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Semiclassical approximations and hyperasymptotics

Research Project

Project/Area Number 10640148
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionMIYAGI UNIVERSITY OF EDUCATION

Principal Investigator

HURUKI Yamada  Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (00092578)

Co-Investigator(Kenkyū-buntansha) TAKEMOTO Hideo  Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (00004408)
SHIRAI Susumu  Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (30115175)
AZUMA Kazuoki  Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (70005776)
TAKASE Koichi  Faculty of Education, Miyagi University of Education Associate Professor, 教育学部, 助教授 (60197093)
URYU Hitoshi  Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (10139511)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordssemiclassical approximation / hyperasymptotics / WKB approximation / resurgent function / Schroedinger equation
Research Abstract

In connection with the theory of semiclassical approximations, we studied the following problem. How can we treat the exponentially small terms in the asymptotic analysis. To do so, we studied some useful tools such as the method of hyperasymptotics, the theory of resurgent functions and the so-called exact WKB-method.
By using those tools, we got some hints and results as follows: (1) For 1-dimensional stationary Schroedinger equations with polynomial potentials, we treated the exponentially small terms as the influences of distant wells. In the research of this problem, we used our previous results for 1-dimensional stationary Schroedinger equations with piecewise constant potentials. In them, explicit representations of the quantum condition and the characterization of then by means of integral forms were shown. (2) We considered the asymptotic expansions of functions represented by some integral. The method to take out the exponentially small quantities as the influences of not nearby singular points of the phase function of the integral was searched. In so doing, we examined the method of steepest descent in a new viewpoint. The essential point is, if we deform the path of integration, it is essential not only to take the path through saddle points, but to take exactly the steepest paths.
Further, points of view essentially used were: (1) In considering the relations between asymptotic expansion and integral by means of Borel-Laplace transformation of the original expansion, the analysis of singularities of Borel transformed function in the complex domain is indispensable, and information of these singular points are include in the coefficients of original asymptotic expansions. (2) To characterize functions represented by asymptotic expansions as exact quantities, it is necessary to take the resummation on complex Lagrangian manifolds. Our researches are now in continuation and some of our results will be published in order.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report

Research Products

(19 results)

All Other

All Publications (19 results)

  • [Publications] Hideo Takemoto: "On a Saito's problem for the generations of von Neumann algebras by power partial isometries"Nihonkai Math.J.. 9-1. 97-104 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideo Takemoto: "A characterization of the power partially isometric operators"Bull. Of Miyagi Univ. of Education. 33. 41-45 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "On Siegel modular forms of half-integral weights and Jacobi forms"Trans.Amer.Math.Soc.. 351. 735-780 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "On the transformation formula of Riemann's theta series"京大数理解析研究所講究録. 1052. 99-111 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "T.Ibukiyama, H.Saito "On zeta functions associated to symmetric matrices I"の紹介"第1回整数論オータムワークショップ報告集. 47-80 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideo Takemoto: "On a Saitos problem for the generations of von Neumann algebras by power partial isometries"Nihonkai Math.J.. 9. 97-104 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideo Takemoto: "A characterization of the power partially isometric operators"Bull. Of Miyagi Univ. of Education. 33. 41-45 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "On Siegel modular forms of half-integral weight and Jacobi forms"Trans.AMS. 351. 735-780 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "On the transformation formulas Riemanns theta series"RIMS Kokyuroku. 1052. 99-111 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Koichi Takase: "Commentary of the paper "On zeta functions associated to symmetric matrices I" by T.Ibukiyama and H.Saito"Report of the First Autumm Workshop on Number Theory. 47-80 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hideo Takemoto: "On a Saito's problem for the generations of von Neumann algebras by power partial isometries"Nihonkai Math.J.. 9-1. 97-104 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hideo Takemoto: "A characterization of the power partially isometic operatus"Bulletin of Miyagi Univ.of Education. 33. 41-45 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Koichi Takase: "On Siegel modular forms of half-integral weights and Jacobi forms"Trans.Amer.Math.Soc. 351. 735-780 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Koichi Takase: "On the transformation formula of Riemann's theta Series"数理解析研究所講究録. 1052. 99-111 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Koichi Takase: "T. Ibukiyama, H. Saito "On zeta functions associated to symmetric matrices 1"の紹介"第1回整数論オータムワークショップ報告集. 47-80 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hideo Takemoto: "On a Saito's problem for the generations of von Neumann algebras by powerpartial isometries" Nihonkai Math.J.9-1. 97-104 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hideo Takemoto: "A Characterization of the power partially isometric operators" Bulletin of Miyagi Univ.of Education. 33(印刷中). (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kouichi Takase: "On Siegel modular forms of half-integral weights and Jacobi forms" Trans.Amer.Math.Soc.351. 735-780 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kouichi Takase: "“On Zeta functions associated to symmetric matrises I:an explicit form of zeta functions"の紹介" 整数論オータムワークショップ報告集. (to appear). (1999)

    • Related Report
      1998 Annual Research Report

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Published: 1998-03-31   Modified: 2016-04-21  

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