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The structure theory of differential equations by the algebraic analysis of singular perturbation theory

Research Project

Project/Area Number 24340026
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

KAWAI Takahiro  京都大学, 数理解析研究所, 名誉教授 (20027379)

Co-Investigator(Kenkyū-buntansha) 竹井 義次  京都大学, 数理解析研究所, 准教授 (00212019)
小池 達也  神戸大学, 理学(系)研究科(研究院), 准教授 (80324599)
Co-Investigator(Renkei-kenkyūsha) AOKI Takashi  近畿大学, 理工学部, 教授 (80159285)
Project Period (FY) 2012-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥13,260,000 (Direct Cost: ¥10,200,000、Indirect Cost: ¥3,060,000)
Fiscal Year 2016: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2015: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2014: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2013: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2012: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Keywords完全WKB解析 / 超局所解析学 / ボレル変換 / 動かない特異点 / alien derivative / 仮想的変わり点 / 無限階微分作用素 / 超局所解析的S-行列論 / 超局所解析 / ストークス図形 / 極大過剰決定系 / Pearcey系 / ランダウ・中西曲面 / ボレル総和 / パンルヴェ函数 / 再生函数 / 陪特性帯 / Whitneyの傘 / Landau-中西幾何学 / 再生函数論 / 完全最急降下法 / ファインマン積分 / acnode / 高階パンルヴェ方程式 / ボレル総和法 / 解析学S-行列論 / 単純極作用素
Outline of Final Research Achievements

We have succeeded in concretely describing the singularity structure of the Borel transformed WKB solutions of several basic one-dimensional Schroedinger equations near their fixed singular points. The main tool for the description is the so-called alien calculus, and the idea is based on the role of a bicharacteristic strip as a carrier of singularities of solutions of linear partial differential equations. Thus our result is an outcome of the complementality of microlocal analysis and the exact WKB analysis. We have also studied higher order ordinary differential equations with a large parameter through its Borel transform ; a virtual turning point is an indispensable ingredient in the exact WKB analysis of higher order equations.
Kawai and Takei, with the collaborator N. Honda(Hokkaido University) published a monograph titled “Virtual Turning Points”(Springer ; 2015), so that this novel notion may be widely appreciated by mathematical physicists and engineers.

Report

(6 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • Research Products

    (44 results)

All 2017 2016 2015 2014 2013 2012 Other

All Journal Article (17 results) (of which Peer Reviewed: 13 results,  Open Access: 10 results,  Acknowledgement Compliant: 8 results) Presentation (24 results) (of which Int'l Joint Research: 5 results,  Invited: 22 results) Book (1 results) Remarks (1 results) Funded Workshop (1 results)

  • [Journal Article] Stokes geometry of higher order ordinary differential equations and middle convolutions2017

    • Author(s)
      T. Motegi and Y. Takei
    • Journal Title

      Advances in Mathematics

      Volume: 310 Pages: 327-376

    • DOI

      10.1016/j.aim.2017.02.006

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] An invitation to Sato's postulates in micro-analytic S-matrix theory2017

    • Author(s)
      N. Honda and T. Kawai
    • Journal Title

      RIMS Kokyuroku-Bessatsu

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On the singularity structure of wkb solution of the boosted Whittaker equation: its relevance to resurgent functions with essential singularities2016

    • 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

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A study of pinch points and cusps in the Landau-Nakanishi geometry2016

    • Author(s)
      N. Honda and T. Kawai
    • Journal Title

      RIMS Kokyuroku-Bessatsu

      Volume: B57 Pages: 175-194

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations2015

    • Author(s)
      Yoshitsugu Takei
    • Journal Title

      Opuscula Mathematica

      Volume: 35 Issue: 5 Pages: 775-802

    • DOI

      10.7494/opmath.2015.35.5.775

    • NAID

      120005663283

    • Related Report
      2015 Annual Research Report 2014 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A study of pinch points and cusps in the Landau-Nakanishi geometry2015

    • Author(s)
      N. Honda and T. Kawai
    • Journal Title

      RIMS Preprint

      Volume: 1840 Pages: 1-39

    • Related Report
      2015 Annual Research Report
    • Open Access / Acknowledgement Compliant
  • [Journal Article] On the geometric aspect of Sato’s postulates on the S-matrix2014

    • Author(s)
      N. Honda, T. Kawai and H. Stapp
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B52 Pages: 11-53

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Visualization of the Landau-Nakanishi surfaces for truss-bridge graphs2014

    • Author(s)
      N. Honda and T. Kawai
    • Journal Title

      RIMS-Preprint

      Volume: 1809 Pages: 1-95

    • Related Report
      2014 Annual Research Report
    • Open Access / Acknowledgement Compliant
  • [Journal Article] Resurgent functions and linear differential operators of infinite order, ― Their happy marriage in exact WKB analysis2014

    • Author(s)
      S. Kamimoto, T. Kawai and T. Koike
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B52 Pages: 127-146

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On the fourth order PI equation and coalescing phenomena of nonlinear turning points2014

    • Author(s)
      Y. Takei
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B52 Pages: 301-316

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Le prolongement analytique de la solution du probleme de Cauchy pour un certain operateur differentiel a coefficients polynomiaux. II2014

    • Author(s)
      Yusaku Hamada, Takashi Okaji and Yoshitsugu Takei
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: (online version) Issue: 1 Pages: 1-36

    • DOI

      10.1007/s11784-014-0152-9

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exact WKB analysis of a Schrodinger equation with a merging triplet of two simple poles and one simple turning point. I2014

    • Author(s)
      Shingo Kamimoto, Takahiro Kawai and Yoshitsugu Takei
    • Journal Title

      Advances in Mathematics

      Volume: (online version) Pages: 458-564

    • DOI

      10.1016/j.aim.2014.02.026

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point. I& II2014

    • Author(s)
      S. Kamimoto, T. Kawai and Y. Takei
    • Journal Title

      Advances in Mathematics

      Volume: 260巻 Pages: 565-613

    • DOI

      10.1016/j.aim.2014.02.028

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exact WKB analysis of second-order non-homogeneous linear ordinary differential equations2013

    • Author(s)
      Tatsuya Koike and Yoshitsugu Takei
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B40 Pages: 293-312

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Le prolongement analytique de la solution du problème de Cauchy pour un certain opérateur différentiel a coefficients polynomiaux. II& III2013

    • Author(s)
      Y. Hamada, T. Okaji and Y. Takei
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 14巻 Issue: 1 Pages: 1-15

    • DOI

      10.1007/s11784-014-0153-8

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A computer-assisted study of the Landau-Nakanishi geometry2013

    • Author(s)
      N. Honda and T. Kawai
    • Journal Title

      RIMS Kokyuroku

      Volume: 1861 Pages: 100-110

    • Related Report
      2013 Annual Research Report
  • [Journal Article] Exact WKB analysis of second-order non-homogeneous linear ordinary differential equations.2012

    • Author(s)
      T. Koike and Y. Takei
    • Journal Title

      RIMS-Preprint

      Volume: 1760 Pages: 1-20

    • Related Report
      2012 Annual Research Report
  • [Presentation] On virtual turning points ― an important ingredient of the WKB theory of higher order ODEs2016

    • Author(s)
      Y. Takei and T. Kawai
    • Organizer
      Resurgence at Kavli IPMU
    • Place of Presentation
      東京大学カブリ数物研究所(千葉県柏市)
    • Year and Date
      2016-12-16
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On some recent results in the theory of virtual turning points2016

    • Author(s)
      Y. Takei, S. Hirose and T. Kawai
    • Organizer
      RIMS symposium “New development of microlocal analysis and singular perturbation theory”
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-10-06
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A unified treatment of pinch points and cusps in Landau-Nakanishi surfaces with microlocal analysis2016

    • Author(s)
      N. Honda and T. Kawai
    • Organizer
      RIMS symposium “New development of microlocal analysis and singular perturbation theory”
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-10-06
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 離散パンルベ方程式のストークス現象について2015

    • Author(s)
      竹井 義次
    • Organizer
      研究集会「第22回超局所解析と古典解析」
    • Place of Presentation
      富山県民会館(富山県富山市)
    • Year and Date
      2015-12-06
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] A remark on the growth order of Borel transform of WKB solutions of one-dimensional Schroedinger equation2015

    • Author(s)
      小池 達也
    • Organizer
      RIMS研究集会“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(京都府京都市)
    • Year and Date
      2015-10-09
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] On the exact WKB analysis of discrete Painleve equations2015

    • Author(s)
      竹井 義次
    • Organizer
      RIMS研究集会“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(京都府京都市)
    • Year and Date
      2015-10-09
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] An invitation to Sato's postulates in micro-analytic S-matrix theory2015

    • Author(s)
      河合 隆裕
    • Organizer
      RIMS研究集会“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(京都府京都市)
    • Year and Date
      2015-10-06
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] WKB analysis for the discrete Painleve equation2015

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      International Conference "Analytic, Algebraic and Geometric Aspects of Differential Equations"
    • Place of Presentation
      The Mathematical Research and Conference Center in Bedlewo(Poland)
    • Year and Date
      2015-09-17
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] WKB analysis and Stokes geometry of differential equations2015

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      Summer School "Analytic, Algebraic and Geometric Aspects of Differential Equations"
    • Place of Presentation
      The Mathematical Research and Conference Center in Bedlewo(Poland)
    • Year and Date
      2015-09-07
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Exact WKB analysis of simple pole type operators ― alien calculus using the differential operators of infinite order2014

    • Author(s)
      S. Kamimoto, T. Kawai and T. Koike
    • Organizer
      Several aspects of microlocal analysis
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-24
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] A study of pinch points and cups in the Landau-Nakanishi geometry2014

    • Author(s)
      N. Honda and T. Kawai
    • Organizer
      Several aspects of microlocal analysis
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-23
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On Voros coefficients and middle convolution for linear ordinary differential equations with a large parameter2014

    • Author(s)
      T. Koike
    • Organizer
      Several aspects of microlocal analysis
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-23
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Exact WKB analysis and multi-summability.2012

    • Author(s)
      Y. Takei
    • Organizer
      RIMS Symposium
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Year and Date
      2012-10-25
    • Related Report
      2012 Annual Research Report
  • [Presentation] Micro-analytic S-matrix theory revisited.2012

    • Author(s)
      N. Honda and T. Kawai
    • Organizer
      RIMS Symposium
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Year and Date
      2012-10-24
    • Related Report
      2012 Annual Research Report
  • [Presentation] On the turning point problem for Painleve equations with a large parameter

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      The 6th Pacific RIM Conference
    • Place of Presentation
      札幌コンベンションセンター
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The fourth-order PI equation and coalescing phenomena of nonlinear turning points

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      Formal and Analytic Solutions of Differential, Difference and Discrete Equations
    • Place of Presentation
      Mathematical Research and Conference Center, Bedlewo, Poland
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] On the exact WKB analysis for higher order differential equations

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      Global Study of Differential Equations in the Complex Domain
    • Place of Presentation
      Banach Center, Warsaw, Poland
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Linear differential operators of infinite order --- another interaction of microlocal analysis and exact WKB analysis

    • Author(s)
      Takahiro Kawai
    • Organizer
      Microlocal analysis and spectral theory
    • Place of Presentation
      Centre International de Rencontres Mathematiques, Luminy, France
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Sato’s postulates on the S-matrix revisited through the Borel resummation of the perturbation series (a joint work with Henry P. Stapp)

    • Author(s)
      Takahiro Kawai
    • Organizer
      微分方程式の指数解析とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] On the computation of Voros coefficients via middle convolutions (joint work with K. Iwaki)

    • Author(s)
      Tatsuya Koike
    • Organizer
      微分方程式の指数解析とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] A happy marriage of resurgent functions and linear differential operators of infinite order in exact WKB analysis

    • Author(s)
      Shingo Kamimoto
    • Organizer
      微分方程式の指数解析とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The fourth-order PI equation and coalescing phenomena of nonlinear turning points

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      微分方程式の指数解析とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The fourth-order PI equation and coalescing phenomena of nonlinear turning points

    • Author(s)
      Yoshitsugu Takei
    • Organizer
      Recent progress in the theory of Painleve equations: algebraic, asymptotic and topological aspects
    • Place of Presentation
      Strasbourg大学, France
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 特異摂動の代数解析学概説 --- 指数的に微小な項の厳密な取り扱いをめぐって ---

    • Author(s)
      竹井 義次
    • Organizer
      2014年度日本数学会年会 企画特別講演
    • Place of Presentation
      学習院大学
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Book] Virtual Turning Points2015

    • Author(s)
      N. Honda, T. Kawai and Y. Takei
    • Total Pages
      133
    • Publisher
      Springer
    • Related Report
      2015 Annual Research Report
  • [Remarks] RIMS Preprint

    • URL

      http://www.kurims.kyoto-u.ac.jp/preprint/index.html

    • Related Report
      2014 Annual Research Report
  • [Funded Workshop] RIMS symposium“New development of microlocal analysis and singular perturbation theory”2016

    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-10-03
    • Related Report
      2016 Annual Research Report

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Published: 2012-04-24   Modified: 2019-07-29  

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