Equivalence entre les procedures de «local tracking» et les algorithmes monotoniques en contrôle quantique
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Equivalence entre les procedures de “local tracking” et les algorithmes monotoniques en contrôle quantique Gabriel Turinici. N° 5564 Mai 2005. N 0249-6399. ISRN INRIA/RR--5564--FR. Thème NUM. apport de recherche.
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(70) [jbjO>ql[KY\`MoxJ Y\c[]`Mqu [jciv_M4Y ZY]O3O p3} ;ivbKwZO^Y\cpr_MY[jJMql[KpiZo3qrpW qu[]O3b%[jJfO []bxqlkmO>o[]irbjW[ji4o3cbxo`fN&rO>_v[[]JfO2Y]cp_fr`fequb%t;ivc_B[jY qr_Mw $ 9Z$ $irbnqY\[]`MwZWiv_ []JfO
(71) Y\[]ivtftfcp_ft;ircp_v[xYqr_Mw tfbjiZoO^wZ`fbjO>Y%[]i cNtfbji¬rO[]JfO>cbnirtf[]cpNqrc[mWr¦ > [ceY4Y]O3O>_[]JMqu[>)Ofo3O3tZ[4irb[jJfOt;ircp_v[jY. de y (t) dt. 2. . . ,-% ,"0 , %($&!.
(72) ,- , . "*+'-,. /$+0 243. 0 $3) "* ! ,-% 0 ,-*. >£_ qu_qrtftfbjivqvojJ9wZc©;O>b]O>_v[b]ivN []bxqroxXc_fN4iv_fiu[jir_fceo3qrpW®oir_XrO3bjrO>_v[quprirbjc[]JfN YntMciv_fO3O3bjO>w cp_ ÈuZ;f qu_MwOX[jO3_MwZO^wcp_ p' (cp_[jJfO q¬rO`f_Mo[]cpir_®bjO3tfbjO>Y]O3_v[xql[jciv_$fqubjO2`MY]O>wcp_[jJfO&oir_B[]O3B[ ir«[]JMO wZO>_MY\c[mW4N ql[jb]cirt;O3bxql[jirbqrYc_ ÈrZXG @¦ZUX`MoxJtfbjiZoO^wZ`fb]O^Yhqrb]Oncp_Mop`MwZO>wcp_ []JfOEbjqrN4O
(73) %irbj ir[jJfOirtf[]cpNqr%oiv_v[jb]iv []Jql[c_B[]bjiZwZ`Mo3O>Y&q9oiBYm[
(74) `f_Mo[]cpir_Mqr @HwZO¥_fO>wqu[q®¥M_Mqu[]cpNO T B []i9;O ivtZ[]cpN c ;>O>w¡¦ _fO.Y]`MoxJOfqrN4tMO2iu`f_Mo[jciv_Mqu$ceY Z @H| B α² (t)dt − 2RehhA, ρ(T )ii. J(²) = Y]irJf`fO>vb]JvO [αqlºce[]YO>q
(75) b&t;`fiv_MY]wZcª[jO>cb
(76) rO [jJf@HOoiro3_irYm_M[xquY\[]_vbx[qucpir_vb[&[]iucpNnO Yjqll[]quceY\bjWXWBcpcp_f_f B %@H O3cp¦ rJB%[>O^¦Xo3IKqr`MJfY\O>O _$vir[]KJf[]OJfo3O b]co[]ivceo3_MqrYm[jbjt;qrirc_Bcp_v[>[jY%q9iu (J(²) qrb]O u q v j b r q f _ rO N&`f[]cptfpcO>b>wZO3_fir[]O>w χ(x, t) cpYcp_v[]bjiZwZ`oO>w cp_[]JfO
(77) oiBYmB [K`f_o[]cpir_qu¡[]JMqu[n_fi bjO>qvwfY T. 2. 0. Z. T. α²2 (t)dt − 2RehhA, ρ(T )ii J2 (²) = 0 (Z ) T ∂ρ (H − ²(t)M)ρ +2Re hhχ, − iidt ∂t i 0. ¾¾#¹GTMÑÑIUxâ. @6 B.
(78) |. FIR
(79) FI R. IKJfO
(80) objc[]ceo3qu¡t;ivc_B[O^dv`ql[]cpir_YKqrb]O [jJB`MYnirZ[xqucp_fO>w . ∂ ρ(x, t) = (H − ²(t)M) ρ(x, t) ∂t ρ(x, t = 0) = ρ0 (x) Mρ 2α²(t) + 2Rehhχ, ii(t) = 0 i ∂ i χ(x, t) = (H − ²(t)M) χ(x, t) ∂t χ(x, T ) = A. i. @' B @R9 B. @m> B. `fcppwfc_f&ir_ []JfO^Y\OEbjO3eql[jciv_MY3u[]JMOENir_fir[]ir_Mcponqrvirbjcª[jJfN Y tfbjO>Yjobjc;OEq.tMqub][]ceo`MpqrbhirbxwZO>b []i&c[]O>bjqu[]O cp_¨[]JfO^Y\Ooiv`ftfpO>w¨O>dB`Mql[jciv_MYnBW¨oir_MY\[]bj`Mo[]cp_fMqu[n[jJfO&c[]O>bjqu[]cpir_9Ym[jO3t k → k + 1 ;q¥O3ew ² (t) c[]J[]JMO2cpNt;irb][jqr_v[tfbjirt;O>b\[mW @mr B J(² ) ≤ J(² ), JfO>_MoOE[jJfO _MqrNOEir E PE N E ! G *EGFIN¦ : Y]cNtfpO OfquNtfpOEir(Y]`MoxJqrvirbjcª[jJfN ceY @HY]O3O vXZ ivbnqrwfwZc[]cpir_qu$wZO[xqucpeY B @\^ B ∂ i ρ (x, t) = (H − ² (t)M) ρ (x, t) k+1. k+1. k+1. k. k+1. k+1. ∂t ρk+1 (x, t = 0) = ρ0 (x) Mρk+1 1 ii(t) ²k+1 (t) = − Rehhχk , α i ∂ i χk+1 (x, t) = (H − ²k+1 (t)M) χk+1 (x, t) ∂t χk+1 (x, T ) = A.. @\ B @\3} B. IKJfceY quprivb]c[]JMN ceYtfbji¬rO^w ÈG [ji·JMq^rO9[]JfO o3ir_XrO3_fcpO3_v[ tfbjirt;O>b\[mWc_¢dB_$¦ @\r ¦ > [ cpY[ji;O _fir[]O^w []JMqu[n[]JfceYntfbjirt;O>b\[mWcpYnrO3bjW Y]`fb]tfbjceY\cp_fcp_[jJfceYnJfcprJfpW_Mir_fpc_MO>qubEY]O[\[jcp_fMMO>Y]t;B O>oceqrpW JfO3_ o3ir_MY]cpwZO>b]cp_f[jJMql[n_fi Y]O>o3ir_Mw irbxwZO3bKcp_Zivb]N ql[jciv_ ceYwfcbjO>o[jpWcp_BrirprO^wcp_ [jJfO
(81) oivN4tM`Z[jqu[]cpir_MY>¦ Pniu[jOn[jJMql[ @m¬ qu_w @m qubjO[]iOY\ivprO>w Y]cN&`Mª[xqu_fO>ir`MY]W4;O^o3qu`Y\OEiu¡[]JMOEcp_v[jO3b] wfO3t;O3_MwfO3_Mo3O ir$[jJfO ¥O3ew ² (t)B qr_Mw[]JfO.B Ym[xql[jO ρ (t) ¦ : _®qu[]O>b]_Mqu[]cprO tfb]ifoO>wf`fb]O ceY%[]i4cp_MY]O3b\[KbjO3eql[jciv_ @\ B cp_v[ji®O>dB`Mql[jciv_ @m¬ JfcpoxJ cph;O>o3irNOq_fir_f@pc_MO>qub
(82) UZoxJfb fwZc_MrO3b2O>dB`Mqu[]cpir_[]i®;OtMb]ivtMquBql[]O>w ivb qrbjwcp_ [jcNOr¦ B B <0
(83) K*K
(84) K 8N E N8
(85) H FINQ E C N8
(86) M8
(87)
(88) N N ?
(89) E <
(90) F
(91) *
(92)
(93) N?
(94) M NQ=M!C *NQRE k+1. k+1. E C N ?
(95) FINQR G 8E 8N
(96) 8 NQRE ?M M
(97) ?MIFL
(98) KP
(99)
(100) G ME C EGF C FIN ?
(101) F EG?MIRK
(102) FL NQREG?M EG N8
(103) E 8
(104) F
(105) *
(106)
(107) EGN
(108) N ? N N M K
(109) MFL =
(110) PF E ?
(111) FIN
(112) M E NA G 7HJ < M 8 FL G8N
(113)
(114) K CEGF N FL !G. #". $,. &% ('. ! ,. 0 * )(%4!
(115) 0. $+,-%( "*. Pniu[jO
(116) [jJMql[ [jJfO4oiBYm[E`f_o[]cpir_quiuhO^dB`Mql[jciv_ H@ | B JMqrY Ofqro[]pW []JfO4YjquNO&Nc_McN q qu_Mwobjcª[jcpo>qu t;ivc_B[jYqvY Z @m^{ B J (²) = α² (t)dt + kA − ρ(T )k , T. 2. dist. ii. 2. 0. ¸ºà¾$¸65.
(117) .
(118)
(119). 6. o3irJf_MceooxpJ `MY]NcivO>_ qrY]ce`fYbjO>[]bjY`f[jO JfOwZ`fwfO cpY\[][ji·qr_M[]oJfO9O iu_f2ir[jbjJfN1O9¥Mo3ir_M_MquY]O3bjwZlO3qu_M[]Y]cpcir[£_W tMb]ivt;O3b][]cp[]O>iY[jiuJf.O9[jJf[jO'qrb]vUfOoj[Jfb ivZtwZO>cp_fbjqur[]O>ivbb O>dB`M¦quIK[]cpJfirce_ Y JfceoxJqupi nY []i ivbj_fcª[jpOWwZc©;O>bnBW qoiv_MYm[xqu_B[>¦ qu_w4[jJB`MY[]io3ir_Mo3`MwfO [jJMql[ qu_Mw ρ(x, T ). A. . Jdist = J + kAkii + kρ(T )kii = J + kAkii + kρ0 kii. IKJfOirtZ[jcN qr«o3ir_B[]bjir«Y\[]bxql[]O>rWiuY\O^o[]cpiv_Z¦È
(120) irt;O3bxql[jO>Y%ir_ q4oiBYm[`f_Mo[jciv_Mqu$wZO¥M_MO>w qu[¥M_Mqu [jcNO T ¦ : Y4Y\`MoxJ$)wZ`fbjcp_f9[]JfOO3rirp`Z[jciv_qu[&[jcNO t < T [jJfceYlqup`fOceY&_Miu[4WrO[qro3o3O>YjY\cpfpO ivb cpNN4O^wZceql[]OO3O>wfMqrox cp_v[ji[]JfO2irtZ[jcNc=;>qu[]cpir_tMb]iZo3O>wZ`Mb]Or¦ Ei O3rO3b^ c[]J®q
(121) ¥MO>pwoirNtf`f[]O>w `ft[]i qEb]O^qrY]ir_MquMO%qu[]O3bj_Mqu[]cprOcpY[]i `Y\Oq o>qu_MwZcewfqu[]O iv_ [jioivN4tM`Z[]O%[]JfOt;O3b]irbjN qu_Mo3O t<T cp_MwZO
(122) ql[(¥M_Mqrr[]cpN4O T ¦ : _qutft;O>qrcp_fEojJMirceoO ivb ² cpY$²[]JMOh¥MO3ew.[t,irTf[j] qucp_fO^w.ql[qtfbjO3Xcpir`MY$c[]O3bxql[jciv_$¦ iv_ OnqubjO%[jJB[]`MJMYhO2p¥MO>qrO3w&ew [ji.c_B[]bjiZwZ`MoOKirbhq.oiv_v[]bjir ² X_f i _4ir_ [0, t] qu_w4qbjOO>b]O>_MoO¥MO3ew ² wZO¥M_MO>w [0, T ] ( @m>| B ²(s) f or 0 ≤ s ≤ t ²(s) = ² (s) f or t ≤ s ≤ T. IKJfceYK¥MO3ew ceY[jJfO.;O>Y\[Eq^uqucppqrfO2o3qr_MwZcewfql[jO.ql[[]cpN4O t < T ?¦ > [jYt;O>b\irbjN qu_Mo3Oc_wZO J (²) cpY Z @m< 6 B α² (t)dt + kA − ρ (T )k , J (²) = JfO>b]O ρ (T ) ceYK[]JfO
(123) Y\[jqu[]O.qu[K[]cpNO T iu[jJfO
(124) Y]WXY\[]O>N @\ ' B ∂ i ρ (x, t) = (H − ²(t)M) ρ (x, t) J. Jdist. ref. ref. ref. ref. dist. t. 2. dist. ². ii. 2. 0. ². ². ∂t ρ² (x, t = 0) = ρ0 (x),. ². : tfbjirt;O3b][mW c[]J®cNt;ivb\[xqu_v[tMbjqvo[]ceo3qr«cpN4tMceo3qu[]cpir_MYKiv_ []JfO.O o3cO>_v[EoivN4tM`Z[jqu[]cpir_iu cpY J (²) vcrO>_ cp_ [jJfO2irppi cp_f l° h°;$²H°
(125) P
(126) N ?
(127) 5CE FIHJ GF K E MINCPNQRE P *CE F N ?
(128) EG8NQF EG ² PK FL
(129) C
(130) F
(131)
(132) 5
(133) K ² dist. M. ref. Jf wd (², t; ²ref ) = +. H 8
(134) FL
(135).
(136) <EG 7<
(137) M EG ρ. ?
(138) . T. @\9 B. t. α²2 (t)dt 2. t. ρref. M N ?
(139) 8
(140) FM
(141) FLE 8 NQREG CFLE. ∂ ρref (x, t) = (H − ²ref (t)M) ρref (x, t) ∂t ρref (x, t = T ) = A. Jf wd (², t; ²ref ) = Jdist (²). ¾¾#¹GTMÑÑIUxâ. . @Hu B. 0. α²ref 2 (t)dt + kρref (t) − ρ(t)kii .. <MD K [0, t] i. . Z. Z. A. H N
(142) 5
(143) K. ²ref . @@Z B.
(144) '. FIR
(145) FI R. l°$° ")IKJfOn¥bjY\[[ %i
(146) [jO3bjNYcp_ qrb]OntfbjO>o3cpY]O3ppW&[]JMOn¥MbxYm[[jO3bjN ir ¦XIi&o3irN4 ft `Z[jO []JfO®Y]O>o3ir_Mw[]O3bjNc_ J J(²) (²,ρ cet;Y&²[ji ;)O O>rirprO>w'bjirN ρ c[]J·[]JfO ¥MO>Jpw ² (²)iv_ [0, T ] []i ivZ[jqrc_ ρ(T ) ¦ `Z[>XY\cp_Mo3O ρ qr_Mw ρ O3rirprO c[]J [jJfO YjquNOn¥MO3ew ir_ []JfO c_B[]O>b]uqu [t, T ] r[jJfO3cpbwZceYm [xqu_Mo3O cppv;O%oiv_MY\[jqu_B[$[jJfbjir`fvJfir`Z[(O>rirp`Z[]cpir_
(147) qr_Mw2[]JB`Y kA − ρ(T )k = kρ (T ) − ρ(T )k = ¦fIKJX`MY %O.o3ir_Mo3`wZO[]JMqu[ J (²) = J (², t; ² ) ¦ kρ (t) − ρ(t)k B <0 M N8
(148) D K EG N M G< G
(149) KGFI N ?
(150) N
(151) FL NQRE ?M E C N ?
(152) E PE N E ! G *EGFIN M ON8
(153) E
(154) FLE 8
(155) FIN# G
(156) +M
(157) K N E EG8N E F N8
(158)
(159) <EG 7NQRE E C N ?
(160) E MNSC PN RE P
(161) N HJ
(162)
(163) NQHJE MI8
(164) IMMI<
(165) N
(166) FL GN REG?M E F ?MIN G
(167) N + M G 8
(168) FL
(169) <
(170) 7 H R ? FIN E CN8
(171)
(172) E 7NQRE E N FI N
(173) IM E FL
(174) N E N?
(175) E PN $ NQRE GK FL
(176) GN
(177) N ++M N E E N
(178)
(179) 8 8 M* M E C E N FLEG . ,-% ,"0 , %($&! "*+'-,. /$+0 243 *+,! "* 0 "! #$% ' 4 ,!
(180) ' )(
(181) IKJfObjO>Y]`f[qu;ilrOKrcprO>Y>uqu[qr_BW&cp_v[jO3bjN4O^wZcequbjW.[jcNO t < T []JMOuqup`fOK[]JMqu[ [jJfOEoiBYm[ `f_Mo[]cpir_Mqr cp([jqrrO&ql[E[]cpNO cª [jJfO&irtZ[jcpN4=c ;>qu[]cpir_¨ceYEY\[]ivtft;O>w¨qu[n[jJfO&c_Ym[xqu_v[ @Hqu_Mw J [jJfO¥M(²,O3ewt;ceY)² tf`Z[)) [ji;O ² ir_ [t, T ]TB ¦rPnir[]O[]Jql[)[]JMOKlqr`fOir J (², t; ² ) ceY)b]O^qrwZcpW&t o≤irNTtf`Z[jO>w qu[
(182) qu_BW¨[jcNO qrY.Y\iXiv_ qrY[]JfO c_XrO3bxY\Otfb]ivtMquBql[]cpir#_ @Hf cpY
(183) oirNtf`f[]O>w ir_Mo3Or¦ : bjNO>w c[]J []JfceY [jiXir@Birtf[]cpN4=c ;>tqu[]cpir__fO>O>w_fir[ qrc[%[jcp;[jJfO ¥M_Mqr;[]cpNO T fB `f[o3qr_cp_MYm[jO>qvwqupbjO>qrwfW4irt;O3bxql[jO qu[%[jJfO o3`fb]bjO3_B[K[]cpNO `MY]c_f iZo>qu«[]bxqroxXc_M4tfb]ifoO>wZ`Mb]O^Y%[]iivtZ[]cpN c ;>O[jJfO.lqr`MO J (², t; ² ) ¦ O.qrb]O _f i c_®t;ivY]cª[jcivt _[ji o3pqrcN-[]JfO2ivp i c_f X° X 8
(184) EGEGN EG8RD E FIN M E NQFL ! PFLE
(185) K FL
(186) OCE F N ?
(187) SCEGFIHJ FLK f wd. ref. dist. 0. dist. ref. ii. ii. ref. f wd. 2. dist. f wd. 2. ii. ref. 2. ref. ref. ref. f wd. ref. f wd. E<MINSCN REG J (² , t; ² = ² ) GN G( N
(188) t N ?
(189) M
(190) ?M
(191) N ? GN E PE N E 8R ! K
(192) fwd FL
(193) <MIk+1 CPref NQRE E C kt E N ?
(194) 8 N
(195) FI< G [0, T ] . ref. Jf wd (²k+1 , t; ²k ). M . ¸ºà¾$¸65.
(196) .
(197)
(198). 9. l°$° "Pnir[]O []Jql[ Jf wd (²k+1 , t; ²k ). . ²k = ²ref. cpNtfW. χk (t) = ρref (t). ¦(O[&`MY&o3irNtf`Z[jO []JfO[]cpNOwZO>b]cplqu[]cprOir. d Jf wd (²k+1 , t; ²k ) = α²2k+1 (t) − α²2k (t) dt d −2 Rehhχk (t), ρk+1 (t)ii dt = α²2k+1 (t) − α²2k (t) d d −2Rehh χk (t), ρk+1 (t)ii − 2Rehhχk (t), ρk+1 (t)ii dt dt = α²2k+1 (t) − α²2k (t) (H − ²k (t)M) χk (t) , ρk+1 (t)ii −2Rehh i (H − ²k+1 (t)M) ρk+1 (t) −2Rehhχk (t), ii i ²k (t)Mχk (t) = α²2k+1 (t) − α²2k (t) + 2Rehh , ρk+1 (t)ii i ²k+1 (t)Mρk+1 (t) +2Rehhχk (t), ii i = α²2k+1 (t) − α²2k (t) + 2²k (t)α²k+1 (t). IKJX`MY J B <0. @Hr B. 2. −2²k+1 (t)α²k+1 (t) = −α [²k+1 (t) − ²k (t)] .. f wd (²k+1 , t; ²k ). . . 8
(199). ceYqwfO>objO>qvY\cp_f`f_Mo[jciv_®iu t ¦. EGEGN EG8RN
(200) DCE EGH M <M EGFLE = F* E C N ?
(201) PFL
(202) $RE +M FLE 8
(203) FIN# E C . Jf wd. MI
(204). IKJfOhb]O^Y\`Mª[qu;ilrO Nq¬WqueY]iY]`fvrO>Y\[¡[]JfOhirp i cp_fcp_v[]O>b]tfbjO[xql[jciv_ irbqu_XWo3qu_wZcpwMql[]OY]irp`Z[]cpir_ [ %i4[jbjqlkmO>o[]ivb]cpO>YKo3qu_;O.oivNtf`Z[]O^w []Jql[nY\[jqrb\[xY%bjirN []JMO.o3irbjb]O>o[cp_fc[]cequ(oir_wZcª[jciv_ ² f`Z[ JfiBY\O¥M_Mqr$Ym[xql[jO ρ (T ) N q¬W_fiu[WrO3ρ[(t);O.Yjql[]ceY\¶qro[jirbjW o3iBY\O [ji4[]JfO2[jqrb]vO[>fqu_Mw[]JMO.qvw¬kmircpρ_v[ Y\[jqu[]O [jJMql[tfbjirtMqrvql[jO>Y4qrox KqubxwbjirN []JfO¨[xqubjrO[ f`Z[N q¬W·_fir[bjO>qvojJ[jJfOoivb]bjO>o[ cp_fc[]cequ χYm[x(t) ql[jO ρ []JfO4cewZO>q cpY []i N qrrO&[]JfO[]bxqlkmO>o[]irbjcpO>Y o3ircp_MAocewZOBW9o3irNtf`Z[jc_f ² Y]`MoxJ9[]Jql[ qrtftfbjivqroxJfO>Y&Nir_Miu[]iv_fceo3quppW χ (t) ¦ >£_·[]JfO®qrtftfbji^ZcpNqu[]cpir_ JfO3bjO[]JfO M`fO3_Mo3Ot;O3_Mqrª[mW ρ (t) R ceYE_fO>rpcvcMO&;OirbjO.[jJfO4oir_B[]bjir(tqub][ [jJfOwfcpY\[jqr_MoO&;O[ %O>O3_¨[jJfO [ %iα²[]bxql(t)kmO>o[]irbjcpO>Y cp)wZO>o3b]O>qvY\O
(205) `f_B[]cpc[jYE¥M_Mqulqupkχ`fOql(t)[E[j−cNρO T ¦;(t)k IKJMOY\c[]`ql[]cpir_9ceY YjojJfO>N ql[]ceo3qrpW wZO>tfcpo[]O^w cp_ &c¦¡r¦ E
(206) l
(207) l .)0 2 & UXcNcppqrbKo3ir_MY]cewZO3bxql[jciv_MY%qrY%cp_v[jb]ifwZ`MoO^wqu;ilrOqutftfpW[]i[jJfO q¬rO`M_Mo[jciv_irbjN&`feql[jciv_$¦XPEiu[]O Jf i %O>rO3b[]JMqu[>O3rO3_¨c)[]JMO&wZO>_MY\c[mW N ql[jb]ccpYnNivb]O.vO3_fO>bjqr«[jJMqu_ q¬rO`f_o[]cpir_(f[]JfOqrYjY\iZo3cpqu[]O>w Jdist (²k ) = Jf wd (²k+1 , 0; ²k ) ≥ Jf wd (²k+1 , T ; ²k ) = Jdist (²k+1 ). k. 0. k. k. k. 0. k+1 T 2 0. . . ¾¾#¹GTMÑÑIUxâ. k+1. k. k. k+1. ii.
(208) ^. FIR
(209) FI R 30. χk(t). 25. 20. χk+1(t) χ∞(t). 15. ρ∞(t). ρk+1(t). 10. ρk(t). 5. 0 0. 2. 4. 6. 8. 10. time. UZoxJfO3N qu[]ceoKcp`Ym[jbjqu[]cpir_4iu;[]JfOno3ir_XrO3bjrO3_oOiu;[]JfONiv_fiu[jir_fceoKquprivb]c[]JfN Yivb _fO>rpcvcMO M`fO>_MoOv¦«IKJfOO3rivXcp_fY\[jql[jO ρ ceY qrtftfbjivqroxJfcp_f Nir_Miu[]iv_fceo3quppW[]JMOb]O3O3bjO3_MoO
(210) [jbjqlkmO>o[]ivb]W χ ¦ : [)[]JfOK_fOX[)c[]O>bjqu[]cpir_ χ cpZb]O>Nqrc_qu[ q o3ir_MY\[jqr_v[)wfcpY\[jqr_MoO%b]ivN ρ ;O>o>qu`MY]O%;iu[jJ`MY\O%[jJfO YjquNOn¥MO3ew ¦BIKJfceYY\Jfbjcp_fXc_fwZcpY\[jqr_MoOE;O3[ O3O3_[]JfOE[ %i
(211) [jbjqlkmO>o[]ivb]cpO>YhO3_Y\`fbjOn[]JfO tfbjirvb]O^Y]Y\cpir_ irX[jJfO%o3ivY\[$²`M_Mo[jciv_Mquv[] i qubxw2irtZ[jcN quvuqup`fO>Y>¦>IKJMcpYirMY]O3bjlql[jciv_ceYo3`fb]bjO3_B[]pW`MY\O^w.c_.[jJfOoir_B[]O3B[ ir O ocpO3_v[tMqrbjqrpO3p=c ;>ql[jcpir_®iu [jJfO_B`fNO3bjcpo>qubjO>Y]irp`Z[]cpir_¨irdv`qu_v[j`fN oiv_v[jb]iv(tfbjirfpO3N Y v{ ¦ >£_ [jJfOrO3_MO3bxqu;o3qvY\Ovv[]JfO2wZO^objO>qrY]cp_f&oxJMqubxqro[]O>bir$[]JfO2wZceY\[jqu_oO ;O[ %O>O3_[]JfO2o3`fb]rO^YceY O3cprJv[jO>wXW [jJfO.¥MO>pw M`MO3_Mo3O&qu_Mw[]JfO&irtZ[jcN qr(o3ir`ftfpO
(212) iu[jbxq¬kmO^o[jirbjcO^Y cpOq4[]`MO JfiBY\O._fiv?_ ;3O>b]i cpwX[jJ ceYKbjO3eql[jO>w[]i4[jJfO
(213) wZbjcXcp_f4eqrY]O3bK¥MO3ew `fO3_Mo3Or¦ . k+1. k. k+1. k+1. k+1. ivMY\O>b]uqufpO>Y&qrb]O cp_fO>qrb>¦ &Mirb[]JfO Kq^rO3`f_Mo[jciv_·Jfi O3rO3b^([jJfOirMY]O3bjlqrfpO>Y
(214) O>_v[]O>b4cp_v[ji[jJfO®oivY\[ `f_Mo[]cpir_Mqr$qvYKdB`MqvwZbjqu[]ceoE[]O>b]N Y> JMcpoxJ cp$cp_MwZ`MoO
(215) Y]irNO2qvwfqutZ[xql[jciv_MYc_[]JfO2irbjNqrceY]N®¦Z$O3[`MY o3ir_MY]cpwZO>b[jJfO
(216) wZbjcXcp_f4O>rirp`Z[]cpir_O>dB`Mqu[]cpir_ @H B ∂ i ψ(t, x) = (H − ²(t)µ) ψ(t, x) 0. ∂t ψ(t0 , x) = ψ0 (x).. &firb.qvcrO>_irMY]O3bjlqrfO «c[jY.q^rO>bjqrrO>w¨N4O^qrY]`fb]O^w¨uqup`fOceY ¦ c[]J [jJfcpY.wZO¥M_Mcª[jciv_ [jJfO2[]bxqroxXc_fivb]N&`feql[jciv_o>quA_;O b]c[\[jO3_®qvYqu;ilrO2qu_Mw[jJfO
(217) YjquNhψ|A|ψi O.oir_Y\cewZO3bxql[jciv_MYqrtftfpWr¦. ¸ºà¾$¸65.
(218) r. .
(219)
(220) ". . ,-% ,"0 , %($&!. "*+'-,. /$+0 243. $O[n`MYnwZO¥_fO2[]JfO
(221) oiBYm[K`M_Mo[]cpiv_Mqu J w (²) =. Z. T. α²2 (t)dt − hψ(T )|A|ψ(T )i,. irO N4cpiv_v_f[]bjiuiZ[jwZir`M_fo3ceo2O quqrpY%rirbjO3c[]irJfbjN On[]Jf$O2' qrw¬kmivc_B[Y\[jqu[]O ¶@ (qrrbxqu_fvON&`f[]cptfpcO>b B 0. -. χ(x, t). @Hu} B. qu_MwrcprOEir_fOEO3fquNtfpO. ∂ ψk+1 (x, t) = (H0 − ²k+1 (t)µ) ψk+1 (x, t) ∂t ψk+1 (x, t = 0) = ψ0 (x) µψk+1 1 i(t) ²k+1 (t) = − Rehχk , α i ∂ i χk+1 (x, t) = (H0 − ²k+1 (t)µ) χk+1 (x, t) ∂t χk+1 (x, T ) = Aψk+1 (T ). i. @@r{ B @@u| B @@*6 B. iv_O4[j_fJfiuO2[j¥MO&_M[jqrJM$qlY\[[jqu[][]JMO O4_fir_fpcp_fO>qr¦ b]c[£W®cp_[]JfO4ivMY\O>b]uqufpO
(222) cp_MwZ`Mo3O>Y2q wZO>t;O3_MwZO>_MoO4iu[]JfOqvw¬kmircp_v[.Ym[xql[jO ψk+1 (T ).
(223) )4$ "*
(224) %(!
(225) O&cp_v[jb]ifwZ`MoOvfivb q bjOO>b]O>_MoO.¥MO>pw ir_ qr_Mw¨q¥O3ew ivYjY\cpfwfpO O¥M[j_fi O^o3w¨ir_M`fY]t¨cpwf[]O3i b qu_9c_B[]O>b\ NO>wZceqrb]W []cpNO t < T []JfOE]o3qr_MwZcewfql[jO2Y\iv²`f[]cpir_ ² [0,qvYTcp_ ] @m^| B ¦?> [ncpYt;²(s) Z @@G ' B J (²) = α² (t)dt − hψ (T )|A|ψ (T )i _fi4c[]OJ ψo3cO>O>_vr[irptfXbjciZ_fonO^wZbj`firNbjOEψceYKq¬ lquc[]cpJ&equ¥MfO3pOnew [ji² ¦ onir iN %tfO>`fr[]O3O b^^wZ`fO[jiEqu[][Jf[]OcpN_firO _Mcp@¶_fiuO>[jqrJfbO3_Mbql[j[jJM`fqubj_ OO3iuXftM[]JMceO%ociv[]pMWY\O>o3b]iruNquftfp`ZOr [jc_f ψ irb %ivb]Xcp_f cª[jJ[]JfO
(226) Mqrox qubxwtMb]ivtMquBqlJ[]cpir(²)_ir[jJfO.irMtY]O3bjlqrfpO A c[]J[jJfO2¥MO>pw ² B ¦ &Mirb[jJfceY4bjO>qrY]ir_(Nir_fir[]iv_fcpo qrvirbjcª[jJfN Y&`MY]O []JfOirp i cp_f c_MO>dB`Mqupc[£W'ivbq t;ivY]cª[jcrO®wZO¥M_Mcª[jO ivMY\O>b]uqufpO A ≥ 0 @@G 9 B −ha|A|ai ≤ −hb|A|bi − 2Reha − b|A|bi, @[]JfO&wZc©;O>b]O3_oO.iu[ %idB`Mqu_B[]c[]cpO>Y;O>c_f −ha − b|A|a − bi ≤ 0B ?¦ >£_®tMqrb\[jcpo3`fpqrb>Zc O.wfO3_fir[]O.BW [jJfO
(227) Ym[xql[jO2O3rivXcp_fbjirN ψ c[]J¥MO>pw ² fir_fO2JMqvY ψ " . $,. &% ('. ! ,. 0 * )(%4!
(228) 0. $+,-%( "* %' ref. T. 2. w. ². ². 0. ². 0. w. ref. ². ²ref. 0. J w (²) ≤. ref. Z. T. α²2 (t)dt − hψ²ref (T )|A|ψ²ref (T )i 0. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i. ¾¾#¹GTMÑÑIUxâ. @ r B.
(229) ¬. FIR
(230) FI R. > [nceY[jJfO3bjOirbjOt;iBY]Y]cpfO[ji wZO¥_fO2[]JfO JH
(231) CN REG CE FIHJ GFLK E MINCNQRE G Jfwwd (², t; ²ref ) =. JfceoxJ®o3qr_®queY\i4;O bjcª[][]O>_. Z. T. α²2 (t)dt − hψ²ref (T )|A|ψ²ref (T )i. @ f B. 0. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i. Jfwwd (², t; ²ref ). IiO3lqrp`Mql[jO. w. = J (²ref ) +. t 0. © ª α ²2 (t) − ²ref 2 (t) dt. @ v B. © ª α ²2 (t) − ²ref 2 (t) dt. @ B. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i.. Jfwwd. c[ncpYnoiv_BrO3_fcpO3_B[K[ji_fiu[jO[jJMql[. Jfwwd (², t; ²ref ) = J w (²ref ) +. JfO>b]O. Z. Z. t 0. −2Rehψ² (t) − ψ²ref (t), χ²ref (t)i.. χ²ref (t). ceY[jJfO
(232) qrw¬kmivc_B[nY\[jqu[]O.qu[K[]cpNO t rcprO3_BW. @ u} B. ∂ χ² (x, t) = (H0 − ²ref (t)µ) χ²ref (x, t) ∂t ref χ²ref (x, T ) = Aψ²ref (T ). i. IKJfceY irbjN&`feqquppi nYqu_ O ocpO3_B[¨o3irNtf`Z[xql[]cpir_ iu ¦Ii Y\`fNN qubjc=;3Or%[jJfO'o3ivY\[ `f_Mo[]cpir_Mqr J (²) []Jql[.o3qr_f_fiu[;OoivNtf`Z[]O^w9O3XtMceoc[]JpW¨cpY(²,OZt;tfp²irc[]O^)w¨[]Jfbjir`MrJ9c[jY`Mtft;O3b2;ir`f_Mw ¦MIKJMO2tfbjO>o3cpY]Otfbjirt;O>b\[mWiu J ceYKvcrO>_ cp_[]JfO2irppi cp_f J (², t; ² ) X° X 8
(233) EGEGN EG8RD E FIN M E NQFL ! PFLE
(234) K FL
(235) OCE F N ?
(236) SCEGFIHJ FLK w f wd. w. w f wd. ref. ref. w f wd. E<MINSCN REG J w (² , t; ² = ² ) GN G( N
(237) t N ?
(238) M
(239) ?M
(240) N ? GN E PE N E 8R ! K
(241) fwd FL
(242) <MIk+1 CPref NQRE E C kt E N ?
(243) 8 N
(244) FI< G [0, T ] . Jfwwd (²k+1 , t; ²k ). M . ¸ºà¾$¸65.
(245) . .
(246)
(247). l°$° " : nY qu;ilrOr O2O3uqup`Mql[jO [jJfO2[]cpNO
(248) wZO3bjcuql[]cprOir . Jfwwd (²k+1 , t; ²k ). ¦. d w J (²k+1 , t; ²k ) = α²2k+1 (t) − α²2k (t) dt f wd d −2 Rehψk+1 (t), χk (t)i = α²2k+1 (t) − α²2k (t) dt d d −2Rehψk+1 (t), χk (t)i − 2Reh ψk+1 (t), χk (t)i dt dt = α²2k+1 (t) − α²2k (t) (H0 − ²k (t)µ) χk (t) −2Rehψk+1 (t), i i (H0 − ²k+1 (t)µ) ψk+1 (t) , χk (t)i −2Reh i ²k (t)µχk (t) i = α²2k+1 (t) − α²2k (t) + 2Rehψk+1 (t), i ²k+1 (t)µψk+1 (t) +2Reh , χk (t)i i 2 2 = α²k+1 (t) − α²k (t) + 2²k (t)α²k+1 (t). IKJX`MY J B <0. w f wd (²k+1 , t; ²k ). H 8
(249) FL
(250). M ². B . ². . @Rv{ B. 2. −2²k+1 (t)α²k+1 (t) = −α [²k+1 (t) − ²k (t)] .. ceYqwfO>objO>qvY\cp_f`f_Mo[jciv_®iu t ¦. % E N
(251) N ? GN N ?
(252) P
(253) 8 N
(254) w ≤ J w (², t; ² )
(255) E
(256) M
(257) 8 7N# N8
(258) 7 N P FLE *8
(259) M ² H +R H 8 J 8
(260) (²) # GNN ?f
(261) wd EG8
(262) F
(263) ref
(264) P
(265) E C N ?
(266) E PE N E R E FIN M. − ²k → 0. . ref. <0 / M E
(267) N8
(268) EGE N E RN# E C N ?
(269) FLE$
(270) K FL
(271) M #MI P
(272) EG?M
(273) 8
(274)
(275) E C N8
(276) K*
(277) FL
(278) <M
(279) E C J w (² , t; ² ) J w (² ) = J w (² , 0; ² ) ≥ J w (² , T ; ² ) ≥ J w (² ) k+1. f wd. k+1. k. k. f wd. k+1. k. f wd. k+1. k. k+1.
(280) % ( & IKJfb]iv`frJ'[]JfOcp_v[jb]ifwZ`Mo[jciv_iuEqmirb qubxw o3ivY\[
(281) `f_Mo[jcpir_Mqrã [jJfcpYtMqut;O>bwZO3Nir_Ym[jbjqu[]O>Y.[]Jql[ [jJfONiv_fiu[jir_fceo4qrvirbjcª[jJfN Yqrb]O opivY]O3pW¨bjO3eql[]O^w9[ji[]JfOoeqrYjYir%[]bxqroxXc_ftfbjiZoO^wZ`fbjO>Y>¦ : Y.Y]`MoxJ$ [jJfOENir_fir[]iv_fcpo3cª[mWtMb]ivt;O3b][£Wir«[]JMOnirbjNO3b%quprivb]c[]JMNYqutft;O>qrbjYqrYq
(282) _Mql[j`fbjqroiv_MY\O^dB`fO3_Mo3Oir«[jJfO cp_MobjO>qvY\cp_f uwZO>o3b]O>qrY\cp_f tfb]ivt;O3b][]cpO>Yir%[]JfO[jbjqvojXcp_fc_MwfO¡¦(IKJfONiv_fiu[jir_fceo4YjoxJfO3NO>Y.qrb]O Y\Jf i _ [ji oiv_MY\[]bj`Mo[4qu[4qrc_B[]O>b]NO>wfcpqrb]W [jcNO>Y[]JfOo3ivY\[&`f_Mo[]cpir_Mqrlqup`fOirEqo`Mb]bjO3_B[2;O>Y\[¥MO3ewB o>qu_MwZcewfql[jOqu_Mw `MY\O4[]JfceY2c_firbjNqu[]cpir_ c_ []JfOivt;O3_ piXirt[]iivtZ[]cpN c ;>O[jJfO4¥MO3ew`fb\[jJfO3b^¦ &fivb [jJfO Y]t;O>oc¥oo>qrY]O&iuh[]JfOwZO3_Y\c[mW®N ql[jb]c¡M[jJfceYo>qu_ queY\i;Ocp_v[jO3bjtfb]O3[]O^w¨`MY]cp_f[ %i[jbjqlkmO^o[]ivb]cpO>Yn[]Jql[ Y\[jqrb\[ ¬O3_w qu[K[jJfO&oirbjb]O^o[nYm[xql[]O^Yqr_Mw JfivY]O.wZceYm[xqu_Mo3O.cpYno3ir_v[jc_X`fir`Y\pWb]O^wZ`MoO^w @HwZO3t;O3_wZc_MqueY]i iv_[jJfO.eqrY]O3b M`MO3_MoO B wZ`fbjcp_f4[]JfO.ivtZ[]cpN c ;^ql[]cpir_tfbjiZo3O>YjY3¦ . . ¾¾#¹GTMÑÑIUxâ.
(283) >}. FIR
(284) FI R. +&0 * !. IKJfO qr`Z[]Jfivbrbxql[]O3`fpW qroxX_fi O^wZrO wZceY]o3`MY]Y]cpir_MYiv_[]JfceY []ivtfceo c[]J R9q ;3WvqubR¨cpbjbjqrJfcNc qubjcpY B qu_w r`fpcpO3_¨UfqupirNir_ @¶g)qrb]ceYD>KQn_fcprO>bjY]cª[mW B ¦ K p
(285) RwZO>¦¡Y]cpR¨r_fcpO^b]bxw quJfo3ircpNJfO>cHb] O> 4_v[n¦¡I(dB`M`fqubjcp_B_f[]ce`foN#c@«quo3_Mirw9_v[jg)bjir¦¡eY3s iv`M
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(293)
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(358) Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(359) . ISSN 0249-6399.
(360)
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