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Research on Functional Analysis and Mathematical theory of Feynman path integrals.

Research Project

Project/Area Number 15540184
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionGakushuin University.

Principal Investigator

FUJIWARA Daisuke  Gakushuin Univ., Dept. of Math., Prof., 理学部, 教授 (10011561)

Co-Investigator(Kenkyū-buntansha) YAJIMA Kenji  Gakushuin Univ., Dept. of Math., Prof., 理学部, 教授 (80011758)
KATASE Kiyoshi  Gakushuin Univ., Dept. of Math., Prof., 理学部, 教授 (70080489)
MIZUTANI Akira  Gakushuin Univ., Dept. of Math., Prof., 理学部, 教授 (80011716)
WATANABE Kazuo  Gakushuin Univ., Dept. of Math., Assist., 理学部, 助手 (90260851)
SHIMOMURA Akihiro  Gakushuin Univ., Dept. of Math., Assist., 理学部, 助手 (00365066)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsFeynman path integrals / Oscillatory integrals / Schrodinger equation / Stationary phase / Selfajoint operator / Quantum mechanics / WKB-method / path integrals
Research Abstract

1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. He proved an improved remainder estimate of stationary phase method for oscillatory integrals over a space of large dimension. And he make the discussion of Kumano-go's results on convergence of Feynman path integrals. He also proved a new formula for the second term of the semi-classical asymptotics of Feynman path integrals.
2. Yajima got results on spectrum and scattering properties of Nelson model, which is a simplified model of non-relativistic QED. He also succeeded in proving that every solution of Schrodinger equation with potentials which grow of oder O(|x|^m), m > 2 at the infinity gains, at almost every t, differentiability of order 1/m compared with its initial value.
3. Watanabe made research on solutions of PDE with dispersive type. In the joint with T. Suzuki and T. Kobayashi he also proved interface regularity of solutions to the system of Maxwell-Stokes equations. He also discussed.
4. Shimomura discussed large time behaviour of solutions to non-linear Schrodinger equations. He succeeded in finding a quite a fine property of solutions which is not foreseen from mere growth order of nonlinear term. Shimomura also proved smoothing effects of time local solution to non-linear Schrodinger equation with electro-magnetic potential.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (45 results)

All 2005 2004 2003 Other

All Journal Article (39 results) Publications (6 results)

  • [Journal Article] Smooth functional derivatives in Feynman path integrals by time slicing approximation2005

    • Author(s)
      Fujiwara, D., Kumano-go, N.
    • Journal Title

      Bulletin des Sciences Mathematiques 129

      Pages: 57-79

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Nonexistence of asymptotically free solutions for quadratic nonlinear Schrodinger equations in two space dimensions2005

    • Author(s)
      Shimomura, A.
    • Journal Title

      Differential Integral Equations 18

      Pages: 325-335

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Smooth functional derivatives in Feynman path integrals by time slicing approximation.2005

    • Author(s)
      Fujiwara D, Kumano-go N
    • Journal Title

      Bull. des Sc. Math. 129

      Pages: 57-79

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nonexistence of asymptotically free solutions for quadratic nonlinear Schrodinger equations in two space dimensions2005

    • Author(s)
      Shimomura A
    • Journal Title

      Diff. Integ. Eq. 18

      Pages: 325-335

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Local smoothing property and Strichartz inequality for Schrodinger equations with potentials superquadratic at infinity.2004

    • Author(s)
      Yajima, K., Zhang, G.
    • Journal Title

      J.Differential Equations 202

      Pages: 81-110

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Local time-decay of solutions to Schrodinger equations with time-periodic potentials.2004

    • Author(s)
      Galtbayar, A., Jensen, A., Yajima, K.
    • Journal Title

      J.Statist.Phys. 116

      Pages: 231-282

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Long range scattering for nonlinear Schrodinger equations in one and two space dimensions2004

    • Author(s)
      Shimomura, A., Tonegawa, S
    • Journal Title

      Differential Integral Equations 17

      Pages: 1-16

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Modified wave operators for nonlinear Schrodinger equations in one and two dimensions2004

    • Author(s)
      Hayashi, N., Naumkin, P.I., Shimomura, A., Tonegawa, S.
    • Journal Title

      Electronic Journal of Differential Equations 2004

      Pages: 1-16

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Wave operators for the coupled Klein-Gordon-Shrodinger equations in two space dimensions2004

    • Author(s)
      Shimomura, A.
    • Journal Title

      Funkcial.Ekvac. 47

      Pages: 63-82

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for Zakharov equations in three space dimensions with large data2004

    • Author(s)
      Shimomura, A.
    • Journal Title

      Commun.Contemp.Math. 6

      Pages: 881-899

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Local smoothing property and Strichartz inequality for Schrodinger equations with potentials superquadratic at infinity.2004

    • Author(s)
      Yajima K, Zhang G
    • Journal Title

      J. Differential Equations 202

      Pages: 81-110

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Local time-decay of solutions to Schrodinger equations with time-periodic potentials.2004

    • Author(s)
      Galtbayar A, Jensen A, Yajima K
    • Journal Title

      J. Statist. Phys. 116

      Pages: 231-282

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Long range scattering for nonlinear Schrodinger equations in one and two space dimensions.2004

    • Author(s)
      Shimomura A, Tonegawa S
    • Journal Title

      Diff. Integ. Eq. 17

      Pages: 1-16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Modified wave operators for nonlinear Schrodinger equations in one and two dimensions2004

    • Author(s)
      Hayashi N, Naumkin P.I., Shimomura A, Tonegawa S
    • Journal Title

      Elect. J. Diff. Eq. 2004

      Pages: 1-16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions2004

    • Author(s)
      Shimomura A
    • Journal Title

      Funkcial. Ekvac. 47

      Pages: 63-82

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for Zakharov equations in three space dimensions with large data2004

    • Author(s)
      Shimomura A
    • Journal Title

      Commun. Contemp. Math. 6

      Pages: 881-899

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Modified wave operators for nonlinear Schrodinger equations in one and two dimensions,2004

    • Author(s)
      Hayashi, N, Naumkin, P.I., Shimomura, A., Tonegawa, S.
    • Journal Title

      Electronic Journal of Differential Equations 2004

      Pages: 1-16

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions,2004

    • Author(s)
      Shimomura, A.
    • Journal Title

      Funkcial.Ekvac. 47

      Pages: 63-82

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Scattering theory for Zakharov equations in three space dimensions with large data,2004

    • Author(s)
      Shimomura, A.
    • Journal Title

      Commun.Contemp.Math. 6

      Pages: 881-899

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Nelson model with less than two photons.2003

    • Author(s)
      Galtbayar, A., Jensen, A., Yajima, K.
    • Journal Title

      Ann.Henri Poincare 4

      Pages: 239-273

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Interface regularity for the Maxwell and Stokes systems2003

    • Author(s)
      Watanabe, K., Kobayashi, T., Suzuki, T.
    • Journal Title

      Osaka J.Math. 40

      Pages: 925-944

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Modified wave operators for the coupled Wave-Schrodinger equations in three space dimensions2003

    • Author(s)
      Shimomura, A.
    • Journal Title

      Discrete Contin.Dyn.Syst. 9

      Pages: 1571-1586

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions2003

    • Author(s)
      Shimomura, A.
    • Journal Title

      J.Math.Sci.Univ.Tokyo. 10

      Pages: 661-685

    • NAID

      110000071241

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Interface regularity for the Maxwell and Stokes systems2003

    • Author(s)
      Watanabe K, Kobayashi T, Suzuki T
    • Journal Title

      Osaka J. Math. 40

      Pages: 925-944

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Modified wave operators for the coupled Wave-Schrodinger equations in three space dimensions2003

    • Author(s)
      Shimomura A
    • Journal Title

      Discrete Contin. Dyn. Syst. 9

      Pages: 1571-1586

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions2003

    • Author(s)
      Shimomura A
    • Journal Title

      J. Math. Sci. Univ. Tokyo. 10

      Pages: 661-685

    • NAID

      110000071241

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions,2003

    • Author(s)
      Shimomura, A.
    • Journal Title

      J.Math.Sci.Univ.Tokyo. 10

      Pages: 661-685

    • NAID

      110000071241

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the asymptotics of solutions for some Schrodinger equations with dissipative perturbations of rank one

    • Author(s)
      Kadowaki, M., Nakazawa, H., Watanabe, K.
    • Journal Title

      Hiroshima Math.J. 印刷中

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields

    • Author(s)
      Nakamura, Y., Shimomura, A.
    • Journal Title

      Hokkaido Math.J. 印刷中

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions II

    • Author(s)
      Shimomura, A.
    • Journal Title

      Hokkaido Math.J. 印刷中

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Remarks on long range scattering for nonlinear Schrodinger equations with Stark effects

    • Author(s)
      Shimomura, A., Tonegawa, S.
    • Journal Title

      J.Math.Kyoto Univ. 印刷中

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the asymptotics of solutions for some Schrodinger equations with dissipative perturbations of rank one

    • Author(s)
      Kadowaki M, Nakazawa H, Watanabe K
    • Journal Title

      Hiroshima Math. J. (In press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields

    • Author(s)
      Nakamura Y, Shimomura A
    • Journal Title

      Hokkaido Math. J. (In press.)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions II

    • Author(s)
      Shimomura A
    • Journal Title

      Hokkaido Math. J. (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Remarks on long range scattering for nonlinear Schrodinger equations with Stark effects

    • Author(s)
      Shimomura A, Tonegawa S
    • Journal Title

      J. Math. Kyoto Univ. (In press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the asymptotics of solutions for some Schrodinger equations with dis-sipative perturbations of rank one,

    • Author(s)
      Kadowaki, M., Nakazawa, H., Watanabe, K.
    • Journal Title

      Hiroshima Math.J. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields,

    • Author(s)
      Nakamura, Y., Shimomura, A.
    • Journal Title

      Hokkaido Math.J. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions II,

    • Author(s)
      Shimomura, A.
    • Journal Title

      Hokkaido Math.J. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Remarks on long range scattering for nonlinear Schrodinger equations with Stark effects,

    • Author(s)
      Shimomura, A., Tonegawa, S.
    • Journal Title

      J.Math.Kyoto Univ. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Publications] K.Watanabe, T.Kobayashi, T.Suzuki: "Interface regularity for the Maxwell and Stokes systems"Osaka J.Math.. 40. 925-944 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shimomura: "Modified wave operators for the coupled Wave-Schrodinger equations in three space dimensions"Discrete Contin.Dyn.Syst.. 9. 1571-1586 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shimomura: "Scattering theory for the coupled Klein-Gordon-Schrodinger equations in two space dimensions,"J.Math.Sci.Univ.Tokyo.. 10. 661-685 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shimomura: "Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions"Funkcial.Ekvac.. 47(to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Nakamura, A.Shimomura: "Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields"Hokkaido Math.J.. (to appear)(発行予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Shimomura: "Scattering theory for Zakharov equations in three space dimensions with large data"Commun.Contemp.Math.. (to appear)(発行予定). (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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