New developments of asymptotics for differential equations

• Project/Area Number: 25400114
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: Koike Tatsuya 神戸大学, 理学研究科, 准教授 (80324599)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 完全WKB解析 / 漸近展開 / ボレル総和法 / 特異摂動 / Voros係数 / 漸近解析 / 複素領域における常微分方程式論 / 代数解析 / Borel 総和法 / 多重総和法 / middle convolution / A超幾何方程式系 / 不確定特異点 / Borel総和法 / Voros 係数 / 無限階微分作用素

Outline of Final Research Achievements

In this research we mainly studied Voros coefficients in exact WKB analysis, i.e., WKB method based on the Borel resummation method. Voros coefficients are one of the most important objects to study global behaviors of WKB solutions. We succeeded in clarifying the following topics: (1) the computation of Voros coefficients of singular perturbed higher order linear ordinary differential equations via the middle convolutions, (2) WKB theoretic studies of boosted operators using differential operators of infinite order, (3) the study of the exponential order of the Borel transform of WKB solutions of singular perturbed second order linear ordinary differential equations, and its relation to the multisummability.

Research Products

• [Journal Article] On the singularity structure of wkb solution of the boosted Whittaker equation: its relevance to resurgent functions with essential singularities (2016)
  • Author(s): S. Kamimoto, T. Kawai, T. Koike
  • Journal Title: Lett. in Math. Phys. Volume: 106 Issue: 12 Pages: 1791-1815
  • DOI: 10.1007/s11005-016-0887-x
  • Peer Reviewed / Open Access
• [Journal Article] On the Borel summability of 0-parameter solutions of nonlinear ord inary differential equations (2013)
  • Author(s): S. Kamimoto and T. Koike
  • Journal Title: Kokyuroku Bessatsu Volume: B40 Pages: 191-212
  • Peer Reviewed
• [Journal Article] Exact WKB analysis of second-order non-homogeneous linear ordinary differential equations (2013)
  • Author(s): T. Koike and Y.Takei
  • Journal Title: Kokyuroku Bessatsu Volume: B40 Pages: 293-312
  • Peer Reviewed
• [Presentation] Voros coefficients in exact WKB analysis (2017)
  • Author(s): Tatsuya Koike
  • Organizer: Geometry, Analysis and Mathematical Physics
  • Place of Presentation: 京都大学
  • Year and Date: 2017-02-13
  • Int'l Joint Research / Invited
• [Presentation] 完全WKB解析とBorel総和法 (2016)
  • Author(s): 小池達也
  • Organizer: 超幾何学校2016
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-09-01
  • Invited
• [Presentation] 完全 WKB 解析と multisummability について (2016)
  • Author(s): 小池達也
  • Organizer: 超幾何方程式研究会 2016
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-01-07
• [Presentation] A remark on the growth order of Borel transform of WKB solutions of one-dimensional Schroedinger equations --- Towards a proof of its multisummability (2015)
  • Author(s): T. Koike
  • Organizer: 超局所解析と特異摂動論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-10-05
  • Int'l Joint Research / Invited
• [Presentation] On a connection formula of WKB solutions of the ``boosted'' simple pole type operators of second order in exact WKB analysis (2015)
  • Author(s): Tatsuya Koike
  • Organizer: Seminar of Algebra
  • Place of Presentation: Universidad de Sevilla
  • Year and Date: 2015-02-17
  • Invited
• [Presentation] Introduction to Borel summation method and some applications to differential equations (2015)
  • Author(s): Tatsuya Koike
  • Organizer: Seminar of Algebra
  • Place of Presentation: Universidad de Sevilla
  • Year and Date: 2015-02-12
  • Invited
• [Presentation] On Voros coefficients and middle convolutions for linear ordinary differential equations with a large parameter (2014)
  • Author(s): Tatsuya Koike
  • Organizer: Several aspects of microlocal analysis
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-10-22
  • Invited
• [Presentation] On irregular modified $A$-hypergeometric series and Borel summatio n method
  • Author(s): T. Koike
  • Organizer: Formal and Analytic Solutions of Differential, Difference and Disc rete Equations
  • Place of Presentation: Bedleqo (Poland)
  • Invited
• [Presentation] Applications of linear differential operators of infinite order in resurgent function theory
  • Author(s): T. Koike
  • Organizer: Global Study of Differential Equations in the Complex Domain
  • Place of Presentation: Warsaw, Poland
  • Invited
• [Presentation] On the computation of Voros coefficients via middle convolutions
  • Author(s): T. Koike
  • Organizer: Global Study of Differential Equations in the Complex Domain
  • Place of Presentation: Warsaw, Poland
  • Invited
• [Presentation] On the computation of Voros coefficients via middle convolutions
  • Author(s): T. Koike
  • Organizer: 微分方程式の指数解析とその周辺
  • Place of Presentation: 数理解析研究所, 京都
  • Invited
• [Presentation] On irregular modified A-hypergeometric systems and the Borel summation method
  • Author(s): T. Koike
  • Organizer: Recent progress in the theory of Painlevé equations: algebraic, asymptotic and topological aspects
  • Place of Presentation: Strasbourg, France
  • Invited