Диссертация (1136178), страница 43
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Îöåíêà íåâÿçêè.Ôîðìóëèðîâêà îñíîâíîé òåîðåìûÄëÿ ïîñòðîåíèÿ ãëîáàëüíîé àñèìïòîòèêè çàäà÷è (0.51) (0.53)îñòàåòñÿ ïîëó÷èòü ïðàâèëî êâàíòîâàíèÿ óñëîâèå, îáåñïå÷èâàþùåå ãëàäêîå ñøèâàíèå àñìïòîòè÷åñêîãî ðåøåíèÿ. Èìåííî èç ýòîãîïðàâèëà íàõîäÿòñÿ àñèìïòîòè÷åñêèå ñîáñòâåííûå çíà÷åíèÿ (0.51) (0.53).Âûøå â 2 áûëî ïîñòðîåíî ÂÊÁ-ïðèáëèæåíèå âèäà1 nS(x, h)g = 5/18 B0 + h2/9 B1 + h4/9 B2 + O(h5/9 B3 ) cos+hh373 1/9S(x, h) o3/95/9.+ h I0 + h I1 + O(h I2 ) sinh(4.292)Ââåäåì ôóíêöèþppdef 2 3/2eS0,0 (ξ, k) = ξ − A−1 ξ(2 ln ξ − 4) + 2A0 ξ + 3ρek 2/3 ξ 1/6 −3−ekπ 2 A−1 ln ξ + δ.Òàê êàê ïðèx−xe−ïîðÿäêàεàñèìïòîòèêè(4.293)gèg− ,çàäàííûå ôîð-ìóëàìè (4.292), (4.247), ñîãëàñîâàíû, òî, èñïîëüçóÿ (4.293), (4.243),(4.244), äëÿ ôóíêöèèSèç (4.292) èìååìZ ε√ h1/9 3Ω− e 1 xS(x, h)0 00= S0,0, k− +S (x , h) dx +O 1/6 ,hh xe− +εh2/3εh → 0.(4.294)Ïðè ïîñòðîåíèè ÂÊÁ-àñèìïòîòèêè ìû äëÿ îïðåäåëåííîñòèk > 0, S 0 > 0, y = (k/S 0 )2/3 > 0.
Ñîâåðøåííî àíàëîãè÷00 2/3íî ðàññìàòðèâàåòñÿ ñëó÷àé, êîãäà k < 0, S < 0, y = (k/S )> 0. Â00ðåçóëüòàòå çàìåíû k íà −k , S íà −S ôóíêöèè B0 , B1 , B2 â (4.292)íå èçìåíÿòñÿ, à I0 , I1 ïîìåíÿþò çíàê. Ñàìà æå çàäà÷à äëÿ ôàçûîñòàíåòñÿ ïðåæíåé. Íàäî ëèøü ôîðìóëû, çàäàþùèå ek− , ek+ , çàïè4/34/3k+ = |k|/Ω+ .  ðåçóëüòàòå ïîëó÷àåìñàòü â âèäå ek− = |k|/Ω− , eñ÷èòàëè, ÷òîðàçëîæåíèåS(x, h)1 ng = 5/18 B0 + h2/9 B1 + h4/9 B2 + O(h5/9 B3 ) cos+hh∗S(x, h) o1/93/95/9+ − h I0 − h I1 + O(h I2 ) sin,h(4.295)ãäåZ ε√ h1/9 3S(x, h)Ω+ e 1 xe+ −ε 0 00= S0,0, k+ +S (x , h) dx + O 1/6 , h → 0,hh xh2/3ε(4.296)êîòîðîå ïðèxe+ − x(4.238), (4.239)ε ñîãëàñîâàíîàñèìïòîòèêîé g+ .ïîðÿäêàñ çàäàííîé ôîðìóëàìè374×òîáû îáåñïå÷èòü ñóùåñòâîâàíèå ãëîáàëüíîãî àñèìïòîòè÷åñêîãî ðåøåíèÿ (0.51) (0.53) ïîòðåáóåì òîæäåñòâåííîãî ñîâïàäåíèÿôóíêöèégg∗èïðèx ∈ (ex− + ε, xe+ − ε). ñèëó ôîðìóë (4.292),(4.294) (4.296) ïîëó÷àåì ðàâåíñòâàcos S0,0 ε√3Ω− e, k−2/3h≡ cos S0,0sin S0,01+h ε√3Ω+ e, k+2/3hZx00S (x , h) dx + O h1/9 xe− +ε1+h0xe+ −εZ00ε1/60S (x , h) dx + O≡ h1/9 xε1/6,(4.297) ε√3Ω− e, k−2/3h+1hZxS 0 (x0 , h) dx0 + O h1/9 xe− +εε1/6≡Z ε√ h1/9 3Ω+ e 1 xe+ −ε 0 00S (x , h) dx + O 1/6 .≡ − sin S0,0, k+ +h xh2/3ε(4.298)ÑïðàâåäëèâàËåììà 4.40.Ðàâåíñòâà(4.297), (4.298)èìåþò ìåñòî òîãäà èòîëüêî òîãäà, êîãäà âûïîëíåíî óñëîâèåS0,0 ε√3Ω− e, k−2/3h1+hZxe+ −εëó÷àåòñÿ èçggèg∗00S (x , h) dx + S0,0+ e, k+2/3hxe− +εãäå n öåëûå.= 2πn + O(h1/9 /ε1/6 ),Íàðÿäó ñ0 ε√3Ω=(4.299)−g ∗ , êîòîðîå ïî−S 0 , y = (k/S 0 )2/3 íà −y .ðàññìîòðèì òàêæå ðåøåíèåçàìåíîékíà−k , S 0íàÂîçìîæíîñòü òàêîé çàìåíû ñëåäóåò èç ôîðìóë (4.208).
Ïîòðåáîâàâïðèx ∈ (ex− + ε, xe+ − ε) òîæäåñòâåííîãî ñîâïàäåíèÿ gè−g ∗ , ïðèõî-äèì ê óñëîâèþS0,0 ε√3Ω− e, k−2/3h1+hZxe+ −ε000S (x , h) dx + S0,0 ε√3Ωxe− +ε= π + 2πn + O(h1/9 /ε1/6 ),ãäån öåëûå.+ e, k+2/3h=(4.300)375Íàêîíåö, îáúåäèíÿÿ (4.299), (4.300), ïîëó÷àåì ñëåäóþùåå ïðàâèëîêâàíòîâàíèÿ òèïà Áîðà-ÇîììåðôåëüäàS0,0 ε√3Ω− e, k−2/3h1+hxe+ −εZ000S (x , h) dx + S0,0 ε√3Ωxe− +ε+ e, k+2/3h= πn.(4.301)n öåëûå; ôóíêöèÿ S 0 (x, h) > 0, à òàêæå êîíñòàíòû k > 0,xe− , xe+ , Ω− , Ω+ ÿâëÿþòñÿ ðåøåíèåì çàäà÷è äëÿ ôàçû; ε = h26/57 , ek− ,ek+ îïðåäåëåíû ôîðìóëàìè (4.245), (4.246), à S0,0 (ξ, ek) ôîðìóëîéÇäåñü(4.293).Ïðàâèëî (4.301) äàåò óðàâíåíèå äëÿ îïðåäåëåíèÿÂh → 0. Ïóñòü èõ ïðîèçâåäåíèå îãðàíè÷åíî íåêîòîðûìè êîíñòàíòàìè C1 , C2 òàê, ÷òî âûïîëíåíî(4.172).
Òîãäà λn (h) = O(1) ïðè n → ∞.(4.301) âõîäÿò ïàðàìåòðûÇàìå÷àíèån → ∞λ = λn (h).è4.25. Ïðè âûâîäå (4.301) ìû ïðåíåáðåãëè ñëàãàåìûìO(h1/9 /ε1/6 ). Êðîìå òîãî, èìååòñÿ åùå ïîãðåøíîñòü ïðè íàõîæäåíèèS 0 (ñì. (4.291)). Ïîýòîìó ïðàâèëî êâàíòîâàíèÿ (4.301) ïîëó÷åíî ñ1/57òî÷íîñòüþ O(hln h1 ), h → 0.Èòàê, ãëîáàëüíûå àñèìïòîòè÷åñêèå ðåøåíèÿ çàäà÷è (0.51) (0.53)g = gn(4.301). Òîãäà ïðègn = gnWKBλ = λn óäîâëåòâîðÿþòx ∈ (ex− + ε, xe+ − ε) àñèìïòîòè÷åñêèåïîñòðîåíû.
Ïóñòüðåøåíèÿçàäàþòñÿ ôîðìóëîé (4.292), ãäå îñòàòî÷íûå ÷ëåíû òà-êîâû, ÷òî âûïîëíåíî (4.188). Äëÿxâáëèçè è ëåâååîïðåäåëÿþòñÿ ðàâåíñòâàìè (4.231), (4.232), à äëÿxe+ïðàâèëóxxe−ôóíêöèègnâáëèçè è ïðàâåå ðàâåíñòâàìè (4.238), (4.239). Âõîäÿùèå â (4.231), (4.238) ôóíê-G− , G+ óäîâëåòâîðÿþò ìîäåëüíîìó óðàâíåíèþ (0.45). Âáëèçèòî÷åê xe− + ε, xe+ − ε àñèìïòîòèêè ñîãëàñîâàíû ìåæäó ñîáîé. Èñ-öèèîòïîëüçóÿ ðàçáèåíèå åäèíèöû, ìîæíî çàïèñàòügn = χ∗− g− + (1 − χ∗− − χ∗+ )gnWKB + χ∗+ g+ .g = gn â óðàâíåíèå (0.51) è îöåíèòü â íîðìåïðè ýòîì íåâÿçêó Rn .Íàì îñòàåòñÿ ïîäñòàâèòüL2 (R2 )âîçíèêàþùóþ(4.302)376 ñèëó (4.183), (4.290) äëÿ íåâÿçêèRnïðèx ∈ (ex− + ε, xe+ − ε)èìååìRn = O(r∗ ) + O(RT ),ãäår∗, R(4.303)îïðåäåëåíû ðàâåíñòâàìè (4.186), (4.182), (4.291).Ëåììà 4.41.ZÏðè h → 0 ñïðàâåäëèâà îöåíêàxe+ −ε Z ∞−∞xe− +ε1/2Rn2 dydx1+1/57=O h1ln .h(4.304)Äîêàçàòåëüñòâî.
Èñïîëüçóÿ âûðàæåíèÿ äëÿ âõîäÿùèõ â (4.176),(4.177) ôóíêöèé, íàõîäèì ∂ϕ 2 2 T19/9 ∂T ∂ϕ19/9 ∂ ϕ20/9O h+O h+O h T 2 +O h T=∂x2∂x ∂x∂x∂x 112+T , h10/9 τ12 T = O(h10/9 τ 2 T ).=O h22(x − xe− )(ex+ − x)2∂2x ∈ (ex− + ε, xe+ − ε).Çäåñü1/3xe− +ε Z ∞Zh(y − y 0 )22χ− (x0 , ε)g−(x0 , y 0 ) dy 0 dx0 T (x, τ1 , h) =02−∞ (x − x )−∞=OÄàëåå, â ñèëó (4.231), (4.232) h13/9β2Zτ2Zxe− +ε−∞χ− (x0 , ε)(x − x0 )2Z∞−∞2g−(x0 , y 0 ) dy 0 dx0 T (x, τ1 , h)+Z ∞χ− (x0 , ε)(τ 0 )220 000g (x , y ) dy dx T (x, τ1 , h) =β 2 (x0 )(x − x0 )2 −∞ −−∞ h13/9 ε5/6 h13/9 √ε(τ 2 + 1) h13/9 √ετ 2 =OT +OT =OT .(x − xe− )2(x − xe− )5/3(x − xe− )5/3+O h13/9xe− +εÀíàëîãè÷íî ïðîâåðÿåòñÿ, ÷òî1/3Z∞hxe+ −ε∞(y − y 0 )22χ+ (x0 , ε)g+(x0 , y 0 ) dy 0 dx0 T (x, τ1 , h) =02−∞ (x − x )Z h13/9 √ε(τ 2 + 1) =OT .(ex+ − x)5/3377Êðîìå òîãî, èìååì:13/9Z∞hv.p.−∞−T2∞ 2 0 0(τ1 − τ10 )2 001−χ(x,ε)−χ(x,ε)T (x , τ1 , h)−−+0 2−∞ (x − x )Z(x, τ10 , h)dx0√ ε11=O h+×(x − xe− )5/3 (ex+ − x)5/32×(τ + 1)T .dτ10 T13/9Ïîñëåäíåå ñëàãàåìîå â (4.182) îöåíèâàåòñÿ ñ ïîìîùüþ èíòåãðè-x ∈ (ex− + ε, xe+ − ε)ðîâàíèÿ ïî ÷àñòÿì.
 ðåçóëüòàòå, ïðè∗r = O(h10/9 213/9τ T )+O h√ εïîëó÷àåì112+(τ + 1)T ,(x − xe− )5/3 (ex+ − x)5/3(4.305)ãäåε = h26/57 . Îñòàåòñÿ ïîäñòàâèòü (4.291), (4.305) â (4.303). Âû÷èñ-ëÿÿ íîðìó ïîëó÷èâøåãîñÿ âûðàæåíèÿ, ïðèõîäèì ê (4.304). Ëåììàäîêàçàíà.Äàëåå îöåíèì íåâÿçêóG± (ξ, η)Zïðèxe− −ε Z ∞−∞−∞ξ → −∞Rn2 dydxRnïðèx<xe− + ε è x > xe+ − ε. Òàê êàêýêñïîíåíöèàëüíî óáûâàþò, òî∞Z∞Z∞= O(h ),xe+ +ε−∞Rn2 dydx = O(h∞ ), h → 0.(4.306)|ex− − x| < ε,|ex+ − x| < ε, íà êîòîðûõ, ñîîòâåòñòâåííî, Rn = r− è Rn = r+ . Çäåñüôóíêöèè r− , r+ îïðåäåëåíû ôîðìóëàìè (4.237), (4.241), ãäå g = gn .Ñëåäîâàòåëüíî, äîñòàòî÷íî ðàññìîòðåòü èíòåðâàëûËåììà 4.42.Ïðè h → 0 ñïðàâåäëèâû ðàâåíñòâàxe− +ε Z ∞Zxe− −εZ−∞xe+ +ε Z ∞xe+ −ε−∞1/2= O(h1+1/38 ),(4.307)1/2= O(h1+1/38 ).(4.308)Rn2 dydxRn2 dydx378Äîêàçàòåëüñòâî.
Îöåíèì âõîäÿùèå â (4.237) ñëàãàåìûå. Ïóñòüε = h26/57 , ω = 1/57, ω1 = 20/171, |x − xe− | < ε.h1/32(x − xe− )∞(1 − χ∗− (x0 , ε))(x0 − xe− )2xe− +εZZZÒîãäà â ñèëó (4.260)∞−∞gn2 (x0 , y 0 )−∞n101/2O(x−xe)+dy dx = h (x − xe− )−0e− )2xe− +ε (x − x h4/9+ω1 o1h1+ω02plndx=O(x−xe),+O+O−(x0 − xe− )3/2 hx0 − xe−Z xe− +4εZ ∞ ∗ 0(y − y 0 )2 1/3∗0h1 − χ− (x , ε) χ− (x , ε)ln 1 +×(x − x0 )2xe− +ε−∞ 0 012 0 020 01+1/57× gn (x , y ) − g− (x , y ) dy dx = O hln+ O(y 2 ),hZ ∞Z ∞(y − y 0 )2 2 0 01/3∗∗ 0h1 − χ− (x , ε)gn (x , y )−0 2xe− +2ε−∞ (x − x ) 0 0120 01+1/57−g− (x , y ) dy dx = O hln+ O(y 2 ).hÒàêèì îáðàçîì, ïðè |x − xe− | < ε2−g−(x0 , y 0 )001/32n121+1/57r− = O (x − xe− ) + O hln+ O(y 2 )+ho4/9+20/171+O h(x − xe− ) g− .(4.309)×òîáû ïîëó÷èòü (4.307), îñòàåòñÿ âû÷èñëèòü íîðìó ñòîÿùåãî â ïðàâîé ÷àñòè (4.309) âûðàæåíèÿ.
Ðàâåíñòâî (4.308) ïðîâåðÿåòñÿ àíàëîãè÷íî. Ëåììà äîêàçàíà.Èç (4.304), (4.306) (4.308) âûòåêàåò ñëåäóþùàÿ îöåíêà íåâÿçêè1+1/57kRn kL2 (R2 ) = O hÊðîìå òîãî,ñòüþgn1ln ,hh → 0.óäîâëåòâîðÿþò óñëîâèþ íîðìèðîâêè (0.52) ñ òî÷íî-O(h2/3+1/57 ),à òàêæå óñëîâèþ (0.53) ñ òî÷íîñòüþÈòàê, äîêàçàíà îñíîâíàÿ òåîðåìà 2.O(h).379Ïóñòü ïàðàìåòðû h è n óäîâëåòâîðÿþò óñëîâèþÒåîðåìà 4.9.(4.172).Òîãäà ÷èñëà λ = λn (h), çàäàííûå ïðàâèëîì êâàíòîâàíèÿ(4.301),ÿâëÿþòñÿ àñèìïòîòè÷åñêèìè ñîáñòâåííûìè çíà÷åíèÿìèçàäà÷è(0.51) (0.53)ñ òî÷íîñòüþ O(n−1−1/57 ln n) ïðè n → ∞.Ñîîòâåòñòâóþùèå àñèìïòîòè÷åñêèå ñîáñòâåííûå ôóíêöèè g =gn óäîâëåòâîðÿþò (0.51) c òî÷íîñòüþ O(n−1−1/57 ln n) â íîðìåL2 (R2 ), óñëîâèþ íîðìèðîâêè (0.52) c òî÷íîñòüþ O(n−2/3−1/57 ), àòàêæå óñëîâèþ (0.53) c òî÷íîñòüþ O(n−1 ).
Ôóíêöèè gn ïî modO(n−∞ ) ñîñðåäîòî÷åíû íà îòðåçêå [ex− , xe+ ] ïðÿìîé y = 0.2.7.Çàäà÷à äëÿ ãëàâíîãî ïðèáëèæåíèÿ ê ôàçåÂûâåäåì ïðèáëèæåííóþ çàäà÷ó äëÿ ôàçû, îòáðîñèâ ðÿä âûñøèõ êâàíòîâûõ ïîïðàâîê. Èç ýòîé çàäà÷è íàõîäèòñÿ ãëàâíîå ïðèáëèæåíèå ê ôàçåS,à òàêæå ïðèáëèæåííûå çíà÷åíèÿ êîíñòàíòk,xe− , xe+ .Ðàññìîòðèì âõîäÿùèå â ïîëíóþ çàäà÷ó äëÿ ôàçû óðàâíåíèÿ(4.261), (4.265), (4.267) (4.269), (4.290). Ó÷èòûâàÿ (4.268), (4.269),èìååìΩ− = U 0 (ex− ) + O(h1/3 ), Ω+ = −U 0 (ex+ ) + O(h1/3 ), h → 0.Òàê êàê äëÿ çàäàííîé ôîðìóëîé (4.253) ôóíêöèè(4.310)H(ξ) ñïðàâåäëèâûðàâåíñòâàτZ√ln |ξ|H(ξ) dξ = 2 τ (ln τ − 2) + O(1),ZτH(ξ) dξ =11√= 2 τ + O(1),τ → +∞,(4.311)òî èç (4.265), (4.267) ïîëó÷àåì1/3U (ex− ) − 2hZn1/3 2/3 e √h Ω− k− e− (ln ε − 2) +xe+ −εxe− +εln |x0 − xe− |×k dx01√ o1/3 2/32/3e× 0 0+ h Ω+ ln |ex+ − xe− |k+ e+ = O h ln ,2S (x , h)h(4.312)380U (ex+ ) − 2h1/3Zn√1/3 2/3h Ω− ln |ex+ − xe− |ek− e− +xe+ −εxe− +εln |x0 − xe+ |×ok dx011/3 2/3 e √2/3× 0 0+ h Ω+ k+ e+ (ln ε − 2) = O h ln .2S (x , h)h(4.313)e± , ek± îïðåäåëåíû ñîîòíîøåíèÿìè (4.257), (4.245), (4.246), ε =h26/57 , h → 0.ÇäåñüÄàëåå, â ñèëó (4.253), (4.254)Ze±D−1 (ek± ) +1 h1/3 1 ek±√eeH(ξ) dξ = A−1 (k± ) + k± e± + O √ ln .2ε h(4.314)Ïîýòîìó óñëîâèå íîðìèðîâêè (4.261) ïðèíèìàåò âèä2/3h1/3 Ω− [A−1 (ek− )√+ek− e− ] +Zxe+ −εxe− +εk dx01/3 2/3+hΩ+ [A−1 (ek+ )+002S (x , h)1√1/3+2/19e+k+ e+ ] = 1 + O hln ,hh → 0.(4.315)Óïðîñòèì, íàêîíåö, óðàâíåíèå (4.290).Ëåììà 4.43.√ (x0 − x3e)Ω− 0−edx +√ ln |x − x |k− Hh2/3xe− +h2/3 / 3 Ω−ZΩ−Ïðè h → 0, x ∈ (ex− +ε, xe+ −ε) ñïðàâåäëèâû ðàâåíñòâàxe− +ε02/32/3+2h2/3 Ω− D−1 (ek− ) ln |x − xe− | = 2h2/3 Ω− ln |x − xe− |{A−1 (ek− )+Z e− dξ 0ξ 0 h2/3√ e2/32/3e√+ e− k− } + h Ω− k−ln 1 −√ +(x − xe− ) 3 Ω− ξ 0012/3+2/19+O hln (1 + | ln |x − xe− ||) ,(4.316)h√ Z xe+ −h2/3 / √3 (eΩ+0 3x−x)Ω++Ω+ln |x − x0 |ek+ Hdx0 +2/3hxe+ −ε2/32/3+2h2/3 Ω+ D−1 (ek+ ) ln |ex+ − x| = 2h2/3 Ω+ ln |x − xe+ |{A−1 (ek+ )+Z e+ dξ 0ξ 0 h2/3√ e2/32/3e√+ e+ k+ } + h Ω+ k+ln 1 −√ 0 +3(ex−x)Ωξ++03811e+ ||) .(4.317)+O hln (1 + | ln |x − xh√√Äîêàçàòåëüñòâî.
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