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Semiclassical analysis of sprectral and scattering problems on energy-level crossings

Research Project

Project/Area Number 18K03349
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionRitsumeikan University

Principal Investigator

Watanabe Takuya  立命館大学, 理工学部, 准教授 (80458009)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywordsエネルギー交差 / 準古典解析 / 量子共鳴(レゾナンス) / 遷移確率 / 完全WKB解析 / 準古典超局所解析 / レゾナンス / 超局所解析 / WKB解析
Outline of Final Research Achievements

The keyword of this research project is "energy-level crossing". Concerning semiclassical distributions of quantum resonances, our series of works (Fujiie-Martinez-Watanabe) showed the existence and distribution of quantum resonances whose imaginary part is a fractional order with respect to the semiclassical parameter. It can be understood that these resonances are created by a weakly trapped trajectories, which cannot be realized in the case of a single-valued Schrodinger operator.
We also studied a problem of an adiabatic transition probability for avoided crossings and, in particular, a problem to which can not be applied an exact WKB method. The problem in a linear crossing model was solved by Watanabe-Zerzeri by means of a reduction to a microlocal normal form and that in a tangential crossing model has been investigated by Higuch-Watanabe thanks to the recent approach introduced by Assal-Fujiie-Higuchi.

Academic Significance and Societal Importance of the Research Achievements

量子共鳴や遷移確率は、スペクトル・散乱理論の研究対象であり、特に準古典解析として、古典軌道といった古典力学特有の性質により量子力学の現象を記述する典型的なトピックである。
本研究の学術的意義は、連立系を扱うことにより、単独の場合には現れない古典軌道をもつシュレディンガー作用素を考察することが可能になり、実際に単独の場合では得られない量子共鳴の存在を明らかにした点である。特に、既存の完全WKB解析や準古典超局所解析の手法では解析できない場合に対する新たな手法を提案したことの意義も大きく、我々の手法を拡張した最近の結果は、もう一方の研究課題であるLandau-Zenerの公式の一般化にも適用された。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (43 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (11 results) Journal Article (4 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 3 results,  Open Access: 1 results) Presentation (14 results) (of which Int'l Joint Research: 2 results,  Invited: 13 results) Remarks (8 results) Funded Workshop (6 results)

  • [Int'l Joint Research] ボルドー大学/パリ13大学(フランス)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] チリ・カトリック大学(チリ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] パリ13大学/ボルドー大学(フランス)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] カトリック大学(チリ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] ボローニャ大学(イタリア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] パリ13大学/ボルドー大学(フランス)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] チリカトリック大学(チリ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ボローニャ大学(イタリア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] パリ13大学/ボルドー大学(フランス)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ボローニャ大学(イタリア)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] パリ13大学/ボルドー大学(フランス)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Spectral Asymptotics for magnetic Schroedinger operator with slowly varying potential2023

    • Author(s)
      M. Dimassi, H. Yazbeck, T. Watanabe
    • Journal Title

      Osaka Journal of Mathematics

      Volume: Vol. 60, No,3

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Schroedinger operator with constant magnetic field and slowly varying perturbation on a multidimensional strip region2021

    • Author(s)
      M. Dimassi, T. Watanabe
    • Journal Title

      RIMS Kokyuroku

      Volume: 2195

    • Related Report
      2021 Research-status Report
    • Open Access / Int'l Joint Research
  • [Journal Article] Widths of resonances above an energy-level crossing2021

    • Author(s)
      Fujii? S.、Martinez A.、Watanabe T.
    • Journal Title

      Journal of Functional Analysis

      Volume: 280 Issue: 6 Pages: 108918-108918

    • DOI

      10.1016/j.jfa.2020.108918

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Resonances for a System of Schroedinger Operators above an Energy-Level Crossing2020

    • Author(s)
      S. Fujiie, A. Martinez, T. Watanabe
    • Journal Title

      Spectral Theory and Mathematical Physics, Latin American Mathematics Series

      Volume: 1 Pages: 153-170

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 複数の接触交差が生成する小さな擬交差間の2準位断熱遷移確率2023

    • Author(s)
      渡部拓也
    • Organizer
      日本数学会
    • Related Report
      2022 Annual Research Report
  • [Presentation] 小さなギャップをもつ擬交差間の非断熱遷移問題に対する準古典解析2022

    • Author(s)
      渡部拓也
    • Organizer
      愛媛大学解析セミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Two-level adiabatic transition probability for small avoided crossings generated by tangential intersections2022

    • Author(s)
      渡部拓也
    • Organizer
      The 19th Linear and Non-linear Waves
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Zakharov-Shabat 方程式系の準古典解析2022

    • Author(s)
      渡部拓也
    • Organizer
      津田塾大学PDE Workshop
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] y for small avoided crossings generated by tangential intersections2022

    • Author(s)
      渡部拓也
    • Organizer
      RIMS共同研究 スペクトル・散乱理論とその周辺
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 2準位擬交差間の非断熱遷移問題に対する準古典解析2021

    • Author(s)
      渡部 拓也
    • Organizer
      神楽坂解析セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Widths of resonances above an energy-level crossing2019

    • Author(s)
      Takuya WATANABE
    • Organizer
      Seminaire: Problemes Spectraux en Physique Mathematique
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Semiclassical approach to a generalization of the Landau-Zener formula.2019

    • Author(s)
      Takuya WATANABE
    • Organizer
      Groupe de travail Analyse semi-classique & Physique mathematique
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] エネルギー交差の上位準位におけるレゾナンスの準古典分布2019

    • Author(s)
      渡部 拓也
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Width of semiclassical resonances above an energy level crossing2019

    • Author(s)
      渡部 拓也
    • Organizer
      第11回 名古屋微分方程式研究集会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Width of semiclassical resonances above an energy level crossing2019

    • Author(s)
      Takuya Watanabe
    • Organizer
      Himeji Conference on Partial Differential Equations
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Semiclassical distribution of resonances associated with an energy-level crossing2019

    • Author(s)
      渡部 拓也
    • Organizer
      HMA セミナー・冬の研究会 2019
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Semiclassical distribution of resonances associated with an energy-level crossing2018

    • Author(s)
      渡部 拓也
    • Organizer
      研究集会 第1回 解析学の壺
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Two-level transition problem for avoided crossings in a non-adiabatic regime2018

    • Author(s)
      Takuya Watanabe
    • Organizer
      Mini Workshop on WKB Analysis and Singularity Theory
    • Related Report
      2018 Research-status Report
    • Invited
  • [Remarks] Himeji Conference on PDEs

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/CHPDE/CPDEen.html

    • Related Report
      2022 Annual Research Report
  • [Remarks] 立命館大学 研究者学術情報データベース

    • URL

      https://research-db.ritsumei.ac.jp/rithp/k03/resid/S000603

    • Related Report
      2022 Annual Research Report
  • [Remarks] Himeji Conference on PDEs

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/CHPDE/CPDE.html

    • Related Report
      2020 Research-status Report
  • [Remarks] Spectral and Scattering Theory and Related Topics

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/rims2019/index.html

    • Related Report
      2020 Research-status Report
  • [Remarks] Quantum Resonances and Related Topics

    • URL

      https://qrrt2019.sciencesconf.org/

    • Related Report
      2020 Research-status Report 2019 Research-status Report
  • [Remarks] スペクトル・散乱理論とその周辺

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/rims2019/index.html

    • Related Report
      2019 Research-status Report
  • [Remarks] 偏微分方程式姫路研究集会

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/CHPDE/CPDE.html

    • Related Report
      2019 Research-status Report
  • [Remarks] 偏微分方程式姫路研究集会

    • URL

      http://www.math.ritsumei.ac.jp/takuwatanabe/CHPDE/CPDEen.html

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Himeji Conference on PDEs2023

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] Himeji Conference on PDES2022

    • Related Report
      2021 Research-status Report
  • [Funded Workshop] Himeji Conference on Partial Differential Equations2021

    • Related Report
      2020 Research-status Report
  • [Funded Workshop] Quantum Resonances and Related Topics2019

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] Spectral and Scattering Theory and Related Topics2019

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] Himeji Conference on Partial Differential Equations2019

    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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