数理解析研究所講究録 1211 巻 2001 年 197-211

197
On the exact WKB analysis for operators
admitting infinitely many phases
青木貴史
河合隆裕
小池達也
竹井義次

近畿大・理工京都大・数理研
京都大・理
京都大・数理研(Takashi Aoki)(Takahiro Kawai)(Tatsuya Koike)(Yoshitsugu Takei)

Local theory near asimple turning point

1

Let
consider aformal differential operator of infinite order in one variablelarge parameter that has the following form:$x$$11\mathrm{S}$with a$\eta$$P(x, \partial_{x}, \eta)=P(x, \partial_{x}/\eta)=\sum_{j,k=0}^{\infty}a_{j,k}x^{j}\eta^{-k}\partial_{x}^{k}$where$a_{j,k}$(1),are complex constants. We assume that its Borel transform(2)$P_{B}=P(x, \partial_{x}/\partial_{y})=\sum_{j,k=0}^{\infty}a_{j,k}x^{j}\partial_{x}^{k}\partial_{y}^{-k}$is amicrodifferential operator of order 0defined outside $\{\eta=0\}$ ;that is, we assume thatwhich depends holomorphicallyis an entire function of
the symbol $P(x, \xi/\eta)$ of$P(x,\zeta)$, where is an open
is aholomorphic function inon . Otherwise stated,
set in C.
Suppose that the system of equations$\zeta=\xi/\eta$$P_{B}$$U\cross \mathbb{C}$

$x$

$U$

(3)

$P(x, \zeta)=\partial_{\zeta}P(x, \zeta)=0$

is said to be a turning point of
has asolution $(x, \zeta)=(x_{*}, \zeta_{*})(x_{*}\in U)$ . Then$P$.
Let
be a turning point with thewith the characteristic value
the operator
is simple if
characteristic value . We say that$x_{*}$$\zeta_{*}$$\zeta_{*}$$x_{*}$$x_{*}$$\partial_{x}P(x_{*}, \zeta_{*})\neq 0$and$\partial_{\zeta}^{2}P(x_{*}, \zeta_{*})\neq 0$.

(4)

we can find two multivalued analytic functions
Near asimple turning point
the equation $P(x, \zeta)=0$ for which$x_{*}$$\zeta_{\pm}(x_{*})=\zeta_{*}$

and$\zeta_{\pm}(x)$of(5)$\zeta_{+}(x)-\zeta_{-}(x)=\sqrt{x-x_{*}}f(x)$hold with aholomorphic function $f(x)$ that does not vanish at $x=x_{*}$ , and we furtherof $P(x, ()$ $=0$ . We also note thatfor any other analytic solution
find thatare holomorphic functions in a neighborhood of .
and$\zeta(x_{*})\neq\zeta_{*}$$\zeta_{+}(x)+\zeta_{-}(x)$$\zeta(x)$$\zeta_{+}(x)\zeta_{-}(x)$$x_{*}$

Theorem 1Suppose that for every $x\in U$ , there is a solution $=\zeta(x)$ of the equation$P(x, ()$ $=0$ andis not identically zero. Then there exists aformalequation
solution of the differential$\langle$$(\partial P/\partial\zeta)(x, \zeta(x))$$\psi$$P(x, \partial_{x}, \eta)\psi=0$

(6)

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that is of the form
(7)$= \exp(\int S(x, \eta)dx)$$\psi$

withand$S(x, \eta)=\sum_{j\geq-1}\eta^{-j}S_{j}(x)$

Here

$S$

is unique up to the choice of abranch of$S_{-1}(x)=\zeta(x)$$\zeta(x)$

.

(8)

.

Asolution (7) is called aWKB solution of (6).
and letbe asimple turning point with the characteristic value
Theorem 2Let$r(x,\zeta)$$P(x,\zeta)=0$
polynomialaquadratic(5).
Letsatisfying
denotebe solutions ofwith holomorphic coefficients in asmall neighborhood of . Then
there exists adifferential operator of infinite order with the large parameter yy$\zeta_{*}$$x_{*}$$\zeta_{\pm}(x)$$V$$(\zeta-\zeta_{+}(x))(\zeta-\zeta_{-}(x))$$Q(x, \partial_{x}/\eta, \eta)=\sum_{j=0}^{\infty}$

rt

(9)

$-jQ_{j}(x, \partial_{x}/\eta)$

and asecond-0rder differential operator with the large parameter$R(x, \partial_{x}/\eta, \eta)=\sum_{j=0}^{\infty}$

so that the following conditions are satisfied:, (;) and(i)are holomorphic inare microdifferential operators of order 0.
.
(ii)$Q_{j}(x$$R_{j}(x, \langle)$rt$\eta$(10)$-jR_{j}(x, \partial_{x}/\eta)$$V\cross \mathbb{C}$and$x_{*}$$Q(x, \partial_{x}/\partial_{y}, \partial_{y})$,$R(x, \partial_{x}/\partial_{y}, \partial_{y})$$P(x, \partial_{x}/\eta)=Q(x, \partial_{x}/\eta, \eta)R(x, \partial_{x}/\eta, \eta)$(ii) $R_{0}(x, ()$ $=r(x, \zeta)$ .
(iv) For each $j>0$ ,$R_{j}(x, \langle)$

is of degree at most one in .
$($Theorem 2above implies that, near asimple turning point , WKB solutions of (6)with the leading term $S_{-1}(x)=\zeta\pm(x)$ satisfy asecond-0rder differential equation$x_{*}$$R(x, \partial_{x}/\eta, \eta)\psi=0$

Hence Stokes curves emanating from$x_{*}$.

(11)

should be defined by${\rm Im} \int_{x_{*}}^{x}(\zeta_{+}(s)-\zeta_{-}(s))ds=0$

,

(12)

and alocal connection formula for WKB solutions can be obtained by reducing the problem to the second order case.

2Study of the global configuration of Stokes curves–an example
To illustrate the general theory discussed in \S 1 and to try to find what occurs globally,
we consider the following example:$P(x, \partial_{x}/\eta)\psi=0$,$P(x, \partial_{x}/\eta)=\cosh(\sqrt{\partial_{x}}/i\eta)-.’\iota.$.

(10)

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-1, and they are simple in the sense of (4)1 and xTurning points for (13) are rof a WKB solution is given as follows:
The leading term$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$S_{=1}(\ovalbox{\tt\small REJECT} \mathrm{r})$$S_{-1}=f_{n}$,$f_{n}(x)=i(2n\pi i+\log(x+\sqrt{x^{2}-1}))^{2}$$(n\in \mathbb{Z})$.

(14)

Hence there are infinitely many phases for (13) and Stokes curves emanating from $x=1$(resp. $x=-1$ ) are defined by${\rm Im} \int_{1}^{x}(f_{n}(s)-f_{-n}(s))ds=0$(resp. ${\rm Im} \int_{-1}^{x}(f_{n}(s)-f_{-n-1}(s))ds=0$ ).

(15)Their configuration is shown in Fig.1; note that all these Stokes curves sit on the samecurves.

Fig. 1and , we need to introduce
Since Stokes curves in Fig.l have two crossing points
new Stokes curves to obtain the complete description of Stokes regions, i.e., the regionsin which WKB solutions become Borel summable (cf. [BNR], [AKTI]).
Let us determine new Stokes curves passing through thc crossing point . To makethe argument explicit we place cuts along $\{x;x\geq 1\}$ and $\{x;x\leq-1\}$ in -plane, and fix
abranch of $\log(x+\sqrt{x^{2}-1})$ so that$x_{0}$$x_{1}$$x_{0}$$x$$\sqrt{x^{2}-1}|_{x=0}=i$and$\log(x+\sqrt{x^{2}-1})|_{x=0}=\frac{\pi}{2}i$(16)along the Stokes curve which
is dominant overhold. Then we can verify thatemanates from $x=1$ and passes through $x=x_{0}$ ; that is,$\psi_{n}$${\rm Re} \int_{1}^{x}(f_{n}(s)-f_{-n}(s))ds>0$$\psi_{-n}$$(n=1,2, \ldots)$(17)

holds along this Stokes curve. (In what follows, this dominance relation (17) is denoted”
by$>-n$ .) We also find $”-r\iota-1>n$ $(n>0)$ holds along the Stokes curve$” 7l$”

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which emanates from x
dominance relation at$\ovalbox{\tt\small REJECT}$-1 and passes through$\ovalbox{\tt\small REJECT} 1\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\mathrm{f};_{0}$. Thus we obtain the following$\ovalbox{\tt\small REJECT} \mathrm{r}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} \mathrm{r}_{0^{\ovalbox{\tt\small REJECT}}}$“$\cdots>3>-3>2>-2>1>-1>0"$ .

(18)This suggests that new Stokes curves should be given by${\rm Im} \int_{x0}^{x}(f_{m}(x)-f_{n}(x))dx=0$$(m, 7l \in \mathbb{Z})$

,

(19)

or equivalently,${\rm Im} \int_{x_{0}}^{x}(k\pi i +\log(x+.\sqrt{\prime r^{2}-|1}))dx=0$$(k\in \mathbb{Z})$.

(20)

Among them $m=-n$ and $m=-n-1$ (i.e., $k=0,$ -1) are ordinary Stokes curves. These
are shown in Fig.2.
curves$(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}-4\leq k\leq 3)$

Fig.2

Fig.3

As [AKTI] observed, each new Stokes curve is associated with (a family of) new
turning points. In our case, we can readily locate new turning points by explicitly writing;they are given by $x=\pm\cos w$ ,down the equations for bicharacteristic strips forwhere should satisfy $w=\tan w$ . All solutions of this equation $w=\tan w$ are real andsymmetric with respect to the origin. Let $w_{1}<w_{2}<\cdots$ be positive solutions of thisequation. Then by numerical calculation we can check that anew Stokes curve definedin (20) passes through anew turning point $x=\cos w_{k}$ for $k>0$ or $x=-\cos w_{-k-1}$ for$k<-1$ . Furthermore no Stokes phenomena are expected to occur near anew turning
point (cf. [AKTI, PP.77]). In Fig.3 we use adotted line to denote the portion of anew
Stokes curve along which no Stokes phenomena occur.
The above description of the Stokes geometry for the operator is akind of ansatz.
We validate it by using the steepest descent method to the integral representation$P_{B}$$w$$P$$\psi(x, \eta)=[$ $\exp(\eta h(x, \xi))d\xi$(20)201of solutions of (13) (cf. [T], [AKT2]), where$h(x, \xi)=x\xi-2i\sqrt{\xi}/i\sinh(\sqrt{\xi}/i)+2i\cosh(\sqrt{\xi}/i)$

.

(22)

Let us first consider the configuration of steepest descent paths near $x=1$ (noticethat is included as aparameter in the integrand of (21) . By its definition, saddle pointsof (21) are give by $=\xi_{n}(x)$ , where$)$

$x$

$\xi$

$\xi_{n}(x)=i(2\pi i+\log(x+\sqrt{x^{2}-1}))^{2}$

(23)

,

which passes throughofand our interest is in the steepest descent path$=\xi_{n}(x)$ .
FigA-k shows the configuration of such saddle points and steepest descentpaths for
$C_{n}$${\rm Re} h(x, \xi)$$\xi$$x=1+0.2\exp(0.1i\pi k)$$(k=0,1,2, \ldots, 20)$ .

(24)

Fig.4-0

Fig.4-1

Fig.4-2

Fig.4-3

Fig.4-4

Fig.4-3

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Fig.4-6

Fig.4-7

Fig.4-8

Fig.4-9

Fig.4-10

Fig.4-11

Fig.4-12

Fig.4-13

Fig.4-11

203

Fig.4-15

Fig.4-16

Fig.4-17

Fig.4-18

Fig.4-19

Fig.4-20

We can observe from these figures that atopological change of their configuration occurs
at three places; between Fig.4-6 and Fig.4-7, between Fig.4-13 and Fig.4-14, and nearFig.4-20 (i.e., Fig.4-0). This corresponds to the fact that three Stokes curves emanatefrom the turning point $x=1$ ;each change occurs exactly when we cross aStokes curve
emanating from $x=1$ . This implies that Fig.3 describes the Stokes geometry for (13)correctly, at least, near $x=1$ . In asimilar manner we can confirm that Fig.3 describes
the Stokes geometry correctly also near $x=-1$ .
Next we consider the configuration of steepest descent paths when runs parallel withthe real axis. Fig.5-fc shows the configuration for$x$$x=(-1.5+0.2i)+0.2k$$(k=0,1,2, \ldots, 14)$ .

(25)

The configuration of steepest descent paths changes topologically at two places (betweenFig.5-3 and Fig5-4, between Fig.5-11 and Fig.5-12), i.e., exactly when we cross aStokes
curve emanating from $x=1$ or $x=-1$ . In particular, no topological changes of theconfiguration occur when we cross the portion of new Stokes curves designated by dottedlines in Fig.3

204

Fig.5-O

Fig.5-1

Fig.5-2

Fig.5-3

Fig.5-4

Fig.5-5

Fig.5-6

Fig.5-7

Fig.5-0

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Fig.5-9

Fig.5-10

Fig.5-11

Fig.5-12

Fig.5-13

Fig.5-14

Let us now consider the configuration of steepest descent paths near $x=x_{1}$ , acrossing
point of Stokes curves in the lower half plane (cf. Fig. 1). Fig.6-fc shows the configurationof steepest descent paths for$x=x_{1}+0.2\exp$ (O.link)

Fig.6-O

Fig.6-1$(k=0,1,2, \ldots, 20)$ .

Fig.6-2(26)206

Fig.6-3

Fig.6-4

Fig.6-5

Fig.6-6

Fig.6-7

Fig.6-8

Fig.6-9

Fig.6-10

Fig.6-10

207

Fig.6-12

Fig.6-13

Fig.6-14

Fig.6-15

Fig.6-16

Fig.6-17

Fig.6-18

Fig.6-19

Fig.6-20

From these figures, we first find that the configuration of steepest descent paths changes
topologically when we cross a Stokes curve emanating from $x=1$ (between Fig.6-2 andFig.6-3, between Fig.6-13 and Fig.6-14) and that from $x=-1$ (between Fig.6-7 and Fig.68, between Fig.6-16 and Fig.6-17). We can also verify that no topological changes occur. In contrast with it,
when we cross new Stokes curves designated by a dotted line inof the configuration can be observed when we cross new Stokes curves
topological(from Fig.6-15 to Fig.6-17). These changes can bedesignated by asolid line in$\mathrm{F}\mathrm{i}\mathrm{g}.3$$\mathrm{c}1_{1}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{s}$$\mathrm{F}\mathrm{i}\mathrm{g}.3$

208
visualized clearly by more precise numerical calculations; Fig.7-A shows the configureof steepest descent paths for$x=x_{1}+0.2\exp(i\pi(1.25+0.01k))$$(k=0,1,2, \ldots, 25)$ .

Fig.7-0

Fig.7-1

Fig.7-2

Fig.7-3

Fig.7-4

Fig.7-5

Fig.7-6

Fig.7-7

Fig.7-8

209

Fig.7-9

Fig.7-10

Fig.7-11

Fig.7-12

Fig.7-13

Fig.7-14

Fig.7-15

Fig.7-16

Fig.7-17

210

Fig.7-18

Fig.7-19

Fig.7-20

Fig.7-21

Fig.7-22

Fig.7-23

Fig.7-24

Fig.7-25

First topological change of the configuration occurs between Fig.7-5 and Fig.7-6, i.e.,
when we cross aStokes curve emanating from $x=1$ . Next topological change occurs, when we cross anew Stokes curve given by
between Fig.7-17 and Fig.7-18,$\mathrm{i}.\mathrm{e}.$${\rm Im} \int_{x_{1}}^{x}(f_{n}(x)-f_{-n+1}(x))dx=0$.

(28)

211
Another topological change of the configuration can be observed between Fig.7-22 and, when we cross anew Stokes curve given byFig.7-23,$\ovalbox{\tt\small REJECT}.\mathrm{e}.$${\rm Im} \int_{x_{1}}^{x}(f_{n}(x)-f_{-n+2}(x))dx=0$

.

(29)

Surmising from these observations, we then expect that on the new Stokes curve definedby${\rm Im} \int_{x_{1}}^{x}(f_{n}(x)-f_{-n+k}(x))dx=0$$(k\geq 1)$(30)

the steepest descent path passing through the saddle point(which is located atthe “top”) goes down and flows into a saddle point. (At the same time, two saddlepointsand $=\xi_{-n+k}$ are connected by asteepest descent path for every $n.$ ) Thus
we conclude that Stokes phenomena occur when we cross new Stokes curves designatedby asolid line in. Similar results can also be checked near $x=x_{0}$ , the crossing pointin the upper half plane.
$\xi=\xi_{0}$$\xi=\xi_{k}$$\xi$$=\xi_{n}$$\xi$$\mathrm{F}\mathrm{i}\mathrm{g}.3$

References[AKTI] T.Aoki, T.Kawai and Y.Takei: New turning points in the exact WKB analysis forhigher-0rder ordinary differential equations, Analyse alg\’ebrique des perturbations
singuli\‘eres. I, Hermann, pp.69-84, 1994.
[AKT2] T.Aoki, T.Kawai and Y.Takei: On the exact steepest descent method –anew
method for the description of Stokes curves, to appear in J. Math. Phys.
[BNR] H.L.Berk, W.M.Nevins and K.V.Roberts:
J. Math. Phys., 23(1982), 988-1002.

[T]

New Stokes lines in WKB theory,

Y.Takei: Integral representation for ordinary differential equations of Laplace typeand exact WKB analysis, RIMS K\^oky\^uroku, No.1168, pp.80-92, 2000