数理解析研究所講究録 1431 巻 2005 年 1-13

1

A fresh glimpse into the Stokes
geometry of the Berk-Nevins-Roberts
equation through a singularcoordinate transformation*

青木 貴史 (AOKI, Takashi)
近畿大学理工学部
Department of Mathematics, Kinki University

京都大学数理解析研究所(KAWAI, Takahiro)
RIMS, Kyoto University

河合 隆裕

小池 達也 (KOIKE, Tatsuya)
京都大学理学研究科
Department of Mathematics, Kyoto University

京都大学数理解析研究所

竹井 義次 (TAKEI, Yoshitsugu)
RIMS, Kyoto University\S 0, Introduction
In the discussion with Professor A. Shudo, the following question occurred

to us:
What if we kill a part of the symm etry of the Stokes geometry for the
given by
Berk-Nevins-Roberts operator$P_{\mathrm{B}\mathrm{N}\mathrm{R}}$

,

(0.1)

$\eta^{-3}\frac{\partial^{3}}{\partial x^{3}}+3\eta^{-1}\frac{\partial}{\partial x}+2\mathrm{i}x$by considering a singular coordinate transformation(0.2)$z=x^{2}$?

Here and in what follows, denotes a large parameter.
As is now well-known ([BNR] , [AKKSST] ), the Stokes geometry for
y7 $=0$ :is as follows when$\eta$$P_{\mathrm{B}\mathrm{N}\mathrm{R}}$$\arg$’This paper is in final form and no version of it will be submitted for publicationelsewhere.
’The research of the authors has been supported in part by JSPS Grant-in-Aid
No. 14340042, No. 15540190, No. 15740088 and No. 16540148

2

Fig. 0. 1, $x=$
Here $x=+1$ and -1 are simple turning points of the operator
is its (unique) virtual turning point, and the dotted line indicates thatof the Stokes curve is inert in the sense that no Stokes
the portion$=0$ . By thephenomena are observed in any solutions of the equationtransformation (0.2), we find$P_{\mathrm{B}\mathrm{N}\mathrm{R}}$$0$$C_{1}C_{2}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}}\psi$(0.3)$\frac{1}{2x}P_{\mathrm{B}\mathrm{N}\mathrm{R}}=4\eta^{-3}z\frac{\partial^{3}}{\partial z^{3}}+6\eta^{-3}\frac{\partial^{2}}{\partial z^{2}}+3\eta^{-1}\frac{\partial}{\partial z}+\dot{\iota}$.

denote the operator that appears in the right-handIn the sequel we letside of $(0,3)$ . With the help of a computer, one can readily find the following$=0$ With $\arg\eta=0$ :Stokes geometry of the equation$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}\psi$$\underline{|z}$3

Geometrically speaking, $z=1$ is, under the transformation (0.2), the imageof simple turning points $x=\pm 1$ , and $z=C$ is the image of $x=C_{1}$ and. The Stokes curve (half line) starting at the point $z=0$ originates; the precise definition of thisfrom the vanishing factor . in front of
Stokes curve will be given in our forthcoming paper [KKT] with the help ofa decomposition theorem to be announced in Section 3 below. The results in
Section 2 will also convince the reader of the assertion that the point $z=0$. In parenthesis, we note
plays a role of a turning point of the operator; this fact canthat there is no virtual turning point of the operator
be readily seen by explicitly solving the Hamilton-Jacobi equation with the,,
Hamiltonian determined by$C_{2}$$\partial^{3}/\partial z^{3}$$z$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$(0.4)$\mathrm{i}.\mathrm{e}.$$4z\zeta^{3}+3\zeta\eta^{2}+\mathrm{i}\eta^{3}$

with the initial condition(0.5)$(z(0), y(0);\zeta(0),$ $\eta(0))=(1, y0_{1}^{\cdot} -\frac{\mathrm{i}}{2}, 1)\rangle$

where is an arbitrary complex number.
Now, in view of the geometrical correspondence between Fig. 0.1 andFig. 0.2, we were tempted to believe that the segment $CO$ of the Stokes
curve emanating from $z=0$ should be inert. This belief is validated in
two ways; in Section 1 we confirm this fact numerically (i.e., with the helpof a computer) by applying the steepest descent method to an integral that$=0$ , and in Section 2 we confirm
represents a solution of the equation
help
of a computer) by reducing thethis fact analytically (i.e., without the
problem to the connection problem for a second order operator with simple
poles in the coefficients that was analyzed in [K1] and [K2]. In either case,the reason for our success is very subtle, or rather miraculous; in the steepest
descent method approach two integrals cancel out, and in the approach of
reducing the problem to that of a second order operator, some parameter
kills the relevant Stokes multiplier that is given by [K1]determined byand [K2].
To show the full scope of applicability of the method employed in Section 2, we present in Section 3 a general decomposition theorem for a class. The details of the results in Section 3 shallof operators that includes
be given elsewhere.
$y0$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}\psi$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$

4\S 1. Steepest descent method approach
We begin our discussion by noting that the equation
an integral representation of solutions; that is,(1.1)$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}\psi=0$

admits$\psi=\oint_{\gamma}\zeta^{-\frac{3}{2}}\exp(\eta(z\zeta-(\frac{3}{4\zeta}+\frac{\mathrm{i}}{\mathrm{S}\zeta^{2}})))d\zeta$

is a solution of the equation(1.2)$(4 \eta^{-3}z\frac{\partial^{3}}{\partial z^{3}}+6\eta^{-3}\frac{\partial^{2}}{\partial z^{2}}+3\eta^{-1}\frac{\partial}{\partial z}+\mathrm{i})\psi=0$

for a properly chosen contour
determined by the equation$\gamma$. The saddle point $\zeta j(z)(j=1_{\rangle}2,3)$ is
,

(1.3)

$z+ \frac{3}{4\zeta^{2}}+\frac{\mathrm{i}}{4\zeta^{3}}=0$

which coincides with the characteristic equation of (1.2). Hence a
is obtained if we choose to be the steepest descentsolution of. Then, as is well known
paththat passes through the saddle point([U]), some topological change of the configuration of steepest descent paths
passing through saddle points is a necessary condition for the occurrence ofwith the help ofStokes phenomena of WKB solutions. Hence wecomputer the topological change of configurations when moves around the, $P20$ ,point $C$ in Fig. 0.2. We choose 20 points, , ,Pi , near $z=C$
as is designated just by a number in Fig. 1.1 below.
$\mathrm{W}\mathrm{K}\mathrm{B}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}\psi=0$$\gamma$$\zeta_{\mathrm{J}}(z)$$\gamma_{j}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{a}$$z$$\rho_{1}$$j$Fig, 1.1$\rho_{2}$$\cdots$$\rho_{21}=$

5

given
The configuration of steepest descent$\zeta=0$
and other dots correspondin Fig. l.l.j below; the tiny dot indicatesto saddle points $\zeta_{l}(\rho_{j})(l=1, 2,3)$ .
$\mathrm{P}^{\mathrm{a}\mathrm{t}\mathrm{h}_{\mathrm{S}\mathrm{a}\mathrm{t}z=\rho j}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\arg\eta=0\mathrm{i}\mathrm{s}}$Fig. 1.1.1

Fig. 1.1.2

Fig. 1.1.3

Fig. 1.1.4

Fig. 1.1.5

Fig. 1.1.6

Fig. 1.1.7

Fig. 1.1.8

Fig. 1.1.9

Fig. 1.1.10

Fig. 1. 1. 11

Fig. 1. 1. 1

$\mathrm{G}$Fig. 1.1.13

Fig. 1.1.14

Fig. 1.1.15

Fig. 1.1.16

Fig. 1.1.17

Fig. 1.1.18

Fig. 1.1.19

Fig. 1.1.20

Fig. 1.1.21

The reader will notice some topological changes at $z=\rho_{1}(=\rho_{21})$ , , ,, as is expected. And, one notices a topological change also at $z=\rho 11$ !
andThus one might think that there should occur some Stokes phenomena acrossthe segment $CO$ in Fig. 0.2. But, if one carefully compare the configurationsof steepest descent paths in Fig. 1.1.10 and those in Fig. 1.1.12, one finds thatthe integral
in the course of analytic continuation from $z=\rho_{10}$ toalong the steepest descent path passing through the saddle pointpassingacquires the integral I2 along the steepest descent path, and at the same time loses another integral I2 along
the sad dle point
the same pathwith the opposite orientation. One might then conclude. the conclusion is erroneous, because thethe net contribution to isbranch of the integrand of 12 and that of are different due to the factor
in the integrand of (1.1). Then the net contribution to is $I_{2}-I_{2}=0!$$\rho_{3}$$\rho_{9}$$\rho 13$$\rho_{19}$$z$$=\rho_{12}$$\zeta_{1}(z)$$I_{1}$$\underline{\mathrm{t}}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}$$\gamma_{2}$$\zeta_{2}(z)$$\gamma_{2}$$I_{1}$$I_{2}+I_{2_{\rangle}}$$\overline{I_{2}}$$\zeta^{-3/2}$$I_{1}$

7

This implies that the segment $CO$ is inert, validating our belief.

Remark 1. 1. The above reasoning tells us that, if we start with the following,
operator instead of$\overline{P}$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$(1.4)$\tilde{P}=4\eta^{-3}z\frac{\partial^{3}}{\partial z^{3}}+a\eta^{-3}\frac{\partial^{2}}{\partial z^{2}}+3\eta^{-1}\frac{\partial}{\partial z}+\mathrm{i}$,where is a complex number, the inert character of the segment $CO$ isobserved only when
$a$

(1.5)

$a\equiv 2\mathrm{m}\mathrm{o}\mathrm{d} 4$.

We will encounter this condition again in the next section. (Cf. Remark 2.1.)\S 2. Reduction to a second order operator
Let $P$ denote the operator(2.1)$\eta^{3}(4z)^{-1}P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$; that is,$P= \frac{\partial^{3}}{\partial z^{3}}+\eta^{2}\frac{3}{4z}\frac{\partial}{\partial z}+\frac{\mathrm{i}}{4z}\eta^{3}+\frac{3}{2z}\frac{\partial^{2}}{\partial z^{2}}$.

Then we can find pre-Borel summable series(2.2)$q(z, \eta)=q_{0}(z)+\eta^{-1}q_{1}(z)+\cdots$

and
(2.3)

$a_{j}(z, \eta)=a_{j,0}(z)+\eta^{-1}a_{j,1}(z)+\cdots$

for some open neighborhoodwith $j=1,2$ on a punctured disc$z=0$ so that the following conditions $(2,4)$ \sim (2.8) are satisfied:$V\backslash \{0\}$

(2.4)
(2.5)
(2.6)
(2.7)
(2.8)

$P=( \frac{\partial}{\partial z}-\eta q(z, \eta))(\frac{\partial^{2}}{\partial z^{2}}+\eta a_{1}(z, \eta)\frac{\partial}{\partial z}+\eta^{2}a_{2}(z, \eta))$

$V$

of

,

is holomorphic on ,has a simple pole at $z=0$ with residue -1,$a_{1,k}(z)(k\neq 1)$ is holomorphic on,
and $za_{2,k}(k\geq 0)$ are holomorphic on .

$q_{j}(z)(j\neq 1)$$V$$q_{1}(z)$

$V$

$V$

$za_{1,1}$

The proof of this decomposition result is a straightforward one, and weare relevant to theandomit it here. Since only two saddle points$\zeta_{1}(z)$$\zeta_{2}(z)$8possible Stokes phenomena near $z=0$ , the decomposition (2.4) enables us
to reduce the connection problem for the operator to that of the second
order equation
$P$

(2.9)

$( \frac{\partial^{2}}{\partial z^{2}}+\eta a_{1}(z, \eta)\frac{\partial}{\partial z}+\eta^{2}a_{2}(z, \eta))\psi=0$

After the change of the unknown function(2.10)$\psi$.

to$\varphi=\exp(\frac{1}{2}\eta\oint^{z}a_{1}(z, \eta)dz)\psi$,\S 2.3]) to
we can use the theory of formal coordinate transformation (cf.
reduce the problem to the connection problem for a second order operator
with simple poles in the sense of [K2]. Then the results in [K1] and [K2]assert that the Stokes multiplier along the segment $CO$ in Fig. 0.2 is given$[\mathrm{K}\mathrm{T},$by(2.11)$2\mathrm{i}\cos(\pi\sqrt{1+4\lambda})$with(2.12)$\lambda=\frac{c^{2}-2c}{4}$,where $c={\rm Res}_{z=0}a_{1,1}$ . (See [K1] and [K2] for the precise statement concerning the dominance relations of WKB solutions.)
Now, it immediately follows from (2.4) and (2.6) that
(2.13)${\rm Res}_{z=0}a_{1,1}= \frac{1}{2}$.

Hence we see that the Stokes multiplier (2.11) vanishes. Thus we have again
confirmed that the segment $CO$ in Fig. 0.2 is inert!
Remark 2.1, If we start with the operator containing a parameter , in, as in Remark 1.1 we can easily checkstead of$\tilde{P}$$a$$P_{\mathrm{B}\mathrm{N}\mathrm{R}’}$(2.14)${\rm Res}_{z=0}a_{1,1}= \frac{a}{4}-1$.

Then the relevant Stokes multiplier (2.11) vanishes if(2.15)$( \frac{a}{4}-1)-1=\frac{l}{2}$This is exactly the same as (1.5);$l$: odd.

9\S 3. A decomposition theorem for operators withsimple poles in their coefficients
In this section we introduce a class (K) of operators with simple polesin their coefficients and then present a decomposition theorem for operators
in class (K). Although we do not give the proof of the theorem in thisreport, we explain its background in a heuristic manner. As our discussion, but rather of a generalin this section is not of immediate relevance tocharacter we use the variable , not .
$P_{\mathrm{B}\mathrm{N}\mathrm{R}}$

$x$

$z$

be an open neighborhood of the origin of , and let$(j=1,2, \cdots, m(\geq 2);k=1,2, \cdots)$ be a meromorphic function on
having a pole at $x=0$ . Assume further that

Definition 3.1. Let$V$$\mathbb{C}_{x}$$V$$A_{\mathrm{j},k}(x)$(3.1)$A_{j}(x, \eta)=\sum_{k\geq 0}A_{j,k}(x)\eta^{-k}$is pre-Borel summable on(3.2)$V\backslash \{0\}$. Then the operator$P$

given by$\frac{\partial^{m}}{\partial x^{m}}+\eta A_{1}(x, \eta)\frac{\partial^{m-1}}{\partial x^{m-1}}+\cdots+\eta^{m}A_{m}(x, \eta)$

is in class (K) if the following conditions $(3,3)$ , (3.4) and (3.5) are satisfied:

(3.3)
(3.4)
(3.5)
(3.5.i)(3.5.ii)(3.5.iii)

is holomorphic on ,and Aitk $(2\leq j\leq m, k\geq 0)$ are holomorphic on
Letting $\alpha_{j}(2\leq j\leq m)$ denote, we fifind$V$$A_{1,k}(k \neq 1)$$xA_{1,1}$$V$,${\rm Res}_{x=0}\mathrm{A}\mathrm{j}|\mathrm{O}$$\alpha_{2}\neq 0$,$\alpha_{m}\neq 0$,

The equation$\sum_{j=2}^{m}\alpha_{i}\xi^{m-j}=0$

has mutually distinct$(m-2)$ solutions.

An important property of the class (K) is that the class is stable under
the decomposition of the form (2.4); to be more precise we have the following
Theorem 3.1. Let(3.6)$P= \frac{\partial^{m}}{\partial x^{m}}+\eta A_{1}(x, \eta)\frac{\partial^{m-1}}{\partial x^{m-1}}+\cdot$. . $+\eta^{m}A_{m}(x, \eta)$

10. Assume
be an operator in class (K) on a neighborhood of the origin of$W$
a preorigin,that $m\geq 3$ . Then we can find an open neighborhoodof theBorel summable series
$V$

(3.7)

$q(x, \eta)=q_{0}(x)+\eta^{-1}q_{1}(x)+\cdots$, and an operator
they satisf the following:on$\mathbb{C}_{x}$$V\backslash \{0\}$$R$of order $(m-1)$ in class (K) on$W$

so that

$y$

(3.8)

$P=( \frac{\partial}{\partial x}-\eta q(x, \eta))R$(3.9)(3.10),

is holomorphic on
is holomorphic on $W$ and the residue of$q_{j}(j\neq 1)$$xq_{1}$$W_{f}$$q_{1}$

at $x=0\mathrm{i}s-1$ .

Once this theorem is obtained, we can repeatedly use it to find the following decomposition:,(3.11)$P=( \frac{\partial}{\partial x}-\eta q^{(1)})(\frac{\partial}{\partial x}-\eta q^{(2)})\cdots(\frac{\partial}{\partial x}-\eta q^{(m-2)})R$$(j=1, \ldots, m-2)$ is a pre-Borel summable series satisfying (3.9)
where
and (3.10) and is a second order operator in class (K); thus we can reducethe connection problem for the operator $P$ to the connection problem for thesecond order operator , which was essentially discussed in [K1] and [K2].
In order to explain the intuitive meaning of Theorem 3.1, we prepare thefollowingof operators by replacing (3.3)
Remark 3.1. If we introduce another classwith$q^{(j)}$

$R$

$R$

$(\overline{K})$(3.12)$A_{1,0}$

and $xA_{1,k}(k\geq 1)$ are holom orphic on $V$,

instead of (K). Let us furtherof operators as follows: An operator is in class

Theorem 3. 1 remains to hold for the classintroduce another class$(K)=$$(K)=$$(\overline{K})$$\overline{P}$

if it has the form

(3. 8)$x \frac{\partial^{m}}{\partial x^{m}}+\eta\overline{A}_{1}(x, \eta)\frac{\partial^{m-1}}{\partial x^{m-1}}+\cdots+\eta^{m}\overline{A}_{m}(x, \eta)$

,

where
is a pre-Borel summable series on
satisfies the following conditions:$\overline{A}_{j}(x, \eta)=\sum_{k\geq 0}\overline{A}_{j,k}(x)\eta^{-k}$(3.14)$\tilde{A}_{j,k}(1\leq j\leq m, k\geq 0)$

is holomorphic on

,

$V$

$V$

that

11

(3.15)$\overline{A}_{1,0}(0)=0$.

In what follows, with some abuse of languages, we say that the series
is holomorphic on when the condition (3.14) is satisfied.
and $(K)=$ are isomorphic by the correspondenceWith these definitions,$\overline{A}_{j}(x, \eta)$$V$$(\overline{K})$. It is also clear that the class $(K)=$ is closed under the operation of
we can find aninconsidering the adjoint operator: For an operator, the adjoint operator of . It issuch that
operator inthen clear from the above mentioned correspondence between operators in. Then the modifified version
that $(-1)^{m}x^{-1}Q^{*}$ is inand those inof Theorem 3.1 that adopts (K) instead of (K) guarantees that$\overline{P}=xP$$\overline{P}$$Q$$(K)=$$(-1)^{m}\overline{P}=Q^{*}$$(\overline{K})$$(\overline{K})$$(K)=$(3. I6)$Q$$(\overline{K})$$x^{-1}Q^{*}=( \frac{\partial}{\partial x}-\eta q)R$

holds for some
(3.17)$R$in$\overline{K}$with $q(x, \eta)$ satisfying (3.9) and (3.10). Hence we fifind$Q=R^{*}(x \frac{\partial}{\partial x}-\eta xq)^{*}$$=R^{*}(- \frac{\partial}{\partial x}x\cdot-\eta xq)$$=R^{*}(-x \frac{\partial}{\partial x}-1-\eta xq)$.

is holomorphic
denote $q+\eta^{-1}x^{-1}$ , we see from (3.10) that
$Q\psi=0$
admits a WKB solution of
on $W$ . Otherwise stated, the equation
the form
Letting$\tilde{q}(x, \eta)$$\tilde{q}$$\psi$(3.18)$\exp(-\eta\int^{x}\tilde{q}(x, \eta)dx)$

with holds orphic on $W$ . Hence the somewhat clumsy condition (3.10)
solution to the equation
corresponds to the existence of a holomorphic
solution to the equaholomorphic
WKB$Q\psi=0$ . But, the existence of ation that has the form (3.18) is formally obvious because is holomorphic.
Thus (3.17), and hence (3.16) also, may be understood as a consequence ofexistence of holomorphic WKB solutions.
Reversing the reasoning in the above Remark 3.1, we can give an intuitive
and WKB-theoretic interpretation of the seemingly curious condition (3.10):$\tilde{q}$$\mathrm{W}\mathrm{K}\mathrm{B}$

$q0$

12

$(K)=$
defifined by $(xP)^{*}$ .
, we consider an operator in
For an operator $P$ in
Then this operator is divisible from the right by the factor
for holomorphic , reflecting the fact that the equation $Q\psi=0$ admits. Although some technical care
WKB solution of the form,inis needed to justify the relation $(R^{*}x)^{*}=xR$ for an operatorby
we can “prove without computation” the modifified form of Theorem 3.1
with. Further we note that the factor
choosing${\rm Res} q_{1}=-1$ is a counterpart of the existence of a holomorphic WKB solution
to the adjoint equation.
Remark 3.2. Theorem 3.1 applies to the equation (1.2) in [AKT] , which playsa central role in [AKT]; this means that the reasoning near the point([AKT, p.636 and p.637]) may be replaced by the reasoning similar to thatgiven in Section 2. Actually the reader will notice the resemblance of the
discussion in [AKT, p.636 and p.637] with that given in Section 1 of thisreport.
$Q$$(\overline{K})$$(\partial/\partial x+\eta\tilde{q})$$\mathrm{a}$$\tilde{q}$$\exp(-\eta\int^{x}\tilde{q}dx)$$R$$q=\tilde{q}-\eta^{-1}x^{-1}$$(\overline{K})$$(\partial/\partial x-\eta q)$$b_{1}$

References[AKKSST] T. Aoki, T. Kawai, T. Koike, S. Sasaki, A. Shudo and Y. Takei:
A background story and some know-how of virtual turning points,
RIMS K\^oky\^uroku No. 1424 “Recent Trends in Exponential Asymptotics”, 2005, in press.
[AKT]

T. Aoki, T. Kawai and Y. Takei: On the exact WKB analysis for thethird order ordinary differential equations with a large parameter,
Asian J. Math., $2(1998)$ , 625-640.

[BNR]

H. L. Berk, W. M. Nevins and K. V. Roberts: New Stokes’ line in
WKB theory, J. Math. Phys., 23(1982), 988-1002.

[KKT]

T. Kawai, T. Koike and Y. Takei: On the exact WKB analysis ofsimple pole type operators, in preparation.

[KT]

T. Kawai and Y. Takei: Algebraic Analysis of Singular Perturbations, Iwanami, 1998. (In Japanese and its translation will be
published by AMS.)[K1]

T. Koike: On a regular singular point in the exact WKB analysis,
Toward the Exact WKB Analysis of Differential Equations, Linear
or Non-Linear, Kyoto University Press, 2000, pp. 39-54

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[K2]

T. Koike: On the exact WKB analysis of second order linear ordinary differential equations with simple poles, Publ. RIMS, Kyoto
Univ., 36(2000), 297-319.

[U]

K. Uchiyama: On examples of Voros analysis of complex WKB theory, Analyse alg\’ebrique des perturbations singulieres. I; Methodes
resurgentes, Hermann, 1994, pp. 104-109.