Garnier systems and exact WKBanalysis

• Project/Area Number: 21740098
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: KOIKE Tatsuya 神戸大学, 大学院・理学研究科, 准教授 (80324599)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 完全 WKB 解析 / ボレル総和法 / ガルニエ系 / パンルベ方程式 / リッカチ方程式 / 一次元シュレーディンガー方程式 / GKZ 超幾何系 / 完全WKB解析 / 特異摂動論 / GKZ超幾何方程式系 / 特異摂動 / 非斉次常微分方程式 / ホレル総和可能性 / キーワード6 / キーワード7 / キーワード8

Research Abstract

Exact WKB analysis is stuidied for second order linear or non-linear ordinary differential equations, such as Painleve equations and the one-dimensional Schrodinger equation. Main results of this project are giving a proof of Borel summability of WKB solutions to such equations, clarifying some properties of singularities of Borel transforms of WKB solutions.

Research Products

• [Journal Article] On the Voros coefficient for the Whittaker equation with a large parameter - Some progress around Sato's conjecture in exact WKB analysis (2011)
  • Author(s): T. Koikeand Y.Takei
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 47 Pages: 375-395
  • NAID: 110008425912
  • Peer Reviewed
• [Journal Article] On the exact WKB analysis of higher ordersimple-pole type operators (2011)
  • Author(s): T. Kawai, T. Koikeand Y. Takei
  • Journal Title: Adv. Math. Volume: 228 Pages: 63-96
  • NAID: 120003272689
  • Peer Reviewed
• [Journal Article] On a Schrodinger equation with a merging pair of a simple pole and a simple turning point -- Alien calculus of WKB solutions through microlocal analysis (2011)
  • Author(s): S. Kamimoto, T. Kawai, T. Koikeand Y. Takei
  • Journal Title: Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. II. (Publications of the Scuola Normale Superiore / CRM Series) Volume: vol. II Pages: 245-254
  • Peer Reviewed
• [Journal Article] On the Voros coefficient for the Whittaker equation with a large parameter-Some progress around Sato's conjecture in exact WKB analysis (2011)
  • Author(s): T.Koike
  • Journal Title: Publ.RIMS, Kyoto Univ. Volume: 47 Issue: 1 Pages: 375-395
  • DOI: 10.2977/prims/39
  • NAID: 110008425912
  • Peer Reviewed
• [Journal Article] On the exact WKB analysis of higher order simple-pole type operators (2011)
  • Author(s): T.Kawai
  • Journal Title: Adv.in Math. Volume: 228 Issue: 1 Pages: 63-96
  • DOI: 10.1016/j.aim.2011.05.010
  • NAID: 120003272689
  • Peer Reviewed
• [Journal Article] On a Schrodinger equation with a merging pair of a simple pole and a simple turning point-Alien calculus of WKB solutions through microlocal analysis (2011)
  • Author(s): S.Kamimoto, T.Kawai, T.Koike, Y.Takei
  • Journal Title: Asymptotics in Dynamics, Geometry and PDEs ; Generalized Borel Summation (Publications of the Scuola Normale Superiore/CRM Series) Volume: II Pages: 245-254
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the hyperbolic Schwarz map at irregular singular points (2010)
  • Author(s): T. Koike, T. Sasaki and M. Yoshida
  • Journal Title: Funkcialaj Ekvacioj Volume: 53 Pages: 99-132
  • NAID: 130000254361
  • Peer Reviewed
• [Journal Article] On the structure of higher order simple-pole type operators in exact WKB analysis (2010)
  • Author(s): T. Kawai, T. Koike and Y. Takei
  • Journal Title: Funkcialaj Ekvacioj Volume: 53 Pages: 249-276
  • NAID: 130000419144
  • Peer Reviewed
• [Journal Article] On the WKB theoretic structure of a Schrodinger operator with a merging pair of a simple pole and a simple turning point (2010)
  • Author(s): S. Kamimoto, T. Kawai, T. Koikeand Y. Takei
  • Journal Title: Kyoto Journal of Mathematics Volume: 50 Pages: 101-164
  • Peer Reviewed
• [Journal Article] On the structure of higher order simple-pole type operators inexact WKB analysis (2010)
  • Author(s): T.Kawai, T.Koike, Y.Takei
  • Journal Title: Funkcialaj Ekvacioj Volume: 53 Pages: 249-276
  • Peer Reviewed
• [Journal Article] Rereading Langer's influential 1937 JWKB paper: the unnecessary Langer transformation; the two h's (2009)
  • Author(s): T. Koike and H. J. Silverstone
  • Journal Title: J. Phys. A: Math. Theor Volume: 42
  • Peer Reviewed
• [Journal Article] Rereading Langer's influential 1937 JWKB paper : the unnecessary Langer trans-formation ; the two h's (2009)
  • Author(s): T.Koike, H.J.Silverstone
  • Journal Title: J.Phys.A : Math.Theor. 42
  • Peer Reviewed
• [Journal Article] On the Borel summability of WKB theoretic transformation series
  • Author(s): S. Kamimoto and T. Koike
  • Journal Title: To appear in Kokyuroku Bessatsu.
  • Peer Reviewed
• [Journal Article] On the Borel summability of 0-parameter solutions of nonlinear ordinary differential equations
  • Author(s): S. Kamimoto and T. Koike
  • Journal Title: To appear in Kokyuroku Bessatsu.
  • Peer Reviewed
• [Presentation] Exact WKB analysis of second-order non-homogeneous linear differential equations - Bifurcated integral and Borel summability (2011)
  • Author(s): T. Koikeand Y. Takei
  • Organizer: RIMS 研究集会「漸近解析に於ける超局所解析の展望」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-11-18
• [Presentation] On the Borel summability of 0-parameter solutions of nonlinear ordinary differential equations (2011)
  • Author(s): 神本晋吾, 小池達也
  • Organizer: RIMS 研究集会「漸近解析に於ける超局所解析の展望」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-11-18
• [Presentation] On the Borel summability of O-parameter solutions of nonlinear ordinary differential equation (2011)
  • Author(s): S.Kamimoto, T.Koike
  • Organizer: RIMS研究集会漸近解析に於ける超局所解析の展望
  • Place of Presentation: 京都大学(京都府)(招待講演)
  • Year and Date: 2011-11-18
• [Presentation] Exact WKB analysis of second-order non-homogeneous linear differential equations-Bifurcated integral and Borel summability- (2011)
  • Author(s): T.Koike, Y.Takei
  • Organizer: RIMS研究集会漸近解析に於ける超局所解析の展望
  • Place of Presentation: 京都大学(京都府)(招待講演)
  • Year and Date: 2011-11-18
• [Presentation] 合流する単純極と単純変わり点を持つSchoedinger 方程式の完全WKB 解析 (2011)
  • Author(s): 河合隆裕, 神本晋吾,小池達也,竹井義次(京大数理研)
  • Organizer: 日本数学会秋季総合分科会・函数解析学分科会一般講演
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-29
• [Presentation] WKB 解析的変換級数の Borel 総和可能性に関して (2011)
  • Author(s): 神本晋吾,小池達也
  • Organizer: 日本数学会秋季総合分科会・函数解析学分科会一般講演
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-29
• [Presentation] WKB解析的変換級数のBorel総和可能性に関して (2011)
  • Author(s): 神本晋吾, 小池達也
  • Organizer: 日本数学会秋季総合分科会・函数解析学分科会
  • Place of Presentation: 信州大学(長野県)
  • Year and Date: 2011-09-29
• [Presentation] 合流する単純極と単純変わり点を持つSchoedinger方程式の完全WKB解析 (2011)
  • Author(s): 河合隆裕, 神本晋吾, 小池達也, 竹井義次
  • Organizer: 日本数学会秋季総合分科会・函数解析学分科会
  • Place of Presentation: 信州大学(長野県)
  • Year and Date: 2011-09-29
• [Presentation] Borel summability of WKB solutions for one dimmensional Schroedinger equations (2010)
  • Author(s): T. Koike
  • Organizer: RIMS 短期共同研究「Exact WKB analysis - Borel summability of WKB solutions」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 一次元シュレーディンガー方程式の完全 WKB 解析 - 基礎理論と最近の発展について I, II, III (2010)
  • Author(s): 小池達也
  • Organizer: RIMS 研究集会「準古典解析における諸問題」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 1次元シュレーディンガー方程式の完全WKB解析-基礎理論と最近の発展についてI,II,III (2010)
  • Author(s): 小池達也
  • Organizer: RIMS研究集会「準古典解析における諸問題」
  • Place of Presentation: 京都大学数理解析研究所(京都府)(招待講演)
• [Presentation] Borel summability of WKB solutions for one dimmensional Schroedinger equations (2010)
  • Author(s): 小池達也
  • Organizer: RIMS短期共同研究「Exact WKB analysis-Borel summability of WKB solutions」
  • Place of Presentation: 京都大学数理解析研究所(京都府)(招待講演)
• [Presentation] Borel summability of formal solutions of Riccati equations (2009)
  • Author(s): T. Koike
  • Organizer: S￥'eminaire Equations fonctionnelles IRMA
  • Place of Presentation: Strasbourg(フランス)
  • Year and Date: 2009-10-22
• [Presentation] On a Schrodinger operator with a merging pair of a simple pole and a simple turning point, II Computation of Voros coefficients and its consequence (2009)
  • Author(s): T. Koike
  • Organizer: Asymptotics in dynamics, geometry and PDEs; generalized Borel summation
  • Place of Presentation: CRM,Pisa(イタリア)
  • Year and Date: 2009-10-15
• [Presentation] On a Schrodinger operator with a merging pair of a simple pole and a simple turning point, II, Computation of Voros coefficients and its consequence (2009)
  • Author(s): T.Koike
  • Organizer: Asymptotics in dynamics, geometry and PDEs ; gen-eralized Borel summation
  • Place of Presentation: CRM(Pisa,イタリア)
  • Year and Date: 2009-10-15
• [Presentation] On a Schrodinger operator with a merging pair of a simple pole and a simple turning point (2009)
  • Author(s): T. Koike
  • Organizer: II - Computation of Voros coefficients and its consequence,New Development of Asymptotic Analysis and Dynamical Systems
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-18
• [Presentation] On a Schrodinger operator with a merging pair of a simple pole and a simple turning point, II (2009)
  • Author(s): T.Koike
  • Organizer: New Development of Asymptotic Analysis and Dynamical Systems
  • Place of Presentation: RIMS(京都大学)
  • Year and Date: 2009-06-18
• [Presentation] 1次元シュレーディンカー方程式に対するWKB解のsummabilityとVoros係数
  • Author(s): 小池達也
  • Organizer: 広島大学微分方程式セミナー 漸近展開とその周辺
  • Place of Presentation: 広島大学微分方程式セミナー 漸近展開とその周辺
  • Invited