Диссертация (1103157), страница 22
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Ïóñòükf kL2 (T 2 ) <1,6π(5.26)ãäå f = f (λ, λ0 , E). Òîãäà óðàâíåíèÿ (5.16) îäíîçíà÷íî ðàçðåøèìû îòíîñèòåëüíî h± (·, ·, E) ∈ L2 (T 2 ), à óðàâíåíèå (5.18) îäíîçíà÷íî ðàçðåøèìî îòíîñèòåëüíîµ+ (z, ·, E) ∈ L2 (T ). Êðîìå òîãî,Zµ+ (z, λ, E) |dλ| =6 0 äëÿ âñåõ z ∈ C,(5.27)Tdiv±à ôóíêöèè Adiv, Vapprè A±, Vapprîãðàíè÷åíû íà R2 , j = 1, 2.appr,jappr,jÎòìåòèì, ÷òî óñëîâèå (5.26) ÿâëÿåòñÿ ëèøü äîñòàòî÷íûì äëÿ îäíîçíà÷íîéðàçðåøèìîñòè óðàâíåíèé (5.16) è (5.18), âûïîëíåíèÿ ñîîòíîøåíèÿ (5.27) è îãðà±div±íè÷åííîñòè ôóíêöèé Adivappr,j , Vappr è Aappr,j , Vappr , j = 1, 2.Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 5.1. Èç óñëîâèÿ (5.26) ñëåäóåò îäíîçíà÷íàÿ ðàçðåøèìîñòü óðàâíåíèé (5.16) îòíîñèòåëüíî h± ∈ L2 (T 2 ).
Êðîìå òîãî, èìåþò ìåñòî îöåíêèkf kL2 (T 2 ),1 − πkf kL2 (T 2 )2πkf kL2 (T 2 )<,1 − πkf kL2 (T 2 )kh± kL2 (T 2 ) <kBkL2 (T 2 )(5.28)127ãäå B îïðåäåëåíî â ôîðìóëå (5.19) (ïðè ôèêñèðîâàííûõ z è E ).Èç ôîðìóë (5.26) è (5.28) ñëåäóåò, ÷òî kBkL2 (T 2 ) < 1. Ñëåäîâàòåëüíî, óðàâíåíèå (5.18) îäíîçíà÷íî ðàçðåøèìî îòíîñèòåëüíî µ+ ∈ L2 (T ) è ñïðàâåäëèâûîöåíêè+kµ kL2 (T )(2π)1/2<,1 − kBkL2 (T 2 )kµ± − 1kL2 (T )+kµ − 1kL2 (T )(2π)1/2 kBkL2 (T 2 )<,1 − kBkL2 (T 2 )3π(2π)1/2 kf kL2 (T 2 )<,1 − 3πkf kL2 (T 2 )(5.29)ãäå ôóíêöèè µ± (z, ·, E) îïðåäåëÿþòñÿ â ôîðìóëå (5.21).Èç îöåíîê (5.26), (5.29) è ôîðìóë (5.23), (5.24) ñëåäóåò ñîîòíîøåíèå (5.27)±div±2è îãðàíè÷åííîñòü ôóíêöèé Adivappr,j , Vappr è Aappr,j , Vappr íà R .Òåîðåìà 5.1.
Ïóñòü f ∈ L2 (T 2 ) ïðè ôèêñèðîâàííîì E > 0. Ïðåäïîëîæèì,div÷òî f óäîâëåòâîðÿåò (5.26) è f ∈ C ∞ (T 2 ). Ïóñòü Adivè Vapprïîñòðîåíûapprdiv,ïî ôóíêöèè f ñ ïîìîùüþ âûøåóêàçàííîãî àëãîðèòìà. Òîãäà Aappr,1 , Adivappr,2divîãðàíè÷åíû íà R2 è óáûâàþò íà áåñêîíå÷íîñòè.
Êðîìå òîãî, f ÿâëÿåòñÿVapprdivàìïëèòóäîé ðàññåÿíèÿ äëÿ óðàâíåíèÿ (3.9) ñ A = Adiv(x, E) è V = Vappr(x, E).apprÇàìåòèì, ÷òî òðåáîâàíèå íà ãëàäêîñòü ôóíêöèè f â òåîðåìå 5.1 ÿâëÿåòñÿdivdivñèëüíî çàâûøåííûì. Ïðè ýòîì òðåáîâàíèè ôóíêöèè Adivappr,1 , Aappr,2 è Vappr èçòåîðåìû 5.1 îêàçûâàþòñÿ âåùåñòâåííî àíàëèòè÷åñêèìè ôóíêöèÿìè x ∈ R2 .Êðîìå òîãî, òðåáîâàíèå 5.26 òàêæå ÿâëÿåòñÿ çàâûøåííûì.Äîêàçàòåëüñòâî òåîðåìû 5.1 àíàëîãè÷íî äîêàçàòåëüñòâó [57, Theorem 9.2] âñëó÷àå A = 0, è ìû íå áóäåì åãî ïîâòîðÿòü.5.2Àëãîðèòì â áîðíîâñêîì ïðèáëèæåíèèÅñëè êîýôôèöèåíòû A è V óðàâíåíèÿ (3.9) ìàëû, òî èñõîäÿ èç èíòåãðàëüíûõóðàâíåíèé (3.12) è (3.14), ìû ïîëó÷àåì ñëåäóþùèå ôîðìóëû áîðíîâñêîãî ïðèáëèæåíèÿ:ψ + (x, k) ≈ eikx , ∇ψ + (x, k) ≈ eikx ik,f (k, l) ≈ f lin (k, l),Zlin−2ei(k−l)x 2kA(x) + V (x) dx,f (k, l) = (2π)R2(5.30)(5.31)128ãäå x ∈ R2 , (k, l) ∈ ME .
Çàìåòèì, ÷òî ôóíêöèÿ f lin èíâàðèàíòíà ïî îòíîøåíèþê êàëèáðîâî÷íûì ïðåîáðàçîâàíèÿìA → A + ∇ϕ,(5.32a)V → V − i∆ϕ,(5.32b)äëÿ ëþáîé äîñòàòî÷íî ðåãóëÿðíîé ôóíêöèè ϕ íà R2 , áûñòðî óáûâàþùåé íàáåñêîíå÷íîñòè. Äëÿ äîêàçàòåëüñòâà ýòîé èíâàðèàíòíîñòè äîñòàòî÷íî âîñïîëüçîâàòüñÿ â îïðåäåëåíèè (5.31) èíòåãðèðîâàíèåì ïî ÷àñòÿì è ñîîòíîøåíèåì k 2 −l2 = 0. Ìû ðàññìàòðèâàåì ïðåîáðàçîâàíèå (5.32a), (5.32b) êàê ëèíåàðèçàöèþïðåîáðàçîâàíèÿ (5.2a), (5.2b) ïðè ìàëûõ A, V è ϕ. ýòîì ïàðàãðàôå ìû ðàññìàòðèâàåì ñëåäóþùóþ ëèíåàðèçîâàííóþ îáðàòíóþ çàäà÷ó ðàññåÿíèÿ äëÿ óðàâíåíèÿ (3.9).Çàäà÷à 5.1.
Ïóñòü çàäàíà ëèíåàðèçîâàííàÿ àìïëèòóäà ðàññåÿíèÿ fíà MEïðè ôèêñèðîâàííîì E > 0. Íàéòè êîýôôèöèåíòû A è V íà R2 (ïî ìîäóëþêàëèáðîâî÷íûõ ïðåîáðàçîâàíèé (5.32a), (5.32b)).linÇàìåòèì, ÷òî äîïîëíèòåëüíàÿ èíôîðìàöèÿ î ïîòåíöèàëàõ A è V â çàäà÷å5.1 ïîçâîëÿåò èçáàâèòüñÿ îò êàëèáðîâî÷íîé íååäèíñòâåííîñòè. Íàïðèìåð, äëÿåäèíñòâåííîñòè äîñòàòî÷íî ïîòðåáîâàòü âåùåñòâåííîñòè A è V .Òàêæå çàìåòèì, ÷òî çàäà÷à 5.1 ìîæåò ðàññìàòðèâàòüñÿ êàê ëèíåàðèçàöèÿçàäà÷è 3.2 ïðè ìàëûõ A è V .Äëÿ èçó÷åíèÿ çàäà÷è 5.1 óäîáíî ââåñòè ñëåäóþùèå îáîçíà÷åíèÿ.
Îïðåäåëèìôóíêöèè Adiv,0 è V div,0 ôîðìóëàìèAdiv,0 (x) = A(x) + ∇ϕdiv (x),(5.33a)V div,0 (x) = V (x) − i∆ϕdiv (x),(5.33b)ãäå x ∈ R2 , à ϕdiv îïðåäåëÿåòñÿ êàê ðåøåíèå çàäà÷è (5.4a), (5.4b). Ôóíêöèè A±,0è V ±,0 îïðåäåëÿþòñÿ ôîðìóëàìèA±,0 (x) = A(x) + ∇ϕ± (x),(5.34a)V ±,0 (x) = V (x) − i∆ϕ± (x),(5.34b)ãäå x ∈ R2 , à ϕ− è ϕ+ îïðåäåëÿåòñÿ êàê ðåøåíèÿ çàäà÷ (5.5a), (5.5b) è (5.6a),129(5.6b) ñîîòâåòñòâåííî. Çàìåòèì, ÷òî ñïðàâåäëèâû ñëåäóþùèå ñîîòíîøåíèÿ:div Adiv,0 (x) = 0,±,0A±,01 (x) ± iA2 (x) = 0,±,0±,0ãäå x ∈ R2 , A±,0 = (A1 , A2 ).Îïðåäåëåíèå (5.31) ìîæåò áûòü ïåðåïèñàíî â âèäå ñèñòåìû:b − l),f lin (k, l) − f lin (−l, −k) = 2(k + l)A(kb − l) + 2Vb (k − l),f lin (k, l) + f lin (−l, −k) = 2(k − l)A(k(5.35a)(5.35b)b è Vb îáîçíà÷àþò îáðàòíûå ïðåîáðàçîâàíèÿ Ôóðüå:ãäå (k, l) ∈ ME , à A−2Zb = (2π)A(p)eipxA(x) dx,−2ZVb (p) = (2π)R2eipx V (x) dx.(5.36)R2Òàêæå çàìåòèì, ÷òî(k, l) ∈ ME =⇒ k − l ∈ B2√E ,p ∈ B2√E =⇒ p = k − l äëÿ íåêîòîðûõ (k, l) ∈ ME ,(5.37)ãäå èñïîëüçóåòñÿ îáîçíà÷åíèåBr = {p ∈ R2 : |p| ≤ r},r > 0.(5.38)Ìû áóäåì èñïîëüçîâàòü íîðìó k · kN,σ è ïðîñòðàíñòâà C N,σ (R2 ), îïðåäåë¼ííûåâ ôîðìóëå (2.13).
Çàìåòèì, ÷òî åñëè A1 , A2 , V ∈ C 0,σ (R2 ) ïðè íåêîòîðîì σ > 2è kA1 k0,σ ≤ q , kA2 k0,σ ≤ q , kV k0,σ ≤ q , òî ñïðàâåäëèâû îöåíêèψ + (x, k) = eikx + O(q),∇ψ + (x, k) = eikx ik + O(q),f (k, l) = f lin (k, l) + O(q 2 ),q → +0,(5.39)ðàâíîìåðíî ïî x ∈ R2 è (k, l) ∈ ME ïðè ôèêñèðîâàííîì E > 0.Ìû òàêæå áóäåì èñïîëüçîâàòü îáîçíà÷åíèåkAkN,σ = max kA1 kN,σ , kA2 kN,σ ,A = (A1 , A2 ).(5.40)Òåîðåìà 5.2. Ïóñòü A1 , A2 , V âåùåñòâåííîçíà÷íû è ïóñòü A1 , A2 , V ∈130C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 3, σ > 2.
Òîãäà ñïðàâåäëèâû ñëåäóþùèå ôîðìóëûðåøåíèÿ çàäà÷è 5.1:f lin (k, l) − f lin (l, k) k − lf lin (k, l) − f lin (−l, −k) k + lbA(k − l) =+,2|k − l|22|k + l|2f lin (l, k) + f lin (−l, −k)Vb (k − l) =,(5.41)2b è Vb îïðåäåëåíû â ôîðìóëå (5.36);ãäå (k, l) ∈ ME , à ôóíêöèè AA(x) = Aappr (x, E) + Aerr (x, E), x ∈ R2 , E > 0,ZZ−ipx bb dp,Aappr (x, E) = eA(p) dp, Aerr (x, E) = e−ipx A(p)√|p|≤2 E(5.42a)√|p|≥2 EV (x) = Vappr (x, E) + Verr (x, E), x ∈ R2 , E > 0,ZZ−ipx bVappr (x, E) = eV (p) dp, Verr (x, E) = e−ipx Vb (p) dp,√|p|≤2 E(5.42b)√|p|≥2 E√bãäå A(p)è Vb (p) ïðè |p| ≤ 2 E îïðåäåëÿþòñÿ ïî çíà÷åíèÿì f lin íà ME âñîîòâåòñòâèè ñ ôîðìóëàìè (5.37), (5.41), è ñïðàâåäëèâû îöåíêè|Aerr,j (x, E)| ≤ c1 (N, σ)kAj kN,σ E −|Verr (x, E)| ≤ c1 (N, σ)kV kN,σ E −ãäå x ∈ R2 , j = 1, 2, Aerr = (Aerr,1 , Aerr,2 ), E ≥c1 (N, σ) =14N −22N −22,,(5.43a)(5.43b)è4.(N − 2)(σ − 2)(5.44)Òåîðåìà 5.3. Ïðåäïîëîæèì, ÷òî A1 , A2 , V ∈ C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 4è σ > 2.
Ïóñòü Adiv,0 è V div,0 îïðåäåëÿþòñÿ ôîðìóëàìè (5.33a) è (5.33b). Òîãäàñïðàâåäëèâû ñëåäóþùèå ôîðìóëû ðåøåíèÿ çàäà÷è 5.1:f lin (k, l) − f lin (−l, −k) k + ldiv,0b,A (k − l) =2|k + l|2f lin (k, l) + f lin (−l, −k)Vb div,0 (k − l) =,2(5.45)131bdiv,0 è Vb div,0 îáîçíà÷àþò îáðàòíûå ïðåîáðàçîâàíèÿ Ôóðüåãäå (k, l) ∈ ME , à Aôóíêöèé Adiv,0 è V div,0 (ñì. ôîðìóëó (5.36));Adiv,0 (x) = Adiv,0(x, E) + Adiv,0(x, E), x ∈ R2 , E > 0,apprerrZZbdiv,0 (p) dp, Adiv,0bdiv,0 (p) dp,Adiv,0(x, E) = e−ipx A(x, E) = e−ipx Aapprerr√|p|≤2 E(5.46a)√|p|≥2 Ediv,0div,0V div,0 (x) = Vappr(x, E) + Verr(x, E), x ∈ R2 , E > 0,ZZdiv,0div,0Vappr(x, E) = e−ipx Vb div,0 (p) dp, Verr(x, E) = e−ipx Vb div,0 (p) dp,√|p|≤2 E(5.46b)√|p|≥2 E√bdiv,0 (p) è Vb div,0 (p) ïðè |p| ≤ 2 E íàõîäÿòñÿ ïî çíà÷åíèÿì ôóíêöèè f linãäå Aíà ME â ñîîòâåòñòâèè ñ ôîðìóëàìè (5.37), (5.45), è ñïðàâåäëèâû îöåíêè√N −22)c1 (N, σ)kAkN,σ E − 2 ,√N −3N −2div,0|Verr(x, E)| ≤ c1 (N, σ) kV kN,σ E − 2 + 2kAkN,σ E − 2 ,|Adiv,0(x, E)| ≤ (1 +err,j(5.47a)(5.47b)ãäå x ∈ R2 , j = 1, 2, E ≥ 14 , Adiv,0= (Adiv,0, Adiv,0), à c1 (N, σ) îïðåäåëåíî âerrerr,1err,2ôîðìóëå (5.44).
Êðîìå òîãî, åñëè div A = 0, òî Adiv,0 = A è V div,0 = V .Òåîðåìà 5.4. Ïðåäïîëîæèì, ÷òî A1 , A2 , V ∈ C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 4è σ > 2. Ïóñòü ôóíêöèè A±,0 è V ±,0 îïðåäåëÿþòñÿ ôîðìóëàìè (5.34a)(5.34b).Òîãäà ñïðàâåäëèâû ñëåäóþùèå ôîðìóëû ðåøåíèÿ çàäà÷è 5.1:b±,0 (k − l) = 1 f (k, l) − f (−l, −k) , Ab±,0 (k − l) = ±iAb±,0 (k − l),A1212 k1 + l1 ± i(k2 + l2 )(l1 ± il2 )f (k, l) + (k1 ± ik2 )f (−l, −k),Vb ±,0 (k − l) =k1 + l1 ± i(k2 + l2 )(5.48)b±,0 è Vb ±,0 îáîçíà÷àþò îáðàòíûå ïðåîáðàçîâàíèÿ Ôóðüåãäå (k, l) ∈ ME , à Aôóíêöèé A±,0 è V ±,0 (ñì.
ôîðìóëó (5.36));±,0A±,0 (x) = A±,0(x, E) + Aerr(x, E), x ∈ R2 , E > 0,apprZZ±,0b±,0 (p) dp, A±,0b±,0 (p) dp,Aappr(x, E) = e−ipx A(x, E) = e−ipx Aerr√|p|≤2 E√|p|≥2 E(5.49a)132±,0±,0V ±,0 (x) = Vappr(x, E) + Verr(x, E), x ∈ R2 , E > 0,ZZ±,0±,0Vappr(x, E) = e−ipx Vb ±,0 (p) dp, Verr(x, E) = e−ipx Vb ±,0 (p) dp,√|p|≤2 E(5.49b)√|p|≥2 E√b±,0 (p) è Vb ±,0 (p) ïðè |p| ≤ 2 E âûðàæàþòñÿ ÷åðåç çíà÷åíèÿ ôóíêöèè f linãäå Aíà ME â ñîîòâåòñòâèè ñ ôîðìóëàìè (5.37), (5.48), è âåðíû îöåíêè±,0√N −22)c1 (N, σ)kAkN,σ E − 2 ,√N −3N −2±,0|Verr(x, E)| ≤ c1 (N, σ) kV kN,σ E − 2 + 2kAkN,σ E − 2 ,|Aerr,j (x, E)| ≤ (1 +(5.50a)(5.50b)ãäå x ∈ R2 , j = 1, 2, A±,0= (A±,0, A±,0), kAkN,σ îïðåäåëÿåòñÿ â ôîðìóëåerrerr,1err,2(5.40), à c1 (N, σ) â ôîðìóëå (5.44).
Êðîìå òîãî, åñëè A1 ± iA2 = 0, òîA = A±,0 , V = V ±,0 .Òåîðåìû 5.2, 5.3 è 5.4 äîêàçûâàþòñÿ â 5.4. äâóõ ñëåäóþùèõ ïðåäëîæåíèÿõ ìû ïîêàæåì, ÷òî ôîðìóëû ðåøåíèÿ ëèíåàðèçîâàííîé îáðàòíîé çàäà÷è ðàññåÿíèÿ 5.1, ïðèâîäèìûå â òåîðåìàõ 5.3 è5.4, ìîãóò áûòü òàêæå ïîëó÷åíû ëèíåàðèçàöèåé àëãîðèòìà èç 5.1 â ñëó÷àåìàëîñòè êîýôôèöèåíòîâ A è V . Ïóñòü çàäàíà èçìåðèìàÿ ôóíêöèÿ f íà ME ,f = f (λ, λ0 , E), (λ, λ0 ) ∈ T 2 , ïðè ôèêñèðîâàííîì E è ïóñòü(5.51)kf kL2 (T 2 ) ≤ ε,ãäå íîðìà k · kL2 (T 2 ) îïðåäåëÿåòñÿ â ôîðìóëå (5.25).Ïðåäëîæåíèå 5.2. Ïðåäïîëîæèì, ÷òî ôóíêöèÿ f óäîâëåòâîðÿåò (5.51) ïðèôèêñèðîâàííîì E > 0.
Òîãäà ïðè ε → +0 àëãîðèòì âîññòàíîâëåíèÿ èç 5.1ñâîäèòñÿ ê ñëåäóþùèì ôîðìóëàì ïðè ôèêñèðîâàííîì E > 0:±A±(x, E) = Aappr,j(x, E) + O(ε2 ),appr,jj = 1, 2,±±Vappr(x, E) = Vappr(x, E) + O(ε2 ),divAdiv(x, E) = Aappr,j(x, E) + O(ε2 ),apprj = 1, 2,divdivVappr(x, E) = Vappr(x, E) + O(ε2 ),(5.52a)(5.52b)ãäå O(ε2 ) ðàññìàòðèâàåòñÿ â ðàâíîìåðíîì ñìûñëå ïî ïåðåìåííîé x ∈ R2 , ôóíê-133−−íàõîäÿòñÿ ïî ôóíêöèè f èç ôîðìóëöèè Aappr,j, Vappri√−Aappr,1(x, E) = − E4Z 1 λλ0sgn−(λ − λ0 )f˜ |dλ| |dλ0 |,i λ0λ(5.53a)T2−−Aappr,2(x, E) = −iAappr,1(x, E), Z0λE1λ0−(1 − λλ ) sgn−f˜ |dλ| |dλ0 |,Vappr(x, E) = i02i λλ(5.53b)(5.53c)T2++ôóíêöèè Aappr,jè Vappr èç ôîðìóëi√+EAappr,1(x, E) =4Z λ01 λsgn−(λ − λ0 )f˜ |dλ| |dλ0 |,0i λλ(5.54a)T2++Aappr,2(x, E) = iAappr,1(x, E), Zλ0E1 λ+0Vappr (x, E) = −i−f˜ |dλ| |dλ0 |,(1 − λλ ) sgn02i λλ(5.54b)(5.54c)T2divdivà ôóíêöèè Aappr,jè Vappr èç ôîðìóë1+−Aappr,j(x, E) + Aappr,j(x, E) , j = 1, 2,2 Z 0 E1λλdiv f˜ |dλ| |dλ0 |,Vappr(x, E) =−022i λλ divAappr,j(x, E) =(5.55a)(5.55b)T2ãäå èñïîëüçóåòñÿ îáîçíà÷åíèå √E00−10−1f˜ = f (λ, λ , E) exp −i(λ − λ )z̄ + (λ − λ )z ,2λ ∈ T , λ0 ∈ T , à z è z̄ îïðåäåëÿþòñÿ â ôîðìóëå (5.9).Ïðåäëîæåíèå 5.3.
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