Диссертация (1103157), страница 24
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 ëèòåðàòóðå ýòà çàäà÷à ÷àñòî ôîðìóëèðóåòñÿäðóãèì, íî ýêâèâàëåíòíûì îáðàçîì. Çàìåòèì, ÷òî èç ðàâåíñòâà (5.65), çàïèñàííîãî äëÿ ψ+ è ψ− , ìîæíî èñêëþ÷èòü ôóíêöèþ ψ + , ïîëó÷èâ ÿâíîå ñîîòíîøåíèåZψ+ (λ) = ψ− (λ) +ρ(λ, λ0 )ψ− (λ0 ) |dλ0 |,λ ∈ T.(5.77)TÏðè ýòîì ôóíêöèÿ ρ âûðàæàåòñÿ ÷åðåç ôóíêöèè h± èç ôîðìóë (5.16), (5.65)ïîñðåäñòâîì ñëåäóþùèõ ôîðìóë è óðàâíåíèé: 0λλh1 (λ, λ0 , E) = χ i− 0 h+ (λ, λ0 , E)−λλ 0λλ− χ −i− 0 h− (λ, λ0 , E),λλ 0λλh2 (λ, λ0 , E) = χ i− 0 h− (λ, λ0 , E)−λλ 0λλ− χ −i− 0 h+ (λ, λ0 , E),λλ 0Z00λλρ(λ, λ0 , E) + πi ρ(λ, λ00 , E)χ −i 00 − 0 ×λλT141× h1 (λ00 , λ0 , E) |dλ00 | = −πih1 (λ, λ0 , E),Z 0λλ000ρ(λ, λ , E) + πi χ i 00 − 0 ×λλT× h2 (λ00 , λ0 , E) |dλ00 | = −πih2 (λ, λ0 , E),ãäå λ, λ0 ∈ T , ñì. [57].
Óðàâíåíèå (5.77) ìîæíî èñïîëüçîâàòü âìåñòî ïàðû óðàâíåíèé (5.65). Çàäà÷à ÐèìàíàÃèëüáåðòà â òàêîì âèäå ôîðìóëèðóåòñÿ, íàïðèìåð, â ðàáîòàõ [53, 94, 93, 57].Òåïåðü çàìåòèì, ÷òî â ñèëó ðåçóëüòàòîâ èç ðàáîò [94, 57] è â ñèëó ïðåäëîæåíèÿ 5.1, âûïîëíåíèå îöåíêè (5.26) ãàðàíòèðóåò îäíîçíà÷íóþ ðàçðåøèìîñòüíåëîêàëüíûõ çàäà÷ ÐèìàíàÃèëüáåðòà, ñôîðìóëèðîâàííûõ íà øàãàõ 1 è 1', îòíîñèòåëüíî ψ è ψ 0 ñîîòâåòñòâåííî. Ïðè ýòîì ñïðàâåäëèâû ôîðìóëû−−4∂z ∂z̄ + a−z (z)∂z + V (z) ψ(z, λ, E) = Eψ(z, λ, E),√+−a−(z)=4∂lnµ(z),V(z)=2iE∂z µ−z̄z01 (z),+00−4∂z ∂z̄ + a+z̄ (z)∂z̄ + V (z) ψ (z, λ, E) = Eψ (z, λ, E),√µ+(z)+++az̄ (z) = −4∂z ln µ0 (z), V (z) = 2i E∂z̄ 1+ ,µ0 (z)(5.78a)(5.78b)±ãäå z ∈ C, λ ∈ C \ (T ∪ 0), à µ+0 è µ1 îïðåäåëåíû ïî ôîðìóëå (5.76). Ôîðìóëà(5.78a) ñîîòâåòñòâóåò ðàâåíñòâàì (5.67a)(5.67b), à ôîðìóëà (5.78b) ðàâåíñòâàì (5.68a)(5.68b).Ôîðìóëû (5.22) è (5.23) ñëåäóþò èç ôîðìóë (5.76) è (5.78a)(5.78b).Íàêîíåö, ôîðìóëà (5.24) ïîëó÷àåòñÿ â ðåçóëüòàòå ðàññìîòðåíèÿ êàëèáðîâî÷íûõ ïðåîáðàçîâàíèé (5.2a), (5.2b), ïåðåâîäÿùèõ êîýôôèöèåíòû A− , V − è A+ ,V + â êîýôôèöèåíòû Adiv , V div .
Îòìåòèì, â ÷àñòíîñòè, ÷òî äëÿ ýòèõ ðàññìîòðåíèé óäîáíî ïåðåïèñàòü óðàâíåíèå (3.9) â ñëåäóþùåì âèäå:−4∂z ∂z̄ + az ∂z + az̄ ∂z̄ + V ψ = Eψ,az = −2i(A1 + iA2 ),az̄ = −2i(A1 − iA2 ).(5.79) òàêèõ îáîçíà÷åíèÿõ êàëèáðîâî÷íûå ïðåîáðàçîâàíèÿ (5.2a), (5.2b) ïðèíèìàþò142ñëåäóþùóþ ôîðìó:az → az − 4i∂z̄ ϕ,az̄ → az̄ − 4i∂z ϕ,V → V − 4i∂z ∂z̄ ϕ + 4∂z ϕ∂z̄ ϕ + iaz ∂z ϕ + iaz̄ ∂z̄ ϕ,à çàäà÷è (5.4a)(5.4b), (5.5a)(5.5b) è (5.6a)(5.6b) äëÿ ôóíêöèé ϕdiv , ϕ− è ϕ+ïåðåïèñûâàþòñÿ â âèäå8i∂z ∂z̄ ϕdiv = ∂z az + ∂z̄ az̄ ,5.4ϕdiv (z) → 0, |z| → ∞,4i∂z ϕ− = az̄ ,ϕ− (z) → 0, |z| → ∞,4i∂z̄ ϕ+ = az ,ϕ+ (z) → 0, |z| → ∞.Äîêàçàòåëüñòâî òåîðåì 5.2, 5.3 è 5.4Ìû áóäåì èñïîëüçîâàòü îáîçíà÷åíèÿub(p) = (2π)−2Zeipx u(x) dx,Zuerr (x, E) =e−ipx ub(p) dp,(5.80)√|p|≥2 ER2ãäå p ∈ R2 , x ∈ R2 , E > 0.
Íàïîìíèì òàêæå, ÷òî íîðìà k · kN,σ è ïðîñòðàíñòâàC N,σ (R2 ) îïðåäåëÿþòñÿ â ôîðìóëå (2.13).Ëåììà 5.1. Ïóñòü u ∈ C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 3, σ > 2. Òîãäà ñïðàâåäëèâà îöåíêà|uerr (x, E)| ≤ c1 (N, σ)kukN,σ E −N −22,(5.81)ãäå x ∈ R2 , E ≥ 1/2, à c1 (N, σ) îïðåäåëÿåòñÿ â ôîðìóëå (5.44).Äîêàçàòåëüñòâî ëåììû 5.1. Ñïðàâåäëèâî ðàâåíñòâîα u(p) = (−ip )α1 (−ip )α2 u∂db(p),12∂ α1 +α2∂ = α1 α2 ,∂x1 ∂x2α(5.82)ãäå p = (p1 , p2 ) ∈ R2 , α = (α1 , α2 ) ∈ Z2+ , |α| ≤ N . Ïîëüçóÿñü ýòèì ðàâåíñòâîì,ìû ïîëó÷àåì îöåíêó− N2N −1|bu(p)| ≤kukN,σ 1 + |p|2 2π(σ − 2)143äëÿ âñåõ p ∈ R2 , |p| ≥ 1.
Èç ýòîé îöåíêè ñëåäóåò îöåíêà (5.81).Äîêàçàòåëüñòâî òåîðåìû 5.2. Òàê êàê êîýôôèöèåíòû A1 , A2 è V âåùåñòâåííû, èç ôîðìóëû (5.31) ñëåäóåò ðàâåíñòâîb − l),f lin (k, l) − f lin (l, k) = 2(k − l)A(k(5.83)ãäå (k, l) ∈ ME . Ìû ðàññìàòðèâàåì (5.35a), (5.83) êàê ñèñòåìó ëèíåéíûõ óðàâb − l) è Vb (k − l). Êðîìå òîãî, ìû ïîëüçóåìñÿ ðàíåíèé äëÿ íàõîæäåíèÿ A(kâåíñòâîì (k − l)(k + l) = 0, ñïðàâåäëèâûì ïðè (k, l) ∈ ME .  ðåçóëüòàòå ìûïîëó÷àåì ôîðìóëû (5.41).Ôîðìóëû (5.42a) è (5.42b) ìîãóò ðàññìàòðèâàòüñÿ êàê îïðåäåëåíèÿ ôóíêöèéAappr , Aerr è Vappr , Verr . Îöåíêè (5.43a) è (5.43b) ñëåäóþò èç ëåììû 5.1.Ëåììà 5.2.
Ïðåäïîëîæèì, ÷òî A1 , A2 ∈ C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 4,σ > 2. Ïóñòü ôóíêöèÿ ϕdiv îïðåäåëÿåòñÿ êàê ðåøåíèå çàäà÷è (5.4a), (5.4b), àddiv , (∇ϕdiv )err , ∆ϕddiv è (∆ϕdiv )err îïðåäåëÿþòñÿ â ñîîòâåòñòâèè ñôóíêöèè ∇ϕôîðìóëîé (5.80). Òîãäà ñïðàâåäëèâû ñëåäóþùèå îöåíêè: √(∂x ϕdiv )err (x, E) ≤ 2 c1 (N, σ)kAkN,σ E − N2−2 ,j √(∆ϕdiv )err (x, E) ≤ 2 c1 (N, σ)kAkN,σ E − N2−3 ,(5.84)ãäå x ∈ R2 , ∂xj = ∂/∂xj , j = 1, 2, E ≥ 1/4, kAkN,σ îïðåäåëÿåòñÿ â ôîðìóëå(5.40), à c1 (N, σ) îïðåäåëÿåòñÿ â ôîðìóëå (5.44).Äîêàçàòåëüñòâî ëåììû 5.2. Ñïðàâåäëèâà ñëåäóþùàÿ ÿâíàÿ ôîðìóëà äëÿ ðåøåíèÿ ϕdiv çàäà÷è (5.4a), (5.4b):ϕdiv (x) = −iZ −2be−ipx pA(p)|p| dp,x ∈ R2 .(5.85)R2Èñïîëüçóÿ ôîðìóëû (5.80) è (5.85), ìû ïîëó÷àåì ðàâåíñòâà −2ddiv (p) = −p pA(p)b∇ϕ|p| ,ddiv (p) = ipA(p),b∆ϕ(5.86)ãäå p ∈ R2 \ 0.
Èç ôîðìóëû (5.86) ñëåäóþò íåðàâåíñòâà √divbk (p)|,∂d ≤ 2 max |Aϕ(p)xjk=1,2 div √bk (p)|,d∆ϕ (p) ≤ 2 |p| max |Ak=1,2(5.87)144ãäå p ∈ R2 \ 0, j = 1, 2.Ñïðàâåäëèâî ðàâåíñòâîα A (p) = (−ip )α1 (−ip )α2 Abj (p),∂[j12ãäå p = (p1 , p2 ) ∈ R2 , α = (α1 , α2 ) ∈ (N∪0)2 , |α| ≤ N è ∂ α îïðåäåëåíî â ôîðìóëå(5.82). Èç ýòîãî ðàâåíñòâà ìû ïîëó÷àåì îöåíêóbj (p)| ≤|AN2N −1kAj kN,σ (1 + |p|2 )− 2 ,π(σ − 2)(5.88)ãäå p ∈ R2 , |p| ≥ 1, j = 1, 2.
Èç íåðàâåíñòâ (5.87) è (5.88) ñëåäóþò îöåíêè √ 2N −1Ndiv∂dkAkN,σ (1 + |p|2 )− 2 ,2xj ϕ (p) ≤π(σ − 2) div √ 2N −1N −1d∆ϕ (p) ≤ 2kAkN,σ (1 + |p|2 )− 2 ,π(σ − 2)(5.89)ãäå p ∈ R2 , |p| ≥ 1, j = 1, 2. Èñïîëüçóÿ îöåíêè (5.89), ìû ïîëó÷àåì îöåíêè(5.84).Äîêàçàòåëüñòâî òåîðåìû 5.3.
Ó÷èòûâàÿ èíâàðèàíòíîñòü ôóíêöèè f lin ïî îòíîøåíèþ ê êàëèáðîâî÷íûì ïðåîáðàçîâàíèÿì (5.32a), (5.32b) è èñïîëüçóÿ ôîðìóëû (5.4a), (5.4b), (5.33a), (5.33b) è (5.35a), ìû ïîëó÷àåì ñëåäóþùèå ðàâåíñòâà:bdiv,0 (k − l) = 0,(k − l)Abdiv,0 (k − l),f lin (k, l) − f lin (−l, −k) = 2(k + l)Af lin (k, l) + f lin (−l, −k) = 2Vb div,0 (k − l),(5.90)ãäå (k, l) ∈ ME . Èñïîëüçóÿ ôîðìóëó (5.90) è îðòîãîíàëüíîñòü âåêòîðîâ (k − l)è (k + l), ìû ïîëó÷àåì ôîðìóëó (5.45).Ìîæíî ðàññìàòðèâàòü ôîðìóëû (5.46a) è (5.46b) êàê îïðåäåëåíèÿ ôóíêöèédiv,0div,0div,0Adiv,0è Vappr, Verr.appr , AerrÈç ôîðìóë (5.33a), (5.33b) è (5.46a), (5.46b) ìû ïîëó÷àåì ôîðìóëûAdiv,0(x, E) = Aj,err (x, E) + (∂xj ϕdiv )err (x, E),err,jdiv,0Verr(x, E) = Verr (x, E) − i(∆ϕdiv )err (x, E),j = 1, 2,(5.91)ãäå x ∈ R2 , E > 0, à ôóíêöèè Aj,err , Verr , (∂xj ϕdiv )err , (∆ϕdiv )err îïðåäåëÿþòñÿ â145ñîîòâåòñòâèè ñ ôîðìóëîé (5.80).Èç ôîðìóëû (5.91), ñ ó÷¼òîì íåðàâåíñòâà (5.81) äëÿ ôóíêöèé A1,err , A2,err ,Verr è íåðàâåíñòâ (5.84), ìû ïîëó÷àåì ôîðìóëû (5.47a) è (5.47b).Ëåììà 5.3.
Ïðåäïîëîæèì, ÷òî A1 , A2 ∈ C N,σ (R2 ) ïðè íåêîòîðûõ N ≥ 4 èσ > 2. Ïóñòü ôóíêöèè ϕ− è ϕ+ îïðåäåëÿþòñÿ êàê ðåøåíèÿ çàäà÷ (5.5a), (5.5b)d± , (∇ϕ± )err , ∆ϕd± ,è (5.6a), (5.6b) ñîîòâåòñòâåííî. Îïðåäåëèì ôóíêöèè ∇ϕ(∆ϕ± )err â ñîîòâåòñòâèè ñ ôîðìóëîé (5.80). Òîãäà ñïðàâåäëèâû ñëåäóþùèåîöåíêè: √(∂x ϕ± )err (x, E) ≤ 2 c1 (N, σ)kAkN,σ E − N2−2 ,j(5.92) √(∆ϕ± )err (x, E) ≤ 2 c1 (N, σ)kAkN,σ E − N2−3 ,ãäå x ∈ R2 , ∂xj = ∂/∂xj , j = 1, 2, E ≥ 1/4, kAkN,σ îïðåäåëÿåòñÿ â ôîðìóëå(5.40), à c1 (N, σ) îïðåäåëÿåòñÿ â ôîðìóëå (5.44).Äîêàçàòåëüñòâî ëåììû 5.3.
Ðåøåíèÿ ϕ− è ϕ+ çàäà÷ (5.5a), (5.5b) è (5.6a),(5.6b) ìîãóò áûòü âûïèñàíû â ÿâíîì âèäå:Z±ϕ (x) = −ieb2 (p)± iAdp,p1 ± ip2−ipx A1 (p)bx ∈ R2 .(5.93)R2Èñïîëüçóÿ ôîðìóëû (5.80) è (5.93), ìû ïîëó÷àåì ðàâåíñòâàbbd± (p) = − A1 (p) ± iA2 (p) p,∇ϕp1 ± ip2bbd± (p) = i A1 (p) ± iA2 (p) |p|2 ,∆ϕp1 ± ip2(5.94)ãäå p ∈ R2 \ 0. Èç ôîðìóëû (5.94) ñëåäóþò íåðàâåíñòâà √±bk (p)|,∂d2 max |Ax ϕ (p) ≤jk=1,2 ± √bk (p)|,d∆ϕ (p) ≤ 2 |p| max |Ak=1,2(5.95)ãäå p ∈ R2 \ 0, j = 1, 2.Äåéñòâóÿ êàê ïðè äîêàçàòåëüñòâå ëåììû 5.2, ìû ïîëó÷àåì îöåíêó (5.88). Èçîöåíîê (5.88) è (5.95) ñëåäóþò îöåíêè √ 2N −1N±∂d2kAkN,σ (1 + |p|2 )− 2 ,xj ϕ (p) ≤π(σ − 2) ± √ 2N −1N −1d∆ϕ (p) ≤ 2kAkN,σ (1 + |p|2 )− 2 ,π(σ − 2)(5.96)ãäå p ∈ R2 , |p| ≥ 1, j = 1, 2.
Èñïîëüçóÿ íåðàâåíñòâà (5.96), ìû ïîëó÷àåì îöåíêè146(5.92).Äîêàçàòåëüñòâî òåîðåìû 5.4. Ó÷èòûâàÿ èíâàðèàíòíîñòü ôóíêöèè f lin ïî îòíîøåíèþ ê êàëèáðîâî÷íûì ïðåîáðàçîâàíèÿì (5.32a), (5.32b) è èñïîëüçóÿ ôîðìóëû (5.5a)(5.6b) è (5.34a)(5.35a), ìû ïîëó÷àåì ðàâåíñòâà±,0A±,02 (k − l) = ±iA1 (k − l), ±,0b (k − l),f lin (k, l) − f lin (−l, −k) = 2 k1 + l1 ± i(k2 + l2 ) A1f lin (k, l) + f lin (−l, −k)(5.97) ±,0b (k − l) + 2Vb ±,0 (k − l),= 2 k1 − l1 ± i(k2 − l2 ) A1ãäå (k, l) ∈ ME . Ôîðìóëà (5.48) ñëåäóåò èç ñîîòíîøåíèé (5.97).Ìîæíî ðàññìàòðèâàòü ôîðìóëû (5.49a) è (5.49b) êàê îïðåäåëåíèÿ ôóíêöèé±,0±,0±,0A±,0appr , Aerr è Vappr , Verr .Èñïîëüçóÿ ôîðìóëû (5.34a), (5.34b) è (5.49a), (5.49b), ìû ïîëó÷àåì ðàâåíñòâàA±,0(x, E) = Aj,err (x, E) + (∂xj ϕ± )err (x, E),err,jj = 1, 2,±,0Verr(x, E) = Verr (x, E) − i(∆ϕ± )err (x, E),(5.98)ãäå x ∈ R2 , E > 0, à ôóíêöèè Aj,err , Verr , (∂xj ϕ± )err , (∆ϕ± )err îïðåäåëÿþòñÿ âñîîòâåòñòâèè ñ ôîðìóëîé (5.80).Èç ôîðìóëû (5.98), ñ ó÷¼òîì íåðàâåíñòâà (5.81), çàïèñàííîãî äëÿ ôóíêöèéA1,err , A2,err , Verr , è íåðàâåíñòâ (5.92), ìû ïîëó÷àåì îöåíêè (5.50a) è (5.50b).5.5Äîêàçàòåëüñòâî ïðåäëîæåíèé 5.2 è 5.3Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 5.2.
Ïóñòü E > 0 çàôèêñèðîâàíî. Èñïîëüçóÿìåòîä ïîñëåäîâàòåëüíûõ ïðèáëèæåíèé äëÿ ðåøåíèÿ óðàâíåíèÿ (5.16) îòíîñèòåëüíî h± ∈ L2 (T 2 ) è ó÷èòûâàÿ îöåíêó (5.51), ìû ïîëó÷àåì, ÷òîh± = f + O(ε2 ),(5.99)ε → +0,ãäå O(ε2 ) ïîíèìàåòñÿ â ñìûñëå íîðìû k · kL2 (T 2 ) .Ðàññìîòðèì ñëåäóþùèå îïåðàòîðû, äåéñòâóþùèå â ïðîñòðàíñòâå L2 (T ):1(C± u)(λ) =2πiZTu(ζ)dζ,ζ − λ(1 ∓ 0)λ ∈ T,(5.100)147Ñïðàâåäëèâî ñëåäóþùåå íåðàâåíñòâî:(5.101)kC± ukL2 (T ) ≤ kukL2 (T ) .Èñïîëüçóÿ ôîðìóëû (5.19), (5.20), (5.99), (5.101) è ðàâåíñòâî √E0−10−1exp −i(λ − λ )z̄ + (λ − λ )z = 1,2(5.102)ãäå λ, λ0 ∈ T , z ∈ C, ìû ïîëó÷àåì ñîîòíîøåíèå Z01λζdζB(λ, λ0 , z, E) =f (ζ, λ0 , z, E)χ −i 0 −−2λζζ − λ(1 − 0)T Z01ζλdζf (ζ, λ0 , z, E)χ i 0 −−+ O(ε2 ),2λζζ − λ(1 + 0)(5.103)Tãäå λ, λ0 ∈ T , z ∈ C, ôóíêöèÿ f (ζ, λ0 , z, E) îïðåäåëÿåòñÿ ôîðìóëîé √Ef (ζ, λ0 , z, E) = f (ζ, λ0 , E) exp −i(ζ − λ0 )z̄ + (ζ −1 − λ0−1 )z ,2(5.104)à O(ε2 ) ïîíèìàåòñÿ â ñìûñëå íîðìû k · kL2 (T 2 ) ðàâíîìåðíî ïî z ∈ C.Èç ôîðìóëû (5.18) âûòåêàþò ñëåäóþùèå ðàâåíñòâà:∂z µ+ (z, λ, E) +ZB(λ, λ0 , z, E)∂z µ+ (z, λ0 , E) |dλ0 | =TZ=−0+00(5.105a)∂z B(λ, λ , z, E)µ (z, λ , E) |dλ |,TZ+∂z̄ µ (z, λ, E) +B(λ, λ0 , z, E)∂z̄ µ+ (z, λ0 , E) |dλ0 | =TZ=−0+00(5.105b)∂z̄ B(λ, λ , z, E)µ (z, λ , E) |dλ |,Tãäå λ ∈ T , z ∈ C, à îïåðàòîðû ∂z è ∂z̄ îïðåäåëÿþòñÿ â ôîðìóëå (5.7).Èñïîëüçóÿ ìåòîä ïîñëåäîâàòåëüíûõ ïðèáëèæåíèé äëÿ ðåøåíèÿ óðàâíåíèé(5.18), (5.105a), (5.105b) îòíîñèòåëüíî µ+ , ∂z µ+ , ∂z̄ µ+ ∈ L2 (T ) ñîîòâåòñòâåííî,148à òàêæå ó÷èòûâàÿ îöåíêó (5.51) è ôîðìóëó (5.103), ìû ïîëó÷àåì îöåíêè(5.106a)µ+ (z, λ, E) = 1 + O(ε),Z+∂z µ (z, λ, E) = − ∂z B(λ, λ0 , z, E) |dλ0 | + O(ε2 ),∂z̄ µ+ (z, λ, E) = −TZ(5.106b)∂z̄ B(λ, λ0 , z, E) |dλ0 | + O(ε2 ),(5.106c)Tãäå z ∈ C, λ ∈ T , à O(ε) è O(ε2 ) ðàññìàòðèâàþòñÿ â ñìûñëå íîðìû k · kL2 (T )ðàâíîìåðíî ïî z ∈ C.
Èñïîëüçóÿ ôîðìóëû (5.21), (5.99) è (5.106a)(5.106c), ìûïîëó÷àåì îöåíêè(5.107a)µ± (z, λ, E) = 1 + O(ε),Z∂z µ± (z, λ, E) = − ∂z B(λ, λ0 , z, E) |dλ0 |T 0λλ+πi ∂z f (λ, λ0 , z, E)χ ±i 0 −|dλ0 | + O(ε2 ),λλTZ∂z̄ µ± (z, λ, E) = − ∂z̄ B(λ, λ0 , z, E) |dλ0 |ZTZ+πiλ0λ∂z̄ f (λ, λ , z, E)χ ±i 0 −λλ0(5.107b)(5.107c)02|dλ | + O(ε ),Tãäå z ∈ C, λ ∈ T , f (λ, λ0 , z, E) îïðåäåëåíî â ôîðìóëå (5.104), à O(ε) è O(ε2 )ïîíèìàþòñÿ â ñìûñëå íîðìû k · kL2 (T ) ðàâíîìåðíî ïî z ∈ C.Èñïîëüçóÿ îïðåäåëåíèÿ (5.22), (5.23) è îöåíêè (5.107a)(5.107c), ìû ïîëó÷àåì îöåíêè√ZEVappr (x, E) = −∂z B(λ, λ0 , z, E) dλ |dλ0 |π2T Z√λλ00+i E ∂z f (λ, λ , z, E)χ −i 0 −dλ |dλ0 | + O(ε2 ),λλ−T2(5.108a)149√ZE+Vappr(x, E) =∂z̄ B(λ, λ0 , z, E) λ−2 dλ |dλ0 |πT2 Z0√λλ−i E ∂z̄ f (λ, λ0 , z, E)χ i 0 −λ−2 dλ |dλ0 | + O(ε2 ),λλT2Z2ia−∂z̄ B(λ, λ0 , z, E) λ−1 dλ |dλ0 |z (z, E) =πT2 Z0λλ+2 ∂z̄ f (λ, λ0 , z, E)χ i 0 −λ−1 dλ |dλ0 | + O(ε2 ),λλT2Z2i+az̄ (z, E) = −∂z B(λ, λ0 , z, E) λ−1 dλ |dλ0 |πT2 Zλλ00−2 ∂z f (λ, λ , z, E)χ i 0 −λ−1 dλ |dλ0 | + O(ε2 ),λλ(5.108b)(5.108c)(5.108d)T2ãäå x ∈ R2 , z îïðåäåëåíî â ôîðìóëå (5.9), à O(ε2 ) ðàññìàòðèâàåòñÿ â ðàâíîìåðíîì ñìûñëå ïî ïåðåìåííîé x ∈ R2 .Çàìåòèì, ÷òî ñëåäóþùèå ôîðìóëû ñïðàâåäëèâû äëÿ âñåõ u ∈ L2 (T ):ZZ(C− u)(λ) dλ = −(C+ u)(λ) dλ = 0,TTZ(C+ u)(λ)TZZdλ=λdλ(C+ u)(λ) 2 =λTZu(λ)TZTu(λ) dλ,Tdλ,λdλu(λ) 2 ,λZ(C− u)(λ)dλ= 0,λ(C− u)(λ)dλ= 0,λ2TZ(5.109)Tãäå C± îïðåäåëåíî â ôîðìóëå (5.100).Èç ôîðìóë (5.19), (5.108a)(5.108d) è (5.109) ñëåäóþò îöåíêèZ√−Vappr(x, E) = i E s(λ, λ0 )∂z f (λ, λ0 , z, E) dλ |dλ0 | + O(ε2 ),(5.110a)2ZT√+Vappr(x, E) = i E s(λ, λ0 )∂z̄ f (λ, λ0 , z, E) λ−2 dλ |dλ0 | + O(ε2 ),a−z (z, E) = −2ZTT2(5.110b)2s(λ, λ0 )∂z̄ f (λ, λ0 , z, E) λ−1 dλ |dλ0 | + O(ε2 ),(5.110c)150a+z̄ (z, E) = 2Zs(λ, λ0 )∂z f (λ, λ0 , z, E) λ−1 dλ |dλ0 | + O(ε2 ),(5.110d)T2 01λλs(λ, λ0 ) == sgn−,i λ0λdefãäå x ∈ R2 , z îïðåäåëåíî â ôîðìóëå (5.9), à O(ε2 ) ïîíèìàåòñÿ â ðàâíîìåðíîìñìûñëå ïî ïåðåìåííîé x ∈ R2 .Íàêîíåö, èç ôîðìóëû (5.104) âûòåêàþò ñîîòíîøåíèÿ√E −1λ − λ0−1 f (λ, λ0 , z, E),√2Eλ − λ0 f (λ, λ0 , z, E),∂z̄ f (λ, λ0 , z, E) = −i2∂z f (λ, λ0 , z, E) = −i(5.111)äëÿ âñåõ λ, λ0 ∈ T , z ∈ C.Ôîðìóëû (5.52a) (5.55b) ñëåäóþò èç ôîðìóë (5.22)(5.24), (5.110a)(5.110d)è (5.111).
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