Global structure of solutions for differential equations of singular perturbation type and exact WKB analysis

2023 Fiscal Year Final Research Report

• Project/Area Number: 19H01794
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Doshisha University
• Principal Investigator: TAKEI Yoshitsugu 同志社大学, 理工学部, 教授 (00212019)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 関数方程式論 / 漸近解析 / 代数解析 / 完全WKB解析 / パンルヴェ方程式 / 超幾何函数 / ホロノミック系 / ヴォロス係数

Outline of Final Research Achievements

To establish the exact WKB analysis for systems of differential equations including nonlinear equations and difference equations we study Painleve equations and hypergeometric systems from the viewpoint of the exact WKB analysis. Consequently the problem of analytic interpretation of instanton solutions for Riccati equations, which can be considered as prototype of Painleve equations, is solved and further the following new results are obtained: the structure of Stokes geometry of the difference equation for Bessel functions is clarified, Voros coefficients of Weber functions are determined and integral representations of Gauss' hypergeometric functions are derived by utilizing a system of differential-difference equations. It is also shown that WKB-type formal solutions of initial value problems for time-dependent Schrodinger equations can be analytically interpreted in several simple cases through stochastic differential equations.

Free Research Field

微分方程式の完全WKB解析

Academic Significance and Societal Importance of the Research Achievements

完全WKB解析が非線型や差分方程式も含む一般の微分方程式系に拡張されれば、その解の大域解析が大きく進展するものと期待され、本研究の研究成果はそれに向けての第一歩と考えられる。例えば、リッカチ方程式のインスタントン解の意味付けの問題の解決はインスタントン解を用いた非線型方程式の大域解析の可能性の証左となる成果であり、ヴォロス係数の決定や積分表示式の導出に連立の微分差分方程式系が有効に用いられたことは差分方程式の完全WKB解析の将来性を保証する。さらに、時間依存シュレディンガー方程式の初期値問題のWKB型の形式解に関する結果は、完全WKB解析と確率微分方程式の思わぬ関連性を示唆している。