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A note on the connection problem of some special Painlevé V functions Wen-Gao Longa, Zhao-Yun Zengb, and Jian-Rong Zhouc 6 ∗ 1 0 aSchool of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 2 510275, PR China n bSchool of Mathematics and Physics, Jinggangshan University, Ji’an 343009, PR China a J cDepartment of Mathematics, Foshan University, Foshan 528000, PR China 5 ] A C Abstract . h t As a new application of the method of “uniform asymptotics” proposed by a Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof m of the connection formulas of some special solutions of the fifth Painlevé equation, [ which have been established earlier by Andreev and Kitaev. 1 v 8 2 Keywords: Connection formulas; the fifth Painlevé transcendent; uniform asymp- 7 totics; Whittaker functions; modified Bessel functions 0 0 . MSC2010: 33A40, 33E17, 34A20, 34E05 1 0 6 1 1 Introduction and main results : v i X In this paper, as an application of the method of “uniform asymptotics” introduced by r Bassom, Clarkson, LawandMcLeod[3], westudytheconnectionformulasofthefollowing a Painlevé V equation d2y 1 1 dy 2 1dy y y(y+1) = + + , (1.1) dt2 2y y 1 dt − t dt t − 2(y 1) (cid:18) − (cid:19)(cid:18) (cid:19) − which is a special form of the general PV equation (cf. [6, (5.4.9)]) with the parameters Θ = Θ = Θ = 0. The special PV equation (1.1) has important applications in 0 1 ∞ ∗ Corresponding author. Email:[email protected]. 1 2 differential geometry of surfaces. For example, if we set y(t) = eiq(x)+1 and t = 4x in eiq(x) 1 − (1.1), then q(x) satisfies the following equation (cid:16) (cid:17) xq 2xsin2q +q +2sinq = 0, ′′ ′ − which was considered in [4] and its solutions are connected with the problem of classifi- cation for rotation surfaces with harmonic inverse mean curvature. Before stating our main results, we first recall some of the relevant facts from the isomonodromy formalism for the fifth Painlevé transcendent presented in [6,8]. The Lax pair of the fifth Painlevé equation, with the special parameters Θ = Θ = Θ = 0, is a 0 1 ∞ system of linear ordinary differential equations for the matrix function Y(λ,t) dY t 1 v uv 1 v uyv = σ + − + − Y (1.2) dλ 2 3 λ v v λ 1 v v (cid:20) (cid:18) u − (cid:19) − (cid:18) −uy (cid:19)(cid:21) and dY 1 uv(1 y) = 2 λ − Y, dt 1 v(1 1) 1 ! uλ − y − 2 1 0 where σ = and y(t),v(t) and u(t) satisfy the following system of equations 3 0 1 (cid:18) − (cid:19) tdy = ty 2v(y 1)2, dt − −  tdv = yv2 1v2, (1.3) dt − y    tdlnu = 2v +yv+ 1v. dt − y   Furthermore, y(t) is the solution of the special fifth Painlevé equation (1.1). In a neighborhood of the irregular singular point λ = , the canonical solutions ∞ Y(k)(λ) have the following asymptotic expansion Y(k)(λ) = I + v− v2(1t−yy)2 uv(1t−y) 1 + 1 exp λtσ ,t R (1.4) v(y−1) v + v2(1−y)2! λ O λ2 ! 2 3 ∈ + uyt − ty (cid:18) (cid:19) (cid:18) (cid:19) as λ in the corresponding Stokes sectors | | → ∞ π 3π Ω(k) := λ C, +π(k 2) < arg λ < +π(k 2) , k = 1,2. ∈ − 2 − 2 − (cid:26) (cid:27) These canonical solutions are related by the Stokes matrices S , k Y(k+1)(λ) = Y(k)(λ)S , λ Ω(k) Ω(k+1). (1.5) k ∈ ∩ 2 Furthermore, the Stokes matrices S can be written as k 1 0 1 s S = , S = 2k (1.6) 2k+1 s 1 2k 0 1 2k+1 (cid:18) (cid:19) (cid:18) (cid:19) where s (k = 1,2) are called the Stokes multipliers. If we now differentiate (1.5) with k respect to t and use the fact that Y(k) and Y(k+1) satisfy (1.2), we immediately obtain that S is independent of t. This is the isomonodromy condition. k There exists a unique solution (cf. [1, Sec.3]) of system (1.3) with the following asymptotic behaviors: 4 t y = 1 +O(t−2), v = +O(t−1), u = uê2t(1+o(1)) (1.7) − − t −8 as t + , and → ∞ (σs2tσ 2)2 1 σ2s2tσ y = − +O(t), v = +O(t), (σs2tσ +2)2 4s2tσ − 16 (1.8) 2+σs2tσ u = r (1+o(1)) − 2 σs2tσ − as t 0+, where uˆ,σ and r are complex constants, and → iσ2 Γ2( σ) σ s2 = − Γ6( ). (1.9) 4π3 Γ2(σ) 2 The above connection formulaes (1.7)-(1.9) have been established by Andreev and Kitaev in [1]. These results were derived by proposed a certain limit procedure of the parametersΘ (j = 0,1, )based ontheresults by using isomonodromydeformationand j ∞ the WKB method in their previous work [2]. It is worth noting that the limit procedure of Θ (j = 0,1, ) tend to zero in [1] seems neither obvious nor easy-to-prove. j ∞ In this paper, to avoid the limit process, we consider the equation (1.1) directly, i.e. the general PV equation with Θ = 0 (j = 0,1, ). By careful analysis, we find that j ∞ (1.7) can be immediately derived from (1.3), while (1.8) may depend on the monodromy matrices fortheregular singular pointsλ = 0 and1, which have been studied by Jimbo [8] via the method of isomonodromy deformation. Yet the main objective of the present paper is to justify connection formulas between parameters involved in the asymptotic approximations (1.7) and (1.8) via the Stokes multipliers for the irregular singular point λ = by using of the method of “uniform asymptotics”. Although the asymptotics ∞ behaviors in (1.8) are not available by virtue of the method of “uniform asymptotics”, fortunately we can obtain the following asymptotics behaviors v2(y 1)2 σ2 − = +o(1), uy21 = r +o(1), as t 0+, (1.10) y 4 − → provided that v = o(t 1) as t 0+, where σ and r are complex constants. It is readily − → observed that (1.10) is consistent with (1.8). Hence, our main task in this paper is to establish the relations between the parameter uˆ in (1.4) and the parameters σ,r in (1.10). The results can be stated as the following theorem. 3 Theorem 1. (cf.[1,Cor.3.2]) If 0 Reσ < 1, then the relation between the parameter ≤ uˆ in (1.7) and the parameters σ,r in (1.10) are given by i σ = ln(3+√8), r = iuˆ. (1.11) π − In this paper, we shall provide a hopefully simpler and more rigorous derivation of the connection formulas in (1.11) via the method of “uniform asymptotics”, which was first proposed by Bassom et al. in [3] and further developed in [7,11–13]. The difference between this approach and the WKB method is that the latter needs a matching process (cf. [2, Sec.7]) in different Stokes domain, while the former does not need such a compli- cated procedure. Therefore, to some extent, the method of “uniform asymptotics” is an improvement over WKB. Although the basic ideas of our approach are taken from [3] for PII, there are some technical differences between the cases of PII and PV. For instance, in the case of PII, the second-order ordinary differential equation(ODE) obtained from the Lax pair has only coalescing turning points; see [3]. Thus, uniform asymptotic approximations of the canonical solutions can be constructed in terms of the parabolic cylinder functions according to [9]. Recently, the method of “uniform asymptotics”has been applied to the connection problems for PV, cf. Zeng and Zhao [13]. However, all these cases also differ from our present case. When t + , η = 0 (see Section 3) is not only the → ∞ coalescing turning points but also a double pole of the second-order differential equation, hence, the parabolic cylinder function is not available. By careful analysis, we find that uniform asymptotic approximations can be successfully constructed by the modified Bessel functionsaccordingtotheideasofDunster[5]. Tothebestofourknowledge, under the framework of the method of “uniform asymptotics”, the modified Bessel functions have never been used in the connection problem of the fifth Painlevé equations, although the Hankel functions are used in [11] for the third Painlevé equations. The remaining part of this paper is organized as follows. In Section 2, we derive uniformapproximationstothesolutionsofthesecond-order differentialequationobtained from the Lax pair (1.2) as t 0+ by using of the Whittaker functions on the Stokes → curves, and then evaluate the Stokes multipliers (see (2.12) and (2.13)) as t 0+. In the → last section, we construct uniform approximations to the solutions of the second-order differential equation as t + by virtue of the modified Bessel functions on the Stokes → ∞ curves. Based on these approximations, we evaluate the Stokes multipliers as t + . → ∞ The proof of Theorem 1 is also provided in that section. 2 The monodromy data for t 0+ → In this section we use the method of “uniform asymptotics” [3] to obtain the Stokes multipliers s and s in (1.6) for the case of t 0+. 1 2 → 4 First, we make the scaling η = λt, then (1.2) becomes dY 1 + v v vu + vuy A B = 2 η − η−t − η η−t Y = Y. (2.1) dη v v 1 v + v ! C A uη − uy(η t) − 2 − η η t (cid:18) − (cid:19) − − Set φ = C 1Y , where Y = (Y ,Y )T is a fundamental solutionof (2.1) andC = v (1 1), −2 2 1 2 uη −y then d2φ C 3 C 2 1C = A2 +BC A +A ′ + ′ ′′ φ ′ dη2 − C 4 C − 2 C " # (cid:18) (cid:19) (2.2) 1 1 v2(1 y)2 1 1 = + − vt +g(η,t) φ = F(η,t)φ. 4 η2 y − − 2η − 4η2 (cid:20) (cid:18) (cid:19) (cid:21) with A = dA and C = dC. ′ dη ′ dη The form of equation (2.2) motivates us to consider the following model equation d2ψ 1 α(t)2 1 1 + 4 − 4 + 2 ψ = 0, (2.3) dη2 η2 η − 4 " # where α(t)2 = v2(1−y)2 vt. Furthermore, we find g(η,t) = vt as η provided 4 y − O η3 → ∞ that v = o(t 1). Noting that (2.3) is the Whittaker equation [1(cid:16)0, p(cid:17)334] with parameters − κ = 1, µ = α(t), and it has two linear independent solutions M (η) and W (η). 2 2 1,α 1,α 2 2 2 2 Hence, we have the following lemma. Lemma 1. There exist two constants C and C such that 1 2 φ = [C +o(1)]M (η)+[C +o(1)]W (η) (2.4) 1 1,α 2 1,α 2 2 2 2 as t 0+ uniformly for η on two adjacent Stokes curves of (2.2) emanating from one of → the turning points and terminating at infinity. Proof. The proof of this lemma is similar to [3, Theorem 1 or 2]. Denoting ψ = + M (η) and ψ = W (η). According to the parametrix variation method for the non- 1,α 1,α 2 2 − 2 2 homogeneous ODEs, we only need to show the following approximation (cf. [3, (3.17)]) η ψ (η)ψ (s) ψ (η)ψ (s) + − − − + g(s,t)φ(s)ds = o(1)(ψ+(η)+ψ (η)), t 0+, (2.5) Zη0 W(ψ+,ψ−) − → where the path of integration is taken along the Stokes curves, η is one of the turning 0 points, and W(ψ ,ψ ) is the Wronskian determinant. Obviously, η 1 2 α(t)2 + 0 as t 0+. − ∼ ± − → p First, we have (cf. [10, (13.14.26)]) W(ψ ,ψ ) = (1). In addition, according to [10, + − 1O (13.19.2)(13.19.3)], it is easy to obtain ψ = (η ) as η on the Stokes curves. 2 Combining these two estimates with g(η,±t) =O(vt) as η → ∞, we conclude that the O η3 → ∞ integral in the left-side of (2.5) is integrable. Finally, noting the condition that g(η,t) = o(1) as t 0+ uniformly for all η on the path of integration, we easily get (2.5). → 5 According to Lemma 1, Y , the second line of Y(k), i.e. (Y ,Y ), can be asymp- 2 21 22 1 1 totically approximated by the linear combinations of C M (η) and C W (η) when 2 1,α 2 1,α 2 2 2 2 η + in Ω(k). | | → ∞ Now, we are in a position to evaluate the Stokes multipliers through Y(k+1) = Y(k)S . k Although this part is slightly different from the approach in [3], they are coincide essen- tially. See details in Sec.6 of [3] or the corresponding sections in [11–13]. Here we only give the derivation of s . To get s , one can repeat the process except for noting that 1 2 the uniform asymptotic behaviors of the Whittaker functions in (2.6) should be changed. For convenient we denote C v (1 1) = β2(t). ∼ uη − y η When η , according to [10, (13.19.2) (13.19.3)] and noting that η = tλ, we get → ∞ C1M (η) c etλλ 1 +c e tλ, argη ( 3π, π), 2 12,α2 ∼ 1 2 − 2 −2 ∈ − 2 2 C1M (η) c etλλ 1 +c eα(t)πie tλ, argη ( π, 3π), (2.6)  2 12,α2 ∼ 1 2 − 2 −2 ∈ −2 2  C1W (η) c e tλ argη < 3π, 2 12,α2 ∼ 3 −2 | | 2    where c = β(t)Γ(1+α(t)),c = β(t)Γ(1+α(t))e α(t)πi/2 and c = β(t). On the other hand, 1 tΓ(α(t)) 2 Γ(1+α(t)) − 3 2 2 the uniform asymptotic behavior of Y and Y can be directly obtained from (1.4), and 21 22 the results are β(t)2 Y21 e2tλλ−1, Y22 e−2tλ, (2.7) ∼ t ∼ as λ in Ω(k), k = 1,2. Hence, it follows from (2.6) and (2.7) that → ∞ β(t)2 0 (Y21,Y22) ∼ C12(M12,α2(η),W21,α2(η)) βt(ct1)2c2 1!, λ → ∞,λ ∈ Ω(1) (2.8) − tc1c3 c3 and β(t)2 0 (Y21,Y22) ∼ C12(M12,α2(η),W21,α2(η)) β(t)2tcc12eα(t)πi 1 !, λ → ∞,λ ∈ Ω(2) (2.9) − tc1c3 c3 Noting that in (2.8), (Y ,Y ) represent the second line entries of Y(1)(λ), while in (2.9), 21 22 (Y ,Y ) are the second line of Y(2)(λ), by the definition of S in (1.5), we have 21 22 1 c β(t)2 4iβ(t)2 α(t)π 2i α(t)π s 2 (1 eα(t)πi) = sin( ) sin (2.10) 1 ∼ tc1 − − α(t) 2 ∼ uy21 2 as t 0+. Similar derivations lead to → α(t)π s2 2iuy21 sin( ) as t 0+. ∼ 2 → 6 1 Since s ,k = 1,2 are independent of t, it follows that lim uy and lim α(t) are exist. k 2 t 0+ t 0+ Without loss of generality, we may take → → α(t) = σ +o(1), uy21 = r +o(1) as t 0+. (2.11) − → Therefore the two Stokes multipliers are 2i σπ s = sin( ) (2.12) 1 − r 2 and σπ s = 2irsin( ). (2.13) 2 − 2 Remark 1. In (2.10), the branch of α(t) is chosen such that α(t) = 2v(1−y) +o(1) as y1/2 t 0+. If choosing α(t) = 2v(y−1) +o(1), we will need to set α(t) = σ+o(1) in (2.11). → y1/2 − 3 The monodromy data for t + → ∞ In this section, our goal is to derive the Stokes multipliers s and s as t + by 1 2 → ∞ applying the method of “uniform asymptotics” [3]. To derive s , let us first make the following transformation in (1.2) 2 ˜ 1 0 σ3 Y(λ) = u− 2 Y(λ). (3.1) 1 1 (cid:18)− (cid:19) As a result, we obtain dY˜(λ) A˜ B˜ A¯+B¯ B¯ = Y˜(λ) = Y˜(λ), (3.2) dλ C˜ A˜ C¯ B¯ 2A¯ (A¯+B¯) (cid:18) − (cid:19) (cid:18) − − − (cid:19) where t v v v vy v v ¯ ¯ ¯ A = + , B = + and C = . 2 λ − λ 1 −λ λ 1 λ − y(λ 1) − − − It is easy to check that the transformation (3.1) does not change the Stokes matrices. Let Y˜(λ) = (Y ,Y )T be a fundamental solution of (3.2), and set φ˜ = C˜ 1Y , then 1 2 −2 2 eliminating Y from (3.2) gives 1 d2φ˜ 3 1 = A˜2 +B˜C˜ A˜ +A˜C˜ 1C˜ + (C˜ 1C˜ )2 C˜ 1C˜ φ˜, (3.3) ′ − ′ − ′ − ′′ dλ2 − 4 − 2 (cid:20) (cid:21) where A˜ = dA˜ and C˜ = dC˜. For t , substituting the asymptotic behaviors of y and ′ dλ ′ dλ → ∞ v in (1.7) into (3.3), we finally obtain the following second-order equation: d2φ˜ t2η2 3 = + +g˜(η,t) φ˜= F˜(η,t)φ˜, (3.4) dη2 4(η2 1) 4η2 (cid:20) − 4 (cid:21) 7 whereη = λ 1, andg˜(η,t) = (1)uniformlyforη awayfromboth0and 1. Furthermore, −2 O 2 1 1 1 g˜(η,t) = , for η 0; g˜(η,t) = , for η ; O η → O (η 1)2 → 2 (cid:18) (cid:19) (cid:18) − 2 (cid:19) (3.5) 1 g˜(η,t) = , for η . O η2 → ∞ (cid:18) (cid:19) Comparing (3.4) with [3, (2.2)], one can find that the two turning points in (3.4) are not only coalesce at η = 0 with the speed of ( 1 ), but also they coalesce with a double O √t pole η = 0 as t + . Therefore, the parabolic cylinder functions are not available here. → ∞ The form of (3.4) motivate us to consider the following model equation d2ϕ t2η2 3 = + ϕ. (3.6) dη2 4(η2 1) 4η2 (cid:20) − 4 (cid:21) By careful analysis and according to the ideas in [5], we find that (3.6) is solvable. In fact, if we let 1 1 ϕ = η−21 η2 1 2 W(z) and z = t η2 1 2 , − 4 2 − 4 (cid:18) (cid:19) (cid:18) (cid:19) then W(z) satisfies the following modified Bessel equation d2W 1dW 1 + (1+ )W = 0 dz2 z dz − z2 which has two independent solutions K (z) and I (z), cf. [10, (10.25.1)]. 1 1 It is worth mentioning that, we can not use a pair of independent solutions of (3.6) to ˜ asymptoticallyapproximateφuniformlyeverywhereontheStokeslines, sinceg˜(η,t)isnot bounded as η 1 (i.e. λ 1). Even so, the following lemma holds for η C [1,+ ). → 2 → ∈ \ 2 ∞ Lemma 2. There exists two constants C and C such that 1 2 ˜ φ = [C +o(1)]ϕ (η)+[C +o(1)]ϕ (η) (3.7) 1 + 2 − uniformly for η on two adjacent Stokes curves of (3.4) with η C [1,+ ) as t , ∈ \ 2 ∞ → ∞ where 1 t 1 ϕ+ = η−12(η2 )12K1( (η2 )21), − 4 2 − 4 1 t 1 ϕ = η−21(η2 )12I1( (η2 )12) − − 4 2 − 4 are two linearly independent solutions of (3.6). Proof. The proof of this lemma is similar to Lemma 1 or [3, Theorem 1 or 2] and it is also based on the parametrix variation method for the non-homogeneous ODEs. Precisely, we only need to show the following approximation η ϕ (η)ϕ (s) ϕ (η)ϕ (s) + + ˜ − − − g˜(s,t)φ(s)ds = o(1)(ϕ+(η)+ϕ (η)), t + , (3.8) Zη0 W(ϕ+,ϕ−) − → ∞ 8 where the path of integration is taken along the Stokes curves and W(ϕ ,ϕ ) is the + − Wronskian determinant. First, simple calculation yields W(ϕ ,ϕ ) 1 according to [10, + − ≡ (10.28.2)]. In addition, since g˜(η,t) = 1 as η , then the integral in (3.8) is O η2 → ∞ really integrable. Finally, by analysis we fi(cid:16)nd (cid:17)that ϕ = (1/√t) as t + uniformly ± O → ∞ for all η on the Stokes curves according to [10, (10.3.1)(10.3.2)(10.40.2)(10.40.5)]. This implies (3.8) accordingly. Remark 2. By careful analysis, we find that (3.7) is not true for η 1. It implies → 2 that φ cannot be asymptotically approximated by the modified Bessel functions in the whole sector region argη [ π, π]. This may be the reason why we cannot use Lemma 2 to − ∈ 2 2 derive s , but should make another transformation (3.17). 1 According to Lemma 2, Y , the second line of Y˜, i.e. (Y ,Y ), can be approximated 2 21 22 by a linear combination of ϕ and ϕ , and it is uniformly valid for all η on the two + adjacent Stokes curves which extend t−o with argη = π and argη = 3π respectively. ∞ 2 2 Then s can be evaluated through Y˜(3) = Y˜(2)S . Here and hereafter, we assume η t 2 2 ≫ to ensure that t η2 1 tλ t. 2 − 4 ∼ 2 − 4 For η qwith argη π, arg η2 1 π. Using the uniform asymptotic → ∞ ∼ 2 − 4 ∼ 2 behaviors of K (z) and I (z)(see [10, ((cid:16)1q0.40.2), ((cid:17)10.40.5)]) 1 1 K (z) (π)1z 1e z, argz ( 3π, 3π); 1 ∼ 2 2 −2 − ∈ − 2 2 (3.9) I (z) ( 1 )1z 1ez i( 1 )1z 1e z, argz ( π, 3π).  1 ∼ 2π 2 −2 − 2π 2 −2 − ∈ −2 2 we get  1 1 t tλ ϕ π t e e , + 2 −2 4 − 2 ∼ (3.10)  1 1 t tλ 1 1 t tλ ϕ π t e e iπ t e e .  − ∼ −2 −2 −4 2 − −2 −2 4 − 2 Hence  (C˜)1ϕ iπ1ete tλ = d e tλ, 2 + 2 4 − 2 1 − 2 ∼ (3.11)  ˜ 1 1 t tλ 1 t tλ tλ tλ (C) ϕ iπ e e +π e e = d e +d e .  2 − ∼ −2 −4 2 −2 4 − 2 2 2 3 − 2 In virtue of the asymptotic behavior of the canonical solution Y(λ) in (1.4) and the  transformation (3.1), we get Y˜21(λ) u−21et2λ, Y˜22(λ) u12e−t2λ as λ ,λ Ω(2). ∼ − ∼ → ∞ ∈ Comparing with (3.11), one can easily obtain −1 1 d3u 2 u2 (Y˜21,Y˜22) ∼ (C˜)12(ϕ+,ϕ−) d1ud−212 d01  as λ → ∞ in Ω(2). (3.12) − d2   9 Now for the case argη = arg( η2 1) 3π. Here we need the following analytic − 4 ∼ 2 continuation formula for the modifiqed Bessel functions K (z) and I (z) (see [10, (10.34.5), 1 1 (10.34.6)]) K (z) = K (ze 2πi) 2K (ze π); 1 1 − 1 − − − (3.13) I (z) = 1 (K (ze πi)+K (z)).  1 πi 1 − 1 Substituting (3.9) into (3.13), we can obtain the uniform asymptotic behavior of K (z)  1 and I (z) for argz 3π 1 ∼ 2 K (z) (π)1z 1e z 2i(π)1z 1ez, argz ( 3π, 3π); 1 ∼ 2 2 −2 − − 2 2 −2 ∈ − 2 2 (3.14) I (z) ( 1 )1z 1ez i( 1 )1z 1e z, argz (π, 5π).  1 ∼ − 2π 2 −2 − 2π 2 −2 − ∈ 2 2 By suitable modification to the deriving of (3.12), we obtain that  −1 1 e4u 2 e3u2 (Y˜21,Y˜22) (C˜)21(ϕ+,ϕ ) − M M as λ in Ω(3), (3.15) ∼ −  e1u−12 e2u12 → ∞ M M   where e = d ,e = 2πid ,e = d ,e = d = id and M = e e +e e = d d . 1 1 2 − 2 3 2 4 3 −π 1 1 3 2 4 − 1 2 By using of the definition of S in (1.5), it follows from (3.12) and (3.15) that 2 S d3du1d−212 ud121 −1 −e4Mu−21 e3Mu21 = 1 −2πdi1d2u , 2 ∼  u−12 0   e1u−12 e2u21 0 1 ! − d2 M M     which gives s = 2iuˆ. (3.16) 2 − Here, use has been made of the fact that s is independent of t. 2 Using the similar method as in the computation of s , we can carry out s . If we 2 1 replace the transformation (3.1) by Yˆ(λ) = 1 0 ( uy)−σ23Y(λ), (3.17) 1 1 − (cid:18) (cid:19) then the Shro¨dinger equation (3.4) becomes d2φˆ t2η2 3 = + +gˆ(η,t) φˆ= Fˆ(η,t)φˆ. (3.18) dη2 4(η2 1) 4η2 (cid:20) − 4 (cid:21) where gˆ(η,t) is bounded for η away from both 0 and 1, and −2 1 1 1 gˆ(η,t) = as η 0; gˆ(η,t) = as η ; O η → O (η + 1)2 → −2 (cid:18) (cid:19) (cid:18) 2 (cid:19) 1 gˆ(η,t) = as η . O η2 → ∞ (cid:18) (cid:19) Hence we obtain a similar result to Lemma 2 as follows: 10