Mika Tanda. Communicated by Yoshishige Haraoka

Transcription

1 Opuscula Math. 35, no. 5 (205, Opuscula Mathematica ALIEN DERIVATIVES OF THE WKB SOLUTIONS OF THE GAUSS HYPERGEOMETRIC DIFFERENTIAL EQUATION WITH A LARGE PARAMETER Mika Tanda Communicated by Yoshishige Haraoka Abstract. We compute alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter and discuss the singularity structures of the Borel transforms of the WKB solution expressed in terms of its alien derivatives. Keywords: hypergeometric differential equation, WKB solution, Voros coefficient, alien derivative, Stokes curve, fixed singularity. Mathematics Subject Classification: 33C05, 34M40, 34M60.. INTRODUCTION For linear ordinary differential equations with large parameters, a kind of Stokes phenomena occur in the asymptotic behaviors of WKB solutions with a change of the parameters. Such Stokes phenomena are called parametric Stokes phenomena. In this paper we study the parametric Stokes phenomena for the Gauss hypergeometric differential equation from the viewpoint of the alien calculus. The alien derivative is introduced by Écalle [0] in 98. Delabaere-Dilinger-Pham [8] and Delabaere-Pham [9] studied the Stokes automorphisms and alien derivatives for WKB solutions of Schrödinger equations with polynomial potentials from the viewpoint of Ecalle s resurgent function theory. In these articles, Stokes automorphisms are described by using the intersection numbers of degenerate Stokes curves and integration paths for Voros coefficients. In this paper we apply these results to the Gauss hypergeometric differential equation with large parameters. We also refer to the works of Sauzin [5, 6], where the relation between the Stokes automorphisms and the alien derivatives is clarified. Parametric Stokes phenomena for second order linear ordinary differential equations with irregular singular points are studied in several papers: Takei [7] studied c AGH University of Science and Technology Press, Krakow

2 804 Mika Tanda then for the Weber equation, Koike and Takei [4] for the Whittaker equation, and Aoki, Iwaki and Takahashi [] for the Bessel equation. In these works, parametric Stokes phenomena are studied by analyzing the degeneration of Stokes curves and then calculating alien derivatives. In general, a Stokes curve emanates from a turning point and flows into a turning point or a singular point. A Stokes curve is said to be degenerate if it flows into a turning point. Then there are two types of degenerate Stokes curves. A degenerate Stokes curve which emanates from a turning point and flows into another turning point is called of Weber type, and one which flows into the same turning point is called of loop type. In [7] and [4], Stokes curves of Weber type are considered, and in [] one of loop type is considered. Note that, in [8, 9, 5] and [6], the authors treat only the case of Weber type. The Gauss hypergeometric differential equation has three parameters, and then there appears two types of degenerate Stokes curves. In our previous works [6] and [9], we considered degenerate Stokes curves of Weber type, and obtained the parametric Stokes phenomena by using the Borel sums of the Voros coefficients. In the present paper, first we consider the same cases as in our previous works, and obtain the parametric Stokes phenomena in another way namely by using alien derivatives. Next we consider the case in which degenerate Stokes curves of loop type appear. Also in this case, we obtain the parametric Stokes phenomena by using alien derivatives. In Section 2, we introduce the Gauss hypergeometric differential equation with large parameters. Then we review the definition of the Voros coefficients for this equation, and give their Borel transforms. The parametric Stokes phenomena are described in Section 3. First we divide the space of the parameters into several subregions such that the parametric Stokes phenomena occurs at the boundaries. Then we compute the alien derivatives to obtain the parametric Stokes phenomena. The main results are given in Theorems 3.2, 3.5, 3.7, 3.9 and VOROS COEFFICIENTS AND BOREL TRANSFORMS OF THEM We consider the following Schrödinger-type equation: ( d2 dx 2 + η2 Q(x ψ = 0 (2. with a large parameter η > 0. Here we set Q(x = Q 0 (x + η 2 Q (x with and Q 0 (x = (α β2 x 2 + 2(2αβ αγ βγx + γ 2 4x 2 (x 2 Q (x = x2 x + 4x 2 (x 2, where α, β, γ are complex parameters. Equation (2. is obtained from the Gauss hypergeometric differential equation: x( x d2 w + (c (a + b + xdw abw = 0. dx2 dx

3 Alien derivatives of the WKB solutions We put a = 2 + αη, b = 2 + βη, c = + γη and eliminate the first-order term by taking ψ = x 2 (+γη ( x 2 (+(α+β γη w as unknown function. Then we have (2.. In this paper, we call (2. the Gauss hypergeometric differential equation with large parameter η. By definition, WKB solutions of (2. are the following formal solutions: ψ ± = ( exp ± Sodd x x 0 S odd dx, where x 0 is a fixed point and S odd denotes the odd-order part in η of the formal solution S(x = l= η l S l of the Riccati equation ds dx + S2 = η 2 Q(x associated with (2.. (See also [2, 2] for the notation and terminologies. Eq. (2. has regular singular points b 0 = 0, b = and b 2 =. A turning point a of (2. is, by definition, a simple zero of Q 0 (cf. [2, 2]. We define a Stokes curve emanating from a by Im x a Q0 dx = 0. A Stokes curve flows into a singular point or a turning point (cf. [2, 2]. If turning points are connected by a Stokes curve, the Stokes geometry of (2. is said to be degenerate. Let E j (j = 0,, 2 be the sets defined by the following: E 0 = {(α, β, γ C 3 αβγ(α β(α γ(β γ(α + β γ = 0}, E = {(α, β, γ C 3 Reα Reβ Re(γ α Re(γ β = 0}, E 2 = {(α, β, γ C 3 Re(α β Re(α + β γ Reγ = 0}. Stokes graph of (2. is, by definition, a two-color sphere graph that consists of all Stokes curves as edges, {a 0, a } as vertices of the first color and {b 0, b, b 2 } as vertices of the second color (cf. [3] and [2, 3.2].

4 806 Mika Tanda Theorem 2. ([9, Theorem 3.]. We assume that (α, β, γ is not contained in E 0. (i If two distinct turning points a 0 and a are connected by a Stokes curve, then (α, β, γ belongs to E. Conversely, if (α, β, γ is contained in E E 2, the Stokes geometry of (2. has a Stokes curve which connects two distinct turning points a 0 and a. (ii If a Stokes curve forms a closed curve with a single turning point as the base point, then (α, β, γ belongs to E 2. Conversely, if (α, β, γ is contained in E 2 E, the Stokes geometry of (2. has a Stokes curve which forms a closed path with a turning point as the base point. Let C j (j = 0,, 2 be a contour starting from the singular point b j, going around a turning point in a counterclockwise direction and going back to b j. We may assume that the other turning point and singular points are not included in C j. Definition 2.2 ([9]. Let V j (j = 0,, 2 be the formal power series in η defined by the following integrals: V 0 = V 0 (α, β, γ; η := (S odd ηs dx, 2 C 0 V = V (α, β, γ; η := (S odd ηs dx, 2 C V 2 = V 2 (α, β, γ; η := 2 C 2 (S odd ηs dx. Here the branch of S = Q 0 (x on C j (j = 0,, 2 is taken as follows: We take a curve in C {0, } connecting the turning points as a branch cut and choose the branch of S (x so that at the starting point b j, we have Q0 γ 2x at x = 0, (2.2 Q0 α + β γ at x =, (2.3 2(x Q0 β α 2x at x =. (2.4 We call V j the Voros coefficients of (2. with respect to b j (b j = 0,, 2. Let ψ ± and ψ (j ± be the WKB solutions normalized at a turning point a (a = a 0 or a = a and those normalized at the singular point b j (cf. [7]: ψ ± = x exp ± S odd dx, Sodd ψ (j ± = exp ± Sodd a x b j (S odd ηs dx ± η x a S dx,

5 Alien derivatives of the WKB solutions respectively. For j = 0, and 2, V j (α, β, γ; η describe the discrepancy between WKB solutions ψ ± and ψ (j ±, that is, we have ψ ± as ψ ± = exp( V j ψ (j ±. (2.5 Since S odd dx and ηs dx have a simple pole at the singular points b j and residues at b j coincide (see [2] for the computation of residues of S odd, these integrals are well-defined for every homotopy class of the path of integration. Explicit forms of V j are given in [5,6,9] (The choices of the branch of S on C j are slightly different in those references.: Theorem 2.3 ([5, Theorem 2.]. The Voros coefficients V j have the following forms: V 0 (α, β, γ; η = 2 V (α, β, γ; η = 2 V 2 (α, β, γ; η = 2 n=2 n=2 n=2 B n η n { ( ( 2 n n(n α n + B n η n { ( ( 2 n n(n β n + (γ α n + (γ β n α n + β n (γ β n B n η n { ( ( 2 n n(n α n Here B n are the Bernoulli numbers defined by te t e t = n=0 B n n! tn. + + (γ β n + 2 γ n (γ α n 2 (α + β γ n β n (γ α n 2 (β α n We take the Borel transforms V j,b (α, β, γ; y (j = 0,, 2 of V j, then we obtain the following proposition (see [2, 2.] for the definition of the Borel transform and [9] for the computation of V j,b : Proposition 2.4 ([9]. The Borel transforms V j,b (α, β, γ; y of the Voros coefficients V j have the following forms: V 0,B (α, β, γ; y = 4 {g (α; y + g (β; y + g (γ α; y + g (γ β; y} + g 0 (γ; y, V,B (α, β, γ; y = 4 { g (α; y g (β; y + g (γ α; y + g (γ β; y} + g 0 (α + β γ; y, }, }, }.

6 808 Mika Tanda V 2,B (α, β, γ; y = 4 { g (α; y + g (β; y + g (γ α; y g (γ β; y} Here g 0 (β α; y. g 0 (t; y = ( y exp y t + 2 t, y g (t; y = exp y 2t + exp y 2t + 2t y. For a fixed t 0, g 0 (t; y and g (t; y are holomorphic at y = 0, and they have simple poles at y = 2tmπi (m Z {0} as functions of y. The residues of g 0 (t; y and g (t; y (j = 0, are given as follows: Res g 0(t; y = y=2tmπi 2mπi, Res g (t; y = ( m y=2tmπi mπi. 3. PARAMETRIC STOKES PHENOMENA AND THE ALIEN DERIVATIVE OF THE WKB SOLUTION Let ω h (h =, 2, 3, 4 be the sets of the parameter (α, β, γ defined by ω = {(α, β, γ C 3 0 < Reα < Reγ < Reβ}, ω 2 = {(α, β, γ C 3 0 < Reα < Reβ < Reγ < Reα + Reβ}, ω 3 = {(α, β, γ C 3 0 < Reγ < Reα < Reβ}, ω 4 = {(α, β, γ C 3 0 < Reγ < Reα + Reβ < Reβ}, and let ι j (j = 0,, 2 be involutions in the space C 3 of parameters (α, β, γ defined by ι 0 : (α, β, γ ( α, β, γ, ι : (α, β, γ (γ β, γ α, γ, ι 2 : (α, β, γ (β, α, γ. Moreover, an open subset Π h (h =, 2, 3, 4 in C 3 is defined by Π h = r(ω h. r G Here G is the group generated by ι j (j = 0,, 2. The union of Π h covers most of C 3 : 4 Π h = C 3 {(α, β, γ Re αre βreγ h= Re(γ αre(γ βre(α βre(α + β γ = 0}.

7 Alien derivatives of the WKB solutions We note that the topological configuration of the Stokes graph is characterized by its order sequence (n 0, n, n 2. Here n j is the number of Stokes curves that flow into b j (j = 0,, 2. Theorem 3. ([4, Theorem 3.2]. Let ˆn = (n 0, n, n 2 denote the order sequences of the Stokes graph with parameter (α, β, γ. ( If (α, β, γ Π, then ˆn = (2, 2, 2. (2 If (α, β, γ Π 2, then ˆn = (4,,. (3 If (α, β, γ Π 3, then ˆn = (, 4,. (4 If (α, β, γ Π 4, then ˆn = (,, 4. We introduce the following notations: ι 3 =ι ι 2 : (α, β, γ (γ α, γ β, γ, ι 4 =ι 0 ι 2 : (α, β, γ ( β, α, γ, ι 5 =ι 0 ι : (α, β, γ (β γ, α γ, γ, ι 6 =ι 0 ι ι 2 : (α, β, γ (α γ, β γ, γ. We denote ι m (ω h by ω hm (m = 0,,..., 6; h =,..., ANALYSIS OF STOKES CURVES OF WEBER TYPE We derive alien derivatives and parametric Stokes phenomena on the WKB solutions for the hypergeometric differential equation with a large parameter for the Weber type. Several formulas describing the parametric Stokes phenomena are proved in [6] by using Borel sums of the WKB solutions. The same formulas can be derived from the alien derivatives, and hence, in this section, we obtain another proof of the parametric Stokes phenomena. The notion of alien derivatives was introduced by J. Ecall [0]. We refer the reader to [5, 2], and [6, p. 78, 28], for the definition of the alien derivative. We assume that arguments of α, β, α γ, β γ, γ, α + β γ and α β are mutually distinct Analysis on the boundary between ω and ω 2 We discuss the case where (α, β, γ is contained the boundary between ω and ω 2, that is, Re(γ β = 0. We assume that Im(β γ is negative. In this case, we consider the case where (α, β, γ = (0.5, ˆɛi, E. It follows from (i of Theorem 2. that we can take (α, β, γ = (0.5, ˆɛi, E without loss of generality. We consider the WKB solutions ψ ±,k = ( exp ± Sodd x a k S odd dx in a neighborhood of a k (k = 0,. Here we take the straight line connecting a k to x as the path of integration. We expand the WKB solutions ψ ±,k as ψ ±,k = exp (ηy ±,k (x ψ ±,n (xη n 2, n=0

8 80 Mika Tanda where y ±,k (x = ± x Q dx. Let us show examples of Stokes curves of those two cases in Figures 3. and 3.3 and a degenerate case, namely, one case of the Weber type in Figure 3.2 (cf. [6] and [9]. Here ɛ > 0 and ˆɛ > 0 are sufficiently small. Here open circles and closed circles are the singular points and turning points, respectively. We place the cut as shown by the wavy lines in Figure 3.h (h =, 3. In this case, we use the branch of S = Q 0 at 0, i.e. we take the branch of S as (2.2. Hence, ψ is dominant (cf. [6] for the computation. Let R I, R II, R I ω h and R II ω h (h =, 2 denote regions surrounded by the Stokes curves as shown in Figures 3., 3.2 and 3.3. a k R I ω R I a R a I ω a 0 a 0 a 0 a 0 R II ω R II R II ω 2 Fig. 3.. (α, β, γ Fig (0.5, ˆɛi, Fig (0.5, ɛ ˆɛi, = (0.5, + ɛ ˆɛi, in ω in ω 2 The WKB solutions ψ ±,k (k = 0, (resp. ψ (0 ± are Borel summable in R I (resp. R II. (See [2] and [3] for the notation and terminologies. Let us denote the Borel transform of ψ (0 + by ψ (0,I the Borel transform ψ (0,I { (resp. by ψ(0,ii (resp. ψ(0,ii y C; y =. It follows from a result given in [3] that is free from singularities on the half line x a k Q0 dx + ρ; ρ > 0 }. (3. Moreover, the Borel transform V 0,B (α, β, γ; y of V 0 is holomorphic at y = 0 and it has simple poles at y = 2m(γ βπi for every non-zero integer m Z {0}. Therefore, the Borel transform of ψ +,k has singularities at y = y ±,k (x + 2m(β γπi when Re(γ β = 0 (m Z. The following description of the alien derivative of the WKB solutions in the formal model is due to [] (for the definition of the alien derivative of the WKB solutions in the convolution model and [7]: ψ +,k = B log ( L L + Bψ+,k. (3.2

9 Alien derivatives of the WKB solutions... 8 Here B and L + (resp. L is the Borel transformation and the Laplace transformation along a path which avoids the singular points from above (resp. from below, respectively. We decompose (3.2 as follows: with ψ +,k = m= y= y+,k +2m(β γπiψ +,k = B [ y= y+,k +2m(β γπiψ +,k (t (m + t (m ɛ n=± ] p +!p! t (m ɛ m! m t ( ɛ Bψ +,k, where t (n + (resp. t (n denotes the operator of analytic continuation which does not pass the n-th singular point y = y +,k + 2n(β γπi from above (resp. below and p + (resp. p is the number of times and ɛ n = + (resp. ɛ n = for n m. Let us consider the alien derivatives ψ +,0 and ψ +,. Since the Borel transform V 0,B of V 0 is a single-valued analytic function with the simple pole at y = 2m(β γπi, we have y=2m(β γπi ( V 0 [ ] = B p +!p! sing m! y= y+,k +2m(β γπit (m ɛ m... t ( ɛ ( V 0,B ɛ k =± ] = B [2πi Res ( V 0,B y=2m(β γπi = ( m+ 2m (cf. [7]. Hence, the chain rule in the alien calculus leads to y=2m(β γπi (exp( V 0 = ( m+ 2m exp ( V 0. (3.3 Since the Borel transform ψ (0,I (resp. ψ(0,ii is free from singularities on (3., we obtain ( exp( y +,k (xηψ (0 + = 0, (3.4 where k = 0 (resp.. Combining (2.5, (3.3 and (3.4, we have y=2m(β γπi (exp( y +,0 (xηψ +,0 ( = y=2m(β γπi exp( y +,0 (xη exp( V 0 (α, β, γ; ηψ (0 = ( m+ 2m ( exp( y +,0 (xη exp( V 0 (α, β, γ; ηψ (0 + = ( m+ 2m (exp( y +,0(xηψ +,0 +

10 82 Mika Tanda and y=2m(β γπi (exp( y +, (xηψ +, = ( m+ 2m (exp( y +,(xηψ +,. Similarly, we can compute the alien derivatives y= y,k (x+2m(β γπiψ,k ψ,k (x, η. Hence, we obtain the following theorem. Theorem 3.2. Let ψ ±,k (x, η denote the WKB solutions of (2. normalized at the turning point a k (k = 0,. Let us consider the case where (α, β, γ is contained within the boundary between ω and ω 2. The Borel transforms of ψ ±,k (x, η have fixed singular points at y = y ±,k (x + 2m(β γπi, m Z. Furthermore, the alien derivatives y= y±,k (x+2m(β γπiψ ±,k of ψ ±,k (x, η (k = 0, satisfies the following relations: ( y= y±,0(x+2m(β γπiψ ±,0 B (x, y = ± ( m+ 2m ψ ±,0,B(x, y 2m(β γπi (3.5 for x in R I and ( y= y±,(x+2m(β γπiψ +, B (x, y = ± ( m+ 2m ψ +,,B(x, y 2m(β γπi (3.6 for x in R II. Next we describe the actions of Stokes automorphisms on the WKB solutions. The WKB solutions ψ ±,0 (resp. ψ ±, are Borel summable in R I ω and R I ω 2 (resp. R II ω and R II ω 2 (cf. [2] and [3] for the notation and terminologies. Let us denote the Borel sums of ψ +,0 (resp. ψ +, by ψω I and ψω I 2 (resp. by ψω II and ψω II 2. In this case, the Stokes automorphism on the WKB solutions is defined by [ Sψ ±,k = exp m= y= y±,k (x+2m(β γπi ] ψ ±,k. Here S denote the Stokes automorphism associated with the change to ω from ω 2 (cf. [8, 9, 4]. Using Theorem 3.2 and the discussion given in [4, 7], we have the following theorem: Theorem 3.3. When (α, β, γ moves ω to ω 2 and Im(β γ is negative, the Stokes automorphisms Sψ ±,k on the WKB solutions ψ ±,k (k = 0, : Sψ ±,0 = ( + exp(2πi(γ βη 2 ψ±,0 (3.7 of for x in R I ω and R I ω 2, and for x in R II ω and R II ω 2. Sψ ±, = ( + exp(2πi(γ βη 2 ψ±, (3.8

11 Alien derivatives of the WKB solutions Proof. We obtain ψ +,k = = m= y= y+,k (x+2m(β γπiψ +,k B [ y= y+,k (x+2m(β γπiψ +,k,b ] m= ( m+ = 2m B [ψ +,k,b (x, y 2m(β γπi] m= = ( m+ exp ( 2m(β γπiψ +,k 2 m m= = 2 log( + exp ( 2(β γπiψ +,k, where B denotes the Borel transform. Then we have Sψ +,k = ( + exp ( 2(β γπi 2 ψ+,k. Similarly, we can compute Sψ,k. Hence, we have Theorem 3.3. By taking the Borel sums of (3.7 and (3.8, we get the parametric Stokes phenomena of the WKB solutions: Theorem 3.4. (cf. [6, Theorem 4.4], [9, Thorem 6.3] Between the Borel sums ψω I and ψω I 2 of the WKB solution ψ +, the following relation holds: Between the Borel sums ψω II holds: ψ I ω = ( + exp(2πi(γ βη 2 ψ I ω2. and ψω II 2 of the WKB solution ψ +, the following relation ψω II = ( + exp(2πi(γ βη 2 ψ II ω 2. Hence, we can give the another proof of the parametric Stokes phenomena from the viewpoint of alien calculus Analysis on the boundary between ω and ω 3 We consider the case where (α, β, γ is contained within the boundary ω and ω 3, i.e., Re(γ α = 0. Similarly, we can describe parametric Stokes phenomena in terms of alien derivatives ψ ±,0 and ψ ±,. We assume that Im(α γ is positive. In this case, we consider the case where (α, β, γ = ( + ˆɛi, 2, E. It follows from (i of Theorem 2. that we can take (α, β, γ = ( + ˆɛi, 2, without loss of generality. Let us show an example of Stokes curves of those two cases in Figures 3.4 and 3.6 and a degenerate case, that is, this case is one of the Weber type in Figure 3.5. We place the cut as shown by the wavy lines in Figure 3.h (h = 4, 6. In this case, we use the branch of S = Q 0 at, i.e. we take the branch of S as (2.3. Then ψ is dominant (cf. [6] for the computation. Let us denote regions surrounded by the

12 84 Mika Tanda Stokes curves as shown in Figures 3.4, 3.5 and 3.6 by R I, R III, R I ω h, R III ω h (h =, 3. The WKB solutions ψ ±,0 (resp. ψ ±, and ψ ( ± are Borel summable in R I (resp. R III (cf. [3]. a 0 R I ω R I R I ω 0 a a0 0 a a a R III ω R III R III ω 3 Fig (α, β, γ Fig ( + ˆɛi, 2, Fig (+ɛ+ˆɛi, 2, in ω 3 = ( ɛ + ˆɛi, 2, in ω We denote the Borel transform of ψ ( + by ψ (,I and ψ(,iii. The Borel transform ψ (,I (resp. ψ(,iii is free from singularities on (3.. Moreover, the Borel transform V,B (α, β, γ; y is holomorphic at y = 0 and it has simple poles at y = 2m(γ απi for every non-zero integer m Z {0}. Therefore, the Borel transform ψ (,I (resp. ψ (,III has singularities at y = y +,k(x + 2m(γ απi when Re(γ α = 0 (m Z. The following description of the alien derivative is due to [7]: We consider the alien derivatives ψ +,0 and ψ +,. Since the Borel transform V,B of V is a single-valued analytic function with the simple pole at y = 2m(γ απi, we obtain y=2m(γ απi ( V = ( m+ 2m. Hence, the chain rule in alien calculus leads to y=2m(γ απi (exp( V = ( m+ 2m exp( V. (3.9 Since the Borel transform ψ (,I (resp. ψ(,iii ω 3,B is free from singularities on (3., we obtain ( exp( y +,k (xηψ ( + = 0 (3.0 (k = 0 (resp.. Combining (2.5, (3.9 and (3.0, we have and y=2m(γ απi (exp( y +,0 (xηψ +,0 = ( m+ 2m (exp( y +,0(xηψ +,0 y=2m(γ απi (exp( y +, (xηψ +, = ( m+ 2m (exp( y +,(xηψ +,.

13 Alien derivatives of the WKB solutions Similarly, we can compute the alien derivatives y= y,k (x+2m(γ απiψ,k ψ,k (x, η. Then we have the following theorem: Theorem 3.5. Let ψ ±,k (x, η denote the WKB solutions of (2. normalized at the turning point a k (k = 0,. Let us consider the case where (α, β, γ is contained within the boundary between ω and ω 3. The Borel transforms of ψ ±,k (x, η have fixed singular points at y = y ±,k (x + 2m(γ απi, m Z. Furthermore, the alien derivatives y= y±,k (x+2m(γ απiψ ±,k of ψ ±,k (x, η (k = 0, satisfy the following relations: ( y= y±,0(x+2m(γ απiψ ±,0 B (x, y = ± ( m+ 2m ψ ±,0,B(x, y 2m(γ απi (3. for x in R I and ( y= y±,(x+2m(γ απiψ +, B (x, y = ± ( m+ 2m ψ ±,,B(x, y 2m(γ απi (3.2 for x in R III. Next we study the actions of Stokes automorphisms on the WKB solutions. The WKB solutions ψ ±,0 (resp. ψ ±, are Borel summable in R I ω and R I ω 3 (resp. R III ω and R III ω 3 (cf. [2] and [3] for the notation and terminologies. In this case, the Stokes automorphism on the WKB solutions is defined by [ Sψ ±,k = exp m= y= y±,k (x+2m(γ απi ] ψ ±,k, where S denote the Stokes automorphism associated with the change to ω from ω 3 (cf. [8,9,4]. We use Theorem 3.5 and the discussion given in [4,7]. Then we have the following theorem: Theorem 3.6. When (α, β, γ moves ω to ω 3 and Im(α γ is positive, the Stokes automorphisms Sψ ±,k on the WKB solutions ψ ±,k (k = 0, : Sψ ±,0 = ( + exp(2πi(α γη 2 ψ±,0 (3.3 of for x in R I ω and R I ω 3, and for x in R III ω and R III ω 3. Sψ ±, = ( + exp(2πi(α γη 2 ψ±, (3.4 Similarly, by taking the Borel sum of (3.3 and (3.4, we can get the parametric Stokes phenomena of the WKB solutions. Then we can give another proof of the parametric Stokes phenomena from view point of alien calculus.

14 86 Mika Tanda Analysis on the boundary between ω and ω 4 We consider the case where (α, β, γ belong to the boundary between ω and ω 4, i.e., Reα = 0. We assume that Imα is negative. In this case, we discuss the case where (α, β, γ = ( ˆɛi, 2, E. It follows from (i of Theorem 2. that we can take (α, β, γ = ( ˆɛi, 2, without loss of generality. Let us show an example of Stokes curves of those two cases in Figures 3.7 and 3.9 and a degenerate case, namely, one of the Weber type in Figure 3.8. We place the cut as shown by the wavy lines in Figure 3.h (h = 7, 9. We use the branch of S = Q 0 at, i.e. we take the branch of S as (2.4. Then ψ + is dominant (cf. [6] for the computation. We denote regions surrounded by the Stokes curves as shown in Figures 3.7, 3.8 and 3.9 by R II, R III, R II ω h, R III ω h (h =, 4. The WKB solutions ψ ±,0 (resp. ψ ±, and ψ (2 ± are Borel summable in R II (resp. R III (cf. [3]. R I ω R I ω 3 R III ω R III ω 3 a 0 0 a a 0 a 0 a 0 a 0 Fig (α, β, γ Fig ( ˆɛi, 2, Fig ( ɛ ˆɛi, 2, in ω 4 = (ɛ ˆɛi, 2, in ω We denote the Borel transform of ψ (2 + by ψ (2,II and ψ(2,iii. The Borel transform ψ (2,II (resp. ψ(2,iii is free from singularities on (3.. Moreover, the Borel transform V 2,B (α, β, γ; y is holomorphic at y = 0 and it has simple poles at y = 2mαπi for every non-zero integer m Z {0}. Therefore, the Borel transform ψ (2,II (resp. ψ(2,iii of ψ +,k has singularities at y = y +,k (x + 2mαπi when Reα = 0 (m Z. The following description of the alien derivative is due to [7]: We consider the alien derivatives ψ +,0 and ψ +, in the case where (α, β, γ moves from ω to ω 4. Since the Borel transform V 2,B of V 2 is a single-valued analytic function with simple pole at y = 2mαπi, we obtain y=2mαπi ( V 2 = ( m 2m

15 Alien derivatives of the WKB solutions (cf. [7]. Hence, the chain rule in alien calculus leads to Since the Borel transforms ψ (2,II ω,b singularities on (3., we have ( y=2mαπi (exp( V 2 = ( m 2m exp( V 2. (3.5 and ψ(2,iii ω,b exp( y +,k (xηψ (2 + (resp. ψ(2,ii ω 4,B (k = 0 (resp.. Combining (2.5, (3.5 and (3.6, we have and and ψ(2,iii ω 4,B are free from = 0 (3.6 y=2mαπi (exp( y +,0 (xηψ +,0 = ( m 2m (exp( y +,0(xηψ +,0 y=2mαπi (exp( y +, (xηψ ±, = ( m 2m (exp( y +,(xηψ +,. In a similar way, we can compute the alien derivative y= y,k (x+2mαπiψ,k ψ,k (x, η. Then we have the following theorem: Theorem 3.7. Let ψ ±,k (x, η denote the WKB solutions of (2. normalized at the turning point a k (k = 0,. Let us consider the case where (α, β, γ is contained within the boundary between ω and ω 4. The Borel transforms of ψ ±,k (x, η have the fixed singular points at y = y ±,k (x + 2mαπi, m Z. Furthermore, the alien derivatives y= y±,k (x+2mαπiψ ±,k of ψ ±,k (x, η (k = 0, satisfy the following relations: ( y= y±,0(x+2mαπiψ ±,0 B (x, y = ± ( m 2m ψ ±,0,B(x, y 2mαπi (3.7 for x in R II and for x in R III. ( y= y±,(x+2mαπiψ ±, B (x, y = ± ( m 2m ψ ±,,B(x, y 2mαπi (3.8 Next we discuss the actions of Stokes automorphisms on the WKB solutions. The WKB solutions ψ ±,0 (resp. ψ ±, are Borel summable in R II ω and R II ω 4 (resp. R III ω and R III ω 4 (cf. [2] and [3] for the notation and terminologies. In this case, the Stokes automorphism on the WKB solutions is defined by [ Sψ ±,k = exp m= y= y±,k (x+2m(γ απi ] ψ +,k, where S denote the Stokes automorphism associated with the change to ω from ω 4 (cf. [8, 9, 4]. Using Theorem 3.5 and the discussion given in [4, 7], we have the following theorem. of

16 88 Mika Tanda Theorem 3.8. When (α, β, γ moves ω to ω 4 and Imα is negative, the Stokes automorphisms Sψ ±,k on the WKB solutions ψ ±,k (k = 0, : for x in R II ω for x in R III ω and R II ω 4, and and R III ω 4. Sψ ±,0 = ( + exp( 2πiαη 2 ψ±,0 (3.9 Sψ ±, = ( + exp( 2πiαη 2 ψ±, (3.20 Similarly, by taking the Borel sum of (3.9 and (3.20, we can get the parametric Stokes phenomena of the WKB solutions. Then we can give another proof of the parametric Stokes phenomena from the view point of alien calculus. Next we discuss alien derivatives when (α, β, γ belongs to E and Stokes automorphisms between ω m and ω hm (h = 2, 3, 4; m = 0,..., 6. Since the potential Q is invariant under involution ι m, the Stokes geometry for (α, β, γ ι hm is the same as that for (α, β, γ ι h. Applying ι m to the relations (3.5, (3.6, (3., (3.2, (3.7 and (3.8, we have the formulas of the alien derivatives. Similarly, we give the Stokes automorphisms between ω m and ω hm ANALYSIS OF STOKES CURVES OF LOOP TYPE We give the computations of the alien derivatives for a loop-type. In [] and [], they will give a concrete form of the alien derivative of the WKB solutions of the general linear second-order differential equation with a large parameter. In this paper, we consider the alien derivatives of the WKB solutions for the Gauss hypergeometrc differential equations of the loop-type. We consider the case where (α, β, γis contained within the boundary between ω 4 and ω 4, i.e., Re(α + β γ = 0. We assume that Im(α + β γ is positive. In this case, we consider the case where (α, β, γ = ( 0.5,.5 + ˆɛi, E 2. If Re(α + β γ = 0, the Stokes geometry has a Stokes which is a loop around. Hence, we can take (α, β, γ = ( 0.5,.5 + ˆɛi, without loss of generality. Let us show an example of Stokes curves of a derivative case, namely, the loop-type in Figure a 0 a Fig (α, β, γ = ( 0.5,.5 + ˆɛi,

17 Alien derivatives of the WKB solutions In this case, a Stokes curve forms a closed curve with in Figure 3.0. We place the cut as shown by the wavy lines in Figure 3.0. We use the branch of S = Q 0 at, i.e. we take the branch of S as (2.3. Hence ψ is dominant (cf. [6] for the computation. We denote a region surrounded by the Stokes curve which forms a closed curve with 0 by R IV. The WKB solutions ψ ±, and ψ ( ± are Borel summable in R IV (cf. [,,3]. We denote the Borel transforms of ψ +, and ψ ( + by ψ IV and ψ (,IV, respectively. The Borel transform ψ (,IV is free from singularities on (3. (cf. [3] Moreover, the Borel transform V,B (α, β, γ; y is holomorphic at y = 0 and it has simple poles at y = 2m(α + β γπi for every non-zero integer m Z {0}. Therefore, the Borel transforms of ψ +,k has singularities at y = y +,k (x + 2m(α + β γπi when Re(α + β γ = 0 (m Z. The following description of the alien derivative of the loop-type is due to []. Since the Borel transform V,B of V is a single-valued analytic function with the simple pole at y = 2m(α + β γπi, we have y=2m(α+β γπi ( V = m (cf. [7]. Hence, the chain rule in the alien calculus leads to y=2m(α+β γπi (exp( V = m exp( V. (3.2 For the Borel transform ψ (,IV is free from singularities on (3., we have ( exp( y +, (xηψ ( + = 0. (3.22 Combining (2.5, (3.2 and (3.22, we have y=2m(α+β γπi (exp( y +,0 (xηψ +,0 = m (exp( y +,0(xηψ +,0. (3.23 Similarly, we can compute the alien derivative y= y,(x+2m(α+β γπiψ,k ψ,k (x, η. Then we have the following theorem: Theorem 3.9. Let ψ ±, (x, η denote the WKB solutions of (2. normalized at the turning point a. Let us consider the case where (α, β, γ is contained within the boundary between ω 4 and ω 4. The Borel transforms of ψ ±, (x, η have fixed singular points at y = y ±, (x + 2m(α + β γπi, m Z. Furthermore, the alien derivatives y= y±,(x+2m(α+β γπiψ ±, of ψ ±, (x, η satisfy the following relations: ( y= y±,(x+2m(α+β γπiψ ±, B (x, y = ± m ψ ±,,B(x, y 2m(α + β γπi for x in R IV. of

18 820 Mika Tanda Using the alien derivatives on the WKB solutions, we compute the action of a Stokes automorphisms on the WKB solutions ψ ± of the loop-type. We discuss the case where (α, β, γ moves ω 4 to ω 4. We show examples which are enlarged near the singular point of Stokes curves of those two cases in Figures 3. and 3.2. Let us denote regions surrounded by the Stokes curves as shown in Figures 3. and 3.2 by R IV ω 4 and R IV ω 4, respectively. R IV ω 4 R IV ω 4 a a Fig. 3.. (α, β, γ=( 0.4+ɛ,.5+ˆɛi, ω 4 Fig ( 0.6 ɛ,.5 + ˆɛi, ω 4 By the definition of the alien derivatives ψ ±,, we have [ Sψ ±, = exp m= y= y±,(x+2m(α+β γπi ] ψ ±,k. Here S denote Stokes automorphism associated with the change to ω 4 from ω 4 (cf. [8, 9, 4]. Theorem 3.0. When (α, β, γ moves ω 4 to ω 4 and Im(α + β γ is positive, the Stokes automorphisms Sψ ±, of the WKB solutions ψ ±, : Sψ ±, = ( + exp ( 2(α + β γπiψ ±,. (3.24 Proof. By the definition of the alien derivatives and the Stokes automorphism, we have ψ +, = = = m= y= y+,(x+2m(α+β γπiψ +, B [ y= y+,(x+2m(α+β γπiψ +,,B ] m= m= m B [ψ +,,B (x, y 2m(α + β γπi]

19 Alien derivatives of the WKB solutions = m= m exp ( 2m(α + β γπiψ +, = log( exp ( 2(α + β γπiψ +,. Similarly, we can compute ψ,. Hence, we have (3.24. Finally, we discuss alien derivatives and Stokes automorphisms when (α, βγ belongs to E 2 and moves from ω h to ω hm (h =, 2, 3, 4; m = 0,,..., 6, respectively. If (α, βγ is contained Reγ = 0 (resp. Re(α β = 0, a Stokes geometry has a loop around 0 (resp.. We can compute them in a similar manner as the computation of Theorem 3.9 and 3.0 and we have the following theorems: Theorem 3.. Let ψ ±,0 (x, η denote the WKB solutions of (2. normalized at the turning point a k (k = 0,. Let us consider the case where (α, β, γ is contained within the boundary between ω m and ω hm (h =, 2, 3, 4; m = 0,,..., 6 and Reγ = 0. The Borel transforms of ψ ±,0 (x, η have the fixed singular points at y = y ±,k (x + 2mγπi, m Z. Furthermore, the alien derivatives y= y±,k (x+2mγπiψ ±,k of ψ ±,k (x, η satisfy the following relations: ( y= y±,k (x+2mγπiψ ±,k B (x, y = ± m ψ ±,k,b(x, y 2mγπi for x inside of the loop. Let us consider the case where (α, β, γ is contained within the boundary between ω m and ω hm (h =, 2, 3, 4; m = 0,,..., 6 and Re(β α = 0. The Borel transforms of ψ ±,k (x, η have fixed singular points at y = y ±,k (x + 2m(β απi, m Z. Furthermore, the alien derivatives y= y±,k (x+2m(β απiψ ±,k of ψ ±,k (x, η satisfy the following relations: ( y= y±,k (x+2m(β απiψ ±,k B (x, y = m ψ ±,k,b(x, y 2mγπi for x inside of the loop. Theorem 3.2. When (α, β, γ moves ω h to ω hm (h =, 2, 3, 4; m = 0,,..., 6 and Reγ = 0, the Stokes automorphisms Sψ ±,k of the WKB solutions ψ ±,k : Sψ ±,k = ( + exp ( 2γπiψ ±,k. When (α, β, γ moves ω h to ω hm (h =, 2, 3, 4; m = 0,,..., 6 and Re(β α = 0, the Stokes automorphisms Sψ ±,k of the WKB solutions ψ ±,k : Sψ ±,k = ( + exp ( 2(β απiψ ±,k. Acknowledgments The author would like to thank Professor Takashi Aoki for instruction and Professor Yoshishige Haraoka, many useful advices and encouragement. She also expresses her heartiest thanks to Kohei Iwaki for valuable comments and helpful advices.

20 822 Mika Tanda REFERENCES [] T. Aoki, K. Iwaki, T. Takahashi, Exact WKB analysis of Schrödinger equations with a Stokes curve of loop type, in preparation. [2] T. Aoki, T. Kawai, Y. Takei, The Bender-Wu Analysis and the Voros Theory, [in:] Special Functions, Springer, 99, 29. [3] T. Aoki, T. Iizuka, Classification of Stokes graphs of second order Fuchsian differential equations of genus two, Publ. RIMS, Kyoto Univ. 43 (2007, [4] T. Aoki, M. Tanda, Characterization of Stokes graphs and Voros coefficients of hypergeometric differential equations with a large parameter, RIMS Kôkyûroku Bessatsu B40 (203, [5] T. Aoki, M. Tanda, Borel sums of Voros coefficients of hypergeometric differential equations with a large parameter, RIMS Kôkyûroku., Kyoto Uni. 86 (203, [6] T. Aoki, M. Tanda, Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, submitted. [7] B. Candelpergher, M.A. Coppo, E. Delabaere, La sommation de Ramanujan, L Enseignement Mathématique 43 (997, [8] E. Delabaere, H. Dillinger, F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier, Grenoble, 43 (993, [9] E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. Henri Poincaré 7 (999, 94. [0] J. Écalle, Les fonction résurgentes. Tome I., Publications Mathématiques d Orsay 8, Université de Paris-Sud, Départment de Mathématique, Orsay (98, 247 pp. [] K. Iwaki, T. Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (204, [2] T. Kawai, Y. Takei, Exact WKB analysis and cluster algebras Algebraic Analysis of Singular Perturbation Theory, Translation of Mathematical Monographs, vol. 227, AMS, [3] T. Koike, R. Schäfke, On the Borel summability of WKB solutions of Schrödinger equations with polynomial potential and its application, to appear in RIMS Kôkyûroku Bessatsu. [4] T. Koike, Y. Takei, On the Voros coefficient for the Whittaker equation with a large parameter Some progress around Sato s conjecture in exact WKB analysis, Publ. RIMS, Kyoto Univ. 47 (20, [5] D. Sauzin, Resurgent functions and splitting problems, RIMS Kôkyûroku 493 (2006, [6] D. Sauzin, Introduction to -summability and resurgence, arxiv: (HAL , 204. [7] Y. Takei, Sato s conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points, RIMS Kôkyûroku Bessatsu B0 (2008,

21 Alien derivatives of the WKB solutions [8] M. Tanda, Exact WKB analysis of hypergeometric differential equations, to appear in RIMS Kôkyûroku Bessatsu. [9] M. Tanda, Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, Doctoral Thesis, Kinki University, 204. Mika Tanda tanda@math.kindai.ac.jp Interdisciplinary Graduate School of Science and Engineering Kinki University Higashi-Osaka Osaka , Japan Received: March 3, 204. Revised: February 3, 205. Accepted: February 5, 205.