Резольвентные пределы квантовой эволюции открытых систем (1104643), страница 2
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K;ª ¦¤ KÞª H@< ¦ >T: ¤ H@`> O_¹ ¦ ¼I½K ¤ ?AE N3Q7J;¼Á?=<;>;H@K ¤ >T:9;K ¤ ? ¦ l2 ⊗ ΓS (L2(R)) ⊗ ΓS (L2(R)) ?@J©¹H: H = H0 + H E+ H L,H0 = Ω b† b ⊗ I ⊗ I − I ⊗ i∇1 ⊗ I − I ⊗ I ⊗ i∇2 ,ZH E = i g dx g1 (x) I ⊗ a† (x) ⊗ b − I ⊗ a(x) ⊗ b† ,ZH L = iα(b† + b) ⊗ I ⊗dx g2(x) a† (x)a(x) + f (t) .¸ EF9LG;º;JLJ 1,2 K ¤ > ¦ ¾/ª ¤ KÞª¬Gk¹ ¦ KjX ¤ :¥½¿«*EF9LG;ºLJ;JÁ<;>LJg (x)→0u KLª KYN3L@HG_MH ¦ ¹WJ;9;JL¨;9LHM_§H'? ¦ G ¤ AH >;: Υ ∈ ΓS (L2(R)) HML@H®9T:®¨;J;¾ ¨ ¦ ½> ¦ P (Υ) <;>;H ¦ G ¤ H@>¬?@J©¹:HP (Υ)ψ = (Υ, ψ) Υ,EK ¤ X T9 :®¨T:KYX@9;H ¦ KMH@K ¤ H®ª;9;J ¦ HAGL>FE!SK ¤ :@?#K;ª ¦§¤ MK HGL@HA¼DKEF< ¦ >;<LHJ;º;JHN2GIHM_ ¦\1ρb =πM 0Z∀ψ ∈ ΓS (L2 (R)).¦ L9 JFªn9 ¦ ª;?#K;ª ¦§¤ KÞª ¨LJ;K ¤ J ¾OÀ:"<L> ¦ ¹L½> ¦ 9 ¤ 9^JB¬K§HAK ¤ H®ªL9;J;¼20P ξI(0,T ) e−|ξ| /M d(Re ξ) d(Im ξ).¹ ¦ dK X M 0 ¢ ¹ ¦ ¼;K ¤ ?@J ¤A¦ KjXA9HJ ¼ <;:@>;:@¾ ¦§¤ >`O T ¢ <;>;H@JF?AH#KYX@9;H ¦ <LHcKTHqS J©½¤A¦ KjXA9LH ¦ ¨;JLKKWH j\ E K ¤ XHO ¤ :GAS ¦ OFH@KMºLJAKHK;ª ¤ HA>D? 9;:¥¨T:KYX@9^J¼Á¾H@¾ ¦ 9 ¤ ?A> ¦ ¾ ¦ ½9LJ"9;:CB H¹J ¤ X@KÞª"?k9LHA>L¾ÀJL>;H@?:9;9LHA¾'GIHM_ ¦ > ¦ 9 ¤ 9;H@¾'KMH@K ¤ H®ª;9LJ;J ρ = |rihr| ² MH _â¹H: O¥?MJ>T:S ¦ 9;J ¦ ¹^K;ª7> ¦ ¹;EFº;JL>;H@?:9;9LHA¼¾·: ¤ >;J;ºHJ{<AKWH ¤ 9;H@K ¤ JJL¾ ¦§¦§¤ ?AJI¹£()†∗†b−r(t)−Q(t)(b−r(t)−Q(t))1: exp −ρ (t) =:,1+M1+M§¦_â¹Q(t) = iαZ0g2eiΩτ − 2 τ a† (τ )a(τ ) dτ,−tr(t) = r ∗ e2−iΩt− g2 t+ iαZt g (1 − eiΩt− g22 t ) 2M = M 0,2g −iΩ +2g2eiΩτ − 2 τ f (τ ) dτ.0¤ H@¼S ¦ ®:C¹:¥¨;J=>T:C®?@J;?A: ¦¤ KÞª7¹W>FEA_§HA¼ç¾ ¦§¤ H¹ µ ¦ Z : ¦§¤ KÞª=K ¤ HBI:@K ¤ JI½¨ ¦ KMGIH ¦ ¹J;«« ¦ > ¦ 9;ºLJT:KYX@9;H ¦ EF>T:?A9 ¦ 9LJ ¦ ¹jKLªz¾·: ¤ >;J;ºHJ <AKWH ¤ ;9 H@K ¤ JzHAK§º;JAKF½KLª ¤ H@>T: Az >T:?@9 ¦ 9LJ ¦ ¹jKLª¬¾·: ¤ >;J;ºHJ<"KWH ¤ 9LHAK ¤ JD®:@<LJ^Z ¦ ¾ ?=?@J©¹ ¦ u KLªbadρ(x1 , x2 )i= − (H(x1 ) − H(x2 ))hλ2+ ((x1 − a) + (x2 − a)) ρ(x1 , x2 )dt4rλ((x1 − a) + (x2 − a))ρ(x1 , x2 )dQ+2n_â¹JT¦+ σb(x1 )b(x2 )ρ(x1 , x2 )oσ− [b+ (x1 ) b(x1) ρ(x1 , x2 ) + b+ (x2 ) b(x2) ρ(x1 , x2 )] dt,2¢¤¤¦¤_:¾JAKjX H@9;J;:@9 ? GIHAH@>A¹JL9T: 9LHA¾<;> ¹WK :?cKH(x)¢ ¹JL«#« ¦ > ¦ 9Lº;J;:#KjXA9HJ ¦ HA< ¦ >;: ¤ HA>HJ?@J©¹:Hb+ (x)d1d1b(x) = √ x +, b+ (x) = √ x −.dxdx22¦ 9LJ;J`OUb(x)E K ¤ Xz?R9;:¥¨T:KYX@9^J¼¾HA¾ ¦ 9 ¤ ?@> ¦ ¾ ¦ 9;J HAK§º;JAKHK;ª ¤ H@>9T:¥BIH¹WJ ¤ KÞª?zH@K½9LHA?@9;H@¾ KMH@K ¤ ®H ª;9LJ;J 21x1 x22T @ U\ρ0 (x1 , x2 ) = √ exp − −22π,:¹ ¦ ¼LK ¤ ?AJ ¤¦ KYX@9^J ¦ <T:>T:¾ ¦§¤ >^J λ J σ B©:>T:G ¤A¦ >LJF§EAN ¤ K ¤A¦ < ¦ 9^X*?¥:J;¾H¹K ¤ ?@JFª H@KMºLJAK^KLª ¤ HA>;: K,H@G;>FE!S ¦ L9 J ¦ ¾nJ K ¤A¦ < ¦ 9^X ¹WJ;KMK§J;<;:@ºLJ;JÁK§HAH ¤ ? ¦¤ K ¤ ?9LH v <L>T:? ¦ ¹jKWJL?:çKK ¦ ¹;EAN3Q:ª A¤ ¦ @H > ¦ ¾À: ¦ ©¼ ½¦ 9I½[IUWdc[Ib*Q `IÇDI üçT Ì ÍLGÏCW"Ø"úó M Ï DÎ@ÐÓ~#Þ;MWYÜóÌÕÏ®ó Ð ÓR1 (t)N (t)(x1 − x2 )2 + ip(t)x1ρ(x1 , x2 ) = √ exp2π+×ÎHôGÏ@§ jÜ3àI R2 (t)(x1 − q(t))2 − (x2 − q(t))22R1,2 (t)!− ip(t)x2 − r(t)(x1 − q(t))(x2 − q(t)) , Tr(t)¡U#ÎqÞFTAØYÙ çØ{ç®óú Ì ÍLcÏWjè YßσdR(t)=−λ+2ΩR(t)R(t)−− σr(t)112dt211+ σr(t)2 − σR1 (t) − σR2 (t)2 ,22dR2 (t) = Ω + Ωr(t)2 + 2Ωr(t)R1 (t) + ΩR2 (t)2 − σr(t)R2(t),dt d r(t) = λ + 2Ωr(t)R2(t) − σr(t)2 + σr(t),dt Þ;WjÜ?óÔ 4ÕWjFÞ jßÌ ÕÏ"ØAóR1 (0) = −1 á R2 (0) = 0 á r(0) = 1jÜØFÏMAØYÙ çØÈô@GÏ"óÜ?ó~Õ Ì Þ;#èTYÜópÏ@ÕWjÞFW¥ó ÍF@ßp(t) q(t)cÍnÎÕÊWjÜó "Í!úó T Wó Ì @ô ©à ® Ì ÎqÞFW"ØYÙ4é="óØ'á ç®óúè Ì ÍFGÏTYö×Î(dp = P dt + ρ dQ,dq = Edt + η dQ,σσP = −Ωq − p − F sin(wt), E = Ωp − q,22rrλ R2 (t)λ 1ρ=−, η=.2 r(t)2 r(t)ÍFGÏ@ pÔË ÍdÎàI®WWWjÜ]èó¥ãT ÕTN (t)r(t)>LJz¹H@K ¤ : ¤ HA¨L9;H.L@H#KYXGZJIB t O? ¦ KWJL¨;J;9HJ R1,2 J r <L>;JL9;J;¾À:cN ¤ 9T:¥¨ ¦ ½9LJFª`ODL#KWJ©®GLJ ¦ GvK ¤ :@ºLJ;H@9T:>;9HJ ¾ ± a ¤ AH ¾KK;EF¨;: ¦ O? ¦ KWJL¨;JL9^J ρ J η ¾qH S 9;HK§¨;J ¤ : ¤ Xç<;H@K ¤ H®ª;9L9^J¾ÀJ Ã\: H@KM9;H@? ¦ ;< H#K;EF¨ ¦ 9L9;HG_MH> ¦ Z ¦ 9LJFª <;H@K ¤ >;H ¦ 9T:,> ¦ ;¹ EFº;J;>LHA?A:9;9T:Cªç¾À: ¤ >LJ©½ º;:=<AKWH ¤ 9;H@K ¤ J j¹ KLª¬KMH@K ¤ H®ª;9;J©ª¬HAK§º;J"K^K;ª ¤ H@>T:7¹jK;ª ¤ :@LG J©B 9T:¥¨ ¦ 9LJ;¼ t l_⹦BN (t)exp −ρa (x1 , x2 , t) = p2(1 + 2r(t)Dq (t))1 + 2r(t)Dq (t)B = at x21 + at x22 + bt x1 + bt x2 + ct x1 x2 + 2Mq2 (t)r(t),at = − 2R1 (t)Dq (t)r(t) + 2Dp (t)Dq (t)r(t)+ R2 (t)i + 2R2 (t)Dpq (t) − R1 (t) − 2Dpq (t)2 r(t)− 2ir(t)Dpq (t) + Dp (t) − r(t)2Dq (t) + R2 (t)2 Dq (t),bt = 4iDpq (t)Mq (t)r(t) − 4iDq (t)Mp (t)r(t)− 2r(t)Mq (t) − 2iMp (t) − 2iR2 (t)Mq (t),ct = − 4R2 (t)Dpq (t) + 4Dpq (t)2 r(t) − 2R2 (t)2 Dq (t)+ 2R1 (t) + 2r 2(t)Dq (t) − 4Dp (t)Dq (t)r(t)+ 2r(t) + 4R1 (t)Dq (t)r(t) − 2Dp (t).± <;H@KK ¦ ¹9 ¦ ¾'?GJ >;:#S ¦ 9LJ;Jv? ¦ KWJL¨;J;9HJ¦ ¤ K§¾]J KK K§> ¦ ¹;½Mq (t) J Mp (t) JL¾ NL9 J©B ®9;:¥¨ ¦ 9;J;¼¬? ¦ KWJL¨;JL9 q(t) J p(t) O;>T:KM<L> ¦ ¹ ¦ KT»M9;9HJBw9LHA>L¾À:#KjXA9LH^Mp (t) = FZt0σe− 2 (t−τ ) sin(w τ ) cos(Ω(t − τ ))dτ2F n (− σ t )e 2 (Ω1 sin(Ω t) − Ω2 cos(Ω t))=ω1+ Ω2 sin(w t) − (8 w 3 + 2 w σ 2 − 8 w Ω2 ) cos(w t) ,ZtσMq (t) = F e− 2 (t−τ ) sin(w τ ) sin(Ω(t − τ ))dτ0:=Dp (t) O Dq (t)2F n (− σ t )e 2 (−Ω1 cos(Ω t) − Ω2 sin(Ω t))ω1+ Ω1 sin(w t) − 8 σ w Ω cos(w t)} ,JDpq (t)¢anK¦¾ ¦9¤JIBi¾À: ¤ >LJ;ºHJ GIH@?A:@>LJT:º;J;¼O+GIH ¤ HA>HJ ¦J¤ :GAS ¦ ?GJ¨;J;KK ¦ 9^Jqª;?@9;HHDp (t) =Zt02e−σ(t−τ )R−(t − τ )dτe(−σ t)−d1 sin(2 Ω t) + d2 cos(2 Ω t) − d3 − ρ2 σ 22ω2d4 − 2 ρ η σ Ω+,ω2Zt2(t − τ )dτDq (t) = e−σ(t−τ )R+=0e(−σ t)d1 sin(2 Ω t) − d2 cos(2 Ω t) + d3 − ρ2 σ 2=2ω2d4 + 2 ρ η σ Ω+,ω2ZtDpq (t) = e−σ(t−τ )R− (t − τ )R+ (t − τ )dτ0e(−σ t)( d2 sin(2 Ω t) + d1 cos(2 Ω t))=σ2ω2ρ2 Ω − η 2 Ω + ρ η σ.+σω2£¹ ¦ KdXç?@? ¦ ¹ ¦ 9HJHGLAH®9;:¥¨ ¦ 9;J©ªR− (t) = ρ cos(Ωt) − η sin(Ωt),R+ (t) = ρ sin(Ωt) + η cos(Ωt),ω1 = (8 w Ω)2 − (σ 2 + 4 w2 + 4 Ω2 )2 ,Ω1 = 2 Ω σ 2 − 8 w 2 Ω + 8 Ω 3 ,ω2 = (σ 2 + 4 Ω2 )σ,Ω2 = σ 3 + 4 σ Ω2 + 4 σ w 2 ,d1 = 2 η 2 σ Ω − 2 ρ 2 σ Ω − 2 ρ η σ 2 ,d3 = 4 ρ 2 Ω 2 + η 2 σ 2 + 4 η 2 Ω 2 ,d 2 = ρ2 σ 2 − 4 ρ η σ Ω − η 2 σ 2 ,d 4 = 2 η 2 Ω 2 + ρ2 σ 2 + 2 ρ 2 Ω 2 .>LHA¾ ¦ú¤ HM_§H^O9;:@¼I¹ ¦ 9^J E KKWH@?AJ©ª > ¦ _§J;K ¤ T> :º;JLJ"¾À:KWH@¼vKMJAKjJ ² :@G©:Cª¬®:C½¹:¥¨T: ?@H®9;JLG©: ¦¤ O? ¨;:@K ¤ 9LHAK ¤ JO<;>LJv<;H@K ¤ >;H ¦ L9 J;Jv¾À: ¤A¦ ¾À: ¤ JL¨ ¦ KMGIH@¼w¾ÀHM¹ ¦ KTJ¹ ¦§¤A¦ G ¤ H@>T: _§>T:?AJ ¤ :º;J;H@9;9HJBÁ?@HcKT9 w] >;d8[>HB[¦§MZ_QVW[ 9T:Cª ª;?A9HJ ¼?@J©¹ç<;> ¦ ¹ ¦ KYX@9;HG_MH,>T:C®> ¦ Z:GNRQ ¦ _MH HA< ¦ ½>;: ¤ HA>;: ¹^K;ª EF>T:?A9 ¦ 9LJFªiT UO9;:CB H¹J ¤ KÞª7?MJ>T:S ¦ 9;J ¦ ¹^K;ª¨T:K ¤ J;¨L9;HG_MH KK ¦ ¹:<L>;H@JF®?@#H KYX@9;GH _MHDH@< ¦ >;: ¤ H@>T: B ∈ H <;H KMH@K ¤ H®ª;9LJ^N H@G;>FE!S ¦ 9LJFª Fu KLªi<L>;HC½J©®?@cH KjXA9LMH _§H GIGH _ ¦ > ¦ 9 ¤ 9LMH _§Hi? ¦ G ¤ HA>;: ψ(h) J© E S >;:@K§KM¾H ¤ >LJ;¾2¨T:K ¤ J;¨L9;H ¦K§> ¦ ¹W9 ¦§¦ ?=«#H@GIHA?@KMGIH@¾ <L>;H@K ¤ >T:9;K ¤ ? ¦ O;ªLc? KL^ª N3Q ¦Mε¦ KÞªaK ¦ ¾ ¦ 9 ¤ HA¾ J© B(H) Pt (B) = (Ut ψ(h), B Ut ψ(h))ΓS . àW"Ø[IUWdc[Ib*QòI`IRÌ YPt (B)Ì ÎC#ÏMÏ@Í^ØÌ ÍFGÏTjÙúTDÎ#ÚÇGÎdPt (B) = Pt (Lt (B)),dt×ÊõÍFçÍàI®çÍö#×Ê A óú§ÏDÎb∗ B Rbt − W ∗ B − BWt á ×ÎCLt (B) = Rttbt = (eiK − 1)(iK)−1 R + eiK h(t),R 1 ∗bRbt .Wt = i H0 + (sin K − K)K −2 − R2 t7Z_QY§@U`ÀQdC/UDEM>LaVI ° ? ¤ HA>~?GJ >;:#S=: ¦¤¤ XiK§?AH ¦ ¾/E{9;:EF¨;9LHA¾/E>©EFGIHA?@H¹J ¤¦ KYN <L>;H@« ¦ K§KMH@>FE °  y,¦ @L H ¤ :>;»§?E ¥:^<;H@¾HMQ;X{?~>;:GL@H ¤¦ J<LHcK ¦ ®9HJ ¦ HGLAKE!S=¹ ¦ 9LJFª 15.6V:;/LcKW:G_§H¹:>;9LHAK"!9%$#751/9(-151 µ JS=:@GIH@?îx ± O v J;9L»M? °  &% ÏMWAØTWTYýÍnÎ¥d×Ê ÞFÕiä3® ®#Ï Í ô@¥jÜ]èfà!Ï@Tç#Ï#Ü]è{cÍdÎCվΧ"ØnTT ýÍHä·õÍöTW®Wqèp× ÍF#ßÏWçW®W èqTçTY Ü l :MEF¨;9T:Cª'GIH@9;« ¦ > ¦ 9;ºLJFª £ r#H@¾ÀH@9;H@KMH@?@KMG;J ¦á¨ ¤A¦ 9;J©ª©½ ¡@A ¯ ÜÏ@# Îà!Ï@Tç# Ï# ×Êçn Ý@=j Þ;§ àI# ×Ê Ì ÍLc ÏW" ØÎA ØÄ ¡ Å µ JS=:@GIH@? x ± 'óú# ÎCÕWTT ý ÍHä·õ ÍöTW®W ×p Î à ÍF¢ × ÍFG ÏTçW¥TY Ü?Ý Ï@ÇWál :MEF¨;9T:CªÁGIHA9L« ¦ > ¦ 9;ºLJFª £ r#HA¾H@9;H@KMH@?½ ¡@@L ¯ %eÍ ßÄ }Å y*¦ LAH ¤ :>;»§? °  O y EF>;GLJ;9 ° ± O µ J8Sç:GIH@?*x ± O v J;9;»§? °  (ô@¥W ó à!Ï@Tç# Ï@ ó G Íöd ÎCÕ ÎA Ø WTçõ ÍAä3õ ÍöTW®Tq è× ÍFG ÏTç# ßW®Wq è^TçTY Ü ± ¦ K ¤ 9LJ;G  x z OöK ¦ >LJFª } ¸ J©®J;G©: ° K ¤ >;H@9;H@¾ÀJ©ª É O ¡@@; O;K ¤ > }M}*)A} ÃFá Ä ÅÄ Å,+&-/.1032'465879.;:=<V?>R OA@CB -/2ED/79FHGJI=< K ?O LNMHO -P58F82*:RQ K TO S GJI/.:U< C>Y1W X8Y[Z'\]^YJ_a`bX*c_YdXeUf'gh\iZ*jlkm\'knjoXip\8Yrqs\'Z8_tc_k_ uk X'gv\ipwcxklyz_{W;km\ipc\ig;k^| } \ip3k } ` Y~j`jlk L D/h F> 5 46- 3 - M Ç"û O ¡ O } ) W T ¡@@ } Uá> OLNMHO -P58F82*: Q K p kyP_ kg6XipHf _W1X8Y[Z8_;p3k< Y_ XebkyP_d yg(X cijpHf_;gdsZ'X'Y } knjoXipkmX{ } \'pk } ` 3kmX y/\ W;knj W XipzZ8_;gf_;p "á> 5 46-P.^5i46G B 5E(2i46.1 O z/ F© ) }G} T ¡A } UÄ Å,+&-/.1032'465879.;:Ä Ã Å y*¦ LAH ¤ :>;»§? °  O µ JS=:@GIH@? x ± % çdÝW@ jÞ;Mà AÝ Ì ÍFGÏTYßØHÝ ØFÏ"ØYÙ4é="óØÁ©WjÜ?ó-ÍWô@ÞW!Ï@T YÜókGÍödÎCÕ¥ó Ì ÍFGÏßWjá èiþÿÍnήTF×ÊõÍF l :EF¨;9;:ª GIH@9;« ¦ > ¦ 9;º;J©ª £ r#H@¾HA9LHAK§HA?@KMGLJ ¦ ¨ ¤¦ 9;J©ª©½á¡@@ ¯Ä ŵ JS=:@GIH@? x ± ¡úAóTç¥ jÞ;Mà ÍüçTAØ à!Ï@Tç#Ï#×Ê{çnÝ#ß Y ÞLMàI× Ì ÍFGÏT"Ø¢# Ì Ï©AØÎAØ"WFAØT ÍFç Ì ÞzÊçWó Î¥ßTWT L D/h F 28£ > 5 46- - M O Ç/ O } O }M à T ¡A U áÄ ÅÄ Åµ JS=:@GIH@?'x ± /¤ WÍDàI#Ï@àI4cÍDÚÇW^ãTAØÎAØÞL@ jÞF×úýÍnÎ¥ß×4 ÚÍF "óúè¥cÏ@ÇjÙ WWÏ4GÍö¥AôGÏ@nÎWTÏ@à ÍjÜ?Ý cÍCRÍFÏ Ou7¦ ?ª ¤ 9;:C¹WºT: ¤ J ¦ ¾ ¦ S=¹TEF9;:@>LH¹9HJ ¦ <AK ¦ BI:@9LHA?@KMGLJ ¦ ¨ ¤A¦ 9LJFª`O ¤A¦ ®JLKdJn¹WH½G K:¥¹H@? 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