Some concrete shapes of degenerate Stokes curves of hypergeometric differential equations with a large parameter

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超幾何微分方程式の退化ストークス曲線の具体形

青木貴史*反田美香**

Some concrete shapes of degenerate Stokes curves of hypergeometric differential equations with a large parameter

Takashi AOKI* and Mika TANDA**

Although Stokes curves of differential equations with a large parameter are transcendental objects in general, in some degenerate cases they become real algebraic or semialgebraic sets and the defining equations are explicitly given for hypergeometric differential equations with a large parameter. Classically well-known curves, such as lemniscate and limacon appear for special values of parameters. Some explicit shapes play a role in a topological classification of Stokes curves of hypergeometric differential equations.

Key words: Hypergeometric differential equation, Stokes curves

1. Introduction

We consider the classical hypergeometric differential equation:

2 w

(1) x(1 – x)d

x2—d+ {c – (a + b + 1)x}—dw dx – abw = 0, where a, b and c are complex parameters. We introduce a large parameter q by setting a = 1 /2+ qa, b = 1/ 2+ 77$ and c = 1 + qy with complex parameters a, fi and y. Since the hypergeometric differential equation contains essentially three parameters, our new parameters are redundant. We regard 77 being real positive and large. We eliminate the first-order term in (1) by taking

(2) = x0.-Fw)(1 – x)(1.-Fixa+,g--mw as unknown functions. Then (1) becomes

We assume a # /3. Then Qo has two zeros ao, al in the complex plane. We call them the turning points of (3). We also assume that ak # 0,1 for k = 0,1, namely, a +fl– y 0, y * 0. If ao # al, the turning points are said to be simple. If ao = al, we say that the turning point is double.

A Stokes curve of (3) emanating from a turning point T (r = ao or al) is an integral curve of the direction field

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AtQc dx = 0

which start on r ([2], [5]). The turning points and Stokes curves of (1) are, by definition, those of (3).

If the turning point r is double, (7) can be integrated easily and we have the following form of defining equation of the Stokes curves:

Hence in some special cases, this defines a semialgeb set or an algebraic set.

raic

2. Examples of algebraic Stokes curves 2.1. Let us consider the case where a = i/2, = y = –i/2.

Then our equation becomes

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*理学科数学コースDepartment of Mathematics

**総合理工学研究科理学専攻Interdisciplinary Graduate School of Science and Engineering, Department of Mathematical Sciences

We have ao = al = 1 /2 and hence 1 /2 is a double turning point of this equation. The Stokes curves of (8) are defined by

(9;

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超幾何微分方程式の退化ストークス曲線の具体形

The left-hand side turns out to be

Im (--i4log 4x(1x)).

Hence (9) reduces to

(10) 14x(1 — x)I = 1, or, equivalently,

(11) 2(u2 + v2)2 _ (u2 _ v2) = 0.

Here we set x = u+1/2+ iv (u, v E IR). This is a lemniscate (see Fig. 1).

3. Examples of semialgebraic Stokes curves 3.1 When a = 1, = y = —1, our equation turns out to be

d2 2 (X - (16)

dx (x(x — 1))2 Qi) = 0

which has a double turning point at a0 = al = 1/2. The Stokes curves are given by

x —

(17) Imf 2x(x — 1)dx = O.

Hence

2.2. If a = i, fi = y = 2i, our equation has the form

d2 712 (x 2)2 Qi) = 0.

(12) dx2 (x(x — 1))2

and we have a() = al = 2. The Stokes curves are defined by

(13) Imf 2i2x — 2x(x — 1)dx = O.

This implies

R 4(x — 1)

e log =0

x2

that is,

(14) 4(x

_x2

— 1)1

^{= 1.}

If we set x = u + iv (u, v E IR), we have (15) 16((u — 1)2 + v2) = (u2 + v2)2.

This is a limacon (Fig. 2).

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or equivalently (19)

Setting x = u

(20) arg

If v = O. we lu

Im log 4x(1 — x) = 0,

Setting x = u + iv(u, v E IR) in (19), we have (20) arg (u —

If v = 0, we have u — u2 > 0, namely, 0 < u < 1. If u = 1/2, (20) holds since v2 + 1/4 > 0. Thus the Stokes curves consist of one straight line and one segment (Fig. 3).

arg x(1 — x) = 0.

v E IR) in (19), we have

Fig. 3 Line and segment 3.2Ifa= 206=y= 4,wehave

d2 2 (x — 2)2 (21)

axe(x(x-1))2 = 0.

a0 = al = 2 is a double turning point of this equation. The Stokes curves are

x — 2 (22) I

m1)dx = O.

f2x(x

This implies

— (23) Im logx—= 0.

x21

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理工学部研究報告第47号

Hence we have

If v = 0, u should be greater than 1. If u2 + v2 - 2u = 0, the real part of (25) is positive if u > 0. Thus the Stokes curves coincide with the semialgebraic set

(26) {xi v = 0, u> 1} U {XI (U - 1)2 + V2 = 1, U > 0}, which consists of a half line and a circle excluded the gin (Fig. 4).

ori-

type of non-degenerate Stokes graphs. A topological type of a non-degenerate Stokes graph is characterized by the sequence (n1, n2, n3), which we call the order sequence of the Stokes graph, of numbers n3 (j = 0, 1, 2) of Stokes curves which flows into the singularity bj, where we set bo = 0, b1 = 1 and b2 = co. By the definition, we have ni + n2 + n3 = 6. We define the set Hj (j = 0, 1, 2, 3) by 1/0 = {(a,/3, Y) E C3 I 0 < Rea < Rey, 0 < Rep < Rey}

U{(a,fl, y) E C3 I Rey < Rea, < 0, Rey < Re/3 < 0},

H1 = {(a )6, Y) E C3 I 0 < Rey < Rea, 0 < Rey < Ref}

U{(a,p, y) E C3 I Rea < Rey < 0, Rep < Rey < 0}

U{(a,fi, y) E C3 I Rea < 0, Re/3 < 0, Rey > 0}

U{(a,p, y) E C3 I Rea > 0, Rep > 0, Rey < 0},

H2 = {(a,fl, E C3 I 0 < Rey < Rea, Rep < 0)

Fig. 4 Circle and half-line 4. Application

Using examples given in the preceding section, we can establish a classification theorem of Stokes graphs of hypergeometric differential equations with a large parameter. (See [1], [2] for the definition of Stokes graphs of Fuchsian differential equations of second order.) Let Ej

(j = 0, 1, 2) be the sets of the parameters (a,fi, y) defined by

E0 = [(a,fl, y) E C3 I

Y (a - )3) - (a - (fi -

x(a +13 - y) = 0), E1 = {(a,fi, y) E C3 I

Re a - Rep - Re(y - a) Re(y -13) = 0), E2 = {(a,fi, y) E C3 I

Re(a - p) Re(a + - y) Re y = 0).

The second author has obtained the following

Theorem 1. ([4], Theorem 3.1) Suppose that (a, y) is not contained in Ecs.

(i) If two distinct turning points are connected by a Stokes curve, then (a,/3, y) belongs to El.

(ii) If a Stokes curve forms a closed curve with a single turning point as the base point, then (a, 13,y) belongs to E2.

Hence each connected component of the complement of E0 U El U E2 in C3 should correspond to a topological

E C3 U{(a,p, y) E C3 U{(a,fl, E C3

Rea, < Rey < 0, Rep > 0) Rey < Re/3, Rea, < 0) Rep < Rey < 0, Rea, > 0),

H3 = {(a , 13, y) E C3 I

Rea Rep Re(y - a) Re(y (3) < 0).

Then the set C3 - (E1 U ((a,/3, y) I Re y = OD can be expressed by a disjoint union of four components:

H0UH1UH2UH3.

The following theorem is announced in [4] as a claim obtained by observation through numerical experiments.

Now we can give a proof by using our examples.

Theorem 2. Let H.; be as above and let us suppose (a,fl, y) is not contained in E0 U E2. Let n = (ni, n2, n3) be the order sequence of the Stokes graph of (3).

(i) If (a, 13, y) E Ho, then n = (4, 1, 1).

(ii) If (a,/3, E Ill, then n= (1, 4,1).

(iii) If (a,/3, y) E H2, then n. (1, 1, 4).

(iv) If (a,fl, y) E H3, then n = (2, 2, 2).

Outline of the proof. As is pointed out in [4], if (a, /3, y) is a point in C3 - E0 U Ei U E2, then the Stokes geometry is non-degenerate and hence any small perturbation of the parameters a, /3, y does not make degeneracy of the Stokes geometry. Let (ao,fio, Yo) be a point in E2 - E0 U El. If we take a sufficiently small neighborhood SI of (ao,fio, yo) in C3, the order sequence n = (no, ni, n2) of the Stokes graph is the same for every (ail, y) E S2 - E2. Hence each connected component of H3 corresponds to a topological type of Stokes graphs. Thus we look at the boundary between Hj and Hk for j # k or that of connected components of Hi. For example, the triplet of parameters

y) = (2, 4, 4) E E0 given in 3.2 belongs to the boundary between 1/0 and H3. Small perturbations of (a,/3, y)

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超幾何微分方程式の退化ストークス曲線の具体形

make non-degenerate Stokes geometries and the order se- quences n of them can be found by using arguments given in § 4 of [3]. That is, (a,i3,y) = (2, 4 + E, 4) E. H3 with a small e > 0 yields n = (2, 2, 2) (Fig. 5). On the other hand, (2, 4 — e, 4) E H0 and n = (4,1, 1) (Fig. 6).

[5] Voros, A., The return complex WKB method, 39(1983), 211-338.

of quartic oscillator, The Ann. Inst.Henri Poincare,

rig. b n = (4, 1, 1)

Hence in the connected components of H3 (resp. H0) con- taining (2, 4+e, 4) (resp. (2, 4—e, 4)) we have n = (2, 2, 2) (resp. n = (4, 1, 1)). Other parts of boundary can be dis- cussed similarly. A complete proof of this theorem will be given in our forthcoming paper.

References

[1] Aoki, T. and Iizuka, T., Classification of Stokes graphs of second order Fuchsian differential equa-

tions of genus two, Publ. RIMS, Kyoto Univ.„ 43

(2007), 241-276.

[2] Kawai, T. and Takei, Y., Algebraic Analysis of Singu- lar Perturbation Theory, Translation of Mathematical

Monographs, vol. 227, AMS, 2005.

[3] Sasaki, S., On the classification of Stokes graphs for second order Fuchsian equations, Master's Thesis,

RIMS, Kyoto University, 2010.

[4] Tanda, M., Exact WKB analysis differential equations, to appear in

Bessatsu.

of hypergeometric RIMS Kokyfiroku

Some concrete shapes of degenerate Stokes curves of hypergeometric differential equations with a large parameter

超 幾何微分 方程 式 の退化 ス トー クス曲線 の具体形

Some concrete shapes of degenerate Stokes curves of hypergeometric differential equations with a large parameter

AtQc dx = 0

Im (--i4log 4x(1x)).

dx (x(x — 1))2 Qi) = 0

(17) Imf 2x(x — 1)dx = O.

d2 712 (x 2)2 Qi) = 0.

(13) Imf 2i2x — 2x(x — 1)dx = O.

(14) 4(x

— 1)1

Fig. 3 Line and segment 3.2Ifa= 206=y= 4,wehave

d2 2 (x — 2)2 (21)

axe(x(x-1))2 = 0.

a0 = al = 2 is a double turning point of this equation. The Stokes curves are

x — 2 (22) I

m1)dx = O.

f2x(x

make non-degenerate Stokes geometries and the order se- quences n of them can be found by using arguments given in § 4 of [3]. That is, (a,i3,y) = (2, 4 + E, 4) E. H3 with a small e > 0 yields n = (2, 2, 2) (Fig. 5). On the other hand, (2, 4 — e, 4) E H0 and n = (4,1, 1) (Fig. 6).

References

超幾何微分方程式の退化ストークス曲線の具体形