b(conclude)e(that)h(for)g(an)-36 b(y)402 b Fl(\030)433 b Fs(=)369 b Fl(A\030)28141 31083 y Fr(0)28667 30884 y Fs(,)407 b(there)401 b(exists)h(a)f(p)36 b(ositiv)-36 b(e)402 b(real)f(n)-36 b(um)g(b)36 b(er)400 b Fl(M)317 32877 y Fs(suc)-36 b(h)433 b(that)g(the)g(b)36 b(ounds)12025 36198 y Fm(k)p Fl(\016)13265 36397 y Fi(\000)14053 36198 y Fs(\()p Fl(\033)15298 36404 y Fi(\000)p Ff(i=)p Fr(4)17346 36198 y Fs(\(\010)18791 36397 y Ff(n)19418 36198 y Fs(\)\))p Fl(\030)20998 36397 y Fi(\006)21785 36198 y Fm(k)295 b Fs(+)g Fm(k)p Fl(\016)25291 36397 y Fi(\000)26078 36198 y Fs(\()p Fl(\033)27323 36404 y Fi(\000)p Ff(i=)p Fr(4)29372 36198 y Fs(\(\010\)\))p Fl(\030)32397 36397 y Fi(\006)33184 36198 y Fm(k)1107 b(\024)g Fl(M)66 b(;)8607 b Fs(\(4.14\))13489 38522 y Fm(k)p Fl(\016)14729 38721 y Fi(\000)15516 38522 y Fs(\()p Fl(\033)16761 38729 y Ff(i=)p Fr(4)18078 38522 y Fs(\(\010)19523 38721 y Ff(n)20150 38522 y Fs(\)\))p Fl(\030)21730 38721 y Fi(\006)22516 38522 y Fm(k)296 b Fs(+)f Fm(k)p Fl(\016)26023 38721 y Fi(\000)26810 38522 y Fs(\()p Fl(\033)28055 38729 y Ff(i=)p Fr(4)29372 38522 y Fs(\(\010\)\))p Fl(\030)32397 38721 y Fi(\006)33184 38522 y Fm(k)1107 b(\024)g Fl(M)66 b(;)13489 40847 y Fm(k)p Fl(\016)14729 41046 y Fr(+)15516 40847 y Fs(\()p Fl(\033)16761 41053 y Ff(i=)p Fr(4)18078 40847 y Fs(\(\010)19523 41046 y Ff(n)20150 40847 y Fs(\)\))p Fl(\030)21730 41046 y Fi(\006)22516 40847 y Fm(k)296 b Fs(+)f Fm(k)p Fl(\016)26023 41046 y Fr(+)26810 40847 y Fs(\()p Fl(\033)28055 41053 y Ff(i=)p Fr(4)29372 40847 y Fs(\(\010\)\))p Fl(\030)32397 41046 y Fi(\006)33184 40847 y Fm(k)1107 b(\024)g Fl(M)317 44168 y Fs(hold)406 b(uniformly)h(in)g Fl(n)369 b Fm(2)g Fh(N)p Fs(.)569 b(It)407 b(follo)-36 b(ws)408 b(from)f(\(3.9\),)413 b(\(4.9\),)g(\(4.12\),)g(\(4.13\))407 b(and)f(the)g(ab)36 b(o)-36 b(v)g(e)407 b(b)36 b(ounds)317 46160 y(that)2451 49481 y Fm(jE)3521 49680 y Ff(n)4147 49481 y Fs([)p Fl(\030)5076 49680 y Fi(\006)5864 49481 y Fs(])295 b Fm(\000)p 7849 48414 V 296 w(E)119 b Fs([)p Fl(\030)9598 49680 y Fi(\006)10385 49481 y Fs(])p Fm(j)1255 b(\024)g Fs(4)p Fl(\013)8 b(M)139 b Fm(k)18204 48405 y Fg(\000)18814 49481 y Fl(\016)19390 49680 y Fi(\000)20177 49481 y Fs(\()p Fl(\033)21422 49688 y Fi(\000)p Ff(i=)p Fr(4)23471 49481 y Fs(\(\010)24916 49680 y Ff(n)25542 49481 y Fs(\)\))295 b Fm(\000)g Fl(\016)28753 49680 y Fi(\000)29540 49481 y Fs(\()p Fl(\033)30785 49688 y Fi(\000)p Ff(i=)p Fr(4)32834 49481 y Fs(\(\010\)\))35291 48405 y Fg(\001)35900 49481 y Fl(\030)36468 49680 y Fi(\006)37255 49481 y Fm(k)14658 51806 y Fs(+2)p Fl(\013)8 b(M)139 b Fm(k)19216 50730 y Fg(\000)19825 51806 y Fl(\016)20401 52005 y Fi(\000)21189 51806 y Fs(\()p Fl(\033)22434 52012 y Ff(i=)p Fr(4)23750 51806 y Fs(\(\010)25195 52005 y Ff(n)25822 51806 y Fs(\)\))295 b Fm(\000)g Fl(\016)29033 52005 y Fi(\000)29820 51806 y Fs(\()p Fl(\033)31065 52012 y Ff(i=)p Fr(4)32382 51806 y Fs(\(\010\)\))34839 50730 y Fg(\001)35448 51806 y Fl(\030)36016 52005 y Fi(\006)36803 51806 y Fm(k)14658 54130 y Fs(+2)p Fl(\013)8 b(M)139 b Fm(k)19216 53054 y Fg(\000)19825 54130 y Fl(\016)20401 54329 y Fr(+)21189 54130 y Fs(\()p Fl(\033)22434 54337 y Ff(i=)p Fr(4)23750 54130 y Fs(\(\010)25195 54329 y Ff(n)25822 54130 y Fs(\)\))295 b Fm(\000)g Fl(\016)29033 54329 y Fr(+)29820 54130 y Fs(\()p Fl(\033)31065 54337 y Ff(i=)p Fr(4)32382 54130 y Fs(\(\010\)\))34839 53054 y Fg(\001)35448 54130 y Fl(\030)36016 54329 y Fi(\006)36803 54130 y Fm(k)12370 56566 y(\024)1255 b Fl(M)16055 56017 y Fi(0)16365 55091 y Fg(\020)17159 56566 y Fm(k)p Fl(\033)18562 56772 y Fi(\000)p Ff(i=)p Fr(4)20611 56566 y Fs(\(\010)22056 56765 y Ff(n)22682 56566 y Fs(\))p Fl(\030)23756 56765 y Fr(0)24577 56566 y Fm(\000)295 b Fl(\033)26644 56772 y Fi(\000)p Ff(i=)p Fr(4)28693 56566 y Fs(\(\010\))p Fl(\030)31212 56765 y Fr(0)31738 56566 y Fm(k)g Fs(+)g Fm(k)p Fl(\033)35407 56772 y Ff(i=)p Fr(4)36724 56566 y Fs(\(\010)38169 56765 y Ff(n)38796 56566 y Fs(\))p Fl(\030)39870 56765 y Fr(0)40690 56566 y Fm(\000)h Fl(\033)42758 56772 y Ff(i=)p Fr(4)44075 56566 y Fs(\(\010\))p Fl(\030)46594 56765 y Fr(0)47120 56566 y Fm(k)47784 55091 y Fg(\021)12222 58890 y Fm(!)1329 b Fs(0)1301 b(as)886 b Fl(n)369 b Fm(!)h(1)p Fl(;)317 62211 y Fs(where)395 b Fl(\013)378 b Fs(=)368 b Fl(\015)294 b Fs(max)p Fm(f)p Fs(\(1)217 b(+)g Fl(\013)14111 61729 y Fr(2)14103 62540 y(2)14637 62211 y Fs(\))p Fl(=)p Fs(2)p Fl(;)444 b Fs(\()p Fl(\013)18589 61729 y Fr(2)18581 62540 y(1)19332 62211 y Fs(+)217 b Fl(\013)21396 61729 y Fr(2)21388 62540 y(3)21921 62211 y Fs(\))p Fl(=)p Fs(2)p Fl(;)444 b Fm(j)p Fl(\013)25728 62410 y Fr(1)26254 62211 y Fm(j)217 b Fs(+)g Fm(j)p Fl(\013)29265 62410 y Fr(2)29790 62211 y Fl(\013)30617 62410 y Fr(3)31143 62211 y Fm(jg)p Fl(:)395 b Fs(This)h(completes)f(the)g(pro)36 b(of)395 b(of)h(our)317 64204 y(assertion)434 b(\(4.11\).)2268 66196 y(Recall)402 b(the)e(de\014nition)g(of)h Fm(E)16259 66395 y Ff(n)17286 66196 y Fs(in)f(\(4.9\).)568 b(T)-108 b(o)401 b(sho)-36 b(w)401 b(the)f(Diric)-36 b(hlet)401 b(prop)36 b(ert)-36 b(y)-108 b(,)406 b Fm(E)41221 66395 y Ff(n)41847 66196 y Fs(\()p Fl(\030)42921 66395 y Fr(+)43709 66196 y Fl(;)221 b(\030)44859 66395 y Fi(\000)45647 66196 y Fs(\))400 b(in)g(\(4.9\))317 68189 y(can)434 b(b)36 b(e)433 b(written)g(as)12940 71510 y Fm(E)13641 71709 y Ff(n)14268 71510 y Fs(\()p Fl(\030)15342 71709 y Fr(+)16129 71510 y Fl(;)221 b(\030)17279 71709 y Fi(\000)18067 71510 y Fs(\))1106 b(=)h Fm(E)22618 70961 y Fr(\(1\))22499 71838 y Ff(n)23876 71510 y Fs(\()p Fl(\030)24950 71709 y Fr(+)25737 71510 y Fl(;)221 b(\030)26887 71709 y Fi(\000)27675 71510 y Fs(\))295 b(+)f Fm(E)30602 70961 y Fr(\(2\))30483 71838 y Ff(n)31860 71510 y Fs(\()p Fl(\030)32934 71709 y Fr(+)33721 71510 y Fl(;)221 b(\030)34871 71709 y Fi(\000)35659 71510 y Fs(\))11222 b(\(4.15\))19679 73834 y(=)1107 b(\()p Fl(I)22982 73286 y Fr(\(1\))24535 73834 y Fs(+)295 b Fl(I)104 b(I)27198 73286 y Fr(\(1\))28455 73834 y Fs(\))295 b(+)g(\()p Fl(I)31747 73286 y Fr(\(2\))33300 73834 y Fs(+)g Fl(I)104 b(I)35963 73286 y Fr(\(2\))37221 73834 y Fs(\))p Fl(;)24864 79850 y Fs(18)p eop %%Page: 19 19 19 18 bop 317 7306 a Fs(where)8580 10418 y Fm(E)9400 9870 y Fr(\(1\))9281 10747 y Ff(n)10658 10418 y Fs(\()p Fl(\030)11732 10617 y Fr(+)12519 10418 y Fl(;)221 b(\030)13669 10617 y Fi(\000)14457 10418 y Fs(\))1106 b(=)18321 9520 y Fl(\015)p 18321 10113 747 54 v 18369 11329 a Fs(2)19201 10418 y Fm(h)p Fl(\016)20294 10617 y Fi(\000)21080 10418 y Fs(\()p Fl(\033)22325 10625 y Fi(\000)p Ff(i=)p Fr(4)24374 10418 y Fs(\()p Fl(P)25718 10617 y Ff(n)26344 10418 y Fs(\)\))27356 9870 y Fi(\003)27882 10418 y Fl(\030)28450 10617 y Fr(+)29237 10418 y Fl(;)221 b(\016)30395 10617 y Fi(\000)31183 10418 y Fs(\()p Fl(\033)32428 10625 y Fi(\000)p Ff(i=)p Fr(4)34477 10418 y Fs(\()p Fl(P)35821 10617 y Ff(n)36447 10418 y Fs(\)\))p Fl(\030)38027 10617 y Fi(\000)38814 10418 y Fm(i)18188 13243 y Fs(+)19333 12344 y Fl(\015)p 19333 12937 V 19381 14154 a Fs(2)20212 13243 y Fl(\013)21047 12694 y Fr(2)21039 13571 y(1)21574 13243 y Fm(h)p Fl(\016)22667 13442 y Fi(\000)23454 13243 y Fs(\()p Fl(\033)24699 13449 y Fi(\000)p Ff(i=)p Fr(4)26747 13243 y Fs(\()p Fl(Q)28283 13442 y Ff(n)28909 13243 y Fs(\)\))29921 12694 y Fi(\003)30447 13243 y Fl(\030)31015 13442 y Fr(+)31802 13243 y Fl(;)g(\016)32960 13442 y Fi(\000)33748 13243 y Fs(\()p Fl(\033)34993 13449 y Fi(\000)p Ff(i=)p Fr(4)37042 13243 y Fs(\()p Fl(Q)38578 13442 y Ff(n)39204 13243 y Fs(\)\))p Fl(\030)40784 13442 y Fi(\000)41571 13243 y Fm(i)18188 15567 y(\000)p Fl(i\015)72 b(\013)21239 15766 y Fr(1)21765 15567 y Fm(h)p Fl(\016)22858 15766 y Fr(+)23645 15567 y Fs(\()p Fl(\033)24890 15774 y Fi(\000)p Ff(i=)p Fr(4)26938 15567 y Fs(\()p Fl(Q)28474 15766 y Ff(n)29101 15567 y Fs(\)\))30113 15019 y Fi(\003)30638 15567 y Fl(\030)31206 15766 y Fr(+)31993 15567 y Fl(;)221 b(\016)33151 15766 y Fi(\000)33939 15567 y Fs(\()p Fl(\033)35184 15774 y Fi(\000)p Ff(i=)p Fr(4)37233 15567 y Fs(\()p Fl(P)38577 15766 y Ff(n)39203 15567 y Fs(\)\))p Fl(\030)40783 15766 y Fi(\000)41570 15567 y Fm(i)p Fl(;)7904 19800 y Fm(E)8724 19251 y Fr(\(2\))8605 20128 y Ff(n)9981 19800 y Fs(\()p Fl(\030)11055 19999 y Fr(+)11842 19800 y Fl(;)g(\030)12992 19999 y Fi(\000)13780 19800 y Fs(\))1107 b(=)17644 18901 y Fl(\015)p 17644 19494 V 17692 20711 a Fs(2)18524 19800 y Fl(\013)19359 19251 y Fr(2)19351 20128 y(2)19886 19800 y Fm(h)p Fl(\016)20979 19999 y Fi(\000)21765 19800 y Fs(\()p Fl(\033)23010 20006 y Fi(\000)p Ff(i=)p Fr(4)25059 19800 y Fs(\()p Fl(P)26403 19999 y Ff(n)27029 19800 y Fs(\)\))28041 19251 y Fi(\003)28567 19800 y Fl(\030)29135 19999 y Fr(+)29922 19800 y Fl(;)221 b(\016)31080 19999 y Fi(\000)31868 19800 y Fs(\()p Fl(\033)33113 20006 y Fi(\000)p Ff(i=)p Fr(4)35162 19800 y Fs(\()p Fl(P)36506 19999 y Ff(n)37132 19800 y Fs(\)\))p Fl(\030)38712 19999 y Fi(\000)39499 19800 y Fm(i)17511 22624 y Fs(+)18656 21726 y Fl(\015)p 18656 22319 V 18704 23535 a Fs(2)19536 22624 y Fl(\013)20371 22076 y Fr(2)20363 22953 y(3)20897 22624 y Fm(h)p Fl(\016)21990 22823 y Fi(\000)22777 22624 y Fs(\()p Fl(\033)24022 22831 y Fi(\000)p Ff(i=)p Fr(4)26071 22624 y Fs(\()p Fl(Q)27607 22823 y Ff(n)28233 22624 y Fs(\)\))29245 22076 y Fi(\003)29770 22624 y Fl(\030)30338 22823 y Fr(+)31126 22624 y Fl(;)g(\016)32284 22823 y Fi(\000)33071 22624 y Fs(\()p Fl(\033)34316 22831 y Fi(\000)p Ff(i=)p Fr(4)36365 22624 y Fs(\()p Fl(Q)37901 22823 y Ff(n)38527 22624 y Fs(\)\))p Fl(\030)40107 22823 y Fi(\000)40894 22624 y Fm(i)17511 24949 y(\000)p Fl(i\015)72 b(\013)20562 25148 y Fr(2)21088 24949 y Fl(\013)21915 25148 y Fr(3)22441 24949 y Fm(h)p Fl(\016)23534 25148 y Fr(+)24321 24949 y Fs(\()p Fl(\033)25566 25155 y Fi(\000)p Ff(i=)p Fr(4)27615 24949 y Fs(\()p Fl(P)28959 25148 y Ff(n)29585 24949 y Fs(\)\))30597 24400 y Fi(\003)31123 24949 y Fl(\030)31691 25148 y Fr(+)32478 24949 y Fl(;)221 b(\016)33636 25148 y Fi(\000)34424 24949 y Fs(\()p Fl(\033)35669 25155 y Fi(\000)p Ff(i=)p Fr(4)37717 24949 y Fs(\()p Fl(Q)39253 25148 y Ff(n)39880 24949 y Fs(\)\))p Fl(\030)41460 25148 y Fi(\000)42247 24949 y Fm(i)p Fl(;)317 28061 y Fs(and)5166 31174 y Fl(I)5844 30625 y Fr(\(1\))8209 31174 y Fs(=)10461 30275 y Fl(\015)p 10461 30868 V 10509 32085 a Fs(2)11340 30098 y Fg(\012)11968 31174 y Fl(\030)12536 31373 y Fr(+)13323 31174 y Fl(;)g(\033)14644 31380 y Fi(\000)p Ff(i=)p Fr(4)16693 30098 y Fg(\000)17302 31174 y Fl(P)18321 30625 y Fr(2)18140 31502 y Ff(n)19142 31174 y Fs(+)295 b Fl(\013)21284 30625 y Fr(2)21276 31502 y(1)21810 31174 y Fl(Q)22840 30625 y Fr(2)22840 31502 y Ff(n)23467 30098 y Fg(\001)24076 31174 y Fl(\030)24644 31373 y Fi(\000)25431 30098 y Fg(\013)26353 31174 y Fs(+)27793 30275 y Fl(\015)p 27793 30868 V 27841 32085 a Fs(2)28673 30098 y Fg(\012)29300 31174 y Fl(\033)30039 31380 y Fi(\000)p Ff(i=)p Fr(4)32088 31174 y Fs(\()p Fl(P)33613 30625 y Fr(2)33432 31502 y Ff(n)34434 31174 y Fs(+)f Fl(\013)36575 30625 y Fr(2)36567 31502 y(1)37102 31174 y Fl(Q)38132 30625 y Fr(2)38132 31502 y Ff(n)38758 30098 y Fg(\001)39367 31174 y Fl(\030)39935 31373 y Fi(\000)40723 31174 y Fl(;)221 b(\030)41873 31373 y Fr(+)42660 30098 y Fg(\013)10328 33657 y Fm(\000)p Fl(\015)12108 32581 y Fg(\012)12735 33657 y Fl(\030)13303 33856 y Fr(+)14090 33657 y Fl(;)g(\033)15411 33863 y Fi(\000)p Ff(i=)p Fr(4)17461 32581 y Fg(\000)18070 33657 y Fl(i\013)19341 33856 y Fr(1)19866 33657 y Fl(Q)20896 33856 y Ff(n)21523 33657 y Fl(P)22361 33856 y Ff(n)22987 32581 y Fg(\001)23596 33657 y Fl(\030)24164 33856 y Fi(\000)24952 32581 y Fg(\013)25874 33657 y Fm(\000)295 b Fl(\015)27949 32581 y Fg(\012)q(\000)29185 33657 y Fl(\033)29924 33863 y Fi(\000)p Ff(i=)p Fr(4)31973 32581 y Fg(\000)32582 33657 y Fl(i\013)33853 33856 y Fr(1)34379 33657 y Fl(Q)35409 33856 y Ff(n)36035 33657 y Fl(P)36873 33856 y Ff(n)37500 32581 y Fg(\001)38109 33657 y Fl(\030)38677 33856 y Fi(\000)39464 33657 y Fl(;)221 b(\030)40614 33856 y Fr(+)41402 32581 y Fg(\013)5166 35981 y Fl(I)5844 35433 y Fr(\(2\))8209 35981 y Fs(=)10461 35083 y Fl(\015)p 10461 35676 V 10509 36892 a Fs(2)11340 34905 y Fg(\012)11968 35981 y Fl(\030)12536 36180 y Fr(+)13323 35981 y Fl(;)g(\033)14644 36188 y Fi(\000)p Ff(i=)p Fr(4)16693 34905 y Fg(\000)17302 35981 y Fl(\013)18137 35433 y Fr(2)18129 36310 y(2)18664 35981 y Fl(P)19683 35433 y Fr(2)19502 36310 y Ff(n)20504 35981 y Fs(+)294 b Fl(\013)22645 35433 y Fr(2)22637 36310 y(3)23172 35981 y Fl(Q)24202 35433 y Fr(2)24202 36310 y Ff(n)24828 34905 y Fg(\001)25437 35981 y Fl(\030)26005 36180 y Fi(\000)26793 34905 y Fg(\013)27715 35981 y Fs(+)29155 35083 y Fl(\015)p 29155 35676 V 29203 36892 a Fs(2)30034 34905 y Fg(\012)30662 35981 y Fl(\033)31401 36188 y Fi(\000)p Ff(i=)p Fr(4)33449 35981 y Fs(\()p Fl(\013)34790 35433 y Fr(2)34782 36310 y(2)35317 35981 y Fl(P)36336 35433 y Fr(2)36155 36310 y Ff(n)37157 35981 y Fs(+)h Fl(\013)39299 35433 y Fr(2)39291 36310 y(3)39825 35981 y Fl(Q)40855 35433 y Fr(2)40855 36310 y Ff(n)41482 34905 y Fg(\001)42090 35981 y Fl(\030)42658 36180 y Fi(\000)43446 35981 y Fl(;)221 b(\030)44596 36180 y Fr(+)45384 34905 y Fg(\013)10328 38464 y Fm(\000)p Fl(\015)12108 37388 y Fg(\012)12735 38464 y Fl(\030)13303 38663 y Fr(+)14090 38464 y Fl(;)g(\033)15411 38671 y Fi(\000)p Ff(i=)p Fr(4)17461 37388 y Fg(\000)18070 38464 y Fl(i\013)19341 38663 y Fr(2)19866 38464 y Fl(\013)20693 38663 y Fr(3)21220 38464 y Fl(P)22058 38663 y Ff(n)22684 38464 y Fl(Q)23714 38663 y Ff(n)24341 37388 y Fg(\001)24949 38464 y Fl(\030)25517 38663 y Fi(\000)26305 37388 y Fg(\013)27227 38464 y Fm(\000)295 b Fl(\015)29302 37388 y Fg(\012)29930 38464 y Fl(j)30543 37388 y Fg(\000)31151 38464 y Fl(\033)31890 38671 y Fi(\000)p Ff(i=)p Fr(4)33939 37388 y Fg(\000)34548 38464 y Fl(i\013)35819 38663 y Fr(2)36345 38464 y Fl(\013)37172 38663 y Fr(3)37698 38464 y Fl(P)38536 38663 y Ff(n)39162 38464 y Fl(Q)40192 38663 y Ff(n)40819 37388 y Fg(\001\001)42036 38464 y Fl(\030)42604 38663 y Fi(\000)43392 38464 y Fl(;)221 b(\030)44542 38663 y Fr(+)45330 37388 y Fg(\013)45957 38464 y Fl(;)4488 40789 y(I)104 b(I)5844 40240 y Fr(\(1\))8209 40789 y Fs(=)1107 b Fm(\000)p Fl(\015)12108 39713 y Fg(\012)12735 40789 y Fl(\030)13303 40988 y Fr(+)14090 40789 y Fl(;)221 b(\033)15411 40996 y Fi(\000)p Ff(i=)p Fr(4)17461 39713 y Fg(\000)18070 40789 y Fl(P)18908 40988 y Ff(n)19829 40789 y Fm(\000)296 b Fl(i\013)22429 40988 y Fr(1)22955 40789 y Fl(Q)23985 40988 y Ff(n)24611 39713 y Fg(\001)25220 40789 y Fl(j)25833 39713 y Fg(\000)26441 40789 y Fl(\033)27180 40996 y Fi(\000)p Ff(i=)p Fr(4)29229 39713 y Fg(\000)29838 40789 y Fl(P)30676 40988 y Ff(n)31598 40789 y Fm(\000)f Fl(i\013)34197 40988 y Fr(1)34723 40789 y Fl(Q)35753 40988 y Ff(n)36379 39713 y Fg(\001\001)37597 40789 y Fl(\030)38165 40988 y Fi(\000)38952 39713 y Fg(\013)4488 43114 y Fl(I)104 b(I)5844 42565 y Fr(\(2\))8209 43114 y Fs(=)1107 b Fm(\000)p Fl(\015)12108 42038 y Fg(\012)12735 43114 y Fl(\030)13303 43313 y Fr(+)14090 43114 y Fl(;)221 b(\033)15411 43320 y Fi(\000)p Ff(i=)p Fr(4)17461 42038 y Fg(\000)18070 43114 y Fl(\013)18897 43313 y Fr(2)19423 43114 y Fl(P)20261 43313 y Ff(n)21182 43114 y Fs(+)295 b Fl(i\013)23760 43313 y Fr(3)24286 43114 y Fl(Q)25316 43313 y Ff(n)25943 42038 y Fg(\001)26551 43114 y Fl(j)27164 42038 y Fg(\000)27773 43114 y Fl(\033)28512 43320 y Fi(\000)p Ff(i=)p Fr(4)30561 42038 y Fg(\000)31170 43114 y Fl(\013)31997 43313 y Fr(2)32523 43114 y Fl(P)33361 43313 y Ff(n)34282 43114 y Fs(+)g Fl(i\013)36860 43313 y Fr(3)37386 43114 y Fl(Q)38416 43313 y Ff(n)39043 42038 y Fg(\001)o(\001)40260 43114 y Fl(\030)40828 43313 y Fi(\000)41615 42038 y Fg(\013)42243 43114 y Fl(:)317 46226 y Fs(Here)569 b(w)-36 b(e)569 b(ha)-36 b(v)g(e)569 b(used)e(the)i(fact)g(that)f Fm(h)p Fl(\030)20925 46425 y Fr(+)21712 46226 y Fl(;)221 b(j)75 b Fs(\()p Fl(A)p Fs(\))p Fl(\030)25462 46425 y Fi(\000)26249 46226 y Fm(i)599 b Fs(=)f Fm(h)p Fl(A\030)31035 46425 y Fi(\000)31822 46226 y Fl(;)221 b(\030)32972 46425 y Fr(+)33760 46226 y Fm(i)p Fl(;)664 b(A)599 b Fm(2)g(M)568 b Fs(in)h Fl(I)42855 45744 y Fr(\(1\))44113 46226 y Fl(;)664 b(I)45816 45744 y Fr(\(2\))47074 46226 y Fl(:)569 b Fs(As)g(a)317 48218 y(consequence)452 b(of)h(Theorem)g(4\(7\))f(of)h([Ara],)458 b Fm(M)p Fl(\030)24787 48417 y Fr(+)25575 48218 y Fm(?M)p Fl(\030)28771 48417 y Fi(\000)29559 48218 y Fs(,)f(whic)-36 b(h)452 b(implies)h Fl(I)39229 47736 y Fr(\(1\))40887 48218 y Fs(=)401 b(0)452 b(and)g Fl(I)46628 47736 y Fr(\(2\))48287 48218 y Fs(=)400 b(0.)317 50211 y(Since)580 b Fl(Aj)75 b Fs(\()p Fl(A)p Fs(\))p Fl(\030)8003 50410 y Fi(\000)9408 50211 y Fm(2)619 b(P)689 b Fs(for)581 b Fl(A)619 b Fm(2)f(M)p Fs(,)f(w)-36 b(e)581 b(obtain)f(that)g Fl(I)104 b(I)30907 49729 y Fr(\(1\))32783 50211 y Fm(\024)619 b Fs(0)581 b(and)e Fl(I)104 b(I)39697 49729 y Fr(\(2\))41574 50211 y Fm(\024)619 b Fs(0.)1018 b(Therefore)317 52203 y Fm(E)1018 52402 y Ff(n)1644 52203 y Fs(\()p Fl(\030)2718 52402 y Fr(+)3506 52203 y Fl(;)221 b(\030)4656 52402 y Fi(\000)5444 52203 y Fs(\))368 b Fm(\024)h Fs(0.)2268 54196 y(W)-108 b(e)530 b(turn)f(to)h(the)f(pro)36 b(of)531 b(of)f(the)g(prop)36 b(ert)-36 b(y)529 b(\(c\).)867 b(Since)530 b Fm(j)p Fl(\030)64 b Fm(j)533 b Fs(=)f Fl(\030)34753 54395 y Fr(+)35901 54196 y Fs(+)361 b Fl(\030)37842 54395 y Fi(\000)38629 54196 y Fl(;)530 b Fs(the)g(assertion)g(\(4.11\))317 56189 y(implies)434 b(that)9236 59301 y Fl(\030)f Fm(2)369 b(M)13087 59500 y Fr(0)13613 59301 y Fl(\030)14181 59500 y Fr(0)15002 59301 y Fm(\\)295 b(H)17318 58752 y Ff(J)18339 59301 y Fm(\))369 b(j)p Fl(\030)64 b Fm(j)369 b(2)g Fl(D)36 b Fs(\()p 24652 58234 820 54 v Fm(E)119 b Fs(\))1300 b(and)p 30674 58234 V 1300 w Fm(E)119 b Fs([)p Fm(j)p Fl(\030)64 b Fm(j)p Fs(])370 b(=)692 b(lim)35336 60098 y Ff(n)p Fi(!1)38011 59301 y Fm(E)38712 59500 y Ff(n)39338 59301 y Fs([)p Fm(j)p Fl(\030)64 b Fm(j)p Fs(])p Fl(:)5596 b Fs(\(4.16\))317 62635 y(Let)405 b Fl(\030)433 b Fm(2)369 b Fl(D)36 b Fs(\()p 6497 61568 V Fm(E)119 b Fs(\))237 b Fm(\\)h(H)10319 62153 y Ff(J)11376 62635 y Fs(b)36 b(e)405 b(giv)-36 b(en.)570 b(Cho)36 b(ose)406 b(a)g(sequence)f Fm(f)p Fl(\030)29156 62834 y Ff(m)30044 62635 y Fm(g)h Fs(in)f Fm(M)34198 62834 y Fr(0)34724 62635 y Fl(\030)35292 62834 y Fr(0)36056 62635 y Fm(\\)237 b(H)38314 62153 y Ff(J)39372 62635 y Fs(suc)-36 b(h)404 b(that)h Fl(\030)45635 62834 y Ff(m)46892 62635 y Fm(!)369 b Fl(\030)470 b Fs(in)317 64627 y Fm(H)376 b Fs(and)p 4274 63560 V 363 w Fm(E)119 b Fs([)p Fl(\030)6023 64826 y Ff(m)6911 64627 y Fs(])369 b Fm(!)p 9338 63560 V 369 w(E)119 b Fs([)p Fl(\030)64 b Fs(])364 b(as)g Fl(m)k Fm(!)i(1)p Fs(.)555 b(The)363 b(inequalit)-36 b(y)365 b Fm(k)p Fl(\030)28650 64826 y Fi(\006)29589 64627 y Fm(\000)151 b Fl(\021)31415 64826 y Fi(\006)32202 64627 y Fm(k)369 b(\024)g(k)p Fl(\030)216 b Fm(\000)151 b Fl(\021)48 b Fm(k)p Fl(;)886 b Fm(8)p Fl(\030)64 b(;)665 b(\021)416 b Fm(2)369 b(H)45714 64145 y Ff(J)46729 64627 y Fs(implies)317 66620 y Fm(j)p Fl(\030)1254 66819 y Ff(m)2142 66620 y Fm(j)473 b(!)g(j)p Fl(\030)64 b Fm(j)495 b Fs(as)g Fl(m)473 b Fm(!)g(1)p Fs(.)762 b(Notice)495 b(that)p 21194 65553 V 495 w Fm(E)119 b Fs(\()p Fl(\030)23088 66819 y Fr(+)23875 66620 y Fl(;)221 b(\030)25025 66819 y Fi(\000)25813 66620 y Fs(\))472 b Fm(\024)h Fs(0)495 b(is)g(equiv)-72 b(alen)-36 b(t)496 b(to)p 38702 65553 V 495 w Fm(E)118 b Fs([)p Fm(j)p Fl(\030)64 b Fm(j)p Fs(])474 b Fm(\024)p 43593 65553 V 473 w(E)119 b Fs([)p Fl(\030)64 b Fs(],)511 b(and)495 b(w)-36 b(e)317 68612 y(ha)g(v)g(e)431 b(sho)-36 b(wn)431 b(that)g(eac)-36 b(h)431 b Fm(E)13697 68811 y Ff(n)14323 68612 y Fl(;)664 b(n)370 b Fm(2)e Fh(N)p Fs(,)433 b(satis\014es)e(the)f(prop)36 b(ert)-36 b(y)431 b(\(c\).)577 b(By)431 b(the)g(lo)-36 b(w)g(er)432 b(semi-con)-36 b(tin)g(uit)g(y)317 70605 y(of)p 1798 69538 V 434 w Fm(E)553 b Fs(and)433 b(\(4.16\),)i(w)-36 b(e)434 b(obtain)f(that)p 9532 72650 V 9532 73717 a Fm(E)119 b Fs([)p Fm(j)p Fl(\030)64 b Fm(j)p Fs(])1108 b Fm(\024)f Fs(lim)p 15692 73930 1807 54 v 321 x Ff(m)p Fi(!1)p 20268 72650 820 54 v 20268 73717 a Fm(E)119 b Fs([)p Fm(j)p Fl(\030)22386 73916 y Ff(m)23274 73717 y Fm(j)p Fs(])369 b(=)g(lim)p 25754 73930 1807 54 v 321 x Ff(m)p Fi(!1)30329 72641 y Fg(\000)31483 73717 y Fs(lim)31160 74514 y Ff(n)p Fi(!1)33834 73717 y Fm(E)34535 73916 y Ff(n)35161 73717 y Fs([)p Fm(j)p Fl(\030)36459 73916 y Ff(m)37347 73717 y Fm(j)p Fs(])38077 72641 y Fg(\001)13552 76263 y Fm(\024)1107 b Fs(lim)p 15692 76476 V 321 x Ff(m)p Fi(!1)20268 75187 y Fg(\000)21421 76263 y Fs(lim)21098 77060 y Ff(n)p Fi(!1)23772 76263 y Fm(E)24473 76462 y Ff(n)25099 76263 y Fs([)p Fl(\030)26028 76462 y Ff(m)26916 76263 y Fs(])27277 75187 y Fg(\001)28256 76263 y Fs(=)368 b(lim)p 29636 76476 V 31443 76584 a Ff(m)p Fi(!1)34212 76263 y Fm(E)119 b Fs([)p Fl(\030)35961 76462 y Ff(m)36849 76263 y Fs(])369 b(=)p 38960 75196 820 54 v 369 w Fm(E)119 b Fs([)p Fl(\030)64 b Fs(])p Fl(:)24864 79850 y Fs(19)p eop %%Page: 20 20 20 19 bop 317 7306 a Fs(Th)-36 b(us)433 b Fm(j)p Fl(\030)64 b Fm(j)369 b(2)g Fl(D)36 b Fs(\()p 8228 6239 820 54 v Fm(E)119 b Fs(\))433 b(and)p 12516 6239 V 433 w Fm(E)119 b Fs([)p Fm(j)p Fl(\030)64 b Fm(j)p Fs(])370 b Fm(\024)p 17200 6239 V 369 w(E)119 b Fs([)p Fl(\030)64 b Fs(].)579 b(This)434 b(completes)g(the)f(pro)36 b(of)434 b(of)g(the)f(prop)36 b(ert)-36 b(y)433 b(\(c\).)2268 9298 y(Since)p 5755 8231 V 524 w Fm(E)119 b Fs(\()p Fl(\030)64 b(;)221 b(\030)8863 9497 y Fr(0)9390 9298 y Fs(\))523 b(=)g(0)h(for)h(an)-36 b(y)525 b Fl(\030)587 b Fm(2)523 b Fl(D)36 b Fs(\()p 21940 8231 V Fm(E)119 b Fs(\),)547 b(it)525 b(follo)-36 b(ws)526 b(from)f(the)e(prop)36 b(erties)524 b(\(b\))g(and)f(\(c\))h (and)317 11291 y(Prop)36 b(osition)434 b(2.1)h(that)p 12229 10224 V 433 w Fm(E)552 b Fs(is)434 b(a)g(Diric)-36 b(hlet)434 b(form.)1880 b Fa(\003)317 15728 y Fn(5.)2152 b(Sp)60 b(ectral)716 b(analysis)g(of)g(Diric)-60 b(hlet)717 b(op)60 b(erators)1618 19036 y Fs(In)419 b(this)g(section)h(w)-36 b(e)420 b(decomp)36 b(ose)420 b(the)f(Hilb)36 b(ert)419 b(space)h Fm(H)381 b Fs(=)369 b Fm(H)32673 19235 y Ff(!)33764 19036 y Fs(in)-36 b(to)420 b(direct)f(sum)g(of)h(eigenspaces)317 21029 y(of)326 b(Diric)-36 b(hlet)326 b(op)36 b(erator,)347 b(and)325 b(analyze)i(the)d(sp)36 b(ectrum)325 b(of)h(generator\(Diric) -36 b(hlet)325 b(op)36 b(erator\))326 b(of)g(Mark)-36 b(o-)317 23021 y(vian)331 b(semigroup.)544 b(W)-108 b(e)331 b(also)h(sho)-36 b(w)331 b(the)f(ergo)36 b(dicit)-36 b(y)332 b(of)f(its)g(semigroup.)544 b(The)330 b(decomp)36 b(osition)331 b(metho)36 b(d)317 25014 y(w)-36 b(e)434 b(use)f(is)h(essen)-36 b(tially)435 b(same)f(as)g(that)f(in)g([A)-145 b(W].)436 b(See)d(also)h(Example)h(5.2.20)g(in)e([BR].)2268 27006 y(W)-108 b(e)434 b(write)g(that)12149 30327 y Fl(D)13229 30526 y Fr(1)14861 30327 y Fs(:=)1107 b(\(2)221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\)\))24664 29779 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)26863 29251 y Fg(\000)27472 30327 y Fl(e)28075 29779 y Ff(\014)46 b(=)p Fr(4)29644 30327 y Fl(a)295 b Fm(\000)g Fl(e)32553 29779 y Fi(\000)p Ff(\014)46 b(=)p Fr(4)34855 30327 y Fl(j)75 b Fs(\()p Fl(a)36657 29779 y Fi(\003)37182 30327 y Fs(\))37688 29251 y Fg(\001)38297 30327 y Fl(;)9379 b Fs(\(5.1\))12149 32652 y Fl(D)13229 32851 y Fr(2)14861 32652 y Fs(:=)1107 b(\(2)221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\)\))24664 32103 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)26863 31576 y Fg(\000)27472 32652 y Fl(e)28075 32103 y Fi(\000)p Ff(\014)46 b(=)p Fr(4)30376 32652 y Fl(a)31059 32103 y Fi(\003)31880 32652 y Fm(\000)295 b Fl(e)33811 32103 y Ff(\014)46 b(=)p Fr(4)35381 32652 y Fl(j)75 b Fs(\()p Fl(a)p Fs(\))37689 31576 y Fg(\001)38297 32652 y Fl(:)317 35973 y Fs(By)476 b(Lemma)g(4.1)g(the)f(op)36 b(erators)476 b Fl(D)18378 36172 y Fr(1)19379 35973 y Fs(and)g Fl(D)23031 36172 y Fr(2)24032 35973 y Fs(are)g(w)-36 b(ell)476 b(de\014ned)e(on)i Fm(M)36887 36172 y Fr(0)37413 35973 y Fl(\030)37981 36172 y Fr(0)38982 35973 y Fs(and)f(also)i(on)e Fm(H)47174 36172 y Ff(f)98 b(in)48670 35973 y Fs(.)705 b(It)317 37965 y(follo)-36 b(ws)436 b(from)e(\(5.1\))g(that)12602 41286 y Fl(D)13718 40737 y Fi(\003)13682 41614 y Fr(1)15351 41286 y Fs(=)1107 b(\(2)221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\)\))24793 40737 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)26992 40210 y Fg(\000)27600 41286 y Fl(e)28203 40737 y Ff(\014)46 b(=)p Fr(4)29773 41286 y Fl(a)30456 40737 y Fi(\003)31277 41286 y Fm(\000)295 b Fl(e)33208 40737 y Fi(\000)p Ff(\014)46 b(=)p Fr(4)35509 41286 y Fl(j)75 b Fs(\()p Fl(a)p Fs(\))37817 40210 y Fg(\001)48037 41286 y Fs(\(5.2\))15351 43611 y(=)1107 b(\(2)221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\)\))24793 43062 y Fr(1)p Ff(=)p Fr(2)26260 43611 y Fl(e)26863 43062 y Fi(\000)p Ff(\014)46 b(=)p Fr(4)29164 43611 y Fl(a)29847 43062 y Fi(\003)30668 43611 y Fs(+)295 b Fl(e)32578 43062 y Fi(\000)p Ff(\014)46 b(=)p Fr(2)34879 43611 y Fl(D)35959 43810 y Fr(2)36485 43611 y Fl(;)12602 45935 y(D)13718 45387 y Fi(\003)13682 46264 y Fr(2)15351 45935 y Fs(=)1107 b Fm(\000)p Fs(\(2)221 b(sinh\()p Fl(\014)74 b(=)p Fs(2\)\))25826 45387 y Fr(1)p Ff(=)p Fr(2)27293 45935 y Fl(e)27896 45387 y Ff(\014)46 b(=)p Fr(4)29465 45935 y Fl(a)295 b Fs(+)g 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Fl(z)59 b(W)181 b Fs(\()p Fl(z)59 b Fs(\))p Fl(;)14254 b Fs(\(5.12\))18911 49298 y([)p Fl(a)19955 48750 y Fi(\003)20481 49298 y Fl(;)221 b(W)181 b Fs(\()p Fl(z)59 b Fs(\)])369 b(=)g Fm(\000)p Fl(i)p Fs(2)28385 48750 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)p 30584 48567 664 54 v 30584 49298 a Fl(z)59 b(W)181 b Fs(\()p Fl(z)59 b Fs(\))317 52536 y(hold)486 b(on)g Fm(W)109 b Fl(\030)7108 52735 y Fr(0)7634 52536 y Fs(.)736 b(Recall)487 b(the)e(de\014nition)g(of)i Fl(W)23590 52735 y Ff(n)24216 52536 y Fs(\()p Fl(z)59 b Fs(\))459 b(:=)f(\()p Fl(W)181 b Fs(\()p Fl(z)59 b Fs(\)\))32277 52735 y Ff(n)32902 52536 y Fl(;)443 b(n)459 b Fm(2)f Fh(N)p Fl(;)443 b(z)518 b Fm(2)457 b Fh(C)487 b Fs(in)f(\(3.5\).)736 b(It)486 b(can)317 54528 y(b)36 b(e)433 b(easily)j(c)-36 b(hec)g(k)g(ed)433 b(that)g(for)h Fl(z)428 b Fm(2)369 b Fh(C)p 14024 56699 820 54 v 14024 57766 a Fm(E)119 b Fs([)p Fl(W)16433 57965 y Ff(n)17059 57766 y Fs(\()p Fl(z)59 b Fs(\))p Fl(\030)19302 57965 y Fr(0)20123 57766 y Fm(\000)296 b Fl(W)181 b 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Fl(\030)37821 75692 y Fr(0)38347 75493 y Fl(:)24864 79850 y Fs(23)p eop %%Page: 24 24 24 23 bop 317 7306 a Fs(Using)434 b(the)f(relations)h(in)g(\(5.12\))g (and)f(\(5.1\),)i(w)-36 b(e)433 b(obtain)h(that)f(the)g(relations)15122 10794 y([)p Fl(D)16563 10993 y Fr(1)17089 10794 y Fl(;)221 b Fs(\010)18610 10993 y Ff(l)18958 10794 y Fs(\()p Fl(n)p Fs(\)])370 b(=)e(\(2\(1)296 b Fm(\000)f Fl(e)27396 10246 y Fi(\000)p Ff(\014)28756 10794 y Fs(\)\))29768 10246 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)31967 10794 y Fl(z)32571 10993 y Ff(l)32918 10794 y Fl(W)34327 9718 y Fg(\000)35131 9895 y Fs(1)p 35068 10489 777 54 v 35068 11705 a Fl(n)35977 10794 y(z)36581 10993 y Ff(l)36928 9718 y Fg(\001)37537 10794 y Fl(;)9489 b Fs(\(5.14\))16423 13361 y([)p Fl(D)17864 13560 y Fr(1)18390 13361 y Fl(;)221 b Fs(\010)19911 13560 y Ff(l)20259 13361 y Fs(])369 b(=)g(\(2\(1)295 b Fm(\000)h Fl(e)26909 12812 y Fi(\000)p Ff(\014)28269 13361 y Fs(\)\))29281 12812 y Fi(\000)p Fr(1)p Ff(=)p Fr(2)31479 13361 y Fl(z)32083 13560 y 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b(implies)h Fl(H)472 b Fs(=)15090 54699 y Fg(e)14796 55034 y Fl(H)103 b Fs(.)2268 57027 y(\(b\))433 b(By)h(Prop)36 b(osition)434 b(5.2)h(and)e(\(5.18\),)h(w) -36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)16811 59816 y Fl(U)139 b(H)103 b(U)20036 59267 y Fi(\000)p Fr(1)21662 59816 y Fs(=)369 b Fl(\034)23608 60015 y Fr(1)24133 59816 y Fl(N)434 b Fm(\012)295 b Fo(1)h Fs(+)e Fo(1)i Fm(\012)f Fl(\034)32222 60015 y Fr(2)32748 59816 y Fl(N)66 b(;)317 62604 y Fs(where)434 b Fl(N)571 b Fs(is)434 b(the)f(n)-36 b(um)g(b)36 b(er)432 b(op)36 b(erator)434 b(on)g Fl(l)21431 62122 y Fr(2)21957 62604 y Fs(.)578 b(Th)-36 b(us)433 b(this)h(completes)g(the)f(pro)36 b(of)434 b(of)g(\(b\).)578 b Fa(\003)2268 65621 y Fs(Finally)435 b(w)-36 b(e)434 b(are)f(able)h(to)g(pro)36 b(duce)433 b(the)g(pro)36 b(of)434 b(of)g(Theorem)g(3.2.)317 68244 y Fo(Pro)42 b(of)498 b(of)g(Theorem)h(3.2:)1302 b Fs(\(a\))433 b(and)g(\(b\))g (follo)-36 b(w)435 b(from)f(Theorem)g(5.2.)2268 70236 y(\(c\))447 b(By)h(\(b\),)i(zero)e(is)f(a)h(simple)f(eigen)-36 b(v)-72 b(alue)448 b(of)g Fl(H)550 b Fs(with)447 b(eigen)-36 b(v)g(ector)448 b Fl(\030)37817 70435 y Fr(0)38343 70236 y Fs(.)619 b(Since)447 b Fl(\030)43301 70435 y Fr(0)44274 70236 y Fs(is)h(a)f(strictly)317 72229 y(p)36 b(ositiv)-36 b(e)435 b(v)-36 b(ector,)434 b Fm(f)p Fl(T)10819 72428 y Ff(t)11215 72229 y Fm(g)11879 72428 y Ff(t)p Fi(\025)p Fr(0)13910 72229 y Fs(is)g(ergo)36 b(dic.)579 b(The)433 b(pro)36 b(of)434 b(of)h(Theorem)e(3.2)i(is)f(completed.)1879 b Fa(\003)3569 74536 y Fo(Ac)-42 b(kno)g(wledgemen)g(t:)554 b Fs(This)380 b(w)-36 b(ork)380 b(w)-36 b(as)380 b(supp)36 b(orted)377 b(in)i(part)g(b)-36 b(y)379 b(Korea)h(Researc)-36 b(h)379 b(F)-108 b(ounda-)317 76529 y(tion)434 b (\(KRF-2001-005-D00010\),)i(Korea)f(Ministry)e(of)i(Education.)24864 79850 y(25)p eop %%Page: 26 26 26 25 bop 317 7306 a Fn(References)317 10614 y Fs([Acc])651 b(L.)434 b(Accardi,)g(T)-108 b(opics)434 b(in)f(quan)-36 b(tum)433 b(probabilit)-36 b(y)-108 b(,)434 b Fe(Phys.)465 b(R)-66 b(ep.)432 b Fo(77)p Fs(,)i(169-192)h(\(1981\).)317 13708 y([AFL])651 b(L.)566 b(Accaradi,)601 b(A.)567 b(F)-108 b(rigerio,)601 b(and)566 b(J.)h(T.)h(Lewis,)600 b(Quan)-36 b(tum)565 b(sto)36 b(c)-36 b(hastic)567 b(pro)36 b(cesses,)601 b Fe(Publ.)2991 15700 y(R)-66 b(es.)464 b(Inst.)f(Math.)i(Sci.)431 b Fo(18)p Fs(,)k(97-133)g(\(1982\).)317 18795 y([Ara])651 b(H.)446 b(Araki,)j(Some)d(prop)36 b(erties)444 b(of)i(mo)36 b(dular)446 b(conjugation)g(op)36 b(erator)445 b(of)h(v)-36 b(on)446 b(Neumann)e(alge-)2991 20787 y(bras)345 b(and)f(noncomm)-36 b(utativ)g(e)345 b(Radon-Nik)-36 b(o)36 b(dym)346 b(theorem)e(with)h(c) -36 b(hain)345 b(rule,)363 b Fe(Paci\014c)382 b(J.)h(Math.)2991 22780 y Fo(50)434 b Fs(\(2\),)f(309-354)j(\(1974\).)317 25874 y([A)-145 b(W])652 b(H.)436 b(Araki)h(and)f(E.)g(J.)h(W)-108 b(o)36 b(o)g(d,)437 b(Represen)-36 b(tations)436 b(of)h(the)e (canonical)i(comm)-36 b(utation)436 b(relations)2991 27866 y(describing)415 b(a)g(non-relativistic)h(in\014nite)f(free)h (Bose)f(gas,)420 b Fe(J.)448 b(Math.)g(Phys.)415 b Fo(4)p Fs(,)420 b(637-662)d(\(1963\).)317 30960 y([BK])651 b(C.)567 b(Bahn)e(and)g(C.)h(K.)g(Ko,)599 b(Construction)566 b(of)g(un)-36 b(b)36 b(ounded)563 b(Diric)-36 b(hlet)566 b(forms)g(on)g(standard)2991 32953 y(forms)434 b(of)g(v)-36 b(on)434 b(Neumann)f(Algebras,)h Fe(J.)464 b(Kor)-66 b(e)g(an)464 b(Math.)g(So)-66 b(c.)432 b Fo(39)i Fs(\(6\))g(931-951)h(\(2002\).)317 36047 y([BKP1])652 b(C.)490 b(Bahn,)504 b(C.)490 b(K.)g(Ko)h(and)e(Y.)h(M.)g(P)-36 b(ark,)505 b(Diric)-36 b(hlet)490 b(forms)g(and)g(symmetric)g(Mark)-36 b(o)g(vian)2991 38039 y(semigroups)622 b(on)g(CCR)h(Algebras)f(with)g (quasi-free)h(states,)670 b Fe(J.)637 b(Math.)h(Phys.)622 b Fo(44)p Fs(,)670 b(723-753)2991 40032 y(\(2003\).)317 43126 y([BKP2])652 b(C.)490 b(Bahn,)504 b(C.)490 b(K.)g(Ko)h(and)e(Y.)h (M.)g(P)-36 b(ark,)505 b(Diric)-36 b(hlet)490 b(forms)g(and)g (symmetric)g(Mark)-36 b(o)g(vian)2991 45118 y(semigroups)433 b(on)h Fh(Z)12418 45317 y Fr(2)12943 45118 y Fs(-graded)f(v)-36 b(on)434 b(Neumann)e(Algebras,)j(preprin)-36 b(t.)317 48212 y([BP])651 b(C.)313 b(Bahn)f(and)f(Y.)i(M.)f(P)-36 b(ark,)337 b(F)-108 b(eynman)312 b(Kac)g(represen)-36 b(tation)312 b(and)f(Mark)-36 b(o)g(v)313 b(prop)36 b(ert)-36 b(y)312 b(of)h(semi-)2991 50205 y(groups)526 b(generated)g(b)-36 b(y)526 b(noncomm)-36 b(utativ)g(e)527 b(elliptic)g(op)36 b(erators,)550 b(to)527 b(b)36 b(e)526 b(app)36 b(eared)526 b(in)g Fe(In\014nite)2991 52197 y(Dimensional)463 b(A)-33 b(nalysis,)464 b(Quantum)h(Pr)-66 b(ob)g(ability)463 b(and)i(R)-66 b(elate)g(d)463 b(T)-100 b(opics)317 55291 y Fs([BR])651 b(O.)509 b(Bratteli)g(and)g(D.)g(W.)h(Robinson,)528 b Fe(Op)-66 b(er)g(ator)533 b(A)-33 b(lgebr)-66 b(as)533 b(and)h(Quantum)g(Statistic)-66 b(al)532 b(Me-)2991 57284 y(chanics)p Fs(,)495 b(second)483 b(edition,)496 b(Springer-V)-108 b(erlag,)495 b(New)483 b(Y)-108 b(ork-Heidelb)36 b(erg-Berlin,)496 b(v)-36 b(ol)484 b(I)g(\(1987\),)2991 59277 y(v)-36 b(ol.)435 b(I)36 b(I)434 b(\(1997\).)317 62371 y([CFL])651 b(F.)556 b(Cipriani,)587 b(F.)556 b(F)-108 b(agnola)556 b(and)f(J.)i(M.)e (Lindsa)-36 b(y)-108 b(,)586 b(Sp)36 b(ectral)556 b(analysis)h(and)e(F) -108 b(eller)556 b(prop)36 b(er-)2991 64363 y(ties)481 b(for)h(quan)-36 b(tum)480 b(Ornstein-Uhlen)-36 b(b)36 b(ec)-36 b(k)479 b(semigroups,)494 b Fe(Comm.)507 b(Math.)h(Phys.)482 b Fo(210)p Fs(,)494 b(85-105)2991 66356 y(\(2000\).)317 69450 y([Cip1])652 b(F.)458 b(Cipriani,)466 b(Diric)-36 b(hlet)459 b(forms)h(and)e(Mark)-36 b(o)g(vian)460 b(semigroups)e(on)h (standard)f(forms)h(of)h(v)-36 b(on)2991 71442 y(Neumann)432 b(algebras,)j Fe(J.)465 b(F)-100 b(unct.)464 b(A)-33 b(nal.)432 b Fo(147)p Fs(,)j(259-300)g(\(1997\).)317 74536 y([Cip2])652 b(F.)591 b(Cipriani,)631 b(P)-36 b(erron)591 b(theory)g(for)g(p)36 b(ositiv)-36 b(e)592 b(maps)f(and)g(semigroups)g (on)f(v)-36 b(on)592 b(Neumann)2991 76529 y(algebras,)434 b Fe(Canadian)465 b(Math.)f(So)-66 b(c.)463 b(Confer)-66 b(enc)g(e)462 b(Pr)-66 b(o)g(c)g(e)g(e)g(dings)431 b Fs(V)-108 b(ol.)434 b Fo(29)p Fs(,)g(115-123)h(\(2000\).)24864 79850 y(26)p eop %%Page: 27 27 27 26 bop 317 7306 a Fs([Da)-36 b(v])652 b(E.)323 b(B.)g(Da)-36 b(vies,)346 b Fe(Quantum)363 b(the)-66 b(ory)362 b(of)h(op)-66 b(en)362 b(systems)p Fs(,)345 b(Academic)323 b(Press,)344 b(London-New)322 b(Y)-108 b(ork-)2991 9298 y(San)433 b(F)-108 b(rancisco,)434 b(\(1976\).)317 12398 y([GL1])651 b(S.)437 b(Goldstein)g(and)g(J.)h(M.)f(Lindsa)-36 b(y)-108 b(,)438 b(Beuring-Den)-36 b(y)437 b(conditions)g(for)h(KMS-symmetric)f (dy-)2991 14390 y(namical)d(semigroups,)g Fe(C.)464 b(R.)g(A)-66 b(c)g(ad.)464 b(Sci.)f(Paris.)i(Ser.)e(I)434 b Fo(317)p Fs(,)g(1053-1057)i(\(1993\).)317 17490 y([GL2])651 b(S.)445 b(Goldstein)g(and)g(J.)h(M.)f(Lindsa)-36 b(y)-108 b(,)448 b(KMS-symmetric)d(Mark)-36 b(o)g(v)447 b(semigroups,)h Fe(Math.)475 b(Zeit.)2991 19482 y Fo(219)p Fs(,)434 b(590-608)h (\(1995\).)317 22582 y([MZ1])651 b(A.)363 b(W.)g(Ma)72 b(jewski)365 b(and)d(B.)h(Zegarlinski,)378 b(Quan)-36 b(tum)361 b(sto)36 b(c)-36 b(hastic)363 b(dynamics)g(I:)g(Spin)e (systems)2991 24575 y(on)433 b(a)h(lattice,)h Fe(MPEJ)f Fo(1)p Fs(,)g(P)-36 b(ap)36 b(er)433 b(2)h(\(1995\).)317 27674 y([MZ2])651 b(A.)417 b(W.)h(Ma)72 b(jewski)418 b(and)f(B.)g(Zegarlinski,)422 b(Quan)-36 b(tum)415 b(sto)36 b(c)-36 b(hastic)417 b(dynamics)g(I)36 b(I,)418 b Fe(R)-66 b(ev.)447 b(Math.)2991 29667 y(Phys.)434 b Fo(8)f Fs(\(5\),)h(689-713)h (\(1996\).)317 32766 y([Mat])651 b(T.)660 b(Matsui,)716 b(Mark)-36 b(o)g(v)661 b(semigroups)e(on)h(UHF)f(algebras,)717 b Fe(R)-66 b(ev.)671 b(Mo)-66 b(d.)672 b(Phys.)660 b Fo(5)p Fs(,)716 b(587-600)2991 34759 y(\(1993\).)317 37858 y([P)-36 b(ar])651 b(Y.)542 b(M.)f(P)-36 b(ark,)569 b(Construction)541 b(of)i(Diric)-36 b(hlet)541 b(forms)h(on)g(standard) e(forms)i(of)g(v)-36 b(on)542 b(Neumann)2991 39851 y(algebras,)620 b Fe(In\014nite)599 b(Dimensional)h(A)-33 b(nalysis,)634 b(Quantum)602 b(Pr)-66 b(ob)g(ability)600 b(and)h(R)-66 b(elate)g(d)600 b(T)-100 b(opics)p Fs(,)2991 41843 y(V)-108 b(ol.)434 b(3,)g(No.)g(1,)g(1-14)h(\(2000\).)317 44943 y([P)-36 b(art])651 b(K.)460 b(R.)h(P)-36 b(arthasarath)g(y)-108 b(,)466 b Fe(A)-33 b(n)489 b(intr)-66 b(o)g(duction)486 b(to)j(quantum)g(sto)-66 b(chastic)489 b(c)-66 b(alculus)p Fs(,)467 b(Birkh\177)-650 b(auser,)2991 46935 y(Basel)434 b(\(1992\))317 50035 y([RS])651 b(M.)346 b(Reed)f(and)g(B.)h(Simon,)363 b Fe(Metho)-66 b(d)384 b(of)f(mo)-66 b(dern)383 b(mathmatic)-66 b(al)383 b(physics)h(I,)f(II)p Fs(,)345 b(Academic)h(press)2991 52027 y(\(1980\).)24864 79850 y(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303281907417--