. : http://sqi.cs.msu.su/store/storage/ctvo0sg_ozhigov2.pdf
: Tue Nov 25 17:44:03 2014
: Sun Apr 10 22:24:46 2016
: Windows-1251


R ??+'+ Ɯ?????? ? Ǟ? ??R '?R?? '?? ??Ԝ+?+????'???+??'????????+??????????????'???????? +????+?????'??+'+?+?'???˿????R??R?'??ž+??????+??R???'???R??????????R???'???+???R??+?R'??R??+???????+R?????R???'?+? ?  ? ????? w uyw~}t|~}t{wxvxt yx~yxy{|x|ytz}|w y|vtzwxx|wtytxwuy|}zzy}wtyy|u~yt{wtw||wt}z~xywxy}wtxu|uyxttwx}tyuw}vy vx y u { } tu u t t u tu t {wu u tu t t uw | yvw}ttwutuzyxt}txtutux{xyw|xwx|zvv}wyt|txyz{w|w|}~yxttut{||xxuxtyxtvwxtuz|}|xzttvtzt}y~utwxttx{ xw}tuty}t x x}tzx tuwv}tyyu {x} tz{tw yyz ut{t |zt v}wy|ut}t ytu ytxtwtz z|u w u vxxu|yt} tt|} yxuyuyx ytuyy} tu{tt yy} wy}yu tt tu wtw ttt uut {x|w v{ uywz|| x{| ytvwtzz||xux x||{ut |u y~ |wxt{| |u yy y { tty} uwu||ztxuu}ttxztu|u{vwxxu}yx}t {uxvyut{uywwxy{}xy~xyxtu|zxwuxtt{t|wttvzx}uy|~ux}t}|w{}zywy}xywxvwu}t|s y w t orqoponm l wusiurqf uqtrtx wkwrr rqtqy rwtuqy qpqqty qpquwrstkxu wswqrtxq twviyxs q yy yt swrwqrqxtxqquq twqyxq q pqr ssytxxwvvssuttxsqrqpwihqiwqwrwxqxqtywsffqtxsqixwwsquxsqqqxtwtjwviwttwiqiq tsvtxtpxrsr q wtqqtssxwstwwqsxrswrwrwusiswtixsrvtssy xtsiw w wqrqtqy sysus qqty wsxsq x s ixwvsxxiy txqwvsrph stsiu rsw t wsisuyyiwquytswwvtxqqtpqqrwwiwswqipxqsvuqsissurswqyqxtqxwrgqixssysxstfqsuqytxqtuwrwipqyt wuf x wrxtsrtuxxsr qutsrqpi ssrqwt s wsgywt swvsrph ysu s etruqpisfevstxxtvqipqusqqtrsiwyqwuguqtrtsisiwrsussiiwpqutriwtqsiyqxvwstuiwrsrwtxquwu w s si q trpws uqt sryiqr irtwd stu s swuwr sutrt suisuqsviq uwvvqpwqwiuwsxirqwtuqi qutqvyxwuvqstxxqsisywqitqqrwrsiwquqqxqqu quwqtxsxsyqxwqvtsutsyrqpqihstx gfedcbaa`


"' YXXT W8V72@6 UT G &9 D9@SR1Q8@9)5&9AC1A& A3A&@2)19@1P8&7'6(5I1((AH@9&2(12)&(F2&1E0 9& 3B 43 ( )('&%$"#"! ? ? ?? ?? ???????



'

? ͜?+ +?? '???? ? ??? ??+? Ǟ??R ???? ????? ?Ǟ????? ?'???? +?????R?' '??? ?? ? ???????++??R?????&ȿ???+?????R+???+????R?????+???+œ???œ????+?'????+?+???'?????+??+??+??????R?+?'??+????+'?&?????'??'?+R??+?????+R+++R?++????R?+????R  ? ? ? ? ? ' ? ? ? +?R++?+?+??+R++???++??+?'?????+?R'????+?+R+?%?+'????˜%+????????͜+????????'???+?+??+???̿???+??????+????????+??+??????R+???????????? ????????'+?'?+???+$+#?? ? ? ' ? ' ? ? ? ? ? ' ' ?? " ?R??? Ǟ????? Ǟ??? ?+R ? +???+?? ?+?+ ? ??? ???????+?͞????+R'????R???????'?????? +??R??R????????+????????? ?????R?R???ɞ+????R???????R??'???????????+????+???????+!??????+? + ??Ɯ??+?R???+???++?R?+R????????R????'?R?+??+????????+????+??'???+?+'???+??+??R?????????'ʜ?????+????+? ?????????+?????????????++R??????R???+'??+? ' ? ' ?? ? ??R????'????+ ?Ǟ??? ??+?+R ? ? '? ??+R??+? ??' ?? ???????+?R?'????+ ???R+???R??'+???????+??'??'?++?R??+?+????? ????+???'??R????R+?+??????R???+???++?R????? ?R?+?????'????ɞ ' ???R ?????? ???? ? ??' ???R?'+?  ͜?+ ??R??? ++?++???????+?? ???R+ '?+??'̜???'?++ɜ??'???????????R+?????'???R????++???+?????R''??????R+????՞?++???Ȟ?????????+?'??+?'??????? ???????+?????+'??+?'????'?+?+?R?????????'?????ž???'+???+?? ??+?????????????R?+?????????R ??+?R???????????'??+? ? z|z x | x w } x| {} x ? yw}x} y x{y | w|{y} y y| t u t u t y z t x u |zt u t tw| u{ w u } u u y twzytttwwzwt}zy||zy{uxxvxvuy}yxy|yxx}zy|tvy|uw}{y|yu|tt|u{tt|yyw|xt|wtz{xx{tyz|wtw|t|zv|}yw}xtuwuttt { t t w y yw w { y w ut{| tyx{yt ttz|w wt| v w} } tw{~ |wyxw~}}x}tuty|xywx}||? zxxyu{wyy}xy}tu~ytxwu|yw~}|t{v|wyu}yutwuvtwy}tx|ww}yywv{ttywxt|zyuxv{y{yxxyyu? w | t t t t | } t t xu t|ut tut tt u }| w||zt xuyy wuytx|z |u{t {yt v{|tz| tyw}x} tyx{yt vxuvtw} xt}vvyxu|}x}tv}{y twttx|tv{tyuxwv|{t|}yx|wxwtt{t|vt}uwxy|xutttwzwyxuty{u|uy~xuu|xwztywvt|s |tw|wuyxy}z wt x}x| yu twtz x wxt{|uy{uxwzy{twyxu{|t}tx twttz |yw |tzttz ttttwz}y~y|xw}}}t|uz{uzy}tttu|}{~wx|ut}y||{wu|y|xuytyytux|uvz|z|}wxwt}t{u|ty u{ {t x t t x u v}tyyxy ttt?wytztvxytz}}ytxzwxytwt{|xxux}~y{wtuxz|ztw~w|wxwvuwt|}}y|utt|tu|yzwt tz u y y u y w u|tzwux|xzv{tt|wtw}zx t}z|w{yttvwyyu{yyux|uy{ttzxuy|uttwtytxwutyz{x~}|{ttt|wu|xzxu||yxw tttutx } v v |t } y z w{ u |ywytt ttuz}{y~wxx}{|}{zyxw|vuuttytuwy~|v}y|w}xzxy{x}xwu|~{wtyttx}yu{w|}wyuyt|xu|wvx}ytv{|vyy x {| y } x~ u w tu u z |w||| ttz}y~xwx{|u| |wywxtx x wxtt} |ywtzttttwxw|z vxyut wxy|u yw{}xt{tuutzxwtyw~tut|uuyxt}t{{|xw}wtwtxz||~|yw|xzttutyyt}wuw|zxz{uttzz}y~{xywtw}xxxzzxwtx||x{||{|~t| tt xz}y~xwx{|u| uutxx|{w |tz |tw }t|{xuyxx vx|wx ?|{t uu|| t xuy{ }tx~y}v}{t{|wyutxx|~|x|t yzxz{yt~xtxyvyu}uttxx{|yww|zttztvxtux ytt tx{ w|y } tt { ywy t w}y tw w|txtux t{ tuz}tyxvxwuxy~xu{||yxtuyyu{{ywxy{}xy~||{z?txvtx ~wx t| txuyw|{wxy{}tx}~uy~tw|yu|t{|t} ~ yy { tuttttx x twxtt{| xxttywtx w|ztyuw vxu}y~xuu|u xw x|uutz| xxz}y~xx }u ytuu|t|u t ywtzzwv}?yv ux~t vxtyw{t}zx~vxuyx~{xxu|ttxty yw|txwtwz|x}| u{t tw~ wyy{} vxuyt t| w ~ } w|zxtx} xx|} }||{vu|{t {| { t tt u w u yx }z{|yt}w}yx} yutwwy}xtz|{ywt|zvy| zx|wtuxuzy}yx~xwxxt}xyy y|y z|tzv}}|w?x|ww|uy}}x } uy{t ywt z?yyx tw}ttu uy~t ttutt|uu|tu}t yzxx tut xz}y~xwxt{ t
T
tot

p

pT

tot

>1




Ǟ??+??R?? RȞ?????? Ü? Ü???? R?R+ ?R ? ?+ ?+?? '???? ? ?'??? ?'?%?'???R'% ? { uu w y } ttwtywz|vx|u|txtwyu|z y}xyu{uuv|xvxxvtxxuyu{yttut|yuy{tyuww|}ytyx{}ttu|wx}ytx|?~yvv}wxxuytxtz x x uu y xw| {yt z|w yv yx}w}ytx| xz}tut{z x yw}x} uutwzy{ xvx|yxx}|u{z ttty}w|{wyx|~w}}}|ytu|w|vut|ww~} wvtw|xv~{}yyw}yttwxzwuyx~| yv}w||zxuuxxyu|yuxtztx|ytxwz|}~v}vxu w t yt }v u} y vtt|wz{}t| y|yuv}ztwywy{u|yytzwy{v}wwtyuyyxtxux|v{x}vt{twy~tw}z|yy{{}|}xxx~utyywxv{y{~xxy~x|yyxu|vvt|w}}t}zyu{wwuw}yuw{|y|~w}}| t u u utw | t t v uw t}ut t tuu u|t u t t w|xux vxu|ytx{yt ttuv{zy{t v{ wxt{| yxw}y} twts tu{|xuyut }wz|xwy}|} wvy}wyxtwt|{u}x~wux|utxtwyywyzxz||wyxx|uy{}xxw~txwx}x}x|w}tv|xutwu}t{}|xtuuy~uxtw}}|z~ ts vt t t tu t tuutwxzxy{ytuutxxx|x~|uuy{{|uw}|t}w}|uyyw|{vx|ztt{?}txwyuv|}uwx|yxwytyxy ttwvt}zwx|zwu}ytut~ut v{ {t t vxuvw}t} yxz| |{}x{ tx~tw wt yxzu tuyux{ wut uyuutt{}x~ x~xuu|t x{ |}tyw}yu||w}ttutwwz}yt{uy|}| wt~w}tyw} tuv|{u|ux|uvyu}t}wz}tuyw}|w v|yw{}|xx}xtuxwwzyy|{zxtutt {xzuyv{ttw}tx} } t u tw| x|xutz uutwzy{xxu|}xt xz y tutwzy{ xz{tz}yu x yw}x} v{ tu xztx|txyu twzy}yy{tx}}|y{zy|v}y{|t wt|z}}xtut{ v}xyyxxxzx| z|z ty vv{|w}y y|z} ~ | |yt w}zy{ yx yyw |w}yx tx}tvxuvzt|wtt}x|yutywuyyt}utyuvu|}vx}z}yt~txuxtwy}xtuuutv|{ztuy{txuuuwwtyw|wtzxutu|tuxtzyw{tyu~{yzw} tu } x | x u wz | u tw {yxz|yxz}y~xx yx|u yxuyyt twu|z } yw}x} tu~xw}|~ttu xx{ yyw z w| } z w x wy|zx w w |z w}x y wxyty|u ttzw|z ttxtutu |u|ttt tttz}yt~xwtxt{| xuu|}txt tttu{|twy z xwtyytyyx~tuyy{|uw}s } yy ytu{t yty{tx tztx x z|z uxwzy?|t y xz|w xzx|ux yxuy{|w}y yt |tuv}{wxx{ywyy|u{yyttwwtzv{xyxw|t{|utvxtwwu}|wz}xyxu|w}txwxt}xtuxyyu xwvxuwyyx x x{t yu|wxu { y | t t }yv{tt wx{||wu|z yyx{u}y{t|w}wyzxtxtww} yy{v? yyxutw}|xwv}tuwvyy} ttwu|z vxuyy~{xx}~|{z vw{y|ut?wxt{{x|yw w|yz wt | z { t w z} } | t }x yttwx{ | t|u ytx{xt|w uz y| | z|wtux{| v{t vxuy~|u tty{uuyu x ttu{|x u u tt u ux yxu|ytww}y}} v}wyv{v |tt tytz}y~xwxt{| yxw}y{} u{|xwxuxs v|z|w vxu||uttwt}wt}ytyw|{t}yt} |}|vw{}x{|txu|} twu|w{t|x|xtuuxtv}xwu|}|zttw~wytw}y}tuxytt t tw wyyx yxu}|}xt tw tw~ tz{tw xwy|? |tt ttz|w |wzy}| xzy}y~xwz| |u }xx~twtx y{x}}t}{twtzwtuw|xw}t|u}tyyy{t yu ? |zxx uyw}yx ytuy ux~xw xz~}y~xtw}x z x|||{t wxt{| utw||x| xz}yxxw|yw| xt|t|xzy|{|x~y{|u||tw||tz|}yt~xt}wyx~xtw|{ywwy|wx}zwxyxwt}v{xttwuttzxx|uy{yvuy|z}yt~xwyx w x t t vxu|}xt tx xt |} v}wv{v {x |t ttz| xx|w}uty tw yt v}w {t}x x |{x xztwtt yt tywtz xx{yt v}wyv{}x~ w x x|tw|w v{yyyxxutvtwt}y}u|yuyxu~xy{y}yyx{zzyy~{xtx? yuvx{} vxuyy~uxu|yxzy|tuy|u~xt||xzt tt|tut{x|z|| } t t ~ {x x wyyx ywtz xx{yt wy|wt| ttwvtz } vx|tux v} |wy~}wt yw}tx} y{ywu |tw{}ttyuu|u|}yywxt{uu|xvywz}|t{yy{tt zx||{zyyy} wy~{|w}}wyy}|tuvuwt{yy|wxw{y}|}xz~z|zx}wwyy~z}x}}|w{{xw|yu | ttuu ~{z {t t z z t xt u } t t t z z u t




+R?'?? ??'?+ ???????? ??? + Ǟ+???? ??'?? Ǟ?'????? '???'? ? ???+??$? ' t |uxw|z v|uw}t{y |xttyu }y? xxtyw ttwu|z |x{|t ttz}y~xwx{|u| ttuut xywx}t|uuwytw}}tt|v{} wyxyx v{yvw}x}xxtztwu|ztzt}uy~~xxww}y|~ttywt~||xxtyuwt}xx}xtwwxt{yyt|~~xw|}?{wt x x u {u vxu||wt u|yz x u ||~x~t | ytu xx{t {uyt wxtw}t wv{tt yu xzxx ttwu|z tz}y~xw}xw|w} ttwy y|z} wxuyuyx xw}tuw tw | x tuuy? ~|| u~xw}|~ ttu xz}y~x}}|{z xxtyw wyw}wu} uwty}v||wt}wttzxww}}ttuw|wt}wtu|ttuw}{x|~xyuvy{t|w}}zt}zxwytwy~ xwu}vttwx}tt}yutw}ut|wz xu~wx}wx}u|~ytt y z w u t u u vyy xw v{ xuy~|u tw}ytu w|wx~} yv x{}y yy| v}wyvu|t} x tu{|xxuxx vyw}{}wytyy{xvt}|yu?yy|xyzu}y~xt|uu{xt x}uy~w|}uy twuwt}tttzy}}x}x|tuxx{xtz|yw|uuxyuxt{tzw|y}tt{tyx}u t u y xz v~{} w |z xw}|~ w} z} x z|w w| x z z|z w| | vx |}xt ttz}yy~x|ux { z|ztvtyu ytw ~wttyt|u x vtw}tttz} tu }tu| txt~{t xu{tt uyw uw|ux xttwz|uxxuy~t|tuz xxu{yy~|yuttuytuztywxv~w?}xtw}}txuwu}yy{yytuyyu~||uxuxw xu}tuttytyyxx|wu}xtttzz} x xz| { t } u x v ut { ttwu}ty}uxuv~}xwwy}v|{~tvu|ttyww}}ut|twz}}tx{x}yw|?uxtttz v{wxtwx}uty}~|u}t}wyv|tuutu}}twyz||z|z|wt}wyyutu w ut w } | tz ytuw}xyuxyzwtxwuvyt|w~}}{}}u~txwwu}||~zu txxwu}|u|twx}{tyttwttuytwwy|~tz{}xtzv}}yw~yx|}{}y|{z|yxw}ttwtzz}|x } t t t t ~t wy|uyxttttzx}uyy~~x|wut|tzwyytutw~xwxzyt|{twxwy}tx~txw{}}yyuwy||yx}wxy|zwx}uutww}|twu {yyww}xx}ux x y t x w tu z} x ytzux{||uxxxutxy~|}}|x{ztu wzxyz|xuwxvxu|}twt tw|t{ytw}u|tzz|uxttttzu}y|~x}|}u||{uzt}|t|{tvxxu~yx{||yt uv{zy{t xxuy~x x xw}z| |uu|wuyy{ tx ttwu|z |wy~ w}t |~ x?t??t y?? y}wuxt}}{x uy|{ tt } xvxz|y xxz}y~xx vxuy{| twy yz|w uw}yx twtw yx|u z|z {t utu}t wy|x ytt }}yt wtw twtz xz|y }}|{z yyut{|tx}? txw}tyu wy| tw}|~ y xz|y xz}y~xx xxuywt}}| x tutwt vxu t vxuyy~{x twzy yxuyvyyuy|tz|uy xv}ut xxuy}t} y vxyuvtw}}| w| v| ~ xxuy{w}xx~t~}txtyuu|t|twt| yxux{tyuwv}utttz} zz{||ytxztxw}|{ tyuuxyv|?|xtxuwyzx}xutvw{ yt~t}tuz|}yvx}w|zyx{w tvxtzw}}y~yx}}v|x{uz|{yt|wvzwy|wtw~y~uyxyywytzyuytwsyx|xuwxutxwtzyy?tw v | | } yw}x}xy w |uzt ~xw}z|~xz vx | x{{y t u u ttu u t t yxwu ttwu{|}yxu w| yu t|t xz|w xut t t}}| yxyx wyz| ytu{t y|{yxtw|t}}t|} wv{|w}y xw}|~ tw~ v}wy||{ty ?  ywt| yx|u yyw}x} t xw}|~ xuyx utt yyxuy~xu|t uyy}x uyu~t } u|v} |y} |xuy{|uv xxuzy|{wv{}y}|t{z wxuttw}tts xzywxut}v}yx|zx?x|uvxtuzyx|tx{xvtxtwzyyu{{ytz}yttywwy u { tw|tx{yt tuttyu yxux yxuy{v ytz|x|xw}|~v|twu|vz |ut z|z vy} wyy |{zy{ v}x }tyut?wz{{ywy}~uyu|twt| xttw{u|zyz{}tyvuu|z|v}xxuvyx{uyv{yutttz{|}wyyy{xwx}yyw}}tts y v yy t v z tw |u yv u{ywt vxu|tx{yzuuw ttx}x|yu |wy~ v{ w|txxxt tut yxuxu xyy {yt w}t w tu |ux |u tzx| x{yt{ |z| u|}xt w wyt yu |uyywt{uuzvy}w}}|yww{yzwtxu|z}|tyx{xy}|y|{{yt{yzwyt{w|txxw|zwxtxv|}tyw{yxtu|yttxtx{yxuzx{|vt}wx}ttwuywvwxzxy|u}y}x{yy}|}|yz{wy| z} | |u uzyw ux t t u tv w { |u t ~ } z u u t tu t
10
-3


(

'????+ '???R ? ?++ +'?+'??????R? ? R???? ??+ Ü?? +? ?+? R? R???????''+?+????????+'????'??????????++'??+''???R?????? +????&??R???+??????????'?????'??????+???+'+ R' ? ž? ??%???+?% ??+ ǜ+??? ??+R' +??& ?? ??'? ?? ???+R? ??? ?????? ?? +'??+?+ž?????'+?R??????'?'R???????+?''??????R???????'+?'???ʜ+'+'??+'++???'R ??????????%?????+????%???????????+%'? ? ' +  +?+? ??++R?????????R? ? ??+??+???????++?R++??R?++!?+R++Ǟ?Rǜ??++????Ȟ' +???+??$??+???R????'?????R+??#??R??+?"+ ? ? ' '' ' yzx x w |z wy y w~ w | v w | w }y| y x w|u zw}yy x} wy { |y |z wttw u|z tttytwxuy|wwt}}|}|uytxvu}|tyt|wx}tyt}ux} utwyutwttzwt~ttwtvxtty}t~ wwx|tt||wuuyxtwtu|ty {| t t yz |}y}ywux |w}t tux~x tu|{ v}wyv{v xwz}y|{t tu{|wuyxy}z xuyx }v tx t u u xtzwwx{}}wwtytuxuxyw}?{ttuut{|twutyuxyyw}}uzxy yxuxywx}ty z|zu|www|ywx~w}}yt}w~yutxzwvx{ xtwu}u|y~{yt |t u t w}yyx |ywtz tttwu|z ttyx|w}| yxu|t} |u twy| xz|zxu ? t uuyw} yyyw{u}}}xwyy{|ywwywz|tyzy{xz?w|y|}wwyt|tztytyxtzw}uytuzyt}xyuv|}twytv}{xzvxztywtt{tyzuxutxwwz}~ytzy}vvxxuzu|t { ywt| | y | x |w tw|z u t? tu { u u tuu t} t tu u v|u~xw}|t~ttu v|tu{t |zxuyw v|z}y~x|y{| ?|z~ty v|z}y~xt |w||| ttz}y~xw| |ywy|wtwtvuxyt x yy|zyw xz}y~xt{ yzxx yxuy{||u ytxttyutuw{t}| w tz tttwu|z xty{tuyw yxwx| ?y}x| tzuutwzy{ zyt t uw|xz|wx}wt}}|{uttwtx|u tyxu{{|w|uwy}xyy}ztyyxu}vtw}}tx}{|w~|uyx|u |zyt ytwytu ?|?ytwuttttyuttwt~ tu {t|wu|xxz}|y~x|}}t|{yzyx{xt~vv |w yw~|{yyy}wt}|ttu tw~|| xy}|tzu}{yxww|}yy wwt|t {y { yt y { y x {| x t w u t t y t{||~t|xt{y|tywttz} tyuttwwxu|tz{|ttuyxwxzy|w}|tzy|xzuxyuttw}xt|yttz}ty~xtwzt|z}yt~txw~wxtutwvuts|?z? xuyt |x{zv} y {yw xtu zxu|y twu|zxw|tx{yttuyt}t} u x|t{} xux z z || w y xw ~t t txuw}{||~yyyw|}x|~z|zz|vytyxwtwvw}v|xyw||| |xyuxyyw}?} yxtuy|yu|x{ytu~v{}|xuytuytwy|tzy}y~uxy{xt} vtxx}|xyw{wz|xtuzyu|xt}w|uu|}ywxvu}~ytuttwzwt}~|y|uz|{wx|zxt|uwuy|u{t}x}~uy~}xtw~~wt|x|xt~{v|||uytzyyuyx{}y}w|xx{{{zx}x{~}uxw~}yyux|yx | x } ~ x ~t } y t v z uy~t ytwtzyu w|y wyt tw}tw} ytz| |ywtz tttwu|z ttyx|w}| t tuuyw~xu|tyu ttuy||u z|w wyxu|tw}wy} wy uttt tu{|xxux tw xw}tuwy| w yty }|yxtuuy}{y|xytuuy~|tu}tzutxuvt}t|}ttyu|z|yw{}ut|w}z}vy}~wxxx{|yuuyy{tyxyt txu{u}yyt~t}tt |{ v y u v{ x}x{x|w|x{y|yttywtwt} | tuuxw|y}tz{vttwt}tywuu||{zvtxzw}tz}wtt{xy}tvxz|{xttt?t|zwwxxxt~x{{||uxtwv}t { tu } w t | uyt t t txz|tzu~uttwt }wtyuxt{xwyy|xzux|u}xttzy|tzzzy|~wxx{tt| ytuu~xw|zuy{u|twuwyyz}ut ?xxzxuxywx}x|tx|wx wy { || t{|} ~|| wy}||y |ut}x tuyy} uy~t v}wyv{v w}tuw v|tw |w yyw}x} ty|xx wxu } w w } |v}t|wt} x}wu||~w|}{t}x~xtwtw}}wtyuw{t|tx}uyyu}ttu}zyxw|}|wxyuyywx}ttuy|u|||wyw|t~|txxz|wy w x t z tvut}w|}{t}t}t~ytw}xt}w xwuxvttww}}tt}}||utw}tuu||w|}zt}t{w}tyuyw|}uxw{|tttw wttttwut||uzxttuz~}xyw~}x|}~}t|{tzu t vxu|}xtt ttwu|z|wt |tyy ttuuvtw}t wyyw yx{|t tz}yw~xwxt{|u| |u ytuxu|tu} w x xyu t ~ tywxx~uy|{ywx|{wyt{t|yu}tvtx~xx{tt|uuxwztx|v}{tt|{|xtuwtut|wzwtu|wu|||w{xtzuytzu{wttyxwyxyu{|}wy~}u {xwtt}wwy~zttt}yx{vuytvy u vu {y | { | | z t v }x






?

?

? ?? ????? ???+???? ? ????? ?????? + Ü?'+ ??' ?? ǜ???? ??R ????????R??'?+?? ?ȞÃ?? {| {|x y }z vx y~y{x vx | } w v{ w y w yu yx | } y z}y~x x ttu u ut u u tu tt tt } y uy ttu tt t | ytxzzx|wwyx{ywwwwwyxyt|wtuzxwvxuy{ytzv tywty}twuy||~z{}w}twzxvt}w}tt xyt|wu yuvxwt}uy{{|yyx yuw{zty y} |u||wt}yyx~ux|{ytzt twytt vvxxuyyywyuwxuxw|wt y~txzxs{yuxt~zx{xy{}yy|x{u||uxtt{v|yy v}zw{xt}tt{} vu u {z y|{w|utxut}wz~xuxtzw|~z |xuu wutt|uwu|tuwutyw}y}ttwyyu{wx wx{t|yuytwttwzu|z z|uwxw| xy }t } t t x t {z t{ y|xx|zu}y~vx|}~}|{zvuuxy{zx yu~tttyy{}twyx}{ }x~{zy{tu txyz|wxxuztz|tuuyyxz}}u}|xwx}wv{yx x w} y ~ w t | w w t ywzy } xx | x x x y wxx| {ywx wy{ wyv{ y|z} x|w } yx}|{ x} ttt{yu wy| uytx tz|uw tw tsu tvu{t tytuy} tttwyt ytw}ut|w}t x{w}tuuy{tyy}|t t ww|yz~xw}|x}}||vut|zyu{xy?vxuyxu|tvtv}uw~yx|w}|t~}tyttuwvzuxyy~ty xtw}|yutwz|ytwt|tuutytx{u}yytu x { w y x x t } t |zxwyw}yxy{txwuxztwu}xtu}yu{xtu~{|utwx|wwwy}tyxzz}uytwwxz|tt{u|{xtzuxxxuwyz{y}t{y|y}}xx}|y?~vx}w|wywyvtt{wvwt~x y u t x x yyxz t t } } x ~x t w ~ s x u ytx t t} tu xwu| | | t t txz|}uy~|xzwxtv{||ut{w|~ytvt}w twx~wty|w|}u{tytwwyx}txwz|ztx{|wuzx}y~x}w|x{zt{|uxzzwyztv}twx} x x xxtyw x uyy w wt xz}y~xwytyw yuuyu}v?t x |wuyxy}z |u ytuuyyt w zw|}|y}utw}yx vtwyyy{vtww}}|tuuyyx} ||wtzuy|u|w{xu|xuyw|z}zz}t||w{~vvxtytwyywuxtywy}uyt|}vw|uyztxutxutvwt wx}t} v{ yxuy t} y x u w y x x | ywz|| xz}y~x}}|{z |u tz{ ?}} vvuy|?wtxttuux tw}tu xwu}tuwuyuyttzy xtxu|tty{}}txx | twtz |xt{|t ttu~xtw}|u~ttu twyut{x ttuutxx|w w}txuyt|}t v|xwuw|w}{yutw |{yyww|{t|zu ttxt|wwuz||zywxx|~x{y|xw}yy|t|uvxw{}|twttxt{ytutu|xw}|tzx|}|uz}wxytytwzytz}ty tu xzx|ux xxu|tx{yt x txw}tyu tw~ v{yw|{|u w|zy{x xw}txttyu |x x } w tvt}xt vxu|}xt ttyz{|w wzyy tuuyw}yx } xx ? w}tvu|wyuyytzy w|~|utttwtywu x }w~|{{ z zxwxyuvytwzx}t xtuxzt xxxtutvuw}tty}xuytw{{v|}uytw|~ztv}|wyy|w}|x{}tut|}xxx{x|xztuttu|{tz t t ttuvw|xu|w} |t | x xx|wx} tz|w yxz~|tvttu uuy~|utt xuyzt uuyt{xt}| ttwtzz~tuyywtvuuyx{yvxxtu|xt|u|tuutytwtuuutw}t | x xw}tuy||w}xtwywxxuvttw}}xtt}x~y{y t t yw|w{y w x xt} yt}yy~{t {} tw yxuyzt x|x{ xt} t} wyw}y ytxuxv|tw}t}ytyuu|zw|w|yxuwyyztxy|xwx{x}ywyytwy? yz|u |tw|yt ty|w| y} x t ttx{yttttwzt|zz xyxxuuvytuwy}}x v}wxt|u wy yxuyzt yy||xw{|wutz|tyw}vxx}u?v|twy}wt}}x}xvv}uwuyy|~x |{ yt }| |wyzx|wyutzz{}tyytw|~x{wx|t{ty tu| tw xxxxuvvtww}}t}} tuwyuwy}xy}x{xxutxwtx}t{x | ttu{u|~|uuty xt{}? vxuyzt ty|x{ yy yxuvtw}t} u t t yw}x} ty|~xyxuvtw} t w} wy|~|utt wxz ty w}s z|w wx|w}y tut yxx{|wyt ttwu|z tu |tu|w}u{|ww}utyuwxuyyy}ztzyx~|u~yxuw||y}tuyuwyw}wytzwtv}wtyw|u||{ztxyw}tu|wu}y|tzywxt|w}ttuuyzt?w w u y x w}xz}tytxwuvvx{uvzt|wztv}yw~wtz|wtvx|w|twttzyuxwz}tuywuvytytzyxtyy{w}tx} }u|twvu|}zwu|yuty}yy x }} y | s
|
env

|

??
ini

?



|

f in

= (|0



sy s

|0

env

|00 . . . 0 + |11 . . . 1

+ |1

1/||2

sy s

|1

| sy |0 env

env

= (|0

s

= |0

sy s

sy s

+ |1

+ |1

sy s

)

sy s

|0

|0

sy s

env

|

sy s

|1

sy s


6

?R??? Ǟ????? Ǟ??? ??R?? + ?R?? ? ? ???'???????+?????+?????'???R??R???+?????+???R???+??????? ???+???R ?R?R? '???'????+ ? ??? ? ? ? ? Ǟ??+? ?+?????? Ǟ?????? + ?+??+++ ?'???R???+'R'+ǜ???+?ʞ??'?+???????R?+???'+???Ü??+???+R?œ???''???+??+ƞ+'+????'??+?ǜ?????R?+R? ???'????+ ?'???? ???' ? ??Ǟ?????+?R???+??++'???????+??+??+??R??????????+?5? ??????????'??&??''???'???++R??+?R??? ??R+œ4+??''++???ǜ??+??????????+??????????+???Ǟ'+???++?? ? ? ' ' ? R ? ? ? ? ? ' ' ? ' ??'͜?????'??? ???? ??+??+'??+? ????? ? +?++ ??????ʜ+'?+' ????? ?++?? ???? ? + ??R?? + R?? '??? ?'??+? ? R??? ? ? +?? ?? ? ?R ? +?++ Ǟ?+' ??+?????+??R?????+???+??+? '?????+?R++?'???+4+ȿ??R?????R?'? ?????????????? ?ƞÜ+?'͞R?+?????+??++R?????R++'??+!3 ' yy |yu|yz ts tvxuyyx xw}t x{}y v}ywxuyx vx|wx} xx{t xuyy wt {t x}ux{|y~u|uvxxuvytwy}} xtytu{~|yut|z wvxxuyuyxwx}|~{x|uy{t{twx}z}zy~txz}}{|t{wz xxww}}y|{xt wyy}tt xutuy}{ t xu u u t t w | yxu x tytywu|}|z xszxuzx|ytut{ttwu|tzt}xyvvx{wv|uwt}yv}wtxw}{|t~ytxu|vutw}ttw}zx?t|}vxxut|}yttvz{}y~xxwu}||{y t tw z{tz}t } xuy vxuvtw}}| x wvuyx v{yu xzx|ux tz}y~x}}|}{z u tz|? tu{|w| v}wy|z y{u w}tuw |w tz|utuxtwzyx}t{| y|~{} yv yyywtu| xuu~ytutzt |z}twwvytyxxuyuyxxty}wy|{uy{t|w~~|uxvuxyuyyuyzxwy{y|?yw}uuy~xxzu}|y~xt|yuxyy~ xvuxyu tw}}|t t |x~uyyw}|yuvwuyv{{|xvuyyy}xtytwz}|xw} xzwywy}t|y~u||ut|y{?y{}ty{|yt}uyyuuwx}{tu~|||uzvv}}wwxvuty} vtu | ~ | uu t xt z t yzxx tz}y~x}}|{z ytt tz}y~txwxt{| x~ts wxt{| wvu{tz| x{|wv} xyw|{t|{|ywvx}uyy~xxw|yxwx|twztyu|utwxwx}y|twwvxwy}y{}wyuuuy|?xzt}uyxz|tyutxut|z{|xtw tx{~tyw{}x w tw {t yx utz| w|tx{t} yt x x{}y vtt y~|u? vtxuyy ttz}y~xwx{ t |u| wy|x yu y|~{} yt {yw xtu yyw}x} xzxt|uxtxxuyu|yw}x}zw~tw}{ywvtw|} } tuuutwx}yxxtwtuxt|ywyutz{y{wtxw{}txu~|vx} xuw|xtt{{ }xuz|x ytxuzu}yy~xxuxyx|x|v}wxvuuyyux vxuyy v{ tuy~|u|uyyttwtz tw}y} ytu{yw|tt} z|z tz{tw uyu wtxut{| t x yy ttuz~|tww|zw|}ztxzyxuxvwvx{|vttyttyut{w|2yv{y{|x~|tx{x|uy|z?x|wx|||twxtzt}y{~|xw|uw|uytttw|yu~||w} rrr ??orr? o oqn?q?? ? w~ ~w w z|xx z}y~xwxt{| wuytt xw}tutut~tt wttv}y|txz{|wztxuuyt~utx|utyutwywt|uvzwt}ytu t u z yx|w}| wxtwy}t tw yy wuytt w}ttuxtt v|uuyw}ux? |yw|tz tttwuy|z t tyyzxxx|w}|wvyxwtyx wux|w|uyw{| wyywx~x{wz|zy w|{|xxwx|u}x}| t|uwtyty|yz | w tzt{tw |txt{|u|xz|zxtutwy|uytttyu ytxu|t|tuttw tzxxt tutz}y~xxt{|utt |y|w}xt w||u yy} {tt ? ?yx{|t tuw|u|w} z|z {| tu~yutz}y x{|u| yu | x|xt{| ttz}y~x}}|{z yxwvxut wxy~{ ytu}t t y tw~ yw ttuw|u|w} wt v}wy|~x{wt tu{|z txyxztxxxttt{w|u|zuyxxwuz|y}tytwzx}tywx}twxtu{txtw}|v~}wx|t|uzt wxytyw}txw} vu{y|~x{}y{ut|wvyxu|| {v w x~} uu{tzyt{tx|ux?xwv}| ||ztw wtyttwyxzuxuvxx{u|x}xtyuz}xy{~x|wxyt}{t|w~ty|xtwx|{{? y zt w |x x y t~ xz } |
10
-8

1$0')%('& %$#"!

?


?

?

?

?

?

R + ¿՜? ?Ҟ+ ????'Ü???+?'????????+?R??? '+??'+??????R?????????+''??????+?????????????!!?¿??'????R?+??'????++??ƿ??'??''+ ? ? + ' 4 ' { t | t ttz}yv~xwyytx |?xxyzwxxtuytxtwtuu|y~z{vtxu|wtu}twyttt|y}x}u|}}x|t|yxtzz|yw~twt~xztxwux{tvxu|}txt x {| u u ttuw}t{y ttz|w w}tu{yw|{y tyx uuyw}}z}x tx{ xz|z vxuyuyx y tttwu|z yx xzxz|uxtyxu|}xt ytz|}wy~x}}|{z twx~{|t wyv{tt t{x| tw ttw yy{t? w}tuwuyvytz w x{x||vw{x|{wt||zuz|zxxuw}tzu|wzvtyyt~tw{uz|zxvzxxuy|{u}xx~t|t{wxu||zytxut|z}}xttuyttuxyxyuyxu}?t ? xy yxyu| y wy| tw}t uy~t vxu|y t{x| ytw}t tw tw~ xx yx? |ywtz tttwu|z ttuu|tx|w}| yxu|t} uttyu vu|y{uyt}xzw}tvuwwy}v{v|~|yx|y~|twxu{|}yzy|~u~{x}w}|t~wttu v}wv{vt tu{t tw {| tu{|x } |~ y t y tz|u xwt} xzty {tw} vxu|xv u u u u xyu~y|uyytxyw{}tuyutyttu|{|xxutxu tttuyuyyxyuyyyxtutxu~uxy{yw}ytw}}txuyuyt|ywt~{|ytuy{utwv}}|wuyv{xu u t yz t yu x{ ~ y x y w |{|wx}z||t}wxx|{xuxyu~wt}||~tttuz|ww|t?x{ytwtzyt}tu|xzzxuyy}w ttuwy{~ywx{}uvt}yt yx yw} t x v }xx v|x t? xw}tu{wyxw}y wyw}y} yttwtz x~ y y vxwuy~|u t w { tuy t t t tyv|xuy{y|utty |wt|w{yy}ttutwt~xtt|wtv{tz|zx|wyutzt~y{u}ttzt}|tuwtzw}vtxuuw|vtxt vv|ytwu|tuzyx}wt|} |{u ux y t t xw uu|twtyu w|wx~} yt tw~ wy|~|ut tw { }y~ tw~ yxz|w twx{| y} {u |}}ytt|wx{~yw}x{y}x~t|tuw}txwvy||tytwuywyttu{uxxt x{|twwxx{t{|| {|tt{}tx~tt{x|{ } u tyzx{} t} vxuvtw}t} w| twtz xzu tu{t |{z wtw y|wx~ yu tw}t ? y{} |wxtw}t} |x{|t |twyw{xx vxu|y ttz|w t}t} uuyw}yw}y | |x{|t|twy{x ttu~xxwx}|~ttu wt vuu|}xttt~z}yx~tx}t}|{z vx{u|yut{zw|t |y z|z tw}t} |vxxuxw|x{y{{t}tyuutwt ytux}tx||ytu {yz|tw?yuztx}t}y|ut utw{yw{}wx~tuxz}xy~yx}}}{zw}wtxuww|uyyyutz|yu y { |tyu~xw}|~yttwu tu|wxu vxu|}xt v{ z{tz}t vxuvtw}t} tyzyw vxuyyx yx|z}vxwx{t} xtzu}~|xxz|vxtuu|x{tt ttutu{uy|w|yvt ywx}ut|}xxttyttyzu|wywzxyxvtx{ut|wtyx{{yxt} tx}uyu} t y tu{ywx{}x~ xz}wy~ux}}|{z }w}tuuy~xu| w}tuwuyytzy z|z vxuy{v ttu{|xxux tvtz|w ztwz|w tu yz|w | xvxuyt{x } xuytuwttx| yy x ~|| tzxx yy x xtw}} yxu|xut |ytttu ytt{wtzyuu wy| ut z|z z|w ttuutxxt|ww wt v}uwuyy|y~}x{~wt tu{|zx|~tt ytzyz~xtyw~xxut{w uw~ytxy}t yyuxtytutwttwz wttz|~u|||wxyz~}}y|~xxtxyx|x{{x vxux|tw}wu}y|ytx}u ut t |tx|t t tw}x{|xy} zxx |u v{ xz}y~xwz| tw}t w}y tt xz}y~xw|x w }t{|uytw~ v}w||zt wyt twts uutz| yu{t wyt zxx} |utv{xuuuttxx{| tvzxx{xyxt}{yttz}y~wxuwyz||tytzu|ywtxtyxytxtzttwu|ywztuutuy~txu|t|yu xw{}tyzyy| } xu| w y t w~ w||z} tut tttw yt xxz|w}| yt}twtwy~ v}wv{|w}y wxtu x twuyxy}z| }|uuy~vyxty|wtwxtzyyz{x|xwu{tuttxyx}y||wztvu}wyxyy{|twzt|wwt}xz y|tzy|zwwxtxz|uvxt t|yztxtxw|ztzt}yt~xxyxwx|xtw|w{ u t }t }tutut t tu| yt t t x tzu ut ttuut wt |yx~x{wt ytu? twxt{| uxwzy tt } zt{}t ut}x w}t x uuyw}yw}y ty|vuttw}|tzwtyztywuxxtwu}ttz||uwuvxu}ttxwyvwt|y~u{zt|zxuu|twzt|tyt}z}z~zzw|w yzxx xuyy tu u t{ } y y |x }xt{| yxwvuxs y twtz |tt ttz}y~xwxt{| vxuyuyx v{ vxw}wvyzyxxtwtzw}yyu~xyz}x|u{|zywtwt}wwu|}xz t


T

1/

2



T

|| <

1 T

T

T




v}wxt yw| twtz x x x xzu uutwzy{ u~xw}|~tut t z vt} wwyxt} t ttw| yv xxuyt{t tuu|| x {zy{t xx|xutz tuutwzy{ t vtx ~y x{}y yx|? |w}ytut tt{| ty tttwtzyu t wy|~|ut tt uy~| wx~txu|twt~ |yw}tx}twtux~zx|wz}|~ut|tvut xuvtyww}}xtx}} ||tww}uu||zw}vtxu|}|xt}twvy{{txwxxww}}ttuuyt}|tuw}t {{ t } t xxw}ttuux{{t|wx}|wuwtxztt{u|xtxzt}y~xy}}|x{zwyv|xzyyyw|}txt~xwu}||~ttv|tu{|x~tw||uzwt}}yx~xx|wx}||{x|x{t|}uyttwt~w}vt|w|{t{y zy t { | t u t x {xtyu |w}tw|} tu{yzw|x{}txu~ttxzy}uy~txw}~}||{}zu|tuz vtxtw}twyy}tuyu{|y|ttyw{u|xxztvwxzuy||}xwtttv}twuyx|xv{|ttux t x}yu wzy?t twtz|z tw~ w|wx~} x x{ u{t zx{yt w}|~ v|z}{yw|t{t tz |z{xtww}x|w{|} txuxuyuxt}t|yuyw|zuywy{}?v|}wxtuyyx|~t{}twyywwy} xxtuytu|xu}ytyvx}wuuyw}y|tu} w yv y z xx t {t yu |utwt tttw xuyyx tw|w{y |zxw}xw|w} |t tt}t} u|~{} y~x w|{y} tut |y} tw |twzywy |t w|y{|ux ttwytz z xw}|~ ytz}{yw|t{t x{t y xtxuty}x vx~wx{|u {|{ux}x} |wuwyy}xyw~}z v}z||w |}|t{wxzyywxtttwwzytwv}xw|t wvyx||yw}yxu yt wy t tw~u u xt{ttyttu{ywxtw}ty xt{yttw}|~ tz}{yuw|t{ttttz{tw }xtwttuttv{yuu| s xu|txwzywy x xuytt t |wuyxy}z vxu|tx{yt v{ tw | wt xxuv tw}}| t{t |u v}wvt|u tutwt vxuyyx v{ |twzywy | x{}? x| s ?tutwt xuvtwt}t} uu|wt| vxuy|tx{wyt yyxw |uwxu}v?t tutttxw}|~ tuw|w}xux| wy}{t {y u xtttwtuzy|u}zt}yty}yty}xts {xy{wtt}y xwv}{||y~tz{}|y{tt|xxtu{{|tzt{zxtwtx}wyt|uwx|u||u|wvtxwut}xytxwut{w|v}yw||vwx}xz|v w y w y t t w|xt tu|{y |t { z u| y z y t z|z z|w xw}|~ tz}y{yw|t{t z wx}tuwt yz|w tu|u |}utz tttwy} xwu |y{ xw} |t~z}t{y|{txtwzz|x}xwu}w|t{t|utwxuyzyw}|tw}uw}ttt}xww}}|y{vt{ywt|yt|x{tw|}}z{tzz}yt~twxw}y|}~ w {t w t t | | w|uxttz yx|? |{x ||zt ttu{x| v{ ywtz yx{yt |xwt ztyu v|twtz vx|ttux v|~t v} tuxw|w}xwux{| z | xuyx xux {wttyw|txt{tz tuytywt}x}} |txyuxx{ywt}txxxu|yxtu}x tuw|}|v~}|wytwxttyz|uxuxwx|w}w|}~xtxz}|{yxw|tz}{wt{tt u ? xw}|~ y |u x{yt yxuy{y| v}wyv{v |tt ttz}y~xwxt{| w}tuuyt}t yy{ yzxu|y w |z w | |w} | t t u w tu y ut twwyu|wz}wvx}uw|t}xvttwxtzv{yww}|tuu{y|uyw|yvu|wwxt{w|tvx~ttz}xyw~txwxtu{|v wyx|z|xzz xxuxzyxy|uxu |ut } y tz|ut |tt ttu~tt vxuyzy xz~tw } |y{yx yu tuuyyt} xw}tuwuyytzy |ztwz|wv|u z wx||{x{xyyyw}zx|{zv||xuvxv{tyww}|v}xt{yw|}uvxwwv}xuyw|}|wxztv{vx{yyytz|zxy}uwv}wxu|w|{ywwx|y{}}x}~|w|uz{}tyz~wxy}w}}|t{uzwxuxwyz}wy|twz|yy~ w x } |w } y x ~ x x u u uvt t} tt t u u u }t t u t tw y tu } tw xzxx x|utz| xu{|wuy|u } yu||u w|xux v}wyx ytt tuu| x {| tx z}y~xwxt{| tw}x~ yxyx vxuy~xu| xz|{wt x wuyw}ux xtwxtz xuuyy{ty}tw||{yw}tx}z}ty~xxwxx{z|u w}uyxuzu||z yxu{tt}yx}yxv}tw{y|xx{xw|t}}w|xtxz{x|x ytvutywux| t x |xuy | t |w}uyyt}yu w||w{y z|zw v}wy|xw|}}| tw}tu~twyu tuwtz x yxu|}xt ytuxuxt yytwytyzzx|ztvw}uwyxux}z}t}yx}|{t}yuy{tttxywx}u|{|ywz~}y|xyxt{wyxyzu|zywtu~ttv}vwxww}xz|yw} xy |y { vtxwvyuxttxtut}zxy{||ywt}yt~}|z| w|y~wyx{t|{||t|tz|}yt~xt}z}}|y{~zx|tuy{|wzxyttzy}xy~vux{|ut|w~uy|xxxz?x w t x wxwx xzxx
H H0 H0


?





'? ? ??'???? ' +??+???? ???'????R?' ? ??+ '??'??? '??? 'Ǟ?+??? ??????????&' ? xw x t|z xuvtw}t} wzy{ vxuyt|u twy yu~x{| vvuyx xw}|~ y{| uut |vu}wvw}xw|t{u}~txwzww}y uxzt}wy|~txtxu{y{yy|t?|zw|xwtx{w|vyxtz}}x~txwyutytx|xu x t x u y y y w z | w t ttu tyx yu{ywxt{}tx ~ |u tutu}t v}w||{t wyt |y|u yyu{| yxu||w ywzuwt{vywx{}ux|ywvyyxw|~}}yywxw}}tuy{tyt{wxywy}{yty||w|{x||y|t~txz{}xy~wt}xy|txy{|xz|ttywttzz}xytz}xw~x{| ~ t x w t x ~ y x x t tut t ~ tw } t tu wt y t y t u t tu zy x w ~ yxtuwyu~yywx|uy}zyxu|u|t}t}uy|ty{y}x~y|{{|u x yu|w}|yzx wtwt}y |u}wvwx{utxz|z wtw t tu u ut |t ytu t u t { |t t v}wv} t|uywt v|~{}xw}x|z~|wvxvtwy}t}} y|uut|}wu| tx}yy{|zyxu wwyzyxtywuyt{wywu||tzyy}yx|{uyzt|w xtu|{tu{}|x|xywx tztywy}}|uvxxy|ttuyw}tt}u{}vtz wv|y~{}xyyyuxz|w ty| yy wt wx} x uyu u tt | x| yw t{{|w}xz wu{x}|x v{ yx|u v}wyvuy yu w}y}t yxuyyt yy yyw}u} t yx{yt t u tw||{ywttt|twtyz~}||twv}twz{twvxzu}~yxw}|{wyytttzxt|tywxyxuwu}t{yw|x{t}ux~~yutzxyuwx}y| v{ y z w w w x |xxty}xtwwxyywx}t|uu|yt|x{ytt| ytwtwzv}yx~wx}wwxwyt}|uxuyyxt{{}tx~yytzu}wtuu|}txyx|w}x{t}?ttzt}vwytv{|'wt}uyy t t t { y yuy|uwy | twy~ y yy| xx | x{yt xv{{|y xw w|{ty txxyttttvy}uytw~txtttx}uxxtuyyw}tuutww}tuz|wy}wt}x}xw}|y~x{|}?uxxttzvwv}ttu(ttwuyu|w{}yt}x{}wx|~ { | }y yxx|u y | tw~ xx ? tu| tuuyx{x uy~t yxwyxxwu|} tz}y~xz vzxut|z|x{y{{t|uw|t{}|xwy~}t|tw|tt{u}xt~u}vty|zyvttw yw|yu} y~uy{yy{wtx{}yxu~w}|vut|wty}xtxtu{y}{x}~x~tw |yu}}ytyxut{tyw {}x~ t{}yxz|wxutyutxtyyuyttyu{twxxtxwu}ytvxwt}t}u|wyt|wu |uvx}~xst x w~ ~t ~ } u } w yx v |w}u|w}t t|tytwwy{txzttttut{wtu|yzxuyuywttu}y}wytx{tywtyyuzuy}|wu}uyxyyx wtyuyuyx{yxyutz~ | twz | u yy{u||~||ux{yy|ut ywut xx{x?|tyuuty|t| |y|}tyzu wyy{}y}tuw~ywtt yyutz|txy{|y~t{ttwx~ vvx}wty{|}~{yts y }} xu y |w}ytut x{t w| t|t u{|xwt tw~ ttw v{ yw}x} tyx{ vtutz|t}tvxwuyyt {yt utx| twtzyu wyx uyu xw}tu{ywxw}y xy t zxu{t xv{yyt x|uy} twtzyu x|{x vvxuyzt vxuyywt {yt xuuytx|twut|wtzyx}uw|ytxyxt{y}xttztyyutty}}tt{ yx|x|w|xu}|}w}yxuu||}uxttx v{w}utu||t|xvz}|wy wx {v t | ~ yyvtu{wyy{|y|~u|tutxyttxuyty|{y}ztwt}uu}wy}xytxtwyutzxwtx}~yyytxuww|u}yw|}txuxv}||wv?y|vw|xxuvty{|ztyttvzx}yvtx{w}}xxt} { | |u | x| w} u t ~ y x| y u~ wyt wx}x| |w}ytuty t tw|~ wy|~|ut tw yw}x} ttwu|tz xzx|utx txxu|tx{ wxtttwxt{tw{x|yutxtwyy}xx||x{x}w?vt}w|y|wxxxwt|xt}}{}|}ww}y|yw~zx|w|xwt}wt|tw~t}z|wx}z||ww}zt}ywtxuzz{ttw{t|}wxwyx?tu|txzw}wy~y z x y t u u t u y t t } t | t tu t t | yv yxuyt{t wy yxuyzt xw xx|xutz tuutwzy{ y|~{} xw}|~ x vxu ytt{ut wt w}y tw vxuytzwt yy wt x tutw|xwwyy wt tz{tw yyu wx}x| xw}|~ tu}t v{x}zxut zy{{ttwxtxzu}wxxz|ut|zu~x~|ww~xt~u|utyxy{|}x|t~ttw}y|tzw}|yuxtwuyyxxtytw{}zyt{wtwty{wywvw{}|{xvy{yyxy} { t ty t twu tw} t| uwu ttu t u t u ~ | u u w w
H0 H(t
env )env

t

env

10

24

t

t

10

9

H(t)0

H(t)0


?+? R ? + ?+R ?????? ?????+?? +???R ? ???? ??????+?? Ǟ?R '????? ? ???? ???????????+?R+??? ??+?+?Þ?++????R????+?????R ǿ?????+???+????'?'??? ž?????????++??'???????? ǜ????' Ǟ'??+?'??"?'?+ ? ? xw}t } xuyuv|{w|ww|}~ }{}w|x~}xuyxwxvw{{u} ttwz}zy~xxvwx|y~v|w}|vxtuwyvtx{|yyxuuvy{w }utw|uxu~wyxt{ww}wyw|t}|tx | }? x } ~ } |ut } w y } w y t x w tt w t tt t u t t ttuu t t t tt x ֿ ? |xw s xw}|~ yw}x} yxuvtw} tt}uytxww}|x~w}xwt|}}|{ytxw}|~ x yw}x} xuvtw}t} uu|w| vx|w}uty tw xw}|~ xux |vxuyy w}txttyu x|w|uzt |~{} tw}t | z |x{|y ttu{|zt{ w|}tu{t{y?tywu}uyx|yu|x{y||u|w}|uz |?yx?y|uxy{xxyz|xux}txt|wwuu|y x |x{}y}ztxtz|uwytxwvt}zwywtu}tt xw}|~ tuxw|w}xux| w}txtty? v{yu w|}xx } w} xuyy{ wxyt} x{yt w}|~ v|uxw|w}{xux x|w}|~ u|wuy yyy{{y}wx}} tx}yy|~~xxxvxuyytx utxwwzy|?ztyxzu}|y}~xxt|uwx|wy|t|}|u|t{tw}{|y~vz}z{y{wt|z}tts{t } z y u t |w}t | yxuy~xux|| z|z w|xw|}}| tut yttwtz uxw|w}xux| x z}{yw x|xuty~{tx ?xtwu}|t~yt y|yu{t{yxt yxuy{y|{ttw x yy|zyw xwt}}| y }y? x{yt y{y| yy{} v}w|}tuxw}}tuuyt}tw tu{ywxx tw v|xw}y{y}{} ytwzzy? {}t| u|wuy|y|t|wxwxt{|{|xw |u|xw|ywv|xu{yu{ytw|ttu|{yw} yt tt} t t t t t } u t uyut u t ttuu y t u t w|tu|twz}|zwxu|wtyuyt|~t{t}zyy~xtwxtt{|wxx~x{{|wtyttux{|xxxwuyx|zyw|ttttuuytttz{}yx~xxwxt{u|ux| tz|tu
?

{ | x w z twtz t}}yt vxu|tx{yt ttuywtztxzwutuywwy}|yxu}xwy twwzzyyv{y}tywuyw|ux w~x|w}yt wywt yxuy}t ytz| |tt ttw y}tytv{z tu{ywx}tuwty xzxx tz} yxw|wx{|xvxxx|{ytyw}txuz|t{wtttw}twytz}ytuzyx{utyw| v}twuvy}wyw}uvxxw|uw}tuy}vytuw}w|yux{zyxx t u|t u xtxt|z} ||w||||tyxz|yuyxxwy|~|uytxyu|v?}wyywuy|z|twwyuy?xtz}{txwz|xyxttywtwut|tzwuyxzttwu}u x |uy tt zw yt z u u }yy}t{t}ty~xxwy|ytztt|xw}yzt}|uuytwy?|xxxy|{zt|wwytttyx|u|wxtztwuy|u|t}t|wx|uyzyt{|u|wwt}zywytzw|}y|xw}ttwwuy{uytx||wx}{wyyxtwyxuxwx|wzy{| |t x}|x t y t u wx y y yt yy { y x{yt xz|{wt }}yts vxuy{|w}yy xwt|u ttuuytw}t |{x y{ywy|tx{t|wt} |y { ttyxuyy{{}t|w}v}yw|yttuw}tuxy|yytt}twwytzyu wwy|xyttuwtuzuxw{}yx}ttvxyuytx{wy}ttvxxuyy|~t{} tyw{z w} | w ~{uytxyu v}wyv{|w}yu t xx tx vtx{ux|sx{}ytwxx{w||wu|zvxxtzwzyw|}uwtzxz}yx~{xxw|xuttwt{ vtx}uxy~xxz|wwzw}|t~yxyvsxuy{y|xuy| | t vxuy{yyt t}tw} x}{x tutwzy{ wxzu |tu{t uuxw}x yy|x{x xzxzu wtu xwt}|~ xtw ?wtw|xwx}|x}x}|| tuvtwytyuu|yw}xy~zw y|w|x vv}wxvuxxu wttwtwwtxztwtyt}yx~} t tt t | {zx y t y{}t twy yx~t y} x{x|uxtz x xux|x wxuyy| x|twyy yuu|x ytxz|w wt|txuxu | ttzwyytxwywx|w?xxxtw|w{uxwy|zwu|w}ztyuty{xtuyxuwt|twttzwytuz xxyuwy~y{w}}xx~ tys{yt}|y~ | x yt w t

?  $?%?( $') 1'0?$'$0$(' $ 1'??%'??&????

| |

2

= |01 0

j

2

+ |11 12 ,



j = 1, 2

|01 0

2

|11 1

2

|0

j

||2 |1j




' ????& ? +?? ?'???+??+???? ??????????? ?+?????R?' ? Ǟ?'?? R'#' tx{yt v{ |w}t | yxuy~xu|| z|z xw}|~ y |u x{yt yxuy{y| x{|txw{zy|twx?w}|~ xtuxw|w}xux| z |w}t | v{yw|t{t vxw}w}wt x wy|zyw ytt tz}y~xwxt{| t xx|tux x~|yy w}ttz} |u yxuy~xu|t ytz}w}xxwv{y t|t xz| w|tx{yt ut ytwtz xw}|~ vxuyyyy w}ttz} ||u yxuy~xu|t v}wyyx xw}|~ tz}{yw|t{t w}y t vy ytuuw|tx}zx v}wxw|w {yw tzw|{z |u x xtw |u xzwyy tuuyw}u|w}t y {xy{}ytyy|~t}}x}t|~ ttzw|tww}y|x~uytttwtz| tww|}x yv{}}wtyytuu vw~ tw t{y |ywtzty tv t ~yt{z|twyt|z w}ttz} tuuy~uxu|~ wy|x|y|t? |xx||zt|uxtxwtxz{twu|tut|txw}|~ x~t|z}yyyx}}v|{ v}yz}xtuuyyxwu}|yt}xt{yytxwt}|w~ w { ut t tu tuxw|w}xux| z v}w||t wt yu x{yw|t{ts x{yt y{yw|t{t v{ uyw}tyu xx x|{ytux xw wtww xw}|~ u~x{| w}tuu|w| wyv{yyt v|twtz vx|tux v}wy|yy vtxw}|~ yt wy ztuuy?| ||z~tw uuy{| tuuyw}u|w}|t z v}wwv}tuwt xut xu{|z xt{|}wwwy|v{|v xuw}|v|}ywyyxwz}txuw~twt~tw~xtw v}wwyxxtt|uuyutz {xtuyyyw}wxu}ywtt yxu{uy|t u y|x{u}yvt|}}tt}y{yz}yw~|xu~xww|w}tyuyvxxt|yuxw}xt}{u|yw~}|~v{wv|{yww|}|xtu{x}{xt|uwu?xw}tt}xy|u~txwx}t|yw~}vxy|{uytxtx||xww}}x|tu~ux{t|z } u w u x x t | y xw u {w t z tt t{{ttt tutt { t }u u t t {{|}u|tz{zwwv|}w|xyttut x{}y tuttyyu vxuvtw}t} ytwu|z yuu|w| x xxzu ttu uv{t ttu{|xuxwt wt yx~x{uwt tw~ y|~w{z| vxtwtus t w wytxyx|~xx{ww}t|~twxxxu}u}||w||uyux xt}uy}|wuttvw}tw|~|x{twztt{utw{y|uxx|uyxxvyxu}ztw}tt}uyyu|wu | u y| xx z| vxuyx{x tty{u tw}y |u tx} t| } y? vu|t|ty xuyux{x yty yt x{}y uy v}wyu|w}t tw tw~ wxyt tut? vx|uxtz vuxvttw}tt}yt} t xw}|~ xtt t}{x vxuy{yy}| yuwvtytu|xw}| y|}{x yxuyywx tu{|~|u y|~{} tu{yw|ty{ vxxw}uxvtxwz}}yt}}tyw}|yxz|~~z|{y}ttu uyyx}xw}zx|||v ywxy}xt|uyw~v|t|yyty}zt|}txwt?z |tyx}uwv}t|tw}tu}y|tuyu|}wy}|vux}|| w u ww x u t y u t y w { w}w{x yxuyyx uy t y w t t t u u t t |u t tu y|twyytyttwtwvtxzuyuyyxxyy{x}tyt~yw{x}yttw~ zt|uwyww}yy?wuxv}wzyu|wt}ty|xzuvztv}}t}yvtuwyxz vytxut|wttz
1 2 1 2 1 2 1 - 2

'

yu x xtywx ||| x|t x t v|}yy?{ytxu}yxxy{?y|ty|ytywxu|yu|xwtuzxtuyt{wu||w}zyv{u}y{z||xz|vxtuyy~|yuvt{ yt xu t u t t t t |uwyz}x x|vu uu| txxzu xtuu{t xuy{|w}y tu}{zx z x}wyyy t|u tvwu}t{|yxw|vv{|wzty}z{y w|zxxuwxv|wt}wx|}~w}|x }t}w|t|uzzxvttwu}t{}yw|twtvyy}{w}xytx|tuwxtvwz|}xtwxty{yyxxwy||~y|ux}wtwvx { x u t }t |tztz t t utt ttuu uy tuu x}t u vux x x|zyw}xt vxuy~|u uuyyx }t| u{t wx~{t u{t ttw t tw yxytz|uxt v}}w|zt |y} }{x y|~{} ty x{}? x y xu| x|ut ww xwz|~ xyt }{x xyx x{}y t? tuttyu wx~x{| vxuvtw}t} | xwz xuywyzwx|wx{|yw w}y tw xxuvtw}t} }|{xt|u yw}x} ||u x{}y x{x~{t t yw{ wtw xw}} ~ w x~{ w { u } w | } txz{ttttwtztxzy{twxyuv}x}v?{v xxtyu}ttzw|tuxtyxuxvwtt}w|}~t}wyt|~xwuxtux|xvttwwttwt}~uuytxxw}w~}xt{wx}tttx xwv|txuuuxvtwwt}|ttu}
|0 |1

1 2

(|00 + |01 + |10 + |11 )

1 2

(|00 - |01 - |10 + |11 )

/2

,

1/2














?

?

?

Ü????'+? ? ??' ???R Ü?' Ǟ??R ?'??+??????????R??R????+?????R??R???+?+?R'??????? ????+?+???'????+????+???? ??'??????????? ?????'?+?????+R?? ?+R+???R Ü??? ??+?R? ? ?'? ? '??'?????R?'? ??????? ????'+? ???ʞ+???' +????????????? ??R R? +???' ?+?œ˞+??+?????+??????+??++??+?????'?????+?̜?????+ ??????+??! ?????'??+????+??+????+???+?+????????+????? ? ????? ?? Ǟ+??? Ɯ?+? ?? + ?+Կ ?#'+ǜ???+?????'??++??ž+?'?՜? ???'????++????+?R??+?+??+??????????? '+'?+? ? + ͞?? ? ?+????? ?+??????+??'??+?????'????+' ????? ?Þ????+??'+?R?? '???' '????'R?&' |x vxuvtw}t} wy| |x{|t |twy {xxttu~xw}|~ttu vxuy xzv~tw }x xx{ut|{ ttwu|z yxuywtw}}| u w{xx|||wyu|tz|uxxyu|ytzwxx{t|}ywu||uzt|? |u|xt|u}ty|xuutyut}wv~?tyxt|  u u u tuu| wt v}wy|~x{wtytu{|xxx v|x|}wyxywx yxu v|uuyyxs ?u? u?  yz|w t} yxy|tu{t}y}x{?twyt?wyx}|w}tuyxwtvxuytxwz~ytw{xtz}}yy~xtxt}twuu|yw}|yww}}|yuytu{|ttuy}u u |v}wyv{v tw~ tt}x u|wuyy{ wx{| ?y{t w}uy| z yvtx ſ xyuvtw}t} uyttxux}y|yywutx |uyt~ut|yutut}wt ytxuuyt{tytxty}tt u}yxuyyy{|y{yuu}yzwt}w}z}tx vxu{}xtz~zx u ut t w xuy tw uu uw y ttzu}yt~x|x{x}tt}uywv}|ywt}wyt|z{xw}tuwvv|ttywtzttxwwu}|tzvvt{xyuy}{yttttuzw}ty~}x|}~}|z{tz{yxx|w}wyt}wt} x ~y | ty u {| y t|{uxyuyy{{t||tuytxww}y|x{u ytxuux|~xtu}txtyttwzzyuu|u?{{xw|u |xyxztw{ttyv}w|u|wzuy x tu {y wuu w x} uxy y|u ytuw|}txttuytuxw}|~xy{ttzvtuwyt|{ty}x tyyxw{t|xz}tyxwtzz||w vxzwx||u{x|tvxwu}|wz}xxu|w}xtyuxxy x ~? } ~ t y t t t wtxz~}y{~tx}}x|w{}ztutzut|ywy|yutw|u|tx}|xw|vtx}}z||z|xz{|v}wyt|xtutxuutyxxx{||ttuytwutuuyzwx} } x z z | | w t { y twt} t x } uxuuyyvy}wxw}yyvt{}yvx|ux?ttw{tx}|y}xys|yxwu}yyyx yt}uxux|x|twy|xu|x|y{w|tzxzy~x|xzw |yytu|vtwtww}zxtyytw}ytyvtywyyx u x vv{ t y} t u tw y t t u


x wyyx xw}tuwvty ttwu|z vxuy{}x~ v{ |ut t{x|s
? $?01?


tt|} ty x {yt wx~{z { tt |}}y u{ywx{}x~ u~tw|w}t t x t v{ywt|wtt{{tyu{tyvv|}uxtz|tw}txw{~}?wy|~|tutxxz}xyu~wyu| xw}v|~v|tuyxtwu|{wy}wxxu~x{z|}xwy|yuyx{xt{{y}t x z } v } v w s t{ |y||u}twtyzwywxxwytxw ' xx{xtuytuxtst}tu||}xxutxtw uwt}txuyy{xt? }z|tt{}tty}|vxvuxvutyw~}xtu} v} t t y yx|wx~ {x| yxuyuyx wy|~|ut |w}t w}tuuytu twtz |ux ux| u~tt{tttu v{t}x w|tx{|t tut tw |w}xu|w}t ttzw}}y~xx tttyx{yt{ | txzy}~x}y{wz}x}ttuywyuy{~wxyuw|}t||v|u{yyyyw}x}}ytyuuxxww|}|w~}tuuxxw||v|wuxuxt|xww}}yttuttwxxttw~ y | } | t t y ux | w } t uy||x{xtyv yy}u|yyyu|w|?{yw}u|ww}t xtxz{}yy~xx{yw||z~tw{tuw|{{|v}wzy|xw}t} wuy } x uyy ~ u w ttyx|utuxttu{tt|{||{t}| yttt w}|~ v|tuxw|w}txux| tw~ txtttytu vxu|t

%? ????0'? 1? '0?? '?$?'? %$0%)???? 1$0%??"'0 %$0'$?!

A

H

p(A) = | A| |2 ,

e1 , e2 , . . . e1 , e2 , . . .

|




xx{t ttwu|z xw}tu|wxu {x} u| w xuvtw}t} wx{| tw yxuy| tw}y yvt}{yytuxuuy{w||ttty{x|wy}xwuxty{{u|t|xztx?wyxtxyttwz|tuxttwy|~twu}y|{ytttww}|wxx~{}wy~|t|yww{u}|z uxwy~xux{y{y|wxt{u}xyt} {y {} t } x w uy yw |x yxuvtw}t} wyuzxut tw} uy~t tyu tw}y xx{t yw|w{y yxuvt{wy}wt} }t{yx x }|u ywz|wutz t|} x x{}y y| w}y t w x tutwzy{ yv tx u xxyx xvt y}}vxuvtw}t}} yxvy|u w}y~ t|u xtuvtw}|t} u}x| } |u zy?xtvuvtywv}ty}wxx}|xuyyu|utyux}|ux~utvtwt}tx}ytwuu|{yzwxyywx}xzt{ytzzw}txyytw u t  
j j

?

|x yxuvtw}t} twzy?t xw ywz|wutz x txw~}t}{ty u}|v{yw|{y~|u t|w{}x}v|yuxwxuwvytt}t}yt}xtz |txyy~twu|~{|yuwxtytx{x}?xuv x y | yyywtxx}y y x}| wv vwu} w} | w |z tw wy}x v } tuu v xu t t tt t u t || tvy|tt xz|ww |wyz t|t}xu|zu|yyw{x{tw|}yttu? uw{xy{wx|ywxu|z|y{x}tv|tyv|uyxxxutww}}x}} ww x w} tt t u tuu w }yt t u}uxy|xx}|xv|xuvt}x}xtw x|t}x xu|wuyy{ yt|? yw}x} tyvyx xvxuvtw}t} x |wuyy{ y yxw}wyw } wtx |w}y x t }x |wuyy{ y} tx{wu|}yytutwwtxz{ytxytttuu}xu|u?|wutyty{}x ywuw|}wyuyttwty{tt{}t}xux~ ywvxttt|{ tu{y}xxw~t} v}wyy t u |{x| ttz}y~x}}|{z xuyuyx z vxuvtw}t} tttwu|z xxuyyx x vxuvtw}t} ttu}x| ttttuuy|{yzyty vxuy~{t xw}tuwvty yx}uy{}x~ xxw}y} tw~ {tw| | uu {y| |xu|x}w|}tuwwtvtvy}v|x|w~{xzw|v}xxtx{}vxx~uyvtwv|}{t}yy{xt||wzw||zxzy|utxuywt}u?tzwy|{t|x y} ~xwuxy { x |? z xz tx yy|}xt txx{|y~{t y|u v}w|~{t wy yttwtz vxuvtw}t} tt{ xzy wxtxt { wx{ | w |z w|| yuww|w}utz w |{y} y z{? t twtzuwxt|tsy||{} y} tvxuuvtw}t}xx}xuy|{u ywxyutw~txuyyyx|zyyutywtt|yu yt
1/
2

x wyx wy yxuvtw}t} t|t xz| tw~ yyw{|twx~y}u ywy}||ut{tx}|~}vt|uz|tv{ vxxuuyv~t|wu}t}tut}xt|tt{t}xuw}yxw}yys||ty|t|{t}wtzyu | t t |w t t~ u xxuy {t t xv|wuyxxtz } yy||{} y} x}twt x{}y v}wy|~{t yttw txzuvw}t}}tuu}~xyu|tztu~|x{|tttww}yyxutyu~y}ytwuy~|yu|t{zt}yy|u v}twyvtu|wx}zt}y~}x|wyxxut|{t x yw}x} }y t
ſ
=
1 N

|e

j

N

| =

N

(

| |

N

suc

l

j

|

j

A

l

j

j,

N

k

tot

|

j



| =

j

p(A) =

j |

j =1

j

j

Nsuc Ntot

j |ej ,

j |ej ,

|

j

j,

k

j

j

l

j

|

|

j

j

=

k =1 l
j

j,k |j, A

k


(

?









??'+??? ? ?+????? + Ǟ?????? ? ? ????? ?R?? ???? Ǟ ?? ? +????????R??'????+????????????????RR??ʜ???????'?????+?????+œ???? +?????+??+????????????????????R??'?+?? +?????'?+??? + +????+??+?œ??+????R?++ œ?+???????+R???+''???+?+??????+?R?++?'?+?? ?Ȝ?? ?????+??'+?'???????+?Μ+??+????+??????&+'?? ? +?+ ?+???? ?'?? ???? ??????+?? ???????? ??+ 5??͞??????R??&???R'œ??+'+'????R??+???'?+??+'?+&?%'+ R? v|{t tu{ywx~|u |yw}x} |uu yw}x} tzyu vxu|tx{yt ttu~tw uy~t v{ z{tz}t t}}y u{ywx{}x~ yxuy~xu|t ytx{ ttyyu |w}ty}ttz}y~xt{ vxw}y{}t| uwvxyu z xw}}yx t{t tw~ vy} ttw|} w|ytxx{{yyttwy|u~y{|z tww~yytuxuyuwx}ux?wytwx{t{y|}tytut|ut vzxuywuywx}xxt|tzyv}z|zyx wyx}{x}}y t? w t t u u ut xwvx}w~yytuxtxtzy|w} x{u}|yv{tyywu|{tt |t}|tttwyxyz|w y|xu{||tutxw{ytwtztyzw}}yx~}xw|xuttw{~|xtsw| {| y|u x w vxu|t}x|ttttwu}xuxytwt}ttz|vw}wwyyv|{yvutx{x|zxt|vu{xyw|ttw{ut|w|uzvzxyuwxw}u||}}xxu|t}vw{zty|wwz|x}z{|||utuv{t|wx}twxt}zu}xy~xw w x|w t x x x wx { w u| t ytx}x|yu wy| tw~ vxuyyx x xx{t tu|wxu yxu|}xt ytuxy wy| tt xz}yu~vxwx v y tt{| tw~ |xx ? wx{| xw}tw}xuy yxw}y{} v}wyv{v xw}tuwuuyytzy x vxtw}t} ttyt|wux|wz|~}}|t |yxuxyw|ytxyuxxxxwu}tuwvytyvx~tx{tw|u|tzwtvz{|x|uutx?tuttz| yytw|}yttwxzy|xuyywz}tx}} z }||xwyyw|tx|utvutywx}}t}}|xxyz|~wy{}||vyxxyuuy|txvx|z|txtzw|zyzy|yww~z|z}wx~wxy}||{wz} tuwwxux{ytw|ztyz|y {|xzy}xzut}{uxvyx |{ uu| |t } ? w t ux{ s t u t wtuu x t u |}|wt z|z~xxzy vxuy{yyt ty|u tz{t~w yxuy~{z}x | xzxu|y ttwu|tz xuy t xtx{uyu}yv?tuytxxz|t||ttx{zx|tytuwx~yy?xvxu}yxu{||t{twy|yzuyw |t{xt|twztyxz}yut|utt} |u | u x tu xxzu ttu{t ztxtu tz{tw x{|t{t}x vxuyt{ty ttuyuyw}yw}ytttw y x } zu ttuty|tw|t{z} twtzuy?t{t}x~yut}}{|t |}uwz|wutzyw|y||{}wz|xyw|}xw}y|||tuw~|txuywytz wwt|wt|wu ? v|uuyw}y~x{tz tz{tw yx{t tu|wxu } xuyu|} t vxuyyx |zxxy} ytt y|u uttyu wy ytt yxu|tx{yt yt tw y tw~ twt tw}t xx{t tu |wxu vxu|tx{yt y| t|z |u x v}twyvu{t vxzy v|z| wx{| {| yxu|} w v|}wy|txz|tz vxxwutvutwwvtt}yttttwtuw|u|zvzxuv|x}uxy{t}xt~z}y|~{xxw|xtt{|tzx}xtzutyyyxxuuyyu{}yv?y{t}tytu|yux|yz}xtzs{tw }} y u} uy y yx ~xx t u tw v y y ywxyut}wv}}uy|yw}|zy|tuwv}utwt}utywt~z|wxtwuyuy?x |ut}xx{?vtwt}}ytyvw}}wxxtuy|uu|}ywt}x}xztwt~xx{xwt{t} | }xu t | x}| t?}x tuxxz{uxwtwtxtxuu{ytyxtxwtwwxx~}x{t| txww}}tuu|wvwt}tyyytuuwt}{t}uwvxtz}yy~xwtz||y|xx|} xx | t z| z x w}tuwxvty ttwu|x }v{ |utut?|tu{txx|ttxw}twuy~tw xt~}{xtuwvyxtyt|||{{} tt{u}xt~ tzw}{yttwww~y xyx{xywztyuxt}x{}yy { t xy||{} y tw}y~x{tz |twxt|?u?|y} tuv{ z}y~x}}||{z wxuyx wz}tuttw wy| { xxtxtzu{yywxxx{uyxytxywxy{wu|ztzux|zxv{xuxtw}t}y}wxw{yy|vt}ywxtuxu~y{xt|yz}z|wt~tzz||u w xy||{} tu |u yty||{} yt|z y|x| tw~ w|~|ut xz}y~xwz| wy tw v }x tt}ux{tx}wuxt|~x}|x~ywwttwttx{u}|yttuv}tzywt|~w~x{yz|vy||zt|w{{}ztz{}x}txx~zytuytw{tyutyx{uyxwtx|ttyxut{}t|wvxt}u?vttw} ty z y uy w y w u t tt s t x || ttu|| xu t x
j,
k

|

j

|

l

j

l

j

|

|j |2




?

ž?? R?R+ ?Ǟ?R? ?????? ?ʜ+?R+??œ??+??ɿ'+?3' v x||z tw~ x wyx u{t vxuywt}}u| xw}tut v{ t{xt xz|w x|wzy?t {ututuwytw}tx yw|twt|zztxvw}}|w~x~xx|uyuttwz~ |tv}ww|xt}tt}u}t{yuyv|{uttu}tt||ux|xzu{|t}zyy}xtsw x y } ww t x ztw}yy wuyt w}|~ v|u|w}t} z|z wyt twy tv twtz xw}|~ xx|yx |u uu|tu}t tt xuy{}x~ v{ u{|}yxuu yy{t yxyt twts x|ywztzy w uy}xxzyu|yyu twtxww}ytt uyuytwttztvuutyxttt?y|uyw}wx}wty|txwy|xu}y}~||u yxtu|{x|wy}|~|xzzy||} w{ t xy |yxw tt{ w}utuu|w| xw}y~ wyv{tt x wx{t| wu|z |u tuuyw}y}tyu v}wy ~wttywtz|xxzx||{} {yxy|~u| xxvx|yx}~ztwv|{ wwyw}w|tx{{t}}xtuut|t|xz{~tw u} t ut}|{t} xtxzututu{tt tu t u u u u t vxuty{yzy~twt t|ututu| |yutux } ttu x v}wy|~{t yy?vtx|wyuv}|wyvuvx{~yxwt}u|w~}tvu}|y|ywx}utyuvy{x|wtw} xxxv}vwxyu|y~{yy{tzy~tw t{}x~ y|ytt xwt}z||~ zu t |{ xx|}zx |t}t} tt{ v{ tw~ w|wyx~} tut ttw y{t? yyw}x} t~y|xw|} }x|z|w xuwy}~t{tx|t|zuyv{{y |x}z|~{tw|w|~tww||wyx~t} tt}yuww}}ttu|v{ xuyytxxxutzytwwt}y|t} } vx t u| y | {t t|t yttwt?xzxzuwx{| |wu|z v{ yxuy~|u ytuu|x ty t{}x~ yyt x tttu{t v{t w||z |u{|utx|t zy~ytxw xw~w}tuwt{ tw~ytzw|}yw yw}x} tu~xw}|~tyu v{ |w}u||w}xt t|tuutxx|{xutz vxux{yzuzytwtttwu}{yt txu{t yu w|xw|}}| y ? y } t xyu{vt}vt}}wyx|}~x{|z|tvzx}u~xtxy{y{|ty{}tw }tvxvu|sxt w y|| y } } t ~ w t } tu }tuy|tx|}y{|x~ytuwuw~t{vtwwwxyv|}}~xyx~v{{wtv{x|uyyy}yuwy|}|uutw{}}|xxxuzx|w|tu|}|{x?}x|u|~vwwtytw~}}|t|}||txtt|wtttzz}ytwy}xytx t y { w u x t ty w tt w x u| ~x {t| { yyw}tuuyt}t x w|zyw xw}|~wtt ywtwtz |t ttw {t vxuy|tt} yxz}y~xw}x tttwtzytu|yyxxz|z?yx|uzvtw}?} xzt|w|z| zx|ux twu|z w|tx{yt ut{ u ywy}x| tuyux{ |u{t |wu|w}yutz v|z|w tw~ wxt{ ut y } t}y|utxuyyww}wx}?x|?twtuz||ww}u|tuzx{tttww~tzutw|uyw~xxtuw|}t}tytwtwt|zu{tyw}xx}utyu~xxwt}|~ uxutvtwy}wtx} twu|z x}x| w}tut{} |u vxuy~xu|t |tt ttz}y~xwxt{| v{ ttuxy~t x wx | tz}x y ? |w}ytut ttz|w |t t}t} wu||z tuv uyu~ } xvuytywxt{u v|x}y{~txwyutyzyt{}}yyx}y?xz|}yy~xuxwxyt?{|vxtuwy}ux~|wvtytux~xw}x|{~tttu xutywywztuxu{yxy{xty x w y| xt y|x tw~ wx~u yz|w tw xzxu|}y zttwu|ztuuutyxwx|w vxuy xz~tyw}y xx|{|wux|tyu{|xt}tu|xwzyyyxxyt{tt vxtu|y}|xxt|wtu|tzzt|w |tw}wt}u|u}xt{z|zt|v}w}wvt{}vyxwz~ w u t yu xt{|w}ytuwytyuu|{ ywxw}|t~z yutw}u|z yyx}|wwyv}{||w}w|ytx{y|tw}yty|u } x zt|w x~ | t t?xxyw|ttx yxuyuyt ytxt|}xtt{tx w|t} tywtztuxz}ty~xt}}t|{z }utt ut |} xxu|t} x} y w}tutt v|z}y~xwytyw vwt v}wyyx |u xw}ttz}} tz|w x tz{tw zy{}tzvt tv|uyux{v}ywywv{vx|w}t }w}tytz} tyy{yxs yu xuxvwt}w|}~t}t|{}wxy~}ttuw}tttzxv{}}wxvw{| t|u{ttytutzt}~txtzwxytu|{}|xttywt}wxz}utux{{y|twuy t rrqrp ?n?qn ?op? ??or?o?
H n


|(t + t)

L=

|(t)

1 g2

H

g

g

|(t) P (t + t)

|(t) , P (t)

n

P = {x1 , x2 , . . . , xL } ?? ?

(t) (t) | (t)

s-1


6



??++????%???????R+??????????????+?'????%???????'?????+??? +??????+???R??R+'???+????+????R+??+??+'?Þ?+???????'????R??R+?????R???++????R?+R??'????????! ? ' ? ' ? R ' ' ' ' ? ' ' R ' ? ?' y | x yx t uv x|xw}~yttwutz| vxyy~|uxuyyuty}xtttut|utywww}xt{uuw|~wu||zw}y~ut{u|y{|twx~z}x|xu{tv{tyt {| yt{w{y|wyyxttut?zy{zvx{ty|uwyw|}tuwttw{ twtz|tuwxyt}|tuyuxv{wzyy{tx xxxzuuttwy|vt{u{ttu w x{t w twtz | ~||v{ tt ut ytt xw}tuu|w| wy|wx~ yu xw}tuwt{ t{|uxzu t yt? yyxtxy|{}zxyzy|ywxww}wyw|tzt}x{}wy}y~xw}y}t|~} yu|v} xuyuyx|x}t zt{ttuv~xtwy|wwx}tz w z w w | uu} t w ~ t y t ttwu y t y 6 t u t }t u wxyw||u |zt|wzyz|yxuy|t|t tvxtxtyvu |uy~utzt}uwzw|xtyztt}w}yyxuy{w}||w}y|~{ytzw|twwtz||u|zt?
6

j

wu ttu tu{t w|}x| wyy{?} xw}|tuxu|wt||u|xtytwytzwwy}~yx xxxzuuyuuxw}zyy{ vtw | | v{ tx| tx| ttttz}uytutt{z |xxwvxtt}}xxttwyx}t|~ v|w}ttwutuz|wxw|}tuuy||w|x~y|wtx~wyyu|~t|uuvxtww}y w w~ t u x w | y{|z{t? |vwxu|yu{xw}ywyy yxwxtu}txxwyy wwxw|uzy?xx wx | yvw{}x} v{ x|z}u|{ }tu~|x}w}|w~yt} w x x | u~ ttu } t tu t u uwt t~ yt t t t} tt| ttuu t t t yt~ | xxz{y~xwz |x}sxw}tt{ u{|xutwxzux}wyytwxyu|txuxt{wv{wxy|{t}txw~z|~|xz||wv{uw}|v~xuvy{{}x ttywtzy|} xz|z v{| xu y{yxyu twyttv wy|{y tw~ x|yt wyyw x|wu}||x~utwtu{xtvxut|}txu~txw}w~}tu~txtzuy{yxuuyyw}u}yyxx|uuy{v}}xw~yxv tw~ xuywyyxx yt| w v{ z t x|| } zwy} uuyw}u|w}t y|xw|}}| x{}? t| |zvt wyyw tw~ |z}xt x~|zvvt xwu|vy{ yxuyu| ytyz}y~xw}xywz|| w|y t|u xzu uuyw}t} y||y{ w|wyyxy? vx|uyu{|}x~ vyxuyy |twy tztuwy}txu| tu~yutz v{ tw}y wyyx ttwt~ut v{ wyu} xw}ytt{}ux~tyytt x t wx}utyuyy||wwyyt|u|xvu{tw|{ x|xx|x{tw|?| | u wxyu|txwvttww}}tt{u xtwty{~}}xyt ~|xw|}|~xwx x yw}x} v{ xw}|~ ' x tyw}x} v}x~ v } | ~ yv wxuyy x {wxyt} t|u xw}wyy tz{tz} } x yywxw~}}ytts xx|uz|uxutwt{xtu|{t xxx{ztutu|uwyxu}txuy|xuty{x}xy~tx?x~|| y |xt|w}t}|yw} | w uy t utwzy{ w }|u v{ xw}tuuyt{ vut ty{tx|u x|xw}|~t?w|~vwyxuvt{wt|xxtuwwtx z| u u }|t~wzy{wyxuywt}}| y|u x~xu|t ? xw}ytx| y|xw||}}|yxu|t{t}x x|uxtw}y|x{|yutx{yy||uvv}{yu|wt|wz}|yx}|t~wtwzyxxw}|u~ytx{xt{t}x}||u{z wxu{yy?t tw|y?xux w y x t} w w x|tu w uu{ywx{}x~ yxz}y~x}}|{z vvuyx wxy tut t|wz |yuxy xuyu| xt tuwywu}yuxxyt|utuxytwytxzuyzuwzyy?tut~x}w}|~wt|uxtutxuvy{w yxx|xxxw}z|u~tttwt~u{wty|y~x|wuvu v y ? | y twtyzxw}|y~ xyuxy z|z w|xw|}}| tut ytwtz xw}|~x|z x{x tuwtwzy{ | yxz} y|u yu{|xyw} yy{t tx| yz|w | {zyy{twx{x t| |v yx |u xw}|~x|ux|ww}u~yt{s ytxxtuw}|x}||~tytuzw|uywy|{z}tzt{tuwyu{t|txutt |tu|tw~tz|xwz|wxw|y~ xz{y yy{t x| wvtw}tx}u|yy~|t t} wyt |wutwt wyt w yxuy~{z}x | |xw}}|~
?

N

2

10

36

N

exp(-

N = 1000

4

N

iH h

12

t)

n

t

j |j .

N2 N

g

2

|k

H

10

N2 N N 2t

2


?

'??+??R ? ?+??? ? ??+ ? ????'? ?R  ????R+? ??R?? ?+ ??? ??????? ? ???R+? ?????????R ?'+??????? R+??? ??'?+?????R??R?œ?+R++??????? +?R???+?͞?R??žR???R???????????+?R????&??? v |z|w w} x |x yywtz tw}ytu wy|~|utt w}s | tw~ z|w yy~ y|~|utwtx w ttwtwtz tz twxt{| uxwzy twtzyu v}wy| |ut yxuy|t{t} z|xz|wu|}txz|}wwtyuttt} |u|wxt{x|ytxuuy|{uytytw~}|y|y|x{uty~|yuytuu|txxz}uzx wytwut{t}}|x{t}w?zyu x | x yxuxy xu| y}wv{twttxyx|uu|ut tt{ttw v{ } t wx
?



| x tw yx wx|w}uy tut |x xzuttu{wt |wvx} }vxwutyw~}|tutvt{xuu tw~y}{zytux}tyxutwy v}wwxz{tx}txytutx}{w}y?v{ |w}y ytwtvu tt~ wz ~ t tyvty{}w}x|w~}|~tyx~|tuut ww}|yuyxxwt}t|z~ytuy|wzx}txu{w}tyw}yyxy|uxw}|~ xz|w| t{}x{~|z}x|} { u x zt x xyx xw}tu{yw|ty{}t tw wuyy{ yy~ | ux{ xw}tuw{}ywstxy}tx~|u vy}wwt u{|~|u yy~ x~|ut xx|yx xut wyyu t||u u{txt w wyt twttz w|uxttz xw}|~ xw }}| |wuy xyyyw}x} t vuuttwxw}x|~ wy|yuxxttz|wvt{yyw}tx}t}u}| yytwuzyy}|uwy|utx{tt}t|zyxu||vuxtw}|~|xxww}|~v|zv} t twx~ }x{xttt z{tz}ts ?? xt tuuyyt} y{y t|} |u wy yxuywt}}| y|u uz}y|~{} txuytut tu~xw}|~tut wy}wywtt} xw}tu~tw twtz yywtz |u ztvt x~x|tz}tzy{ uuyw}yw}y w||xw|}}| y| }|~y~ tu{|utx tt ytw}t xw|x}}|zuxvtut{t txuw}~|~tyy}xuyw~|uuxt}tyz?vx uy|~uywttztutuyt{w}x|ux}{x|w}v|u~xtyw}x}w}yy|
?
s
k

x x|w|yuxxttz xu{yzwux}uwvt}uw{t|}x|u| }wtyw|x|zt}wvutytxtz |zy~tw v|{y||uyuw|{u|x ttz tt u~ w yvuyw}x} w|uxttz ? yx|t vut xw}|~ w| w}s x|t }x tzxtw tyy{} xw}|~ yw}x} }}| |wwu|yuxxt|twzuyxuutytzw}u|wxw~t}tutt} twwy}y|~xw|vuxt | u t | w|uxttz tux} x w v|utxwt}~xuvy|x|ztt|wtztzxzw|zy?|tyv}uxw|y~?txuu{|t ytxuw~|x~w}v|x~utyu{ytyxxxttzutzwytw|twu{{tyyx { y uy } | tw~ tw tz{tw wxtw}t} vx|uytzy v| xx|yx tw vxuyuyx v{ t{x| yx{ xtvu}t ttt|z}xyyy{ut{|wv}{| xx{yt v{yw|t{t ~|| y| ty |u tt} wyv{|w}y t x ~ |zwtx} |tt}t} t wy|~|utwv|ut t xw}|~ xt}~ |tt t xwz|xv utvutty{uxw}x~}|x~ xxzvztwy}}utz|uyy{tttx}|uwyvw{||}yu yvutvutt |ttw|y yt t x t|w|xtw}t|u wxuyw{t|uwtvuuuytw}y}tyu uw}t ytwtz wzy?t yw w}y tw {zy{t}yu t xtxtzuy{uxt|uxt}|v~xuwy|{yy|utx}|{ vxxw|}|~ yxx|z{}tx|w? t{w yw v{ x t t u t u tt{xty u xx|{|xutt|xtyuv{ty|ty{}yx~ zttz|{t{x}x}~y~xtwwz|xsw~xxuyzvttv}| wyy|xuwyv}yy||xyutxwv| w x z t ytu{|x}z| | tywtzy} x ttw} tt|} t| t{}x~ w}y y u| w|wx~}
n n a n
1 T

r ?

rk

?

j=n

?? [rj ]

k = 0, 1, . . . , n r ? k j s 12 rk = (rk , rk , . . . , rkk ) rk

sk = 2

r ?

n-k-1

fj (rj ) = rj ? ? r ?
r

a1 , a2 , . . . , as

k-1

1

T

r ?

r ?

r1 |r1 ?

N

K
r
7 3 2

j

r2 |r2 ?

rk k

j

n

r ?

rj ?

r |r ??

Ca

xa , ya , za Ca n+1

...

r

n

rn |rn ??

f j

j

a

...

rj ?

j = 1, 2, . . . , n - 1

n+1

|(r) ?

rk ?

rj ?

s

a

Fj

(r) ?

(r ) ?

k k

a


?

?

|w yy{ x|} w x tuxw}{tuzuy|{wt| wx|w yxuw|t~tuy}|w yy{|yy{x x vx|wwx}~} xyww}}|x~} vu } v { tz{utw yu yxu|tx{ytu yttwu|z w|tx|wt}| w}tuttuu{|xxxt wty| xuvtw}t} u u~xw}|~ttu yxuy{|w}y ytz}y~xyyx yyuy yu y {|x y }z w y{ xwv u tw|uyyu|t y u t } |t{w}tttvwtyw|t yxttwtx{|u twtuy{t{yxxuyt|uw~}xwy}|~tyttzu}y~xuu{xxtzwtws wytu ywyx| } z z x zt { w zy xuyyw} {yw||zt y~x vut ttu{|x}z| xw}|~ |{}x~ wt vxzu v|uuyyw} |z xw}| wy |x vxuvtw}t} ttzv} x}x| v{ v|xttyu wv| tw yt|u t{ |xw}|~ {}x~ y {|x}z| x v { xz yw} t} w}y~x{ z y y x yw} uw}ttyx}tzux| x{x yw}uu|tw}tttzy~tw tu{}x~ yyuut t{t}x~ x~xtu|ttutxwt{u}u?{| t uxwzy {uu|x tw}ytu x{xt vxuy{yyy}| yxxw{y x xxu|tx{yt vtxwu|xtx{ywtxt|txvvxutyw{yyyyv}u|yx|u v}wtv|ttxtxz|wtwt}zxvxux|tyttvw~ttzv{utwttu|t x tu xt} xw}|~ v{ yttuw}tu| tu~yutz yx|u |twy ttuuy{}x~ tx{ ttz|z tt } yw} | xxut{yzy}yy~xtxywx||~uvxu|vv}txww}t}yzw{yxtx|x~|{vtu~tvt}uwvx{|~wxuwx|{||wy|xuty{xy{|yy}t|uwtx{yyt|tt xz| y y v x w w {t u t } w t vut t yyu wt v|ux~|u {xw}|~ | u |tt u yt |vutxw}~uuyut{z}|z}|tzv}wwt|{~{tyy{|vuz|twx xw}|v~utt{x|ww}||u~xv{ttzu|vx|uutww{tx}| wx|w}t} xxzu yuuyw}t} ty v|u vut xw}|~ v{ u|xutw{x|u| wx|w}t} t|t v uxz|w yt vut xw}|~ v{ xxzu yuuyw}{|tx}yuw|t|wt|w{|w~x{y}}y|ut|x{t}w?yt u t xw}|~ | wyw}y w z t u x } t tt y y w yz|w |{|xuywyw}xx}uwyu|~v}wtyuy|xw}x}{}?t {z~xtuyt{ttywy}yxu}yt}wx|uy|y|tt} xw}{t|uxw}u|y~wt w ֿ {x| ttwu|z tu}|{t} v}wyv{yyt yxuy~|u yyuy} | t wtuyw|{|}xty| yyv{|vtuwtw}ztxx}vwx}|~ wwy}yw~xx}ty}xu}v|ty~ww}tyx}yyywx}uu|zw}txzxuyt{yw{|twyytxw}t|w~wx|ux}t|ytz |t}s w } } } t w w w w t x t t u t x | u t tu t t t t wx}x| vxuy{yyt x tux z|z v|twtz xzu tu{t yy~ x~|ut u yyw}x} y|u xuvtw}wt} |}yx|t{ xxwxtvwt~w}zt|}}wxuuyvttww}}t} }}|v{zuyt{yyy|~{}|{t} x{t}w?wvuxyuvyt{w}t}ty{uuu|ywuu|xytuttzw}{tw w}}wyy w z x y x t t t t tu wux yxuv w}tx}w|}u{|x}{zsw{ytwxux{tw{||t}v}||zyxtyzt{|uyv}twwy}twt}yuwxvz}|}xw}||{zwt}uy{tuyv yyxyx|yzuyyt} } t t ttzu t zu{ tt v wut t u t t | w t wv| wyv{tt y~ wx{| x|w ut|zxxtx{|uytztw|xut|} tw||wutt|wt}twuyttwwyx|{w|~|wu|wxz{xvv|xuy{vyxuyu|| ytxuu|yw}tyt} w w| y } t t } t t xyuywyw yu xuv|tw}t}wt}}|{z uuyyxx } vxuy{}ux~ tz|u v|wvx}{| xw}tw}xuy}|u yxuyu{ty{tyvxux|~yxutvtyuu}ty}{}ty}{xt}}}|{tzw~twxy|{?tyyywxx}{ttyy}|x}ww{}|vvxuvxtww}}|t~}|x{x~ } u wttww}xyttw{| wtuxwwy}zxy|ttyutx{uwyu~yxuztvw}}tw}||x|}|w wxtvxu|vztwv}xtu}|yxtzx|xwtyz|uyxx{t}yt~}xwy | y utv z t } uv w yt u v{uzy|tvxt}uzv~x|u}xt}uytz|~wyutzxz{zuwww|t}xuwyxuu|}|||| wz{twzt?stwxt{| xuw~wzxyzu|vxt}zt| x y twt u xu tw t xu t t t zy | } y t t u u
A Fj K
j -1

Fj

Fj (rj -1 ) = [fj ] ? j = 1, 2, . . . , n

rj ?

-1

Fj

rj

-p

, rj

-p+1

n

, . . . , rj

-1

?

1, k - 2, . . . , 0

k-1

A

rk ?

Vk (r)

L

-1

S

j

|

rk ?

L

n-2

A



V (rk ?

rk ?

= T r (A|

-1

r

r ) = V + Vk (r)

k

rk |rk ??

k

n-1

rk ?

rk |). ?

k

rj ?

N

p

S

k

A

k

n

S

k

k

V

k- 0


?

'

{xt t x{t xw}|{t uttw} t xxzy x wuy~}x yy||{} yxw}wywt } y w}x{| wyx wy xtv w}tt}u|}x | uw}|~ v|{ z|z z|w w | y x{ t}ttuxw}||ytux} twxtwu|~yvt{u||w|}uw|}uw|}tw}tutttuutuxtxu|xxuztz vxuy{|ytxz~tw xt{tt}| u x{ ?x tyxuyyxttux{yyt yy {|xy}} tz|w xt|u zw x x y u u {tt x tyyuxx{ttu|ttwu}ttytxuuwu|uy}?tyyvuyt{tywwytxuyt{yuwyxvuyxvxx{uz|y|zyyxu|u}v}}ywyt| yxuuy|u|xxtxyxyztytxu t txu{ywxw}wx}{t}uxv~v}x}yt|uu{ttvz|tzzy|}xuzy|tzut|wwtytuzuy}yxy}xzwyyw}twttttwzt}yzxy~uxyytw|~|xuxyyz|tu}txtz|swuy{x|tu~|{yu t }yv}yw }t }x z x ~tw } w w t{x }}tu tu u t yyt tu{|y tz|zxu wy|~ywt yu x xuxx tu{ywx{}x~ |ztz|zzxuuw{yw}wywt t}tzutxuxtzyw}|{yttxxww}x|xu ututuy|{uyxyxztuwty}t~xt{uw{tt x|xzyytt v|w}tt{||~z|{x||t w y} ty tuv x } wutyu{tttz tz{tzyu |uy v}wy||}| w}y tw uv}yu v}wx txuuy}y? } w t|xwtv|w|t~tztvtxwutyzyyuxxxzuyttu{{|yxwxy}t}u xw}uyuw}u|w}t |tz wuyt wtw v}wx u xy}z|w x{t ?y??????? w |uxttz yw}x} |{|~|u vxuyy| y{}t w}tuu|wyw}z}yt~uxu|w|y|xyuv}ywy|}y| wtuzy}y~ | } | y w }u} x yxu txu}}{|w|{~z|uxyt{xtvt|wut|wtzz }ytwux{|wtu|~xw}z|~{ttzt}utw xyuxx|wyut{ywxt{u}uy~w}}yt}tt}ywuyvx{|ww}y|z|t}z x t x{ |uttz}|yzz|wv|xzx||txx}{xyy|txuy|uxxt{yyxytyyv}twyxtutxtttz{}yy|{txxw}||u~v}wvyw{}vw~xtxwuu||wwy||zwyt|wx}{y}|yus|wz} u x | t z} y }x w uy tt t yx tu~ t ty t } tt xy~xw}|}u|x{zttzwxy}vx{y|yyyxwu}xx t|ut xw}txtwz}}|x~ w|xw}xt}tzxzxu||u|xtxwxu|v}tw|x~{y{tt xxws|~?xz} ut }zz x ?xywxz t wu|t{t}x tut x|~|| xu~xw}|~ttut tz{t|w }}xwtt xwt x{}? |vxxvu}|wytvx|{yt}wt}tu~|xww}||~t|twux tvxt|vxw{yyyw wx}yw~xwz}ww |wt| vxwzx{tz|}wy}u yxxxuvy}vw}tx w uw}|~ uxtw}tyutx| yuw}xt}z|z tx xwt}u|~t uuyw}x} tu u t t wxzx|ux vuxxu|tx{yt u x yyxuyy~yw utyvtwtyz{}} wxtwx}wt}}||~| tttuz|tw wyuutyx~xv}{yvytv}wyx|~yx{z }twv xt ||{xxw}t| | | u t t x}xuu|wy|yux{uyyw|yt}}yytttztzx||xuwy}|~w}uyt|uwx}xtuu||wyx| txuw}|y~xyux|wvtu|st}'u| v y w yx } x t y u ֿ |x yxuvtw}t} w|xw|}}| t|u }y? |utwt yxuyt{t } xx|wx} vxu}ywt}}| w u~xt{|u wx|wx} uu|}xt wyx y xxzy x x yty| |{ytuytxuvtvw}}w}y||{|xyzwyu}|uxxxxy}y{twyw}t|u|wxxuzuxx u|~xyww}|{~tu|ztt?xxxxvxwu}yt~}|u |xutxuzyy} v uvt w { t |tz}t}uxt{{yzx uyxw|}~|yutty |{uzy{xt{?wt|u|tuwt ttuuyt{t x{x |utx tyuy}t} vxuyvwx t vtwzy{w wt w|y}xt tut t|t xz| | x|xw}|~ xx } w|tw}ytx| wyut|x}t|u~|x|uwtw{xy| }xxxuyw{}ywyywt}t|}uwvxu{t tztw}yyt yw}wywtt} } vut yxw}|~ xw}|~ w yv{v?t |t vxuyy yxxutv ttu~uyw yx y v}wyv{xy y |x xzu x t|z vut yxw}|~tx ytut xu |twx}t}vxtuwvtw}tt}wvz{ty{twy~xxv}{vwtxyvzyysvxxuy|uywx}{yxwu||tvvt|u}xxyxxy{xtztzux|}yw|~xyxuxu|x{yt xu t t u u | uu { |yt | w t |z} y y

j { photon in state j , no photons} a k-1 a k
j

j |j , j ,

a

b

|

k-1

a

rk ?

-1

b

k-1

D1 , D2 , . . . , D b

k


' {x z| yy~|w|u y|xtwuyy||w vxtz|z |z?|zvt}w|yww}|tu}}{yw|tvxy{z}utv}vw|y|tuy{tyֿ|ywy}}x} vx{z} | x | s t u ֿ xuyu| uuytuzt yyw}x} tuwuy{|xz xxzu ttu{t vx|x| yxuyu||twx 'ֿ xuyu| yy{} v}wvux~t tzxxvxzuu|yxuutyw}|tt}wytw~t|twzv| tvxtuuywu}yyxxxwyxytw}xxv?w}}xwu}y|}t}}|y~xuyy}?tw t | t {y u t t tu} w x v}wxt|ut twty~ v{ wyt x{ v tw tu uw|uxttz z xzu tu{t vxu wy|{|w}y t z}y~xwy yu w wx yy v}wyy | t| xw} uuy { v z } xw}|~ uxttzutw{t}|xxuy{ytyttxs tu|xutw{x| vu{ vxtuvtw}tt} yuuutyw}ttt} t| w|uxt tz ytuuy|w}u|w}tx yu wu|xut} u{t } xvxuvtw}t}~ xu}x| ut| x }v y} tz v} x w}y}uwtyu|txuytvwu{y|u tywxvtvwx}utvtywt}utu}ywx}wtvutyw~ ttw~twuytxwutyw{z|}zx}|}xvx}xxw|t~tytxzz}|yw~x|{tyzx} wy y } z t wx|ux|txtwz}tyuu|ytu{tyuxy~xwzyt}yux|w}u|x~ywtzv|xwu|xy{|tywzyyy{~w|}ytuxuy|{yutvtt}wy}}{t|y}yyxut|u}ytxwzy|{ y u u y tz} x | | x y u t u tu tu y~ vxu|ttx{yt xxt{vtyx wyw}wywtt} y}{t tw~ y uytu{tz}|u w vxuyx| x |z x~tw xxwz}|tzuwt{|y{}{xt~ yw}t|{tzt|w wtyuy|wwt}u|xxwt{tttyuxuut|xt|x{yxt|xxyxvu|tt'uy|txuw}|~ | tt w|uxttz yyw}x} yy uxtyu x vut xw}|~ |x} yxu|tx{yt x vut y vxw}|~ tuxtyu x vut ' xw}|~ vx{t ?t| yxu|txywx v}wyx{yt vx{{ytxzyv~|uw|tw{x}ux w w|ywx'{vwu|tyx}ww}|~w|w}}|ww}ttz tvtyu~txw}|~txuw}t|t~tvu{ut|xw|}|xw}utz xy ~ w w y u tt}tuuuxt|txutz vtxuy{y |z~t x{}? tt ttwt}y~x{z uyt x x ywtw} t yuxuyt|{ t |u } xx|w}wywtwt} uy{wyy}| vuyt| xw}|y~|xw}}u|w}{t ttuutx|xutz v{ vyxuyt{yxzv~utwttx}x|u~yux||xzt|zvxuutwv{'wx}w}y|~u|w~|~{}|}wxt~w}tty{t |~{} |u yxu v{y}xx}wt}}t|w|{z w t u tu~xw} t y|~ttu yxu}y{|w}yyytytz}y~xx|yx |}u ux w|x}tt ttut vxuyx| zy~tw xw}tuwxx ztuuyytyyyt?yxwxw{}w|wt}{|yy~tz{|xu{}xvw||w}|vxuwyt{yztyw~ttzwxwtwv}xuvttwu}tu}y|ytywzyyx w|tw xz}y~xtw|tw| |uyu|t} wy xxtzu ttu{t |ztxtu vxuy{}x~ y{|~|u v|uu| xxuyy|{~|{w}}yw tuytwuy|uw}yuytz xxtztuwzyut}tu{t yxuy~|u yxtx{{}}?v}|wwv}uu{|twttutt{ututxt }w x|uxutz vxuy{y wzy~tw w}tuwt{ w} tuuy? t|vtt}x u|wuyy{ x{|tw}wywtt} u{ tzt}~xt}w}~|{zz|wxxzw~xttxw}x xw{|}y2xx|t{t|u ytx|uwxu} xxxwu}|twxyy|ttxtwu|}z}vy{x{uy{tywx{t}xt~u y u ? w|txxxux tw~ tttwv{ xxzu ttu{t wt wy}x| zu{t vxuyx||zy~tw vxuy{y yyyuw}||tw}{ttuw}tx{x{z}|wy~yxu}uty ytuyt|tt{|twywxuxx|yx}ttz vxtxuuy{yyux{y~twxyuxvutyw{}yt}yt}uxyt|uytu|y w u } tuu { zwy|t{|t{|ywtuxw}twytty|t}xvxwu|yx}}|| vx{|xy|wuxxzx~ uwxtw| xuyyxx yx{w}||w~}xy|tytxtz | y | x y u| t w w~ { uy x u y ut t u ( u z|w}y ttuwyvxx{yyx{|u||z~twt|t{w|vuxywuxwtvxtyxyy| xs txutyy}wxyzu{tx|uyxu{tvxu|yu wvx{ xw}|{t xz} tuuyx z|z z|w y|~ uyt{t}| w u{t} vxuyx|uxz~tw wx{| w
j E () = 0, E () = (x) (x) = C |(x)|2 E () = 0 j [1/ 2 ] |r ? (r) H (r) dr (x) x


C

x1 , x2 , . . . , xk

j




''

? ??? ????? Ǟ++????R+?œ?+????Þ?????R?????????????+???R???????'?????+œ??????'????+R??????????'?????+!"+ ? ? yx uy{}x~ z{ z}ty x{ |} | y t x y z{ w }y x w }}| ? vx v w} } }x | twu xw}|t~ stu yu uuwtuyxxttzt wt twuyu|ty x{x zy{ywxu{ytytttyxtuuy{}x~ y xy|~|ut tw y yt ttwx}yz uu|| z x y{}x~ tw vxu tu{t { |zwuy|xxw tut xx{|xxwyxx z| xx z z|z{ w|x}xx||x twuytx} wx|w|{xyxzu yyytw~}}x~x|xutwtytux}|tw||uxxwwttttzxuxu}ttvxx}zxu| tuw{uty||utuuxu{tyxtwttw|uvx|uuytwyt v}{wxyvv{|uvutxyy x ut t w t u u y t } u ֿ y zu v|tu{t v}wx~{t (ֿ xx|xwyx} y{}ts y||{} t{}x~ yt{t tu{|xyuxut}vzx|uv} wyvu|w}t}| |} y|~{} yuy{}t z{tz}t yx y~ yyu~xtutz t|t (ֿ vxx yy x || yu t|tut}uy|wxyzxtu|wxyxyzuw}tutwyt~u{ttutyxxuyyuy ~{{}ywxw|t~|utut~ttxtztu|w}yxuyuyy}tuttttv} |u y ~{ } x | y ywyxwt|tu xyu|tx} y vxzu |u|| x{}? uuyw}|u|xw}zwutwv{xw|txywyyxu} tx yy | w|uxttz tux tz}y~x|tzx}zy{ yx|u yzvt tuuy{yyt ttw} v}w xx ytw vxuy~|u xw w|uxzttz vwxuy~|u t }t} yx xxzu tyu~xw}|~tut tzv} yxuy~{xw|}w|}~tyut tvxtuwyxyzxt| yuvttuytuut|w~tvx}}zxx{twtt{zyutv}wtw}yxuw}|wyyw}tt}xxtzuu~tw u t v}wvtx xxu|tx{yty xyxzuw tuu| |u xx?|yt tw~ tw xt v|y|t { txyzuxyxtzu{t twvxxuu|{|wy}ytwxtxtsyttx{|ֿtzt{ztx{tuu~tuw|{w|}tx}z}|}xy|tyy x u tw w |xyvuxx|vxxuvxtuwx}}tz|uw}xx| zw|v}vw}xt~uwtytw~twztwyw |tzyx{tw t|txxy|z|{x vtytxu{t|ux|x{|x } yt v ? v{tt yx uxt }y yxt ?|zyt}}yyx|u|}tyx{yxut|twxx{wy}tttttuuvyw}yv|}xy v x x{| v| yz}y~xwxt{| |z{tz}ts |wzz}w} uv{zy{t x |tw| xuvtw}t}uutwzy{ttu utxx|wu}|v~}wxyz|x}zutxw ztttvwut|zxyxuy|~xtx|{yyxtxwy}tyvtuuwyyvw{}y|ww}}yyv{ywu}tw{t}z v{ w|}zy?v t | t } t tu| | tu|t t t tw } x|xxux y|ux~|u x x z w}y tw xxxuvztuw}}tuu{wtxuwy{|{w}tyyxuytxxtz yz xtyy u~y|u xzu xw |uxyw xux z|ztttw y{}ts xxyyu yxuy~u|u vt|ut{|{xuvx| vt|txwz}wttwyxxtx{}zu}|{|~y|u}xv}w|yvuyxtw~ xwxy|~y|uuxxtw|xxuxxtwwyyxvuxu|z}x ut t ֿ t ttutuyw yx v}wy|{y tu~t tw z|z w| ty} yt x?xzuu~xw}y|~utu{|t~v|xuuxy~|y|x } xy y }txu~~xyyw|~ttuxw}v|{~ t{}x~ |z|vw}tu|x}t|yttttu{~}xw~}|y~utwv|xuytxw~| zyz~ttvt{t}x~yyx t| t{ ww w tt yx||yw}tu||{w}}xt~wyttyuuwtx||txtut|zztvyx|txvxyux| zx~twzuy vxu|w{}}|x~ttttuy{|u{ttxx|tutyxu | {| y yx }
k

M =N

N

a

N

( )

|

M

1 3

ind

(r1 , . . . , rk ) = i1 (r1 )i2 (r2 ) . . . ik (rk ). ? ? ? ? ?

1 D(|1 , | k



S y m|

D

a

= |1 (r1 ) ?



? i

(r

ij |j

E ()

(1)

S y m(| , r
2 (2)

. . . , |k ; r1 , r2 , . . . , rk ) ?? ?

|2 (r2 ) ?

, . . . , r

ind

)

|

(k )

)(-1)

...

s j

( )

(r1 , r2 , . . . rk )

|k (rk ) ?

k



s

N

ij

k+

1 3

j




'
s=1 k

?

?

ֿ | } } x |u| vxuvtw}t} yx |x yxuxyxzu |yxuy{|ywt}yy|s|{6w}|~xtuuyt|w|xx{zyvxvu}wy~|~u{t u x u t yxy{}t y} yxw}yy y} w|yt} ytwtz yy||{} yw x y x wyyx vxzu ttzv} v{ w}s | t y yx|uy~u ww~yw|}wwy{wxtt|}yy|tt6u||zty||{} {w}x|t{uytuzyuvutuwy}tw}}uy|tuw}}xt| z u x x ttuu t tt z|x{yxzy6yֿ x ||uu|| vvxxuuvvttww}}tt}} ytxuy{u|ywx}|uyyw}||ywt}wz}}yy~xyuttzu}|yz~xvxu}tw|uy|yz xwx|~uz{tvxwt}yxysu |x z z| | u t ttuut x } ut u y yz~|tw x{u}yx~yt yw}| tu{|xxuyut}z ux{ vxuvtw}t} yxuy{|w}y w}|t xx|}zx x xt utuyw y{wz}| ytw}t tw~ tw |u vwt}yxu vxutvtuwy}u} xxuy{|w}y ttz}|y~xtutux|zz| twxv{}y|yyxttt|tu~tt{w}|xw~}tytzy}}y~xw|xy|yt{ | wtvytuy{|uxyywu}t|}zywt| ywyv |wx{| v|}xw{y ֿ vxuytu yw|w{y y tw| x y{t|x vxu| } zy t 6w yxxyu{vtyw{t}x~ w|x{x|}zyxuux|}xy~u{vyy} w|~xxyy tyw}tx t|utu{}{xyw~xw} xx tx t ytz}y~xtux|t{ | w|{ytu}u tutt xx|uxtuz tz|w ttuu tu t uwx{t| wtu|z xu ut { tw ty{t}ux}~x| v|{wxuu||~ |u y~ {xtxtuy~y|uxyx|yuxwytz yxz|w w| yt vxuv|tw}tt} u u ֿ tw~ ttz|w vxuy~|u ttu |t yxy{t yu wyw}y} tw u } x{}y yw}x} ty|xw|}}| y xxyw}yuutz |twutyuxy{} xt{}xt~uu|yuywx}uy|~xwut|tyttu{{|vx{txwx}{ttuw{y||ut|{t|wuttvxtt{}xzt}yt~xwx{tx}{ t u |x yxzu ttu{t tuu|txtu tu~xw}|~tut v}wyv{v v||z x w |xw}|~ v||z ywt}xuyyy{|u yxxuvwtxwt}wt}}t}ytu}x|tx{ x||wxuyyyw{|uxtutttz xtuwuyt{wy}}yt } xt t yxuy{|w}y wyyx yxuvtw}t} yt tx{ ttzv} v{ y~x t} xuu|txtu xu{|uttwt tux| xu~xw}|~ v}wv{v vxuvxtwv}xtu}vyt}w } ? vxt{} yxz|w uyu{t y 6ֿ x wyyx tut x{}y xz}y~x t uytxuu||zx|xuzuxwy|ttu|{tyxxutyw{~|w(}ֿy|xytx|z?u x uu|w|yu y|~{}zt} xuyu|} t wv|xw| u{ywxu{tt xz|zxu xzw}y~xwz| wyyw yu w z|zzzu|wxxtu|{wt |xv}|ww yvx{uyvu|vx yt? xz~uwyyz}xyt~x| tytu{|t}uxyx yzxxwxx|ww}}y|wv|{y|u |x tty{u xvuuzu u t t |xxxuzu ytut{utu yt|?u }w|uxww}xt{}yu yw{xt{|x{yy~|x{yvt}wyv{xxv|w|xwyzx}xyt~ywxtt{y|x|~x|{u|vx{wtzwyww|}xx~||{y}wxy|~ ux tt u |u uu u t xu u|z t tu t tu |xtu t |
| 0 rs 1/g |r = |r1 , r2 , . . . , rk ? j
j,hs (j )





H (j )

+r r r ?

j,h1 (j ) j,h2 (j ) r0 0 1 2

j,h1 (j ) j,h2 (j ) r0 0 1 2

|

j,hs (j )

g

|

|

can

j,h

s = 1, . . . k

|

j

r| ? g

. . . r |
j

. . . r

j,h

s -1 0 s -1

can

j,hk (j )

0 k

r| ?

(j )

1/g

00 0 |r1 , r2 , . . . , rk . 00 0 r1 , r2 , . . . , rk

00 0 |r1 , r2 , . . . , rs-

k

can

|

S y m(|1 (r1 ) ?

can

| |r ?

j = j r,1h

|

j,h

= S y m(

j,hs (j )

|

1

j

H (j )

1

(j ) j,h2 (j ) r2

j J

k

(|

h H (j )

|2 (r2 ) ?

r| ?

j |j ),

j,hs (j )

j

can

. . . j,kh r

|

r1 , r2 , . . . , rk

j,h

|

+ | =0 | ...
k

j,hs (j )

k

can

(j )

j,hs (j )

|k (rk ) ?

+

d d+1

)

j,h (j ) rs s

b=s+1 k

h1 (j ), h2 (j ), . . . , hk (j ) j,h |l l

0 |rs

g

|

+ |

j,hb (j ) l

jJ

j,hs (j )

J

d+1 J j


'
, l, m , n n

x wyx ttyy wuyy{ yu~xw| | tutt v{
, c+ |n1 , n2 , . . . , nj , . . . j cj |n1 , n2 , . . . , nj , . . .
2

x wyx vut xw}|~ vxuytw~xu x vxuyt tw|yt y x wyyx yw}u|w}t tw u|xutw{x|?
H=
k ,l



?'

w w { xuvtw}t} |w}u|w}t }x| uu|txtuxtwtyxxuyyyy|ttwtxxwz|utt|tut{twx|xxt tz|w xxzu? vut w}tuuy{y}|u twtz v{ tytu xz|w {}x~ tu|xzuy ~ y yyw~ zw|~|utt y t|w~{}yt|wz} ? uvxtu|tut|}yz~txwyyut wxy{yux xzu tu{t v|yx |xw}|u ~ |tu}~xxw yty uuy{|wt}y txu|zu ytu{t tv{ wv ytx{tty{}}yttu{xyxxuyuwx|xyxwxx}uuutyw}z{utu|t}uxtv{xwwtsttuu|x|uztwy{xxuu|yxxw}uytyv|s x w
1$0'??0'? ???0)$?? %(? ? %$0'??$?%?( %?(1?
d f Ed - Ed

vxw}ytx|vyx tuu| x xyu x |u xuvtw}t} twu|z u xyw~}xtwuyu|wyu||uxw~}xt{y t{}twwyz|yy|ztxyzt}yt~wxtwzyyxuuvx{ ttw |tzxw}xywz|| v}wyv{v twzyy xz} |x |ux~x{yt yt|u yu| w u |txx{tyy|uuyxuxtxtxu{y|uzt{y}vxuvtw}t} wt v|ux~|uw} sutwx{w{||wuy|xz|xxxyx~u|xu ttuuu{|y y} u ~{t } |x yw}x}xty|yxw|}{| yxuyvtwy}wt} ytu|utx|w} x{|u tw~ yw|wx~} yt } | uz||w {v}xuvtwx}xtz}|ztyy xyxxx{|u}vyxuyyu~yxtwy|z yyyu{yx| yv}|w|u{xx}|xyx|xux}y|ztu x twt~ xxy?u uxxtuxtu twxtuxt z|wwyvt{uttu}t}yttuuu||z twx vxuvtw}tt} xuyu vyxx z wyy vxuy~|u ttu||{ tttu tt } xxyu vx|xxux tttwtz |u| vxu|vtw}tx}t|tuxuy~{xtuyy~x|uy{wu}|y{yyt}z|wxxw||}zx t|uytyt~w y{}xts ywxx{x|xxxw}wywwttz} u t tw u yy w tu xtt } u ux ttt t wx|tyuxx|x{ux||yvyx}uvx?}t|x}wu}wyuyzx}|xuyv}wyu|y|}tww|yzw|? }xx|zuu|xyx|uvttuwx{tt}|y}}vz|ytw~y|uxxxvxyu x wxu t t tu t } tt u t w | xxtxux v|z|w xuy~|u tw wyw}wywtt} v|twtz ywx{|y t xxyyu x|xxux xw}y { |yw~uxytyut{|zztyv{}twtuyuvtv?xwxxtzwut}w}t|~tuy{}ttwyw|xuzxyv{{|twt}zxyyuxyu~y|~ut|zuytwyut|uwxu}yvtxyytz|uwwyxt|u~yyu~w|t{{xutyyw~~t|uw|y?|zytx} t{ x } x tu v wwz u ~|t u t u xxzu ttu{t y tw x tut x}x| yt tu| t wyx ~y tw~ yuzy~ts
?
j =j
?

d+1

n

j

d+1

S y m(|1 (r1 ) ?

|2 (r2 ) ?

d

j

l

j . . . |k (rk ) ?

j

|

d

j , h s (j )

f 1 2 Ed , Ed , . . . , Edd

n ?

1 2

'

j (n) = n1 + n2 + . . . + n ?

j (n) = 1 ?

j -1

vk vk

,l

vk,l c+ cl + k

p = k | 2m + V1 | l = l , k | V2 | m ,

x tutxy v{

n |n . ??

k , l, m , n

il = j

vk

= (-1)j = (-1)j j d j

++ , l, m , n c l c k c m c n

(n) ? (n) ?

H (j )

(1 - nj )|n1 , n2 , . . . , nj + 1, . . . nj |n1 , n2 , . . . , nj - 1, . . . j r(j ) ? |n = |n1 , n2 , . . . , n ? j d j Ed
L




('
''
S 1 , S 2 , . . . , Sj , . . . ,
0 L

?

?

|x xuvtw}t} w}tu{yw|ty{}t z|z wyv{ wy xxuvtw}t} v}{xt|u tu{|~ |uty twtz tutwzyy{ |utwt tvxuy~{x }}yt yx|?t |utwt vxuyvtw}t} yuuyy txuuuvytww}}tu}|w}ut}x|x|yutw}wyz|{ vx|uuvttww}z}y{yxz}zyx~x|wuyxytuzy{tuuyw|w}x}w|w}|}|xyt|wux ~z w tt tyyu{xt{y~v|wux~twtut}w}t|vw|}wyxt~w|tuttwtttyxtuwy|~{ux{wywz|yw{ty|{y|u|~y|ux}yxwtyu}}xw}tyyuw}t} | y w | y x tut tut t x xxv{{x}t yut{} yu~yutz}y tw}t w}| yxu|tx{yt | v}w|t|t yu wyt |{zy{tw xxt{} u{|~|u wt wx}x| tw xwtxt yu x wyt xx|yx yxuyuyx yttt tz| xws xtuu|tx}zx utwt v|wx~} tutwzyu{tv{ |twy ttuw}tu| tu~yutz t|uttwttu vx}yzuxv|tyu{tt z{ttzu}ttwzy{xytt|t{yyuuuyu{|}yxtty|txvy}wxwvuyy~ yw{y}ttutwt s xtzu t z xx|uuxu|vtxx{ty}t}yxxtwwywtwutwt}}tv|xuyt{uy~yzyt~ztw |wwuxtvv{yy|{yy}t|y~yt{u}ttwstt{xxuy|t{yt xu|{t t } y u y u t } y} v{ |yt w yv{w}yu} tutwz{ txzx|uux vxu|zxtx{ytt |twy ttuw}tuz| u wtxu~v|y{t|zvyuwx}x}t tyzv|{wy}xx~z|w|}u|x~tyxtw|twxv{xyuxytt{u|tztwyttu|ywxt|uyx~}{xwtzuwy|{ytuy|{uty?vxtx{v{| {zw|{ x } yw t } u t tt x tuut tt t z| tut | u t? ttt xwy}|~ utwzy{ w vut }tywyw x|xw}| tt tz{tz}yu wyv{ wy vx |x|z|ty~txt {yzwyy{twxxuutvtwwzty}{|tuvu|uttx|'w} xw|}u|t~wt|~t|uytt}t|}}}vwuxtyt{|tuwx~w|xwux~x} w t {| ttuwt tutwzy{ yy{yvw tv|t utwt z{tz}ts |{zy{t v| w}y tw vtutt tt wu|twtxw}|v~xyutxwzxy}w{xxwxz|xyvxutxwt}yt}w twzy}y}?|ytxw xtt|ywx}|~|t?uw{||{{~x|u|tutytx{|zy|{}t|{vu}|tytytytttw} x y w x u zy w t u u t u t y t u tu t u t v}wvuyx wyt x yw}x} tuu| xx|xutz wt wx}x| vut t xw}|~ ytxuy{yy}| w x{|s xux yy~ tz{ttw v}wv{vt w tutwzy{ ux} | y{x}xywxz}ut{z tz{t w|y|wxx~w|y}}|x|xt|}uuxyz|ttt{|xxuyywuttuw|x{uz|tywyzy{yuu|twz}yuyzt|wt|y yx}w}|t~ u t xyx wtw }y w|xw|}}| y yu ? |{{y}z|? vxuyu| v}wyvux~t yttwtz v{t zx}|ux |wy~ wyyw x~|| tz|w yxuywt}}| y{t tuwxu|twzy{ yuyu tuuyyy vuwuy|xxt wvy||twxtzzx|tuutwzy{ x tutwt |xyw}x} tw |ttty {|zy{t v{y tt } wxzytyu|wxxzu}y~vxxwuy|tyux{wyyvt{|ww}t}tx{v|xutvzt|wz}t}|xw~u}xxt}xyuu|z|ts} t y |xxyu ttxu|{xwxxuu{xttwy||ux~tx|{ u'u||u|xutw{x|? vxuvtw}t} yuuyw}t} tuuyw}wywtt} x x y z}xt vx|yw? yy~ y|~|vut t yxuvtw}t} v}y yxuyx~||u xz|w {txu|xwx~}yutx{x|xzx|xvu{x xtxt wx{tw|tzx{y{t vxu||x { {ts y t y z t yxz|vuxttz }uyvu{x { u{|xxwyu z|vxtyxyuxyyuyxx|xyx|uy|u|xz~||u?x v'x u|yux{t|v x {y}x~ yw}u|w}t }x xxzu xw yt|u tuu| {| y|xz}x vutvtw y{v}ts x xuyw}}xuxxvx|ux{||zxt{|?{y x xzutw|}tt}xwtyuwx}xt|w~ytv}twyt|zx tvuvtw}t}xtztutwtzyu x u t } yw|w} ? z|tw wyv{ wy xxyu xx|xx~ux }}yt |u yx{|t tyw z|t}v{yx}wtywv|uyyvx y{y|xxuywt uw~xwy}w|x~xyvx }uxxw}|}~ytut | wuwyu|t|twz}tx}xx|} v ~ t xtu|tx} xuttt}tvutyuyt}uuuuyw}u|tw}t t yxu|ttxtywuxu yyu v}w|wtxu~}x
Aj |n + |n ? ?
j

|n ?

V1 , V2

0 , 0 , . . . , 1 2

|n = c+ c+ cm cn |n ? ? kl |d+1

2s d

| |n ?

d



=
j

n ?

d |n n? ?

p

k , l , m, n

n ?




? t|t xz| wt| yvtw}|u yxuyt{xs yz|w } x?? } tu|uy? tuy~y w wtt|y{|tww|t}z}|yv~x{}t|}|{z|u{|txwxut|txt|u{vttuyyuw|}zuwy|}{w}swyttutu}uyvxtw|w}ty} {}uyyuy||~}{t}t|zwxz{yy?txxztxwz|yu|x ? xu t } y yt ? } x xu }xxt } z x wyyts uuyw}uxy yu wzyw w xt~w} tz|u? |?x}x|zx|z|zxyww ytwwzy| v}xu|{ wv|}{v}{v tx{z}|uy y}tz}tuwtxtww~y|wy|u z y zw wx t}twtw}u|t{{tz z|tz |tw xw}ttwvt vtxuuyt{}x~ t{xt|u ytz}ttut x tzut ttutt z}y~xz} u } xyw ut t x tt|wt{utz~}y|twtyyv{tw|wy~|~x{wtzxuxty{w|txttxu{wutux|xzx|ztywvtu|x|wvtxwyxvy{|z}xy|~xx{wx|xxt{zxx{t|}utxux{uuyu{y|{u}yt|y{{wt } y y t| v| x
? ? $$??% ??%??? ??%)$($???' &$ %$#?'%?? ? ?%


'

x~|| tw v}w|}|z y}y xywy|yzuxtvx|w}t|w~tzu~~||t xutu{|xxy}yyvvt{w}|tw}}y{{|w}xxuyztuttzx}wy|yx~w}xxx{}||xz|xyu}x|u|tx}xxy{|t} w tx }~ w u u uu t u x yw}tty{t wy||{t tvt xz}y~xwxt{| tw~ xx ? x|xw}|~ vwy~ } y w}tuu|w|{} yuxu|{x} vxuyxxtw xzx|ux tz}y~xw}}|{zz |tuxxyw w|} x|txvx{wyu wytww}twzywt{t}zyytwvzuyzy{x|wwyzwy}yxxzu|xw{wyw|x{}wx|~t tutyt ||wuv?uzyxxuyyxu tt t {yw||ux y|u{yw|y{tz |}| x yxu|t|t yxyv|~{z xzx|yuwx tz}y~x}}|{| | uxyw xxw|}x}t|txtyyttwtz|{tzyt{t |{}z|y{t?}}yvxuwy w|x yw}y|wy|xu|ytyxtywt| ?|utwwt tytvuxyt} |ztwtt yuxyy}t v}wy|y|t|uxuttztt{|~|u ?|tuuutzy{ xuuvttwt}t} tuuyw}t} vxuy{}x~ v{ |ut uy~t vx|yx v|z|w tw~ xwy|? |ttt {zy{t vxuv tw}t} ttu||}utx|w}yxx|yx yxu|txt wy|y|x} tw~ vu}t x|xw}|~ v}wvtu|uw} vu vw}wwx|zyt{u|wx}xx|twxxw'x}vu{tx}|yxxuwv}t|w~}zt|}zytvu}uwy|w}xtw}|x}t|twwyyt{t|s|' vuxxttuu|xwx||tyx{x}} xut x x|utwt x|xt}xt} v} zw}y wvyw utwzy{ tutwt xxuyx{} xs z|w twu|yxxw|}}w|xt}yxtttu{tt xxz||wyx|x{zyyx{utyuy|?tttuttwzty{tw|xtt xxvuxyuuvxtwy}?t}ty{x|~s|u w } ~yyuywtut| tutwt yxuyyx x v{t tyuy}u wyu tz{tw x{t}uytwttutzuyu{|tyxxu}vztxwt}yy{y{xtux|utu|xzyw{}}| t||yuw|}|x}x}v}wtyw|~xv}ww|x~}}{|ttu|~uxt{t|wu~ z|w tw x |xuvtw}t} v}wxtt|u tw| x ttxtw yxuy~{ x tw~ wxwx~} yt tw~ z|w tzx{y tu~tw u vw}xtw}|w~yyuuyuvt|wzvy}{wv{x yt}tttzw}tu{||xutyuw~tyutwz|t|}yxwxxuyt{}txu~uytx{|zx{tz}yttwu}tywtt xu|tx{yt t| ttu w|tw}wywtt} tu{t tutwzy{ vxuyx vxu|tx{yt | tu w}tuwvty tu| } ' y||{} x tut w| y u{t xxzu vttu{ty xxzy {{x} tu{ywxw}y xxw}tut~yutz}y zw v}wxyw} tu{|utxtt yttwtz tuyy{yx{}y y z v}w}wxywv}xwyy~{vxxuvy~y{x |wy}{uwyvty t~~wt|uxwv'x vxyuy{|w}|yxxzt|z|uxx y tt u tu t u w} ut yt tu }t ut t u tt{t v|xxzuv{vw|yz}yw~}xwwy}xwtytw}z|{y|wxtyut} xu{yyw{x}tu||w}t|wtwt}zy} yxu|v}twx}t|}ytvu}tzt|z}y~xyy} x wyyx |utwt x |tw| yxuvtw}t} ytuw}yt} yt|z y
S
j

Sj =

1 j

(|

|0 +

j

|

|r ,

'

1s

|0

1s

j

r Rc

|r = exp(ir )(c rt)

2s

r =1

(|0 + |1 )

(R)

1s

j


6'

'??+ 'Ǟ? ???????? ? ? ??? ͜?+ ?'? ? ??? ????' '?? ? Ü+??R ? ?R+ +????? +??' ???????+?+???????'?+$ #?+? u x{t y| } tutwt u~x |xwy}uxyu|z}t}yxyvx{uy|tt{tv{} ~uuw||wv}}yt{|t vxuyuww}}y}} yz||w?t}ytxxuv{zyw{t xtu t }t u t u |t tt t t y ts tut t x xxzwu}| uuuw{t|wtyuv}z|wz|vt}w||xw}w||}y|xutxt{w}~y xw{yx~|xuxt|xzy}xy~xwvzu|yxuxytx{utz}yt txw{ ~ yt| | yxwvut wxt}xt |tz y|~{} tzyt{ttwy|tu|{ vut t v}w|ytyy wt uttuwzy{ t v||z}t}xz|yt?||{uvtvxu}yuv|}wt|}uyztu}uxyxtwu}wytwtxt{}xxtyyuv|t{vtxuvxt{wx}tu}twuttwxx|w} x u } ux | t}v}vwxvu|t}x|uttx{ xx{t tu|wxu vy t utwzy{ tw~ w|wx~} tut tutwt xuvtw} } x xw}tu~tw t v}wxwt| tzx{} yu yt |tz v|~{} xtu t vxuyz{}wx~t twxwtyxutyutt}{tt}xuytyu}t|twytwzy{ut|zytt?twtztwyxtu{}t|{t}tuw||wxtuxxwwy~txt} tw||{uy}y}yv|tuy y zyw | |yuwv}v}xxv~xtu||ttuttwx{xytxuvttuw~}ttw}|wt}ut|vz}u|t{tx{yytzttz}xyx~xw|z|yxty{tywu||zw}xuu{ttvv{ wu x | xv w tw xs utx x {z{t xy{}t x tttw tw| vut tty x|txw}|~ utu xt|xxw}u|t wzzyxw}xwt|wt}w|t|twvu|wz|wtxu~}}|{xxtu}|vt}yxv{uyyt}txuxv{|zyyx{tt xutuuyuxt~tvuuttty{x y { w y u xu xw}|~ yx|yt}{t ytuuyw}u|w}t | yw}u|w}t xuu|tx}zx| v}w|wx~} yut xx}|~|xw}~ uy~xwt}w|~vtuuttz tz{txww}}|y~v{v}wxxvuy{txtyuw}|twuxu| |u~yyuvt{z}x~twytwx~{}yxtxw s | y? wxtt}{| yuxwuzyy yyuwz|w}| x|u yy||xw||}}|y y~ }}y yuyt{ywx{}x~ |xy{yt{xu|uu|}xxvtxxx~xux|{|txtxzz}wy}~yxwzy|y{ttyyxwu~|y|{txttwttzz{t|z}yywuzwyu|{{y|y y u t |
m

1$0')%('& %???%?

?

t xz}y~x}}|{z tuuyx w|}xt y~{ yw}x} tuu|w| tu{x} ttwu|z tu~xw}|~ttu| t xw}|~ yu{ywt |tz xx|wx} utt|x}w}y t  y z|z yu | xxyu tuxw{|xwuyw y xtw}t|u{xxuwx}zz|yzxww}}yyt{w} z|yzwxxxwz}}|y~~x}}{zxt|tyuvv{{yytttuwzy|w}|x}vtxuu~|xwt}|x~{tyutuuw x}utvut}tt} | wtyu z xw}yxxwyt tw yyx|? xw}tuu||wx| x} w}yt} utw}u|z ztxztv{ tt} yyvx{uy|wy}y| }xuvy{yuyyyt|wyyxuxwu}y|~ttwyvxwv}t|w~}twxz}x|yxxw|y~twxvxuuytw}|t} xx ~t u t |}zx t{ x y|~{} tw x wyyx x yyt x yxuvtw}t} u|w| xut y x{}? |x vxuvtw}t} ttwzy xt} v}wy|}xt v | z |x|z u|ww| ywu xyw{}~ yx x{}y tu{ywxww}yy x{yt tuzv}y~xt|yuxu| }v tut x}yw}y|yw|~u~xw}uw{tyu|~txyw}|}|~ttuyut|uv}twz{|xzx{x}ytwwuz}y|y{twywx}yxtuw}|tztxtv{wxyuu|utzww}x}xttwxzttw{uw||wzzy|yw}tyzyz tw } | {t y t xz y u |u |tw ? wyxzt{ ty tw~ twt wyt yu wxt}xt vxu|tx{yuuw ttz|zxu yt}t tz} y~xwxt{}| xy tw tzttyy t}zt} uy~xt}x|v u{|xuywt uuyt{t|} tv t v}wxwt|xwt|~|uxuwttw~}}z||w tzx}{y}~?x}|{|zztxzx|uxt{ytwtu|xzxuw|t}xt|uyxywtyu|{|xxz}yy~xw|twtw|
N j |
c om

=

j,j =0 N

|

k

j,j |1 j =

j =0 N

k |k , k = 1, 2 jj

|2 j


?

'

xy } xtt x xxu|tx{xyxtw x }}y yu{ywx{}x~ w|tt}| tututz wy|tt twz~|z{}z} wwtww tz{xt}wxw|yyyux y}|x}yxy}v |u xw}|~ yxuy{y| w|y{|ux xut }v w ~x?{yt x|xvt x|xw}|~ xy} y |xu tt t t |tt t t | y x w{}{xw}tuuw|xw{| wwzyy{y}u yy||wx~w}x yxutyw~xxst{|x{txzuxt{}} xuvttwwt}zt}vxzwu}yu~||uw}yt u w y {x yx yy x y yw yy~ xw}|~ yw}x} xzx| x vx |}x| xt}|y~xw|y tttwu|z yxuyu|tu~twtuv}twuv{v x{y vxuytw}t |{x|ux|u yu uutwtxt{{tu|z t t ty||wu v}xu}y~|uy tttu{yytttwtzyu yuy {t t} twtz |wx{w|xxxttuy w}}uvxwuzvyt}yt yxxwx}w|z~||wy~x{u}~vx|xtwtzyw|yxtxyxz{zy|ywttw}tytt|tzvxwyy|||xx~tywv~|}yxwxt||zxvx|t~t{x|wut||t~uux| v t x y uxzyywt t u t ut w u t u | y u u } tu t t u t |z{tw v}wyw{t}x yw}x} tz|w xx{t tu|wxu vxu|}xt v{ vut tyy{} |xw} ~ |ut z|z v}wywz|w |xxuvtw}t} tuu|w|} v}wxvt|uu vut tttz|uxtt xw~}|~ }yv |u {uy|tut}tttwzxxt{|wxtww}~{xttuuxuyyww}}yx}xwuv|t}tx}wzytw|yzxxtuwyu{||zw}ytwztzv}{yxy|w}xx} u~xw}|~ttu yxuyyt w|tx{yt ttuu|x |wxt{|ttz}y~x}}|{z x u{ywxx{ tx x{|}xt ? yzxu|y ttwwu|zy tuutxx|w yuyy| {tw{tz{}||ywutz t|xttwu|z t xtw~yxt{xtw|}w|}|ywyy~xxuu|wtz~wy}xyt}xyw}~txy}|}|y{uz|t|utttutz||wu}xy}y}z|uutuyywxtuxw~}wyttt}vtuw}|{u | t tu t x tx z |u xw}tut{} t{ yw}x} vxu|tx{yt w}tutt { } u|yt|utt|tuwv}ywxyuyt|{tytwtyztuyz|x{?xzw|xttt|ttttzz}}yy~~xx}|w{xztt{t|uxxywzxyyxtxxwuvyu~ty|utux{||?w x orqo?? wu|w}utz t|u wxu| tw~ z|w wzw} tu{|zt{ tyx{yt wt v|xv}x| |wu|w}utz |w|uxttz v|uuyw}zu|w}t y |{|xuywt ttvt wyv w {|xuuyw xtvyxxuzv}tyw~}xt}}}|{wzxwwy{ty{xtxxuuv|tuwt}xt}|wx}|uwu}tt}wzy{xxvxtz|wyyv|{ t}y|{x|uywxyuw} v | x t{ xz zy~x}}|{z t uu| xzu| wx|w}t} tu|tt}y{y |t |yyuy |ut? xxuyx{x ||uz yv yxw}ytx| ytz}y~x}}|{wz xv{yyt {|xuywt z|z tz{tw w|xw|}}| t w tuuty}t|}~ vxuvtw}t} yuutwzy{ wtt | yuv |xw |wzy?t {yvw z v}wtx}tuwt tw xw} t x}x|z~}y~xu}}u|y{zvx|z}yx~wx}tzx|{twyuuyxyttx}|||w{}}uxx~t{tt{|yxvux|u}vxx{tttwwy~yttz{tw w { } t ywxv|{w}t}xtu|xtv{tux|t}xz{u|xxz|xtxuutvwttwt}vxxuyxzt|{wtxy|z}yv}xx{xxw}ttuutx}wuzyyw{ux |yzy| vu t u u y | u v }} }w |t|xuv}yvtuxt|ut{{x?tuwttyyzwx||uvvx}uvtwutw}uy{uuywx}|wtw|~xxxtuttwyzy{} xvuvx}vt} | | u } } tw v x xw}wywtt} x|{{|w}xz x ||{zy{t yv vxuyt{t x~tw} t uu| xzu| x}wv}tuwt xw tw  |vvu~tw tuuyyw}t wxtu| {}} w}y txx t ytwtz uu| zvu}|w|wt}yxtz|tuwvt}wxuttw~w|x}yutwyxwxz|}y|~}x}y|ux wtuu|wtt}zut|tutu|vxu|wtx{xyw}t|~ yxt t x x u xz x w yw}|~tuywyz{yt{ yyutyxuyvy}x} }w}{t~||uyv|{ tuutu| xvzxuu||z}yz}|xw |yw}xtuvxxu}vutyww}utx} xuxuz|| t t x wt t x~zu| yv xx|}zx x xuvtw}t} uutwzu{ uu| vxxzuu|| w|t{t}x t|u x~||u wxyxwwy v|tyw{txz|||ywt|ztsxt zyxtuyyttywu|yzz{yt|ywttyy|xxxx|yw{}t|v}xwx~x{t|xxx{ywtyv|{ytu|txzyvt}tx~v yt{ yu { w y t t u ttu x t ut t u v u { u
t
2

k x2

x

t

k k


'

?

?

?

??

?

?

?

?

?

?

?

??

?

?

?

?

?

?

?? ?

?

?? ?

?? ? ?
?

?

?

?

?

?

?

?

?

?

?

??

?

?

??

?

?

?

?

?

?

?

???

|z|? ? v{t ytwu| tzx? t{tt? ? ? ? 6 6 6 ? ? ?? ? ? ? ? ?  '  ? ? ? ?  ?  ? ? ? ? ? ? ?? ? ? '  ? ? ?? ? ? ?  ??  ?   6''  ? ? ? ? ? ? ?   ??  ?   ? ?? ? ? ??? ?? ?? ֿ  ( ??? |zx|uxtwzy{ v|twu| xzyw}yy? yyx ? ' z}y?|z}t? w}tu{|y x y|u ?ywtz ytwu| uxzt yx{| ??  ?or ??r? yyu|| {yt x vt yu ytt tz}y~xwxt{| x x } y tuwx xw wvz{|w}y tut z|ztttu||zt vxuyt{xyz yyyxzxx uw}yx tttz x tu| xuytxt}z|uwu|yuxw}ytwtyz x{xyw}xw}}uy{y|{xtyxuxutyutwt xt}~yw|xyw|xzxwwywtttyxu{yxtuxy} xu vuu|t} x xxzu ttu{t xx|wyywux tuw}tuwvty ?txux v}w|}|z xu x }tt tuuyyw}uxy v}xwtv{v yu ytwtz}yx uyu{ywx{}x~ x{ ty} tw |txu|y ty|tzy}~xxtxxytzt|zwytuuy~xtu|}|yzw|}zx}w|xwt|{}}z|xvuyu{yyxyxtzy}~uxww}|~vxyyuyttx{xxzxwtux wy}| ~ v|u{|~|u yytz} tw xxtyw ttwu|uz yx|wyywux v}wyv{yv yuz|tt uu|}xt}tw~ yu zty~u|t}styttzwt tz}y~xwxt{| x xzxwy ttx{zyty} vxuy{|w}y w ut x |vz ' xxu|yt{xs v xxuyty{xswu|~ux|uy? vxtwzy|w t t{|ywux y} u t|vxuy{ttw}}xtw{x{xttwu|tyzu }xwy|vw{yvyuwxx{|x|wuy|y{zttxx| yyuu|y yu y |wx { v{ywx|{|{|xuvxyuw}{w| x {} xx} z y wyttzxx||uxyutwytwu|yzy|xwu|y}xxu|t}xtuty{t|z}|wyxvuxuywyy|xtxxx|zu}xy~xwu|tw{|u|u vxuyy| yy tz x x t|xxtyx{zx|x tu||zt wy tw~ xxuyt{ty u {ywxu{t xzv} y ytz}y~x}}|{z xzx|uxu vtxu|}st t ytytv{twu|{xz|xwyyt|z{}ytuvtzxyyy}yuttzt|u?xtxzu}yy~xt}{}x|{s zxxw}ytuzw|vwtu||tzttwuw|yzyxtuwy~yyvx}uy{}uxvw~{ z ty | y|y t tyuuy}|w}yyy}tyyxuyuz|wvwxzty}|}wy}u|ywu}t{ywxt{t}x}~y~xtxw xtvww}|{yxtuyx~{uz}|x |vuvy}wzt x z ~ t x z u t vxu|}xt ttz|tzxu wyyw yu x vx|tx{y ttuy|t y{}} w}t uy~txw}tuwuyyt~ y t t t w yyzuyyutzw|zvx}ut|uwtvtx{yytvt{}xvyyu|xwxu{vxty|}}|w}yy~|wuyv{tv}yytx|xtx{yyt y uy y w v u |y|t{||{yus~txw}}uwuuvytwy}tt}zyyxuv|xu|}|txw{tywt}y ttwtuxx{zv|{|uy~ttw|w}|t|yux~xyxt?x{ y tx y yx ~xt}w|u~ x{ xz}yw~x t u w tu t
1 T

T


?


? ?? ?

?

?

?

?

?

?

??

?

??

??
?

?

?

?

?

??

?

?

?

?

?

??

?

?

?

?

?

?

? ? ? ?? ?

?

??

?

??

?

?? 

?

?

?

?

?

?

?

?? ?

?? ?

?

?

?

?

?

?

? ? ? ??  ? ? ?

?

?

?

?

?

?

???

?

?

?

??



?

?

?

?

??

??

??

?

??

?

?

?

?

?

?

??

?

?

??

?

?

??

?

?

?

?

?

?

?

?

??

?

?

?

?

??

?

??

?

?



?

??

? ?



?

?

?

?

?

?

? ???

?

?

??

?

?

?

??
?

6  ? ?  ? ? ?? ? ?  ? ? ?  ? ? ?? ??? ??  ( '  ? ſ ?  ? ??  ?? Ł ?  ? ??? ? ? ??? ? ? ? ?  6'(  ? ' (6 (6 ?   6( (    ?  ?? ? ? ? ? ? ? ??? ? '   ? ? ? ?  ? ? ? ??? ' ? ?  ? ? ? ?  ? ? ???  ??'   (' '  ? ? ?? ?  ??? ? ? ? ? ?? ? ?? ?? ? ??? x } t yxuy{|' } yxvxxzz}|ytxz?zy~xtzx|y?u {|x|yytw}txw}ux wy|z?xsxtuv?z?{ ? xy | t u x|{ y ( ? ? Ł ? ? ſ ?   x wx twu|z uyux{yu xuvtw}t} uwuyytzt y6} t wy? u yw6{6yz?x? wtx||y?} } ?  u ' ' ?? ? ? ?? ?Ł  ? ?  ?  ' ('' ( ( ? ?? ? ?? ? ? ? ?? ?  ?? ? ? ? ? ?  ' ' wx{w|x? ? ? w}tuwuyytzy x vxuyyx ytwu| xz}uy? ?  ? ?? ?? ??? ?? ( ? ? ?? ??? ? ?? ?? ?? ? ? ??  ? ??6 ? ?? '? ?????ſ ??ſ Łſ   (' (  ? 2 ? ?? ?? ? ?? ? ?? ? ?  ?  ? ? ? ? ſ ? ? ?ſ 6  ֿ ( 6 ? ? ?
? ? ? ?


vx uww}}tyu|wuytuy~{yutzxv}wwtyy}}x|}|uxzxw|}uxs tzt}yt~wxu}|}|z{zz xxzuxyu|u|x vvxxuu|y}xyt vt{tz}y~}t}||{z }wtt|wyt x yyy v{ t{x| ytz}y~xwz| y|~t{t |}w xxtwzy|w xxuuyuywx wt{|ux| v}wyv}uy u tyzx}yw~}x}y}|{zw}zy xtwwyy tzy{xwwvyy{xtxxttww~zy|yww }|tuz}vy{~x}}u|{tzyxuvyx|tzy|xzxt|x{yxwuu|?x t x y x v xxxtwzty|vw v}{wty|x|wu}yvyxtzz}uy~?}}|{z ytu~t |}|{vxuxyuv{| tttu{ty{|{xu|yuwxt?tyuxvz{u|z} { ywx}wx}}|~|yx|{|} |{yyx|u xxyuz tu{|xuywwtxxtz}yy~xwyuxzvxw}twuzy|| wwy y} } u ~ u t t u t t t
1 2 i S [x] Dx, S = h

'

?? ?

?

| x {|ywux z|z v}wy|}x| yz~tw z xz~tw wt xw}|~ |tyy |wx{| |uxyxw xwt ?? } vxtwzy|w t {|ywux yxz}tu|uy yy~ xzxu| yytttwu||uyzt|wz}t{|{ tww}tuwuyy{yytzxy||uvv{{yw||z~tw v|tw|w} v|~{x|u tw~ v}wy||z { {{|u vxw}w}x| wt t{y}x|yyu xzx|ux |yxu|}|xtttzt}w~ yy|zwwz|ywutvz||w}u ? wx{yx|v{}t|w|w |xzzxzz| xxw}tuwvtv{zvyx{uy{}~x|~| {x yw u tut t u t t t uu ttuu t vy{yxyw}xtz|y{ tyuy|wtuy~ttwx~w}ttuww}||~t xywt}ttuwu||zzu~yuxtwz}|y~tvwxyu||}xxttuz}|y~txu}}}t|{xzw|ty|xt x u xvw{ywx{}x~ } xuyvu|} t |w}yxy {yx twtz xx{t ttwu|z vxu|t vx{uytt}t}twtytuw{x|~w{tx x{y } wx{| twu|z yxwvut yxxxt ywt| tuu| }wv{uyt ttuuyw}}z}x tu{tt tt } |uyu{t |{ |~|| |w yxxu|tyz}{twt}xtyutvtz}wy|~x}|{z}|uxyw yw}tuwvty vxuy{}x~ |ut? {t ttwu|z x u tywt| uyy| x{ wx{| wu| y~ wxv{ xx { |wx xvxzy{y|w|}yu|u}t tyzy|w zuw||tz|zt|wt|wztzuuvyxw|}}yxw}ty t utz}ty~x}}|{z wt tyty ttuwt xuxu|u}xt tz}y~x}}|t{z zttzx{ uy~t wtx{u | twu|uz vtt } xx{t ttwu|z yxu|}xt ty} txts vxtwzy|w t |{|ywux xz}tu|uy |u w x}{y|tu}t twtz x u~ xx yy{tx|u v}wyv{|w}y twtz w }y y|z ? w {| wu|z xw}y w}y twxxty} twu|z tyvt vxuvtw}t} wx{| y| tu wyut||xuxytwx~|xwu|z|zt twyz||zztyyuuy|uyuxw}ytu{ywxw}y tuuy{}}t yy{t |u|{y} w twt|? yzxxwttwuz|z tuutxx|w y wy|wu} xwz}}yw~txwz|tw|tuztwtyuuy|x|xyywttzttzt}yt~wxu|wxzt{yxxv||wu}y|}yywx}uy||tz}||wuvuwxy|wwy|w|t{z| t u yywtz|w||wxtxu{v|twt}ty}||{u}xw~}tu}txuwuyy~x x tu{yw|~ty{} | xuvtw}t} x}x| ww}|~ } w z u|t wy|xzy}yt |yt|w tuuuy? xuvtw} t t t } u}yvw}}yy|x~{{tttt{wuz|zwvxu{yyx|wyw}yt{}w}tttwuttxxtzzty}t|uyyt |uvu}ywyy|xxst tu t ?? ?? ? ???? ?? ??? ?? ? ?????? yxuytt} ytuw}|~ yyy? 
?

x

L = Ekin - Epo

t

t

Ek

K (2; 1) =

in

=

p2 2m

x(t)

ex p

S

p Epot = V (x)

1 = (t1 , x1 )

t0

t1

2 = (t2 , x2 )

L(xt , x, t) dt

S x

K

=0


'

?

? +?????R?' ?R?? ? ????? ?Ǟ????? Ü'??? ??Ü+ ?++????''????R?????????????R????' ++??????R+????R??'??????????ž????+????R'?????????'??+????????R?+'?++?????Ü??+?????+??????R'????ɞ???'??'??????'???4?????+R? ??? ' ' ? ? ? ? ? ??????+??? ??????+???? ????????+??+?+????R?' ?++R+??????????+? ??+Ɯ?+???+????+??+?????????R??' ???+???????+??????????? ??+???R ????????? Ü+????? ???+??? ?'???????????+????'?+'?'??+??'++??'??????'?? ? ? R ???????+?'??'?R'%+?????+?%'+?+ʜ??+Ǟ???????Þ???+? ?ž??'?+R???+R????R+??R??+'R?'????????+??+???????R?????????!%ǜ?? ? ? ' ? Üʞ??+ ? ???? '???? ' ?+R?? ?+? ?+ ??+' ?????? Ü+??R? ??? ? +?+?' ??????+?? ????????R ????? +?Ȟ???'+????+'+??R+????+????!+??R?????'????????+???+???? ???R??????Ü???+???'???????Rǜ'?+'R????!? vx z wuy~|u w|xux wyt twtx yx|y|wy{||z|yww{}w}yy xtzuxuw|}xvy}wz|wu||ztyuxz{||wxwy|}|uttyzyx|zy}|utxt}|xww|tt}}|{t} |w | u } tu t x t t y yy~ y|~|utt wx{|yy~y{}yx|~y|u|t|tz|yw}ttwtz|} w twttzt yw}y~|z yxz|xw}xxwz}|| yu{|wxy} x t t v|}uwx|y|x{tz?wxyzzyx?w}ytxttwwztz}w|xyuvyy{z|wy}~wxyuy|ut|xt}uy{||zw|y zyyy{wty}uw|}uw|}ztw}twt{uyy}|ywuw|xvxxwttt}xz}|ux| t t y |y | x z } w
? ??? ??????


yx vxuytuz{tw} t | tz|ux }}t| x y} tw~ xt{ty |{|~|u} tu xuytuz{tw} xy x }}| xxwvu xyxwx}txwt}zt}uxtt}xxuyytyuzw{}tyw}}|xux vvuuyxwzxtwu}|xzzv{}tzw}|tsz{|?x}zxww}xtywxz|y||yut{|y{uyyu} xw |wt|wtzyu wyy~txw vxxuwy}{|y~yyu|wxzwx}ux~w|tu|twv~xuy~yw}ytt v{ ??xuy~w|xu{utwuvz wv|xw|zuy } | | |{ t } tx x u z ut |{y|~|uxtt|}twxtztw|zyx|w}w|~xvz||uwxtw{z}x~ tvw|~uuyw{|ywxy~}tyv}wyx?xwu{x} tt|~zxwt}|~xtututy~zxv}|yu |~ u } w y xt{?}yyy~w}{x}}tzu~~xxww}}||~~ttu yxx{w}ywy|tt}}(v'xuyxx' {xtz||wxxw{||xutt yxtwyy|{w | | wt z x ty ttt ~xw}|~ t t|u ttu u | wx{ | twt|ztxwv| y y w vx y~{y ty{ywx~| yu tyzu{wwyytyuwutt{}tywy{txxuyyy|z}tt tuuuyuxyw}yt}tywu w}u|tuuxv{wyu}|xu||xuwuy|t|{uttxuy|uzuuvtx |t xuw
('
t
1

u xzu tu{t yy~ v}wy|| xuyy wuyt xw}|~y{y|tuvtxzuwyv|ttu{yt tywy tyv ' yt{t t x{x {}x~ x{}? xxz}y~x}w}|{z wu|w} |{z ty{uyu yx| xxtw z|yw|~y{txwwu|zy}{t|wu~xtzwzytw}twxtwyxytxvxvuxy}|{yt}vxxu|yu~|tutwvxu{t{xx} tw ??wy|xutvtxw}yt} x u |yzyuyw|u{xxxt|tuuzxt{}ywt}yytwv|}twzzx}ytu|}w|}{z wttzx{ztyw{|z|zvx}twwxzyt|?wxxty}wyut{wvtyw|zy}zy~z}}{tz}{zt|~|t vw w { ~x } yx t{ x |{ z s u t t t t u x u t u }wttt|u ? |u vxzu v|tu{t tw~ z|w u|t}z| wxv tut tw x w|uxttz yxuy{yy}| ytu{|~ y|x?|}xxzxty|tuzx}y~x}u||{}zxt ttwx{? z|wtytuyu|t|xuw}wyxt{t}}y uxw|z uy{wx}x|}ywy|u x} } w u
h S = Lt S h
p ? (x) exp(i h x) ? ?

t

(t2 , x2 ) =

t

x(), v ()

t

K (t2 , x2 ; t1 , x1 )(t1 , x1 )dx1 .



S

t

2

x

() S

h

x(t)




' ?'

R? Ǟ??+? ??? Ü?' ?R ?? ?+????? +R?? ?????? ??R ? ? Ü+3 xxzu yxuy~|u v}yyx~{ts yz v}wvt|u wuyt ytwtz |z y} v{ |{}x~ u yy} yyw yx}ts t{ z yx{ w t{{tyxxuy|t} tuxwuzyw}y wzw|{wyx}}yttuzwtu{xt|twwxz{ywxz}|xyy xx|{{x}vyv}}ywxuw|||xx|}|{|zu}xuxtt{t}y ?y}y?txx{|wv}wyy|xyuyyyyxuxt|uxtw} { w x w x z z } yw|x~} uyx y | } tyw~tt}utxwwtx vux}yxuy}w|xu{yyyut{uytztwtwuu|zuxyw{}}tu~tz} yttw?{u|y|wtuu|z{ztwt}tux|yu|{xuwyt}y y t |tuy ~ u x t x vx y


| xz|y yx xtt vxuytuz{tw} xw}|~ |z zxw}xywz|| yxu|t|tysx tz}y~xt{tut y{}} tu tw}uy| y tw xuyy wt tyv}x| |u|xutw{x|yw|u~yu{t}}z|
C (M )t
3

|| t t y|~{} w~ x|t{} xuwx}uy||z y}}t{vtu{{}tx~wyw}tu~tw tw }tuxu}ttwu~xw|} |tytuuuxvtw}? } w {t y u t vxx{uy||xxw}ytxutyzyv}wt|{ty tuuyx zx|uzy~|wux{yy}yvt{twt~w~twvxuwyxttuwzt{}tw|}~|x| ||z|?w } } yy~ w|~|utt y wx{}x~twtzyxtzu tux{ty~xux w|wx{xz|xyx|uy{yy}| yyw}wywtt} y{t tw t w y |{ } |y yw x{} v}w|z} w}|~ wy }zyux t u t} ? t}t yxuyxtuyyw}ttzy}xy~w|uwxuytttz xzxwt}wxy~ywz|xw|}|{tut{xy|yx}w|y}}}y|z|w||zyw}ttu|~utwx}~|tyv u{
j, k , s {0, 1, . . . , r - 1} |
K ,

'
m

ut yz xw}|~ tu{|y vxuyt|u |wx{| tw~ y|wx~} tw |z~tw xw|}| y v u y} t wx{| twu|z y}|xyxuzyy~tt{wtxtu|x|vtw|}t}x{}y t|ztwxz|{| vxuyt{t xw y} ut w | }~ t |twtx{?y|t{tu|tz xuytw{}tu~tw } v}wwvtyw}wxywtx}w}w|y~tu{xwy|w|tuux{|tytz wx}uys twt|uxutz { tt t t w ?y t| ww|xu}xx|ttzvvt{yxuyt~t|utwt}wytztu?t|t}wxuz|wyz|}yv|zu{u||z}wxx}|twy}wtx}{ww}|x{w}|yt|w}u|xz tw}u|y vx t t wt } xw xzxw|}{x|~y|wuz| | yxzy}yy~~x|zux}xywz||x|u{|xyxz}}y~ty|u x|u~tw} wz||~z|uv}twt|~|yutt||z z x
() = () x(), v () Cl,,n,
m

M

: x()C

i ex p - H t , h

6'

v1 , x1 , 1 , t1 ; v2 , x2 , 2 - v1 , x1 , 1 ; v2 , x2 , 2 , ?? ?? ?? ??

? xz|w wy x |{}x~ xw v} ww|ty t|t xzy|zx{yyt yy~t |vwxuu|vtzw}tt}ttttwtxut|tzxyxuyv{xyuyytutz{tw{}ttu|{y'xuy{ytt}

z yxuy{yy |z xxuytu{tw|} y yy|tzxvxy|}{}x~wwx{|yx yx
t1 , t
2

()



=

j

Sj = Lj tj , Lj = Ek

j

|

t

j = j e

K(t) ,

i h

S

, l,n,m

j

, j = 1, 2,

().



in

t |

Cl,,n,

- Epo

, ,

K(t) ,

m

( / )3

t



l + j, m + k , n + s

=r , r

t

()

K

Cl,,n,

()

x






? +? Ü?'? '???? ? R?? ? Ü??+? ??+??R ??R??????? ???+? ??????????R??R????+ +? ? '?+?+??R?? ! ? '??'?'??'? +????????R?????'???+R??4?++????R?''??? ̜??+?????????R ??' ?????"5 ?? ???????͜??+5 ????Ĝ?????+?R??+???R??+?????????????????+ǞR??R?????5????+?R???????5?+??Ü????R!????????????++?R????RR?''???!?'??"+ ?? + ? ? ? '?? ? '? ? ?????????????????Ü++????+'???+???+????+????Rʜ?R? ??Ǟ???ɜ???R?'?+?+4??+ʞ??'R?R???????'??????' ??????????????????+??????"+ ? ? v~ w }wx{xtw}||~utt uwut}xzzu~tw txuw{}t|~xt{w}xv~xuy(|'ut{xw}ttuvtyyxu x ty t y
2 ih(t2 - t1 ) m
-1/2

v | ~ ~|{ tw ? xw y?tu|?{|utxxw}|t|u~ttuwxy{~}|tyxu}yw{xy~y}{t}|twtxtt}}v{|yyy'y{w|xtzyttztww}tx}u|}{t|xu}t?wy}v}yv y x t u tuy tu t tu tt v{ t t u xwt utt t txz tt}}| x wyyx x tttwtz yxuyy v|uuvtw}t y
C t x 1 C u xx , 2

?

?xw }}tuutttt{yw xuyu| v}wy?vux~wt w}tuwt{ |w |tw wuyt yz~tywx|xw|}}|}~y |{}x~ wzt}wz{|w}wy} x w}|sz|zw?|x|}}v}x|uyuxttz}ywt}ux}tz|{wyttzxz{|yz tt ttuu xt tt t ttu tu t x{}xxu~y}xt}yy}tuxytxuz|{tyw}|{|xxu}ywxuvy}uwy|v{}v|tzx{|yz {w}|~t{||xtuyt|ux{uuy{}w?}u xxy|t~wz||~ vx y w x wx } w yxxuytx tzy}y~xwzty|{ v}yxvt|u xw}{|~ yw}x} xwt}w}|x yxxy{} wvtw}t} {t y y{ytv|tuy{wxtwttuz?twt}{y|w~yv{u}yt{w}}ux|txtt}w~}ytwtuwv{ytt{}x~wuyxytu}tzz yzwuxuyt{}xx yu w y||| yy{} v||uuytt}}v||uw{y|x}vzx||wx}|ww{| wu|z v}xwyy|xwuw}xywt}t|wwy}xuy||twwvxx{uyw{|}xt~w~wwx}t{tt x w ~ tu~ w t tt vyxtxzy}|?w t ||{{|}x~ywx{ttt}ttuy|uuyyyt~ty{u| wuwwyyz}xvxvu}ywvx{{vx tw|t{ywt}uxzx|wz?' w{ | y y uz u t u t ttu{t yy yxuyx{x v}wyv{v tw~ xxt{} x wuyt xw}|~ xxzu xtu{t yxuy~|uytytuuywx{x |u w}|u w}y tw ('x{|t uyxu{y|x|{tx wy |vuy{}}x~yz|~twy|}|wuy|wttt}w ty?zyxxuyy|tu{t|}||{{wuytzyu{|~|~uuy|zy}v{x{}y y|{~y}x{~} x tw ' yz~tw xw}wywtt} z x|t |z y} t {y}xy~ yxu|tx} z|z w||xww}}t|y~utu|u|~yxw{w}}}t|ut{uxttuy{yyy{|{t tywx|v{w}||yxtuwyyux|}xyzxy{}y|xx~x{twz|wywyx{w}uywy{yy|w~x{w}}yt tx t y x{} |x xu ' t t}u t t v u { u~ } t t w z
| |
K(t2 ) , K ,

()

t

1

()

t

2

1 x - x0 ex p - u= 2C t 2 C t





exp - x2 2

u(x, 0) = (x - x0 )

ut =



|

|()| = g

K(t1 ) ,

u(x, t)

.

im(x2 - x1 )2 2h(t2 - t1 )

'

K (x2 , t2 , x1 , t1 ) =

1


x t

(y , t) Ct

exp -x2

1+

2 h2 t2 m2

ex p



.

u t

x x+y u(x, t + t1 ) =

g

.

R

()

u(x - y , t)(t1 , y )dy t1

?

w tu{| y}|tuwt{twx

wutxtt vv|{u|wy}tts


(

Rœ ? + ???R?? +? '??????? '++R???? ? Ü?????? ? ? ? ?? ÜR??? +?? ??+ Ü?'ʞ ??!"?" t yx{yt t| {}x~ ytu| x |tw|w}xux| v{ tz{tw {}} yyyx vy tw tu yuyx x vy yt wy|xu||{x ttw |z ty?} t? t{|z t} xxtw}wywtt} w|y~yw vy w||ztxutwtuxtxtw|w}yxuy~|vyy|yt|uztywyztxw{txz |tw{|xw}xwuxxw| }v{yyw|vtxu{?tttxw{~yzt|w wt x w vt xuy | zyty~ w|~|uttt y y vy ytz}y~xx v{yw|t}{t vy yx|{ tw|w} w xtx{}|yt{?yxuu|~wy}|xvxuvy{wt|{txt{ytyuu{|w}tv?xywt}u|u~yywyx{y}ty~yt||{u|~|uw}vyy|{|y}x{ { u x vxuy{yy}|yy x wx{tu|{ywt|wut|yz{x}zx|wu}x~ xxwwxtyw}x}{yxxtzuwtx{ut|tvxwuuy{}x~ tx | t v|u t{ t |  ? ???? ? ?????? uxw|z uutxuyyywux xxuyt{} x ww ||wx~x}ytwtz || tuy~{z| xxu y yx~x{| y} |tw x {t t xx||t} wtut wx{| twu|z xw}|z~ yuwx{| y} x{yt tuwx{| ttwuw|z ttwu~{txyxuwy xxxuu{|t}wt}wyyy{wttt}vwxyuytyu{tww}}ttuvwxtu{y|u}}u|ywx}~|xu{twzy|{yx tww~x{xx|zutu|tz { yw} v yt x{uxttt}w yy|u~|{}twzx|{u| t wu|y{wt}utwt{v}w|~uu{ytw}yx~ux{txzyttztw{zwxxx{|zuw~tt}uytyxuzy| z t y x { z t x w u zz|yw|yw{ywxyxy{vuy~}t txu~yyut|z }vxuy}}}y| xw{ty{wx|{xy~wut?xttuwtyz ttywz|yzu} wx|t } } xt| t | tt u u t t u |u{t txxxw t tw~ wx{| twu|z xw}tuwt{ x u{yx|w zw}y|u|yt yy} t y~}zzxt|wzyz|xw{uxx}|v~x}wtvyu{}tvt|z}x{y||~yx{}vy|uy{yyyxxytuuzx|{wt}zw|}~ttw~xwt|wz|v|{xxvx}uwtyv|uvytyy{y}y}t|w} y |t } uy t y uu u w tu tt } zt v t tx}u uu yx y v yxu|w |u tz{tw uu xutx~yxuy{yy}|yy x{xwz|ty xtt y xw~t w v{|tu|u|zw|}utuzt{wwxxtt}ttyu wyw|x{|}yttwu||zxzy|xuuxy{vy|u}y|xyuz{xtwx}xxz||ytt{w}~xtu|? x y y wu ywx t {t |vwwtz}yu v ty} xwu } w||t} wy |ut tw x t}}| wyyx yy w}tuwt{ x t{tz}xts xw}|~yxxzuy|ttu|{t xw|tw}wywt|t} wy yxu| tz ty|xw|}}| |uwxw{}uw|t{s|w2u|xuzy{yy}|xyzytyzw|xw}xwtx{yu twu|z w}tuwt{ } wy|txt tw t | xw}{t xwu | wtv{|t}txut}~tw|xwx}tz|y xyxyuy{ywy}|yy ytz|w y~x ||yxu| |?uyxuwy}{yt t}tzv}}yw~yxvuyy}t t xtwtzyvx||v}|w||t yywx twyu|zv{uwxw{x||wzy{tw~xu|y}}x xv|{w}yu |xx|xw}|y~ vy |s twu|uuy} |yt v|tutttsuttwx{| ttw|tztuvutyxu|zyw}| ytwt} tzx{} wtxw|tu u yxtt{~y|t{w|yyuv{wtv{y|ytww}|{yywv{}u|ywt}wu}|xtuvu}t||uttxx}z{y|t|w~}ty{} xyzx|}wvux}|wzxxtxut{z|wwxyt{}yyxwxw{y ux t | t x |xuxt ttt z w{y u u t{t t } } |w wu|z x{}y? |w}u|w}t ttz}y~xx ttyx{yt xvw}|{t xu| y twu|z yxuy{ w yy{y}|xy|}wxxt}xt tuuvtw}t wy wx{| x|wu|z y xz|y yt tw~ t t ztw|w}tyu wtw tz|u x wx{x| ytwwu|t{z xtwzy|w v{xw{y{t tzw}tut| w{t z|xyuwwxyx{txtxxwt}w|~xyxzy}}tt{uw|ty| v{wyuu{ywyx}v{ytw|x|zt } wuyxxxtz| tu{||z x t t ux t ut u u t u y{|~|u x{}y tu{ywwxwy xwx}|}~ tutt} v{ }wtxu{wt{| ttuwu|xzt}|wyyxyxyw}wt}uwt{|uyytw~ y{||~{xz| v}y|ֿ{x}| {w|~txvz|}xtuu|t w } ~ t
t {r : |r| < R(t) - } 1 - B t2 {r : |r| < R(t)} t {r : R(t) - A < |r| < R(t)} B |(t)|2 >0 t 1 - At










?'??'??? ???????? ? R ?͞????+?? ?R??????'???? ++???'????'????+???Ȟ+?'++??+?????????????????+??ž????????????????????????+?+?????+???????????+??R+??????R?+?R??+??????R???????œ? ? ? ? ? ? ? #" ( z y ?twu| | ywx}|w}tuwwxw{}t|tz}x xy|y tw|xux}ytzuuty~xtt wy||{wxt{wyx{tu|xwwu}|yz t|zz{tt}z}{ts ~ y t {
ſ
|
K ,

? y? vxu|t|ty xxxuyuyyx x twux{ttw}ytx}uyw~|uwuvw{}|x~uwz|xxuyy{||zwtyx t?u}x{w}|~x|}{x yvxxuuyv~t|wu}t}yttut{utu|xtzuyt | z } y |zxwz tt{|w ttz }|zxwxutw~utz|wx~}ttuu|ttxu{ytttxz||uuxttutu|tuyxxyuyt{uyz{t}}|xwtux{t||t{u| { w| { u w } ~xw x} xv}x{wv { |wy}{t}yttwtzyu|uyy{x|~txxzzu{|ttttuu{tv}xyuyt~w|~ut{ttwztz{vt{}|xuxy}x}|u~|? t } w | w y uu z w~ y| |{ y ? y{ t v}wyv{}x~ vxw}|~ vxzu v||twu{t t t w w z t |w | v||twuw|}wzyw|tutx}twy~w z|xxzux{t|ttuw{ut|xuy{tt}yu} |tu}y{y}xwyzytx}y|uyxw}wts} x{ } ?? ?? ? ??? ??????? ????? ???? ?????????? |  ??? || x{|t}}yt ty}x{yt ywtw}x~yttwz|w y|u{|u}txtxtz tt~y uuxyu~txz{yxuty~xtu{txwvxu|uttxxuwy}tytz}ytxuzyt~xtxtuvx{u||tx}zt|{|tu{v|{utxttwt v|tww}tt u vtxz}|xx{{yt uvy {vt}zw}ywy|wtx{yxtyy|zw}yytw~ twy||~|ux}tzwx yxyutyu~y{w|utxtw}t z } y t u t} uv}tyxt xw}|~ vxu|ttx{yut v{ |t t{tyuutu tyzwyy y|u z wxxytt}xt u |yy} xw}txttyutx yt ?tuw~yutz}y ttu{|xuywt tw}u|w}t tuwy~ yy} x|w}ys wxyw}w|~ xw}xtuytzy}}utz{}|yu~txxxtzttwyytytywtwztw x uxxuwytyw}ytttuywxtyzw|twt|z}y~xux| ztwyt | |xw|~ wyy{tty ytttwtz xuvtw}}| t~ twu{|utxtt wy vxu|t z} tt } x{yyw|ttuvy{}ytt}xtww}|~t{uytt|~tu{ttwy|xyt|uz~t|wzw}tz?{txxtuytt{{t zu{ttzyuyyty~ tztxzw}}y|~yw|y|tz}xwztywyyu { x t {xw}tu{yw|ty{}t yx v}wyv{|w}y |tw xw}|~ tt{ut~y~ytwwtyzxu{ytxx xxzww}yyt tww|wxx~}|y} w|t|w~tz}|wyvtxwutyz{|xw}||u~ yy|y}xtw|}|tzv|xuuxy w } y } t zw y wywttt}| tux{tu{t|w}uu|y~wt}stwttzwy}uu|}xwztwyyv}wytvu{yvxy{ywtyxxw|v~}wtvw{|wyxwutxwuxx|{w?} } } v } t y u tw twutt}u}t ~y xw}ty y} y||| tuu| x x ty|{|u }tu{|y w|u ytxw|}{tyyx|{yt|uw{~tw}zy|xw{wyuu|wx|wt~z||{yyvw}|wxvt}|wzy{tvt{vvx|uwuyyw}t {{x||?{~||}ytwyw~xuxy}~|y|~u|ttu~xwv|~w|w|ywtwy|{ y | z| u | } }y t u w tt{tt t t u wu tw utt xt |
v v t |
K ,



(k ) = A



= e
R

(r)e



-ir k

K

k0

-ir k

.

dr. t m
ax

t

x


6

?

?

v}wyx |z ut{} xy tw|uxttz } y?t }twtzyu |utwzy{ yxu||t wy} yywux }|uz w}s {ywxyx utwzy{ yw}x} tut{ xz{t uw}yt~}tw|tx{yt wu|u{xttt tw}yy ytxwuytyw}xtzw}{tz}tu{y~wyxyyuyx} ttuyytxw|}utytwyx{|yytxu{{xt}uytuuwyy|w}|{tt}yy yw}y|tuwtz{x{ttu{ywt |z wx{| yuy ty|t tw~ wx{| xuy~|u yuyx|u w yw }| |wu|z yux~x{y u| tuyx x ww||t} y|ux~x{y tuw{t}| tw||z uttw{z}y{y}yxuxyw|t~{tyuytuwuxy{w}u||tww}~ty|yt|{ytvyx |tuu~|tz||zz?u w tw~ yw w}tywxw}}~ |u xw xy } w~ z|w w v|~ {z w} w} y {ywx y x x v xw}|~ |z w |~| tt |ywttz ttw wuuytuyt|{ yu{|w}tt| |uttwzy{tu|z wy|~|utt xtw}ttuuy{yyt v{w}us |utw|zyx{ ttwwxw{z|w|utzwwxzy}tutt{w}~v}wwy~utttwtw|uxtwztyw{~ zt|wtu~xxyw}yxy}xty~tx{wt}|~ytutuuywz}} t x | || u u u t t u vux|uyw{}}tx~ytv{xy|xytz}ytuxt|uv}wyw|}~s{twx{|v|{ttw|w|z||uzxz|yzwxw}zt|uzwvtxywt}}t|wx|}|~ wx{ u w t x v |y y xu wyx|uwxy{w}|} {w |zu{x}xy~? {yux| yw}wywtt} xzx|ux tu|wxu {yt |u ww xx y {|x yw y x |w |z {} ttw yw}|~ut|wu}t} y} xxtyu ttz}y~xtyuxz twyu~ }tutu~ut zt|z tvt}wy|v{yuytttuwyxt{ t ||u|}wu|ywz|t~tz|uwvxwuy{|vxxtz|ytwytut{x||{tywxx{w}|ytywu| z'uu}t{{t}vx}zw|ywv{vy{y x t } x x xu|tw | u| } } x tvwxy{xyx||}xwu}t|z~|ztwu}ut|vw{}tt|yxtuwt}z||{wt}u|zwy}ut|vwxuu|y{zyztyu~t{x}w}tztuwwt{yxut|yx|u|t~yz||?w w z x t tw t | } w t}twv{yy{}ytstvxyuwy}xt}uy{}tw}v{t|wtx{uuytwwuy|w}zy v}yyv|{yy|utwz|yw?zyt?wtt|t|wt vx{w}|v~xtxwwzy||z
?

?

|x t|u ytwt }zyu tww~ w|twx~} y y?}yv}wyv{tw{}x|}t z|xzuy{y}|w}vtuttuwxx{w}~| ytutw{uy|wzt xw}|~ xtuwysy}tx}|ztyuyusyw} w y x y | ?? ???????? ?? ????? ?????????? x} | xt t vxuy{}x~ w}tut|{}tz||utwt xxw}tu~twuuxt{t{txwxu|zuxwttwux{~t{ztu}{x{txuxwx|uwxytxz} z v{ ( wx{| x| x} xuy{|w}y tu{x z |tyy v{ wxyt} t|u yttwtz yxxw}y ytuuyw}uz? |xw |u ttxuzu wwx{| twu|z yxu|tx} ytw}t tw~uv|~{} ttutw|ux6tyt{zz zt| twxtytuy}{tzy{ txtzz||ux t yw{x{t|ttwu|xxzzu|tt|utwut{ttw~xwuy|y{||w}ttyw
6

y u w } y | wt~{|y|t~u~{ttw|ww}ty|xywtxu{uyx{wx|ztww|~}}x{|xzxy|xtxyxy{xw}}wu|yx~wxt{}yyyy{||{|{tyx}yyxu|y}t{wux}| }u t t | t u t u y y t xu t t u u


y|~{t ( v{t}x x



x xxzu ttu{t v{ yxuy| ſ vv{|w}ts

a2 , . . . , as , s < k

S1 , . . . , S

a

K

1

k

(k0 ) = A

(k0 )

1 , . . . k

: |k -k0 |<

V


e



ir (k -k0 )

.

|k - k0 | >

.

x

k=2


?


?



?

?

? ??'?+ ƿ' " ?R? '??? ??+???'RR+???????? ???ž?????R ?R???????+???R?+ ?????̜??+ǜ????+?+?+ ??+?+'???????Ȝ+???????????+!%"? ? ' ' ?+ + ' ' "" ?' ? Rœ?'+ ??+?? ???R???+???? Ǟ??????' ??????̜??+??????R??+?Þ+??'?????R'?'R?'??????+?"?? w |z vx | x{y w} wy| x | z|z |z w v| {ywx |y v|z| xw}|~ tty t u u t t tu tz t { x t t t }u t wytyuyuyy{~ytw|yyt{t |utww|x|y|w}w|y~tutuyw|tt{yxw}ttutzt|wwtxwy|y~ ytw{tyw|yt{wt|{yv}ywy y s xw}uty x yyu| | v}wxtwt{x tz|w tw~ tw}t w|~|ut {|ux} yu~ttxtu t v{x |uxyw w twuyxy}z yv tuuy{tu|w} xw}tu{w|zt{yu tztwu|z vxuyu}v?t { t t{uut|xt|txyxu|yyt}xu}xtzt|{z|xuux}wx}xtzu|ywyyx~wx{t| ytuz|u{|ux?} y{uy~|tttx{tt? xyu{~ytw|ttxtu u yt|u {|ux} xwy ?wx{y| wu|z v}wx y~ yyw} v}wvu|w}t}| ytwtz |{|uxt} xttyu vxuy~yt}ytvyu vu{{x xxzu ttu{|t xzztxutu y{ytu |uuyw}t xyxuyytw~|xwu |{y}x~x{t|t xtuxwyx|t|} tv{twuv|}wzxt|vyttwt v|w{x|{zty|utvw|u|uyzwy}xuuxy?t ?? ? ??? ????????? ? ??? ????? wxt{| y |xz}y~xwyuy wvuyx yz|w tut wx{| twu|z u~xw}|~ttu ||}| x vxu|t|t x{yyw|vxtuyyu{~}ttw xv{ v}v}wy|uxt| twtz t vx|tux |z u~xw}|~tut vxuytuz{tw} u ~w|||t wxu|z {y}uwt{} yxz|w tw~ w|wx~|} yt y} |u ~txw}x~tut||wxz|wutwzx{y|{} w zyw}} tu~x}w|~yttu zxxuw||txxtzzx?}s 2 |zyu~yxww}?~tu | u z t u u t u ut t t{|u z tt u |uy twu|z xu|xtw}xt{y{t|tz{yyuttttwu|tz{{xx}s txuvy{ywx|yut{tx}yx|wuyyxu|ttt{utut}xxw}tx} v yux z }yxty yu yw}xt} twytxw{yt yuyy} }yxvu|w}t}| {|ux} zwvuyx t|u t|~|u{twtv{{ tuttxwuyyxx|}xyy|}yu|wy}xttwwt}zwyxwtt}xww}x|{~t|utwuv|xu w|t{t}x yyxx |}x| yyxuyyuyxyxuy|t}xw}t|u xtuvu}yvxy|||u{y{?|wwt}uut|?yu yxuy|u wxw}uty ytwtz|x|vsvtw}xtu} x w yz y u w yxuyy~y}xx }x{utx wv w|~{wtzyt{utx{t||t wx|z|zw wx}yytw yy~}vxx{w tzwuxzxx x v} } zy | w } | w| t tw|uxxuyyx x wx{t|u ttwuut|z uxwtut vxtuyutuz{tw}u tttuy wt ttzy{|yuuyt{ t }x{|}yvwx{| wu|z w|y~}x tw x tw |tw| |v yxuyt{t yxuyyx xwt}}| yxxw|? wtuwt}ytx| tu{x} yu xw}|~ |tz y|~{} tw y{|}u| y} t yxu||}uu|wy|}x|w~}|~tut yxuyyx ytz| y tu|}xt tw~ yw } wy||t} yxu|}xt tw y{ xuxyyx|yxu|}xt w| tzy ?{tw}u|w}tut yyww}wywtt~} |u x|x~tzyttu yxuxvtw}tw}}yyxtuwyyyx{|wv{tw}tytwtyu{xwtuv{}twytvyyyttttwtz|zwxz{||tywt}wxu}|tzu xw} u }ty vxuy{}x~ t{x| ytz}tut v|| |y} v|tu v|z}y~yxz}x}u||{yw}tyxtwtuw|z wtyw}xtwvvx| z yu twu{yxxxtz } |u{|utxttut wyw|utwzy{ xuyu t{wt {tuu| tyw}wy wwut|t}zvyuwt{yxyyyxuy~|u||wux~{tx{}ywt}w~wwvyt|~y|utt tw{tywxtyw ttuzwyxuv}ywwy}vx{}vvxwx{{t| } }x| v|yx{yt z|zz|w tu{|utxtt w tu{t t{}x~ yyt x tuutzt tu{| zx|uzxz|z xzxv{x xy {tz|uyxt{xt twwxx{x| x x ttu uzy~{t z}xwx{t|wx{tw|| zzutx{w} w? { | |} tu|v} ytu tu~yuutzt tt{}x~u?t twy||tu |wu|z ttuutwzyt{ tyuyt{t twttz v{ yu
x |00 . . . + 1 |11 . . . 1 + 2 |22 . . . 2 + . . . l a
s+1

, . . . , ak

a

1

V

|(x)|2

|(x)|2

l





B1 B1



? ????ʜ Ɯ?+??????????R ??+???Ȟ???+R+???Ȟ?? ?+??+???? ???圿???'+??+??+++++?R????+???R????+R?????? ??R?ž???????'??+Ǟ??+???R???????? ? ? ? ? ? ' ' # ? ?????? Rœ?'+ ??+?? ??? ????? +????? ??+ ??????+R? ??+ ?+?+͞??+ʜ????+?R? ????????+??+??+?????'??????????+ʞ+R?????+?R???????'???+?+R??'??????+????R+????????+?+??????'????+??R+???+?Rž++???????ž?R???????垜?+?????++?????++??'???? ? ? ? ? ' '?' ? $ # ??? ?? ? ? ՜?Ҟ ?М????? Ϝ??+?? ?????????????????''?'?????R'?????R?+????'+???+'+???++'?'?˞+?R?????3?"? ?+ w txwzywy t wuyxty}z y}x|u yv{t ttw|} yxuyyt x tu y{t xw}|v~ yxx}yttx||v w{||u}xx}t|wyw}}tuw}tttz} wxut{|xyxuxw tzy{tyu w|zuww}}| tzw}xtuuxyw}t|u~y}wwvu|w}x} v t t t u zt tz t tt u t t{u}tt xtzutuz|}wv{wv||wy}yyv}t{yyutwtztwxx{}|zt{wtu|}t w{yyxw~|x{twt{ txzx|xuyttuxwtyt{y|w|w}{tt | t}uyu~yxtxut}tz|{z|xuuxw}yyxu~ytu{x|xt}uxxtxwt}y|u~ vxtuuyu|yxtwxt{u|y|wyuu|xzx|y}x{t|zytv{ }t t u w v yyt y tuu|}xt tu}|{t} ttu t }wvy{yytt}w|tz wux|w}zuyxv~t}|wuy}yutwtxz{|w|v}twu|z uxwtut yxuytuz{tw} t{txwt t uwy|x v tttu yw}y~|z y|~{} ttu w xytwutyzu}ytx~xxyz|tu{v|}y{ux|uyvyxxtwt||y}v|xwtyzu}|v{wy|twt}uutytuw yxxzu}xy|yux v}tww|u|tz{yx}|x} t y ytzt w | vxu z t tuy t{ ? |u | ? t w x?y}xuywxytx wx~zvt|{}|w~yy|wtyyt ytvt w}t v v ut t ut yutywtyztvxuwv|t{wy}t}t|utttwy~|ztyyxzyw uyztt{}xyttux|{uxzw|}ztw}|tu~xyw}txuxwwytu{wyxw~|yv}yttyuyuy } y u uy t vut x|wu|z xutt} t} ywt| xs t}{x y xz|w } y~x |wu|z ttw xtz }yw||y uy{t wx{| wu|z y|tw~xu y{t t tuu|wx~}t y wx{| wxxuy{yy}| } yxww}wywtt} vxuy}t{t yuuyw}u|wy}t yxww |u vz}wv{yy}v|{ ytwuz yxu|{t|{tw}ux|ytzuyxtt|y~y}xy?xxzu|}ytttut{ttuyz{ytw|x{tyuu ttutuy{v|xuxy|tww~tywx{||yw|| tt{t xw}txuyt t xu|zyww}| yx |x uwx{| twu|z |{|}x~ xuy| tw} tz tx}{}zxvw}}yywxyt xtw}}t|~} wttuty{{wyxtt{|xtwuyywtwtwvxxw}z~vy{ y tu yw}u|w}xt w vt ? ut{ttxwt tux| xut x{}y wx{| |wu|z |t w|tw~xu ttw ytz x vxuytu z{tw} {|tw}|~ yu xuyy ttu{ywx{ tu~tw|w}t xxuyvwt |u twtz wx{| wu|wxz{t{ w|tw~xu v}xttt ? |{}x~ tyt x vxuyu|yt} v{ |xttyu v|twtz | twu|z vxuytw~}x}uu |yt tw w|}xt t|u tyw~ x}{ tw~ z|w |vxuytuz{tw} x u|twzu|ztzxu~tzw|}zxtv}twwyywtxwv|x yuu}uy|vvx}uxy{xt}}u|txt|}?? |wxw||}x}||wz xwxu{{y|wttw} w u wxwt}xt tw~ w|u t|u xzx|ux ty vxu|}xt v{ yw}x} tu~xwt}|t~tut v{ vxuvtw}t} tt ttuu|ywz}w}v{yyuty~tttuwtuw}|t{yy|wuyvv{}twyvt{v|zwtxw{yxxz}|yw~ux|xz x|ux{|}t|uutywu{yt|{ux|x|s } | z uy u x~txywt}utxux{yuttyy~ xwxx~{{ttttyw~ytwv|x{uvvttww}}tt}}twtx{tw||ztwtu|yzvxwuyxwy}tuwyu{yyytxz|yuyu|} zy t t t y }zt|}| xz}~xt{ y}xu|y|wty } tu|v} x txz |uw}{t|wu|{yuwxu~xyuy|xtttuw xxx{t|tutxx~||yuxyxyww}ttyzyu yxxuzy|~zxu|w|ttwyx|{~|uttx}t|uzt| y| y w t |tyux{zy|wxvy|ut{ytyyxwwu}w}wwwttuu}yttut|v}y{y|{yy|yt{t{|ux}x{w}}?y twtywy}x}tuutxw |w}xux| x{||u x u twtz yxwy} tu{ywx{}x~x tzw}y~x}}|{z tt } xw}tu{|zt{yu
xs x-s , x {, } r t>t
0

B2

?




?



R ?? ? ??? ƞ? ? ??? ?? ? ?R ????? ?????????+R?+???????RΜ????R+?????Ü++??+?R????? ?'???? '?? ? +''??˞ ''?'?? ?+?'?+?? ?''?+'? ??+???? ? ??? Ü?+?'?? ??? ?#? w w }y{x?}wywtt}yw}tytywt|?zwyxv{z|twy}}xuyvtww}w}ty}|yzxuw}|xwwu||zwytwvx{t{yw||uw}|y~v{yzy|ztwyv{yy wuyy{ xyuy x} t yzt w x y uu u t t
x {, } s {+, -} j = 0, 1, . . . , N - 1 q xs , r (q ) r |
j



x wyyx xw xzv} yy~ xw ty y|~|utt |z ttuu| v{ |twwt xzx| x w|z yx |}x y} | xw z}y~xxxuyvxx z|xy|zxw|}y|} uwxvt{xwv}twx{ ~ u|z yuuttt t uwx{|u twut|z uu|tvt} ytu}t |u wxtxt yyut ttttxx t ywxuy{t}}x~ ytz}y~xwz| twt vxu|tx{yt tttwx yu | tttt{|u| xxt{{yyx wwz} ywzwytyyy{tux|wzzyy{{uut|tytxtz|w xywxu}yu~xywu}|~twwtxt{wxx|u|y|twxx~{}yxt{}yx|wxw~x~{t{tvtuyy|uwtwwy|tz } u t w xz t ywuuy||zv|}zu|zt|tz}}yt~wyxxx{v{xt|z}x|?}t||w}wx}uuz|y{w}wtyxuvy|}|}xu|u~zxw{v|t}wwy{xv|zy~ux||zwyy}{u|{tvtu||}x u xw tt w wy u t t } tu}t t tv w ut {t z } yu ytt xw}tuwvty ttwu|z x|wyywux z}y~x}}|{tz w| tut |z utt} |u yxyw wx{|x x|wu|tz xuwtt} yt|u x ywtz yy||{} yw{ | |}u wut|z w y utuw|vux|v|z} vy{xxx|zt~{x}|tt{uy{||tz}z|~|x}}}w|x{~z{w|txu{ wz|||zyuwy|w|~zzt wx{|y||}x|||}|xtyvx}x||xxwx{tuuwv?uxy t |u {y}y y u tuy tu t y t ut uu v y| u u t y t | t ? ??? ?? ?? t{|ux} u~ttxtu u{|zt{yu xz|y xz}y~xx u{|zt{ wxtw}t} yw}x} tu~xw}|~ttu v{ }}yt x? xw{xy{t|tw~wuw||z|xutyu|tyv|xttxwwts x t z} | u {yw v}wyv{v wx{ | w |z |uxwty|w y|vxtwx|wuv{xyz|t|}wyxxt|}x v|wt~wtz y~|| t } xxwyw v{t tttuwtxtu|z|twz?yxwwx}{ty|wtutt|uyz|} yv}uwu~xywtyv~|yywxw~}ttuxtw|ttvuxxtww|tvu|yyy wyyv{tt t { uu } y y {u ytuuxut|uu|ttxutxzu|} v}wyv{v yw}x} ty||tt y|u xv{yw|t{t y |uyt t x w }uxx t}t} tz| y tu|}xt t{ tw zz|z zy|w wxt{y| twu|z xvxu|ytuz{tt} t}{yuux|xv| vuyxyxwyx yx{y~ tz{tw}wxy{x|x||wu| tx xxy|yuyx yw|~wy{t} wy}t x|{wyyw|| t ts t u u t tt t u u x t t u tz}w}xxwv{ty vy|xv|{uyyyutwt{|tuz}x}tt{w}tsv}wu||ytyxuyxy{wyytxuyy|u{||uwyt} yxtu}xxx v t } { y{y}{xy~w|tw{ y}xyz}{tzz}{wwwyyuywy}wt}{x|uxx}{xu|u x}v}ww}y|yt| x w~wxy~|txwu|wzxyv x }x{ w t tut ttw wtyuzxuttyu xx|tux x~|yyt w}ttttzt} |u vxuy~xu|utt ttzt}w}xxuwv{y tutu?|t uy|u xuy|u yxz|zxutw~ z|wyxx|tux y{yw|t{t tuu|| tz|zxu wx|}tuy yw|yty}xtx{x|x}wtt{u xv{yyuw}vxx}uytxy{ytttxw|}}yytytxwtxuuywu}v}wv{vs{uxux| | z| z | t{


r = j r1 . . . rk








?+?? Ǟ+?? ' ??????????'???+?R?'???+????'???'?'+'+???'+? ????+R???????+???+?+?????????????????????R'?R'??+??+????????+???? ?4?#?#? ? ?+ ? ? + ????? ?  ?+?? R ?? ???? ???? ??? ? '?? ???'??????????+R??'?????̜?????????' ?R?+????????R?'???Ɯ+?????+?+??????????R????+??%?+??+?????+?% +?!'"?#? ++ |w u |z|wt |wx|z|zyv{xw}|~ wt||tt ttzw}{ytw wy}t|u|yu w'x{|tw~twu|tz tuwuuyt{|tx}zy|~y{z}w| w t t t y| ww }u|x tz|u } v}wyy tzw{tw tvutxtu~| x zz?y zt|w v}wy|wx~} {y?t ttu{yw|xyut y|uu|{zywxwxw{tx|yzt|ww |xys?|tvy{wuzxwvyt tz|u }yxuwyx~u||uttzuttwvy}wvt{wytzyvxtuyw~| x ty|t xz|w xw}|x~ yy tw twu|z x } |wu|z ttutu| vxuytuz{tw} tyy y uyw?x{tw||t}twxxzyxttwu}twu{ywwxw{zxt}x|y|wttwxu|xzyx|}ytuwwt}{ wtx}t{y}xxw|w~y|yz||w |twwu ||zty?tt|xz x y t u| tu t t zw tu u ywu u | |
- ei s s j = 0, 1, . . . , N - 1 pj = |j |2 - s , -s , - s , s ,

'ſ

u v}xx|xz{|w} y xz|y tzx}wxv{}wv|{v{w}y|w||z xx|w||yxxxwx{}|xt||w?|z } y |t {ywwux tuw}tuzwvty yyxwu}|tutwuv}ttyt wvxxy{{}}xt~w~ v{ |{x| yu~t vvuyyx}sv{ tw yx}xyt~|y~|~{}{|tw y u
j, k = 0, 1, . . . , N - 1 x, y A | = | = B
B

{w y} v{ |tz |tw tz{tw x ||wt}wuy|yxxy}x}{wyyw|tt}} v}w x }y|y|ux } yxuvtw}tt |~ wy }zyyuxxuxuytwtzyu ſ yxuyvtyw~}t} ytuu|txxt~u|ytv{t|wyw}?}vtwy||z}xwt|w}}}| u ſ t w{ vxu|vtxww}tt}ttutuu|vx{| | w w tw x yu|z|vxwuyvzxtuvzy{wt}w}} wwu{{{y|wyy|tvxy|{}~xtu{ty}ywx}wu|vwwtv{}t}yyz||uzz{w||yx wt|x}{|}y tx{{}xxz~|z z|{ſ x z v u t t ttu uy t tu t } tu t t yt
[j ]2 + [j ]2 . [xk ]2

?ſ

|{} ~ { | x x{yy yxzy v}wy||u txw |z xv{xxxuyu|uyuwttwtyy}uyw|uwuy{w}yyy xt{t v}w|z}t tw}|~ w x xxt yu{yw|tt} y}zyux yuxu xs ~ u t x y | |{yxyx yy}~|wt| z|x|ux||yzw|t{x|}wxu~|yzyxxtytw}ywyuyw{yy~w}|vtxy|v}yw|yxyw|zz}|uxw|tu}yvyx{uyuv|t{|vw{} | | u u u y ttuu ? u ttz tuu tt }w t u tt tt w {| uu|t
B [xs ]B j xr , x- r
1

v ()

x A, s {+, -} j = 0, 1, . . . , N - 1

s {+, -}

p

j







0 v () = v t

pj =

v0

| =

Re j , Im j

[xj ]B xj, = . [yk ]B yk ,

xA,0kN -1

N -1

j =0

-s

j |

dt

j

B

j, , j,

s



r

j

0 v

B [xj ] = [x+ ] - [x- ] j j

B



(xs , xs ) j j






'

?'??? +?'?' ? +? Ɯ+?'?+'????????+'??Ǟ??R???ž?+????++??????????????Ǟ?+??'???????????ʜ?+???RR????????? ?R???Ȝ??+?'????''???'+??????R???R????????R???????+!R????? + ' ? Ü?? ? ? ? ' ? # ++R???? Ǟ????'??? ?ǜ????R?+??+??+???RǞ???+??? ?? ?? ????+??????R????++???R? ????'?RR??????'+ ? ' ' ? ''? ' ? ? # |twv{{x}t v{ | y tz|w wxt}ty tu|tuz|xuvxuyyyxw}}|~xuyuwx|{}| twu|z vxu|tx{yt | tuuyw}wywtt wx{| |w y t {| txyyuy |zxuwv|w |y| xuyy wuytty{ wyw}|~ yu z|zyx~uxwvxx{z||yx|wwyu|wz}|x~xx|u|x}vux|uy|t{uz{xtv}}wwyuy|t xzx|zyw}u||wv}xtz|y x |y}wx|{|ytttyvw{t{yxxu}yyttywuxy{tu|{wv}u|utyytutzy{wt}wxwy{}{yy~wxtw}t }ywyxxu{yy|yttwwu}|t|zyxt|x2wv}xtwuyzww|tyy}tx} w tt{ t x z t{ yv y xt w|u ty t x x x|w|uxttz w u{t x tw~ v|wx~ wx{| twu|xuz x xxuytuz{tw} x |twv{ {x}t x xw}|~ w|uxttytz w|t}|{ty} zw|x xzx|uxtwzy{ vuywt}}|tx w| txttywu u zxtuwytzx|uswywtxw}}t|~vxuuv||uux{|txwu|yztt }|yt?wv {t{xt}t xxwwx}|{~|w|tutwu|ttzz y~||x ty
?

??? } x wyyx twv{{x}t |xw}|~ yw}x} u|xu|| v{t |~ } y |t~|t|~u|w|x|u|w|t~wyxzwyx}{w|}~|x~}xwyz}|twu}uyytv||vwxx{yw}w|yv{t{xxw|}|tztz{}yyy~txxutvwyy|vw{}tyv~||~u|t|twx~wuw}|||uu||w} x u ttu t t ttu t t t u | z t w} ??????? ?????? ? ???? ?????? x}x| v{ ut yy{tx|u z|z t uu|v} tuuyx w|t{t}x yv yyu{| x twu|z w?u tt} vxuy{|w}y v{u tt tz}y~x|y{| v}w |v}{}ytwxt{u{|yww{u}|x~zyuwut|v}tuz|{|x?uyutwx}{zz| wxtwtu|xztu|tvy}ttttwyt~}x}xxuy|u|tz}}tyz x } v ~ w xwuy x| x z y| v|yx yu tuv x|{}x~ } y|wt| }y tw~ tw wx{| twu| vtt} |twy ztw|w}ty? 'ſ x wyx x uuvtw}t |y} xxz|y z|z |tw ||{|xuywtu wttww{}xx||ttwux{t{}|tutwtu|z{z}y{|tyuxuy{xtywxxu{y{||twu}|zwvtty}x|tuyxuy|}txtuz|tuz|}xy~ x}}|{z z w}tuwvty twu|tz vvt} tz{tw yu wx{| ywu|z yutt} tw~txx w ?wy yt t yu tz|w x t}}y u{{yw|x{}x}~xxwv}xwuy|wtttt}}|txwt}u|~{xt|ut yvy{{xx|u|yxxu|tz|wt}wu}|tw}twy t x y w y ~y vyxu?{yyxzuyy~utw|xw}tuxwxt{w}wyyxuwytuty}xxxtzuw}tut|tu{tuvxutz{ttvw}yw~t{yty}||y w}txu{~}y?ut|z}yyuzx } t x ux wx{| twu|z t{}x~ | {u z yw}|{t xtw y|| xyw} |z x{}? | t{tzt ~x{| vxuxytuutz{|tww}}u|xw|}tx|w|zwyu?ytxyxu}u~tyz~t|tuwtxux{{xwx{w{ut|~twv}ww}ttw|~yxutyw~tywxwuxttxtww}x~}ttw}tvx}uxy{t|tuz z|w} z t y y y u z | {t {|wu}x uyu twtz wu|z xzv} w|txxzzx{ }x{xttt z|z z|w vxuvtw}}| yx w x | | tv{{}t}xx ~|xyxtſyt{|w|txzxx|vwz{zyv{v}xwvyxv}wwx}y{yyxt}uyvxwtuy{ttyw{~vwt|yu}t{|w|u|txzy}}ttyyyt{ 'ytw{{x}{x~tt|wttuv|t}w~y u | v vx z| w wz { ' u
2

mx 2

MX 2 + 2 X 2 + g (x , x, t)X (t), 2 V

ſ

e

ij

,e

x

i(+ ) 2

X

X

j = 2j /N , j = 0, 1, . . . , N - 1

L=

X0 + X

- V (x, t) +

X

j J

0

j |j ,





S

x

X

J

(x) = C |(x)|2



Dt

|X0 |

t

|X |




w}tzx txttyu w||{tx uy|{t yxuxw ttt uwx{| ttwuu| |{{yw}z|? z t wvyxyuyyyuw|y xtzwx|wuvx}ttuywwzyy{tuytz||}u{|y{{ttu|vy}||x|y}w}||yututtwttuuzy{tt}zt}ttuyx}tyut}xyuu w vyuw{{xwtytwz~}wt}utt{{|}z||{uttttyv|uzyt}wwt|~ytu{u}|z~yx|wt|wxyu|xu||t||u{tywyxx~uy{yztv}}}x|y||wyuz|tuuy}}w|xt}}wy}tyz| x | yx w t t { | u { t tt u { { x ? ??? ???? ?  ?? ???? ???? xuytuz{tw} t{}x~ t xw}tuy~twy wvxuyxw}ty v{xyt}u~tw|}wy}x wy|vxy}~wyy}uyyxtw}z||~zxy|utywutyv{w{xxtw}w}y|twtwwxxv{z{}{yx|~x}wttw|t|wvuz}|wzxxxyxuyzx{wxw{}txxx|uxut|{|tztywv~{tyu|ww~{|~xxyy}|t?? v x w }y t t u t t t u u tu t | u t t w uſ } xxt{|u| t vxu|t|ty ttwv{{x}t vxw}ytx| v{ w|}x|u x{x ?? y} w|uxttz |yw}x} tuy| xw}ytx|yu z xw}yxttut uttwv{v{{x|}t xw} |~tx|wxy{|~vx{u}|tw|tw}t|y~tuw|wtwwy}~y}ttux~|xty{u|ux|uvt}wxy|z{}txz}y twxt{w{xt}wtu|x{z}?yw} tyt{}tw}yt}{tyw|zx{xyty|tuwttt{yxyxuw|uyytxzxy{t~zx|wxuwuy|y|zwy}w|ut|tww{}tyw|t{xxzt| w y z ww|y| t t y uu yt{x x yw} x t { x } uyt tu tz z t | wyt vxw}ytx| yty||{} z{tz}ts xw}|~ tu wx{ | v{ vx z|y wxv{ ~x {| | x { } yxw}wyw } wx }xt vxz|y tw~ w|wx~} tut xw}|~twux{t| utwu|z }tv}wy|txz{|w} |twv{t{tx}t twx{| wu|z x{}? xxuytuwz|{tw} x x tz{tw v}wyv{vt |twv{{x}t xvxw}|~ wx{| tw|u|z yxw}y y tyx|u u}{txux||tt~txt{t|uutytwvtv{u{w|x}vt}|wwzxz}{{}|xx|tuw|tuw|yv{{zt{zx|xtyx|utuzz||vu}ww|uy||wzztwu~|tzwtwvx{wtx}{t}s |xw} z v xu |w t y } ut wx{ t|z | w { y x t ? yzxx ttwu|z x~|| tw xxuywt}}| x x v}wyyx x tw tuuy? v{t zx~}tuyy xw}|~ z|z w|xw|}}| wyy{} ty y{|xuywt xw}|~ x{|xw|}| z|z z|w w|x w|}}| v{yu t|t xz|w |twv{{x}t wx{| wu| |twvt{{x}t vxuyx v}x|}|z vwxx{t{}| u|{zwvx}uw{txvu}twzy||uz||~z|t|tu}uvuy|u tt{wt|wwtzw|xzzw'tyxy{x|u|tx|}x yyy| wu y t t v| ~ |t w zw w x|zxut x {}z} xz}y~xx xyx }tt uyy} tw vxuyuyx utzx| w}y t xw}|~|w|uxwv{t{tx y}w} u||xww}}t|~vx|w}twyv{{xx|}t tvxty|}yvxuvy{yt?yyyzx|zu|xztyyxxuyzt|{wtwy y x tu u ut t t x u u zt|wyv{{xy}? ytwvx{{{|x}ttx{xt|z}xyw~}x|~x|xw}w|y~?v{w{wy|twyxuztxu{yxtuz|tzvxxuywtt}zt{tw}~xxyx|x }t u|| wt y { | tuuyyt} wyv{ wy xw}|~ wx{| |wu|z xw}|~ tuwx{| yxu|t|tutyts u w |u|tw|txy} z} wv}xyuytz{tw}tyuyyy}{}}t w|uwy|t vx yxzw{|~}ywx{y twwuy|z w}}tyttu|uyy } | u ut t u z } u ytt u z u tu tw t u t
ſ ? z|w wyv{ wy twu|z utt yy} } wuyt vxuytu z{tw} t z t t y'wuytttx}z||t}w|v{{xx{|}twtx{uu||y|wu||z|t yxxwu}||~ttxu{ywtx{|{||~y|xuu}|x|wtt}y}|s t
t j jo
sc

t

0

= osc e j





i h

S

g

osc j



, S



osc j

X

=

0

M (X (t ) - X (t0 )) + 2 X (t )2 + g 2(t - t0 )

t

1

1/2

g



xj (t )-xj (t1 ) t -t1



xj (t ) - xj (t1 ) , xj (t ), t t - t1

, xj (t ), t



xj

Xj (t )



X

0

X

0

Xj (t ) dt,

X

0




t |}{x |twzy tt|z v{ u| xx|txv{wttwvyuy{x}zx| x tw|t u{tx x }||u tw~ |wx~|}utyw t? xxut|xtvz{ytx|vxu|y{t|w|uyxvu}ywt|{||u vxuy{||u xtwy~x?w?wywtt}} w|ywxu~}{tutt|tw ? uy u zy
(R, t) A(R, t) = =

ſ ?
,

|{{y}z|? vxuyu| xw}|~ ux~x{y vxuy{|w}y yu}{x xwt}}|
E =- - 1 A . c t

vxuyu| x v}wvt|u v{t ttuwxu|twzy{ {|xuywt uv{|z} x utwzy tw xs
E B E B j = = = = = 4 , 0 - 1 B , c t - 1 E + 4 j , c t - . t

(ſ

z| u u|zu tz|u xx{?t|v|tutvzx|uyu|utt}uy}xuyvxx||{w{y{y}|v?uxyyuy|w yyw | yw}x} v}wyv{v yw}x} z|z }txz {t xxuyyyy ty x yz~tw |v| |ztw xw}tuwt{ twzy x{yy t{uuyywwywtt} x xw}tuuyv|u v{t| tuwxu| x tz}y|~xwzy{ w}tuwt{} } tv|x xxzt}~xwzy{tw~ytw{}}xx}~xxwyttu}~}t|w|w}tutwtw|u v{twy|yxyuy}t{| xwxuy|ux| x |w||txx}tvyz t t u t yuut y| xz~twtuu|tyx}zx tuu| xw}tuw}yz{ tw~ twx}| z|w v}w||yy wx{| wu| uut{wtxtw~ ||twuxw~|}xtt|zt|ywu|v|}wytyx|yyutwt u|{vx{t|yxuxyt{|vtyw{~ yyy{tw}u|yww}ty u tu} }xww}||~x}w|x}|wt}|yutuxt|wttz|tuv{xxw}}ttuxvt}~yxuttt|wu|yywxx}yuy}vx?t yxtzzx}y~xx}y}|}{zxwy~ } { wz y z }u t | w {uytwttvv}u{vy{wxvtvu{zt{twt}tux|uu{||twvw|zyz{wwux|{zz||z|utwtttzvw{xxuy}xxz}yuu{xtt|w|tu w w u xu| z u w { t ~x txttz txuyt{} yw v}wyv|~t t{t |{x|xuywt t{tutuwzy |ux~x{y zx|z tw tv yxuy|{t} wxz}yz~x{zx}ty}wvxuuxtv}wuy||tttyt}twttzyy|{w{tyw}z|v?txyyuxu|y|xtxxw|zy{w}x|ytuwytz|{tt} ttwv{{x}t xz}y~xut|tyw}x} } yw}ytx| xw}|~ yx wv{|w}y |~{} |xz|y y{t |xw}|~x |utz}yw}}x} v{ y v}wyv{vt vxuywt}}| twtz}}y~x|uxttwzy{ vzx{x|y}vx} xw}t|w~tszy{tytyuv|{xt|tz{utw{z}|vzx|y| z||w xw}|~ uu}y}yxw|t~w ztu|vw|}~yy{} vx uv}wy|~t{txw}|~ xz{tz}uu |~{ v{tt xzxuuxttwty~ }txwt}|~ uuyvt|u ttyw}x} u w u}yw|xu| y}x ? vttwv{{{xw}t xz}y~xutu| } xw}|~ xxw}|ytx| ~t ~|| x|u u |uvw}ty}|}|xyt |uy}||x|t|tt twuxtuwt|t|wxzyt{y zw|xut|wt}tty}|ux{twyyuttwu~}tttwwxvy{
?

j (R, t) = eq (t) 3 (R - q (t))

t

k





3

(R, t)

j (R, t)

ak = (a ? e = =

4 c

ikR d3 k (2 )3 d3 k (t)eikR (2k3 , ) d3 jk (t)eikR (2k3 , ) d3 k (t)eikR (2k3 . )

1,k

,a

R, t

ak e ?

2,k

)

E

a

k

B

q (t)

k


(

? ? ? ???? +???? ???+?+ '???? '+??????Ǟ?????žR'ž+????+??++????R??+???????++??????'R????????????????+????+!3?#?+? ' ? ? '

A R A(R, t) =
j



? ??'?? ?R??? ? ??R???? ?'??? ?'?' ? ??? ? +?+͞?? ??? ????????????+?'?????+??+??''+͜??+??R??'+??'+?+?++?? ?Ǟ??? ???? ????+ ???? Ü???? ???????? ? '??R???  ???????R ???? R?'???'+R ???+?????R+?????R+???????+??'????+?'??Ȝ??R?œ''+ +??++R??????Ǟ??R??'??????+??+ʿ?͞??+??++???????R??? + ? ?+?'??+???'?'???R?+????+?????????'+? ???œ+????R?+?+Rœ?++?'?+++???????+?????????????+??R? '????'? ?Ǟ'+??? ??# ' ' Ǟ???++? ???????Ü??? ?R????????????????+???? ???˞??? ?+??????+??+??????R?????'??՞??+?????????'???????++????ƿ?'???????+???????????+?! ? ? ? ? ?? ? ' ' ? %# uutwt x xxuytuz{tw} yxx tutwt |}wu|z yw}||~ uwx{| vxu|t|twys ~ yx{|tt}{ttxw}y{xxw}|tuu|w{|~|t|wu|z|u uyx|wt}}t}utwt|y}wx{y|u|wu|xzz|}s t}}| } xw}|~ v{ wyyx tut twtz |{}} ttw tuyx{ |utwt xxzu ttu{ty?y{w}} xu | vy ytuw|uxttz tw~ yw } x yz|w | 6ſ yxuy| } v}wy}|{t} tw~ tutwt wx{| xxuyuyx y}}yt wyw}|~ yu }{x x|t{} xu? t } vxuytuz{tw} x v}wvuy yu z|uzxu wwx{| twu|z uutwtt }{x x vxtwzy|w tw~ w|wx~} y {?tx|xuw}{|y~wx}vt{tz{tvwx}}uywt|xu||w} zwx|txy{t}}xxu|t|wt yxuyuyx x{tyu xut } t y y|tw y{}|ts {ywxtutt |wx|{uy|w}w|t~wu|zwxxu{y||w}xyxuyttuu}t{wy|~z|uxwtywyy|y{|~u}tutuyu{ytt{tuywxutu}|t{wz}y{tw~ u { y x | x|w}|w~||{xyt}|xww}y|uxxxt{t|txwvxuutwtvw{xx{{{xyt}|t wxv{| |?}xx{x||{}xxx~vt{xtzuy}yzxy~{x{|tz}tvw}|uwxv{tvtzw}zxytwtxwy{} z | w | v x tz t ut t u | u t t u z tuu t |txtwvt{ztyu |wt|wz|y{xut z|zz|xw ttuwt{uttwut|wzyyvwx}{||}{wu|x tytxw x vxx |zy }|wz utwt tw~ ttw twy~ } y uu|}xt tt uv~{xtw}|x~t|ztx~}ytyyw}yx} xt|wxzw|x~uyv}xs{v u u z ? | y x {tx | wx{v|{txtwtuu|twtzytuutyuut{wttuwxv{| wyu|xzxv}xwwy}v|{~|wx}wxy|{{wwu}|}z?yyy|uywyxuvutwx}tw}y | z | t uu t 6 ' xw|}}}w|vvxvx{tw}yv|ytwwu}|y~|z xxx{|}yxv'{t(vxuy|{{|t| |tu}|{|tt}wvz}ywy|x~x{ztytyw}x} tyy| uu
S
par ticles

?

y

z|z v}wyv{yyt yw}x} tz|w v{ yxw}y

=

mqj2 2

+

l

6ſ

q1,j , q2,

p

k ,1



j

p

k ,2

S

S

int

f ield

e

?? -ikx



=

1 2 (a = 4

X

0,

q ?

1,k a1,k

j

k

(a



1,k q1,j

4 c

- k 2 c2 a

t

[t,t+ t], ( )CR

ej el |qj -ql |

+a



1,k a1,k

2,k q2,j

S=S

dt,

k

+a

)e

d3 k dt 22 2,k a2,k - k c a2,k a2,k ) (2 )3 ikqj (t) d3 kdt (2 )3 ,

par ticles

? X( ) aik

k

B

+S

( )R

k



f ield

,

?

+S

int



,

S

a

1,

j = k ,a
2,








? ? ? ?' ??? ?R???????R+???++'R??????+?+?????+!+????'???R???'+?+?'???? ??R+R? ???+?+??+?'?????? ?Ɯ?R+?R??+???????+?'???+?+Ǟ? R
?

? x wyx ytwtz ztt{t vxxuwyxx}t}y|yux|x|u|xv||tu||xuuxx|ttu{~txtt||{ttt|uuu~xzy{~tt|uutwywxxxuty{{y|ywuxy{|y|txu{|{|ut?ttzy?xxt| w tw { |z z u t t yt xv t | t t | w y y


wy t y ||x| x| u x|yw{}}uvt}|tu|uy~yytywt}x}tzt}uy~{x|x}wxyytu|x{|v}txuwu|w{|z|ww{wwv}{|wwwt}w|}yxw} z{yw{t}yuw}{t}x~zz w yy tw u vu t ttu t t u tt t y x tu{| wxu}t tw~ ttw v{ xxy{}xyttw~wy~u|x~x{y xw}wywtt} yxuy~|u x~|utt yuuy|yy yuu|txw yy~ | }ut w}tuwu|xy|ux uyt{ |t yxuxy |uz| |{ |wzy} w}ttz} tw~ z|}wsxuxy yw}xw|yu|xxtz wyx x xxtwzy|w tux{ } xz|yuv|}w}yt|{||xuwyy~t |zttyxw{tzwyttw y t u tw~ w|wxx~|}wt|u xvxuyu| xu{|xuyyx utz| yxz|w y|| x{}y tw xs x|uxtxtwz } xuyy yw}u|w}t xw}|~ tu~y~tw xxtwzy|w yzyswt ttwtzyu |xuyxtzvt}yw}tx}u|xt}yyxuux|}xtuttww|tyxuttyyyxxuyxy|ut|z|w}xuxz}}ty~xxxzxwyt ttzzwy|u t y} tyx y | } wyt|~|{ | xzx x z}w}xxwv{y tz~tw tu||wwt v}wyv{v w}ttuwu|xx{|ux u u u t z u|tt u z| t v |{yt}tt|tuytx}y~xxuwx|t{t|yztut{tywwx}xtu|wtw}uttutyz|v}wxyz}{y~vxvxxyt{wxtuwu|txuu||ux}{vy} wy} v u } t ????????? ??????? ? ? ?????? vxuyuyx y yxz|w tu~tw w|{y} yw}|u~xt|wv{{xx}{t||~t{} x tw~ yy} tuuyt{y x { t y { xtz}y~xx zt |tyy v{ u vxw}y wuyy{ tw xw}|~ wx{| x|wu|z xxyy|}|}ywytzuwy}uwxywy?t}t wxwu||tz xt{}|?
S
3,

? z|w |u|t|ty wy w}|~ v|uwx{| |t xw}|~ t wx{| t twu|z } x } ywtz uyuxy?t ut yx|u w}s |xw y ttw twu|z } xxuytuz{tw} xy yxxy |ww}uw|yzw t}tu~wxw}||~yw}||~ vxu}yw{wy|yy{xu|tw|xtwy}|~ xy|wyw|zxxwtx }}| }ywyt|z{} x wx y | y uyux?t wu|zttx{}y y~xtt u x {tu |twzty vxuyt{| uwuyuttz
x1, , x2, S S
2, 1,

aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl - aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl ; Sr1 , Sr2 , . . . , S = = = e
4 (2 )3 1 2 a1, (t )-a1, (t0 ) (t -t0 )

sign(e ) a

i h

(S

1,

+S

=

= e

2,

1,

msign(e )(x (t )-x (t1 )) 2

)

,

ds2 = dx2 + dy 2 + dz 2 - dt

x

2

1,

+

i h

(t )-x1, (t1 ) t -t1

(S

a2, (t )-a2, (t0 ) (t -t0 )

3,

+S

2,

x

)

+a

,

2,

2

2

+

x

2 - k c2 (|a

p

2,

k ,1

e e | x (t )-x (t )|

(t )-x2, (t1 ) t -t1

2

ds2 = ds

p

k ,2

1,

|2 + |a

) dt,

2

e

ij x

2,



(t - t0 ),

|2 )

dx, dy , dz , dt

(t - t0 ),







r

l

'?0$?? ?0$'( %?0)?????????0?

? ?

x, y , z , t



x wyyx twu|z


6

y } xuzx}~xxuuuw|u|w} wvuy xz twz|yu|xyyw}xxtu{|xyyux tuuu| xxuyyx||wy~}wt v{ uxy||tutxt||zy|v~{{vxv|{xwtzt{}txwxuwt}twt}utyuy|yyztutw{t}ttyuxuwyvwyztvuxuuuyxuw|wzyy|tz}y~uxxv}xwy|vz{tv{ytu | y y xut yt{|tyy tuy~ywt t{ z| w|| tu{t t{x| tw tw~ ttw {t ttzuxxwty ytytxx{z? txyz}}y~yxx}u}y|u{|zytu}ttuuuuyy{yx yt w tu{t vxu|t}|{t} t{x| w}y t yw}ttz} vxuyt{} t{x| txuyxyu wy t}y yx|u twt x ytwzy} tz}{yw|t{t xw}|~ vxuyyyy w}ttz} xw}tt u{|x}z| |u yxuy~xu|t xw}tuw}|~ |yw}x} xtt xyzx|ux} yxut|}yxtt ytz}y~xwxt{| w yw|xxv~xuut|{{tt}|{tx{x||yxzw|||u|tywwy}x} xxztxtwy|}}tyttx{zytttwyxtyxuu|vx}ut|}}txwtwtw}ytyz} ~ |} u t tuuvtw}t } wttt}u{ywx}?tuwty v}wyxx twtz zx} |ut yw}x} yxz|w y xvwyyyxw}|u x{}? xt}ttww| txu{xyxv}ttwu}wt v}twxzy|tyyywv}ww}y|v~{vyxxy~|xuxuy|{z|ywu}x{yv|uyu{zy|wzxuyxy|}wx} twyw} |ywytzuyxwtx|w}ytutu{~|txxt{|ytuxttxtuxwu|z}|x|ztw}xuv|{x|}zwtu{tyyut{yw|wty~|txtywxy}wzy}|~xw|}zwvuwy{|wxvt|yu}xyx?||yzwv}}xx } w t } z ? xz ux w w }yyx { xy}w y } { u
???' ??$) %?0)$''? ?0 ?0 1$0'??$?%?( 1? ?(%)?? ?

wyw}y {} y xx xw} yy vx yy | } v} t tw}t ttu tytz|zx? ztt{t uuy| uuuytw}u|w}tt ttu|uu z |{xtu|t xxuuyuyxt t} t wy|x~w{tz}|xw}wt}utuyuytutttzv|tzw|z|wx}|u~uxyww}tuxwyw||xwuyx|zwvu|xw|tz y{zyx~u|y ttwu|z y u w }uu| | y t yxutttx{wt}|t~tut?xw u t t ||xwu}|xtuwtx}zt|w|}v|{twtt|{xv}|wyv{vw|}tuxtuy{utx|wu~wtyt{|ttuw|v{{xt| vzxt|tux{t|ywt vu}xty| u z vw u z |{|ux} txtuz{|xy} tytz|zxu wyyw yu xtxt |} yxuytuy~}x twu||u xtz||wtx txtzu|txywuyx~}|t wyvttwxtzwtxw{tz|xttw{}|zxuy{|uzyt{t{ytxu|ytutu}yxw}u|wy}tx|u |u t u w|y|x|z|}zt|tu{yt~{z| |xtvz|z |zz|wtutwt uu|w| | v{ yx|? ux| v{ |{x u }w}}tu{|zwt{yu ztwut|z w|~}xt tzy{t|ux} ux| x |w|u~ttwt}{wtytwtttuwy|uxtw~ywy tuxutvtwxtux|tu|tz|u~xww}~|~tw|twux}vt{uxt{ytz|uttzy~tx|x|y } u xw|}u|~wyvx{t yu z|zxu |ux yux| ztt{t t{}x xxuvtw}t} tuu|w| v}xvt|u {}x~ tu|~ yu ztt{t|zt{}x~ tw~ z|w |uxw ux| |tut| wyw}wywtt} wx{| u wtu|xzz| t~uxxw}|tttu||tz{ttww~y|wy|wxt~}t tutt?tzw|zty~wv}yxx|ywttztutx{|y}tv{ ux| y y|w}uwuywtt} tytu xzt{t ytuw ?|x | x xvtw}t} ywttz y

aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl ; xh1 , xh2 , . . . , xhl - aj1 , aj2 , . . . , ajl ; qk1 , qk2 , . . . , qkl ; Sr1 , Sr2 , . . . , S ?? ? S
r

? z|w wvv{ |{x| yuuyut{} xw |zy~v v|ux x{x |w x{ v}wy|ytt ux| xz}|z x|zt{t xxz|z ttw wt x tu |{ux| wtuu| ztt{t vxuvtw}t} x wuy{ zyy~v t {x twxx}y|twuwwyt{tyzt{t|wuyuxxtzvx}xt|utx|uuvtut?||ux|{ty}yuyx~xtyy{|uxx}u|x|tz}{txzu}yy~ yw t x ? u uw | u u ||{x|xtxuzvxtu{yyu|yxxyuxxvy{wx}}t}tutx{ztx{y{|yywwxw|yy~|tu}tt|tuy{tyt~tvwtxyz}yyxxuytyxwyw|~t|w|w{yx|~||utvxtuyuyy x t t t u t t x t~ t t ty| tt xz t u|yt} utt
ajt , qkt
t

xh ?

t

t

t

t

t

r

l


?



+ ??&???????

? ?Rǿ5?Rœ+?ɜ'?++????&???????+??????????Ǟ+????+????+?ƞ?+R??+?'?+'?????++???+????Ü?????????R???'+???œ??????'???????Ü+?????'?????????!'???' ?? ? ' ? ' ? ? ? ? ' '

tx tut{z| xw}tuwu|yx|uxuyt{ wx~ytxwt yu tt z ~ x wx}yxww~ttw{||utwwy~uwy|wy|ztyuyyt{wt~ tywtwx vttzw{}tvw}wtyv{x{vyyxu?w}y}uwy{ty|t|}twutu|yytzzw|}yxy~y x t u t uy w x xu x u x|xwyz}x x |wy~}t ttz|w v{ xuyx x{}y |u|uy? xtwzy|w twuy t xx zu ttu {t |wy~}t |twy x wy|zyw xuyy x ux{ xuy~|u xz}y~xx wt |wxt{| tt| |{}x~ w}tx}x| v|u~xw||z z|w? t z yw}x} tut wt ytyy x xzxwy |ut{tx{zxwy}ttuy} yxuyyu|t} |u wy| x tw~ yw}x}u{|xyux u| v{ |tz|uxt w zw ? w }t{} |u |w tw~ }tw yv}wyv{vt |ttw yw}x} u{|xyux wv}tyuwuxy{|xt tu |yxu|~xyy{~uy}wtyvx}yuw|xy~wtxx}t{y}t||v{wv}txyu|t|xty{t}yxwy~}xtw}w}ttuwtt}{y}wxty{vu}uwyyvw{tuuvtxuy|wy}txy}{tyty|xw|}|yyxt t { u x ~ } t xy y x tu u tyu {}txw~~ xw}|y}y|ut |{}tx~tyut{ywxx{w}}xy~~t{t{zt}|uyxww}tx} yu{y}u? y|xwuyy~}ywttyww|}xt}x{ut| | zu w x yuyy txuuywu ytu | vxu|tx{yt |utz x |{|~|u twuyt v}wyv{yyt x wt wx}x t |uyu|wxuwy}u{{|tu{t}xx~tyt wtuyxxtxyutztw xs tu~xw||z wt wx}x| yttwtz x|y{ y y tx wuyu x xuyuywu x t}}yty y}vxu|tx{yt v{|w}wy}y tyzyxuzx{yuxt{u{tyw|}yxw}x|~yyx|uuuyuxw}x|~twy~uwyuuy{y}|u ztywy~vxys xzwyx{ tut{}ywvx}wyvx ut yx x y|u tu{ywx}tuwt vxu~t{t yu| wyx|{ xu| tw| wtw |tz v}w|~{x wt wx{| w u|z yu| utwt wy|uv{x{x}|twwu|yyz{yuzvtuy{xyuwvxtwy}}|x |uxvtwt yutz|wwvtxw{}}wyt wytxy|? | | t twttzzytutx{t|}||~v}}wxvxv|}xty}}yxtvxwtzt|}w |xsxuyzyux|{u}}utwxyxwxt}|uu|yyx x v xw x { y{y } x|yt tu | tuwuvyxuyvtw}t} |yz}vwt|tw}y{ttw}x tuyu| yvwuyy{||u yyyw}x}w}tyu{|ywuxxyxyux||tt}u}~tt | { u y ~ t{ xuuywu vxu|tx{yt v{ xuyy wuyt xy{} xxzu tt|u{t ty ywy}y yxxx}tz|wy}u|ytwwztuy{wxt{wwzx|y}x|w|u|yzu|yxzx~|tuw|xttuxw~}zyzyuxtwx|uxy|{tx~y|wyutw}wy}~|w}w|w|y|zw{ytt~t}|y|tv|z}t|wv|ytwtw|} x w x w y x | z tu{|w | z t w tu t y tt t u t t u tw u y w tt }| xyy~tt u{t wxuu|yzy |zwyt xxu|tx{yt x wzx{| twu|xz |uxw x tywy|xyw|x}}||u x{ztxxutuytt{t}x|wt}tyw|} vz}~wty|y }xyty~yx|uuyt{wtvtwwy{t{tzy}txs|ut}}wyzyt| |
c1 dS 2 - c2 dt dt
2

dS

dt

dt

dt

dS =

dt

dx2 + dy 2 + dz
2

2