WKB analysis and small denominators forvector fields
Masafumi Yoshino1
Graduate School of Sciences, Hiroshima University
Hiroshima, Japan
1. Introduction
In this talk we shall study the exact asymptotic analysis of a so-called
homology equation and the normal form theory of a singular vector
field. A homology equation is a system of partial differential equations which appear in linearizing a singular vector field by the changeof independent variables. We shall introduce a (large) parameter inthe homology equation by taking the eigenvalue(s) of the linearizedvector field as a new parameter. This new parameter agrees with theone introduced by Aoki-Kawai-Takei for the Painlevé equation if werestrict the variable in the homology equation so that the homology
equation reduces to the symmetric form of a Painlevé equation. (cf.
[7]) By constructing the WKB solution of a homology equation, we will
see that the classical Poincaré series solution of the homology equation
coincides with the WKB solution via resummation procedure (and analytic continuation). Next we discuss the small denominator problem
via resummed WKB solution and the appearence of a natural boundaryof WKB solution.
2. Homology equation
Let x = (x1 , . . . , xn ) ∈ Cn , n ≥ 2 be the variable in Cn . We consider
a singular vector field near the origin of Cn
X =n
X
j=1

aj (x)∂,∂xj

aj (0) = 0,

j = 1, . . . , n,where aj (x) (j = 1, 2, . . . , n) are holomorphic in some neighborhood ofthe origin. We set
X(x) = (a1 (x), . . . , an (x)),
and write∂∂∂=(,... ,),∂x∂x1∂xn∂, X(x) = Λx + R(x),∂x
R(x) = (R1 (x), . . . , Rn (x)), R(x) = O(|x|2 ),X = X(x) ·where Λ is an n-square constant matrix.
1e-mail: yoshino@math.sci.hiroshima-u.ac.jp,
Supported by Grant-in-Aid for Scientific Research (No. 16654028), Ministry ofEducation, Science and Culture, Japan.
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We want to linearize X by the change of variables,(T )x = u(y),u = (u1 , . . . , un ),namely,µ ¶−1∂y ∂∂x∂∂X(u(y))= X(u(y))= Λy .
∂x ∂y∂y∂y∂y
It follows that u satisfies the equationµ ¶−1∂u= Λy,X(u(y))∂ythat is∂u≡ Lu.
∂y
Now we introduce a parameter η −1 in the homology equation by regarding the eigenvalues of Λ as a new parameter. It follows that weobtainη −1 Luj = λj uj + Rj (u), j = 1, . . . , n.
In the following we write x instead of y for the sake of simplicity.
If u = x + v(x), then by the same argument we obtainΛu + R(u) = Λy(∗)η

η −1 Lvj = λj vj + Rj (x + v(x)),

j = 1, . . . , n.

Remark. If we restrict the variable of the homology equation to one
variable appropriately, then the homology equation is reduced to thesymmetric form of the Painlevé equation. Clearly, one can introduce aparameter η in the Schorödinger equation as the inverse of a Planck constant. Then by the monodromy preserving deformation, Aoki-KawaiTakei introduced a parameter in the (symmetric form of ) Painlevé
equation.
η −1 U10 = λ1 + U1 (U2 − U3 ),η −1 U20 = λ2 + U2 (U3 − U1 ),η −1 U30 = λ3 + U3 (U1 − U2 ).
The above introduction of a large parameter in the homology equationagrees with that of the Painlevé equation, if the homology equation is
reduced to the symmetric form of a Painlevé equation.
3. WKB solution
A WKB solution (0 - instanton solution)
A WKB solution (0 - instanton solution) v(x, η) of (∗)η is a formal
power series solution of the form
∞Xv(x, η) =η −ν vν (x) = v0 (x) + η −1 v1 (x) + · · · ,ν=03

where the series is a formal power series in η with coefficients vν (x)holomorphic vector functions in x in some open set independent of ν.
Definition (turning point). The point x such that(3.1)

det (Λ + (∂R/∂z)(x + v0 )) = 0

is called a turning point of the equation (∗)η .
Remark. If(A.1)

λj 6= 0,

j = 1, . . . , n,then the origin x = 0 is not a turning point of (∗)η for any v0 , because
det Λ 6= 0. Then, we have
Proposition Assume that det Λ 6= 0. Then every coefficients of a
WKB solution is uniquely determined as a holomorphic function of x
in a neihborhood of the origin x = 0 independent of ν.
Definition (Resonance condition). We say that η is resonant, if(3.2)

nXλi αi − ηλj = 0,i=1

for some α = (α1 , . . . , αn ) ∈ Zn+ , |α| ≥ 2 and j, 1 ≤ j ≤ n. If η is notresonant, we say that η is nonresonant.
Definition (Poincaré condition) We say that a homolgy equation
satisfies a Poincaré condition, if the convex hull of λj , (j = 1, . . . , n)in the complex plane does not contain the origin.
4. Summability of a WKB solution in a Poincaré region
For the direction ξ, (0 ≤ ξ < 2π) and the opening θ > 0 we define
the sector Sξ,θ by
¾
½
θ
(4.1)
,Sξ,θ = η ∈ C; |Argη − ξ| <2
where the branch of the argument is the principal value. Then we have
Theorem 1. (Resummation) Suppose that(C)| Arg λj | <π,
4j = 1, . . . , n.

Then, there exist a direction ξ, an opening θ > π, a neighborhood Uof the origin x = 0 and V (x, η) such that V (x, η) is holomorphic in(x, η) ∈ U × Sξ,θ and satisfies (∗)η . The function V (x, η) is a Borel sumof the WKB solution v(x, η) in U × Sξ,θ when η → ∞ in the following
sense. Namely, for every N ≥ 1 and R > 0 , there exist C > 0 and4K > 0 such that¯¯N
¯¯X
¯
¯−ν
(4.2)
η vν (x)¯ ≤ CK N N !|η|−N −1 ,¯V (x, η) −¯¯ν=0∀(x, η) ∈ U × Sξ,θ , |η| ≥ R.
5. A Poincaré solution via analytic continuation of aWKB solution
We shall make an analytic continuation (with respect to η) of a
resummed WKB solution to the right half plane. We note that there
exist an infinte number of resonaces on the right-half plane < η > 0
which accumulate only at infinity. The solution may be singular withrespect to η at the resonances. We have
Theorem 2. Suppose that (C) is verified. Then the resummed WKB
solution is analytically continued to the right half plane as a singlevalued function except for resonances. If the nonresonance condition
holds, then the analytic continuation of a resummed WKB solution
to η = 1 coincides with a classical Poincaré solution of a homology
equation.
6. WKB solution in a Siegel domain- small denominators
In this section we assume that we are in a Siegel domain. Moreover,
we assume, for the sake of simplicity
λj ∈ R (j = 1, 2, . . . , n) are linearly independent over Q.
Then the set of all resonances is dense on R. This implies that theresummed WKB solution has a natural boundary on the real axis.
We will study the small denominator problem from the viewpoint of a
WKB analysis.
References[1] T. Aoki, T. Kawai, T. Koike and Y. Takei: On the exact WKB analysis ofoperators admitting infinitely many phases, Adv. Math., 181(2004), 165-189.
[2] J. Guckenheimer, Hartman theorem for complex flows in the Poincaré domain,
Compositio Math., 24:1, 75–82 (1972)[3] Y. Il’yashenko: Divergence of series reducing an analytic differential equation
to linear form at a singular point. Funct. Anal. and Appl. 13, 227-229 (1979).
[4] T. Kawai and Y. Takei: On WKB analysis of higher order Painlevé equations
with a large parameter, Proc. Japan Acad., Ser. A, 80(2004), 53-56.
[5] T. Kawai and Y. Takei: WKB analysis of higher order Painlevé equations witha large parameter — Local reduction of 0-parameter solutions for Painlevé
hierarchies (PJ ) (J = I, II-1orII-2), preprint (RIMS-1468).
[6] S. Sternberg: The structure of local homeomorphisms II, III, Amer. J. Math80, 623-632 and 81, 578-604 (1958)[7] Y. Takei: Toward the exact WKB analysis for higher-order Painlevé equations— The case of Noumi-Yamada systems, Publ. RIMS, Kyoto Univ., 40(2004),709-730.

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Present adress: Graduate School of Sciences, Hiroshima University, HigashiHiroshima, 739-8526, Japan. e-mail: yoshino@math.sci.hiroshima-u.ac.jp