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Mathematical Analysis of Schroedinger equations

Research Project

Project/Area Number 16K05242
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionGakushuin University

Principal Investigator

Yajima Kenji  学習院大学, 理学部, 教授 (80011758)

Research Collaborator Tamura Hideo  
Adachi Tadayoshi  
Ogawa Takayoshi  
Kato Keiichi  
Nakamura Shu  
Arne Jensen  
Horia Cornean  
Gianfausto Dellantonio  
Alessandro Michelangeli  
Raffaele Scandone  
Heinz Siedentop  
Marcel Griesemer  
Abraham Soffer  
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsシュレーディンガー方程式 / シュレーディンガー作用素 / スペクトル理論 / 散乱理論 / 初期値問題 / 波動作用素 / 点相互作用 / ルベーグ空間 / point interaction / 閾値漸近解析 / ディラック作用素 / レゾナンス / 本質的自己共役性 / 準古典近似 / 散乱の波動作用素
Outline of Final Research Achievements

I studied various mathematical problems on Schroedinger equations and obtained following results during the period 2016-18: 1) I obtained a sufficient condition for the existence and uniqueness of the propagator for quantum many body systems which are in an external electro-magnetic field whose potentials increase quadratically-linearly at spatial infinity and which interact each others by potentials which carry very strong local singularities. The propagator preserves a large subspace where energy observable may be defined and computed in a natural way; 2) I decided exponents of Lebesgue spaces in which wave operators of scattering theory for Schroedinger operators are continuous when they are regular or singular at threshold; 3) I studied spectral and scattering theory for Schroeinger operators with point interactions in 2 and 3 dimensions. I obtained the asymptotic expansion of resolvent at threshold and proved wave operators are bounded in Lebesgue spaces of certain order.

Academic Significance and Societal Importance of the Research Achievements

シュレーディンガー方程式は量子力学の運動方程式で初期値問題の一意可解性は運動の一意存在と同意義であるがそのための最も一般的な十分条件は未だに不明である。この研究の成果は問題解決への重要な第一歩である。散乱の波動作用素はシュレーディンガー作用素の関数のルベーグ空間での連続性をフーリエ空間のかけ算作用素の連続性に帰着する。従ってこの研究の成果は線形あるいは非線形問題において様々に利用されている。点相互作用をもつシュレーディンガー方程式は低温量子力学において広く用いられているが、そのスペクトル・散乱理論は十分には理解されていない。この研究の成果はこの分野に新たな知見を与えるものである。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (19 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (2 results) Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 5 results,  Acknowledgement Compliant: 2 results) Presentation (10 results) (of which Int'l Joint Research: 7 results,  Invited: 8 results) Book (1 results)

  • [Int'l Joint Research] SISSA (Treieste)(Italy)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Aalborg University/Department of Mathematics(Denmark)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Two-dimensional Schroedinger operators with point interactions: Threshold expansions, zero modes and Lp-boundedness of wave operators2019

    • Author(s)
      Cornean Horia D.、Michelangeli Alessandro、Yajima Kenji
    • Journal Title

      Reviews in Mathematical Physics

      Volume: 31 Issue: 04 Pages: 1950012-1950012

    • DOI

      10.1142/s0129055x19500120

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Instability of Resonances Under Stark Perturbations2018

    • Author(s)
      Jensen Arne、Yajima Kenji
    • Journal Title

      Annales Henri Poincare

      Volume: 20 Issue: 2 Pages: 675-687

    • DOI

      10.1007/s00023-018-0746-7

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] L^1 and L^∞-boundedness of wave operators for three dimensional Schroedinger operators with threshold singularities.2018

    • Author(s)
      Yajima Kenji
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 41 Issue: 2

    • DOI

      10.3836/tjm/1502179271

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] L^p -boundedness of wave operators for the three-dimensional multi-centre point interaction2018

    • Author(s)
      G. Dell'Antonio, A. Michelangeli, R. Scandone, K. Yajima
    • Journal Title

      Annales Henri Poincare

      Volume: 19 Issue: 1 Pages: 283-322

    • DOI

      10.1007/s00023-017-0628-4

    • Related Report
      2017 Research-status Report
  • [Journal Article] Existence and regularity of propagators for multi-particle Schroedinger equations in external fields2016

    • Author(s)
      Kenji Yajima
    • Journal Title

      Communications in Mathematical Physics

      Volume: 347 Issue: 1 Pages: 103-126

    • DOI

      10.1007/s00220-016-2582-2

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Remarks on Lp-boundedness of wave operators for Schroedinger operators with threshold singularities.2016

    • Author(s)
      Kenji Yajima
    • Journal Title

      Documenta Mathematica

      Volume: 21 Pages: 391-443

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] L^p boundedness of wave operators2019

    • Author(s)
      谷島 賢二
    • Organizer
      Q-Math 2019
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximation of point interactions by regular potentials2019

    • Author(s)
      谷島 賢二
    • Organizer
      Spectral methods in mathematical physics in Mittag-Leffler
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Two dimensional Schrodinger operators with point interactions2018

    • Author(s)
      谷島 賢二
    • Organizer
      Recent results on qunatum many-body systems
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Schrodinger operators with point interactions in two and three dimensions2018

    • Author(s)
      谷島 賢二
    • Organizer
      Fundamental Problems in Mathematical and Theoretical Physics (Waseda University)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The L^p-boundedness of wave operators for three dimensional Schroedinger operators with point interactions2017

    • Author(s)
      Kenji Yajima
    • Organizer
      Center for Advanced Studies, LMU-Muenchen
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On wave operators for Schroedinger operators with threshold singuralities in three dimensions2017

    • Author(s)
      Kenji Yajima
    • Organizer
      Work-shop on Analysis of effective one particle equations and their derivation
    • Related Report
      2017 Research-status Report
  • [Presentation] Schroedinger 方程式の propagator について2016

    • Author(s)
      谷島 賢二
    • Organizer
      第27回研究集会「数理物理と微分方程式」
    • Place of Presentation
      かんぽの宿富山市婦中町羽根5691-2
    • Year and Date
      2016-11-26
    • Related Report
      2016 Research-status Report
  • [Presentation] L^p boundednedd of wave operators for Schroedinger operators2016

    • Author(s)
      Kenji Yajima
    • Organizer
      Mathematical challenges of zero-range physics; rigorous results and open problems
    • Place of Presentation
      SiSSA, Trieste, Italy
    • Year and Date
      2016-11-07
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence and regularity of propagators for multi-particle Schroedinger equations2016

    • Author(s)
      Kenji Yajima
    • Organizer
      Contemporary trends in the mathematics of quantum mechanics
    • Place of Presentation
      INdAM headquarters, Piazzale Aldo Moro 5, 00185 Roma
    • Year and Date
      2016-07-04
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Remarks on Lp-boundedness of wave operators for Schroedinger operators with threshold singularities2016

    • Author(s)
      Kenji Yajima
    • Organizer
      Solid Math. 2016
    • Place of Presentation
      Aalborg University, Denmark
    • Year and Date
      2016-05-26
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 基礎数学11 数理物理入門 改訂改題2018

    • Author(s)
      谷島 賢二
    • Total Pages
      402
    • Publisher
      東京大学出版会
    • ISBN
      9784130629225
    • Related Report
      2018 Annual Research Report

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Published: 2016-04-21   Modified: 2020-03-30  

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