Semiclassical Analysis of Schroedinger equations

• Project/Area Number: 18K03384
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Ritsumeikan University
• Principal Investigator: Fujiie Setsuro 立命館大学, 理工学部, 教授 (00238536)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 行列シュレディンガー作用素 / 半古典解析 / 超局所解析 / WKB法 / 量子共鳴の漸近分布 / 量子共鳴 / 半古典極限 / 退化したエネルギー交差 / 固有値の分離現象 / シュレディンガー方程式 / スペクトルシフト関数 / 量子共鳴の不安定性 / 固有値のスプリッティング / シュレディンガー作用素 / エネルギー交差 / 固有値の漸近分布 / 超局所理論 / WKB解 / directed cycle / 準古典解析 / WKB解析 / スペクトル理論

Outline of Final Research Achievements

I studied with M.Assal and K.Higuchi the asymptotic distribution in the semiclassical limit of resonances for matrix valued Schroedinger operators. We consider the case where one of the corresponding classical trajectories is periodic and the other one is non-trapping and they cross each other. The eigenvalues created by the former periodic trajectory transform into quantum resonances by an interaction with the non-trapping one (Fermi's golden rule). We succeeded in describing the asymptotic distribution of these resonances in terms of the geometry of the classical trajectories, especially the contact order of the crossings.The point is the computation of the connection formula at a crossing point which describes the asymptotic behavior of the microlocal solutions.It is essentially reduced to a degenerate stationary phase method.

Academic Significance and Societal Importance of the Research Achievements

二つの古典軌道が横断的に交差する場合の接続公式は、作用素の標準形に帰着させることによって計算することができる。これは特に新しい発想ではなく、A.Martinez、T.Watanabeとの共同研究でも利用した方法である。しかし、交差が退化して、接触している場合には、標準形は知られておらず（あるいは存在せず）、全く別の発想で接続公式を計算する必要があった。我々が採用した方法、すなわち厳密解を構成してそれらの間のWronskianを計算する方法は、新しい発想に基づくもので、停留位相法に帰着されるという事実の発見は、この分野の研究に新しい知見をもたらすものである。

Research Products

• [Int'l Joint Research] Univ. Bordeaux(フランス)
• [Int'l Joint Research] Univ. Crete(ギリシャ)
• [Int'l Joint Research] VIASM(ベトナム)
• [Int'l Joint Research] Universidad Catolica(チリ)
• [Int'l Joint Research] ボルドー大学/パリ11大学/パリ13大学(フランス)
• [Int'l Joint Research] クレタ大学(ギリシャ)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Int'l Joint Research] Pontificia Universidad Catolica de Chile(チリ)
• [Int'l Joint Research] Universite Bordeaux/Universite Paris 13/Universite Paris 11(フランス)
• [Int'l Joint Research] University of Crete(ギリシャ)
• [Int'l Joint Research] Universita di Bologna(イタリア)
• [Int'l Joint Research] Crete University(ギリシャ)
• [Int'l Joint Research] Universita di Bologna(イタリア)
• [Int'l Joint Research] Universite Paris Sud/Universite Paris Nord/Universite Bordeaux(フランス)
• [Int'l Joint Research] University of Boras(スウェーデン)
• [Journal Article] Semiclassical WKB problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential (2023)
  • Author(s): S.Fujiie, N. Hatzizisis, S. Kamvissis
  • Journal Title: Journal of Differential Equations Volume: 360 Pages: 90-150
  • DOI: 10.1016/j.jde.2023.02.019
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Spectral asymptotics for the Schroedinger operator with a non-decaying potential (2022)
  • Author(s): M. Dimassi, S. Fujiie
  • Journal Title: Asymptotic Analysis Volume: 130 No.3-4 Issue: 3-4 Pages: 335-365
  • DOI: 10.3233/asy-211754
  • Peer Reviewed
• [Journal Article] Eigenvalue splitting of polynomial order for a system of Schroedinger operators with energy-level crossing. (2021)
  • Author(s): Marouane Assal, Setsuro Fujiie
  • Journal Title: Communications in Mathematical Physics Volume: 386 Pages: 1519-1550
  • Peer Reviewed / Open Access
• [Journal Article] Widths of resonances above an energy-level crossing (2021)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential (2020)
  • Author(s): Fujiie, S., Kamvissis, S.
  • Journal Title: Journal of Mathematical Physics Volume: 61 Issue: 1 Pages: 011510-011510
  • DOI: 10.1063/1.5099581
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Barrier top resonances for non globally analytic potentials (2019)
  • Author(s): Jean-Francois Bony, Setsuro Fujiie, Thierry Ramond, Maher Zerzeri
  • Journal Title: Journal of Spectral Theory Volume: 9 Issue: 1 Pages: 315-348
  • DOI: 10.4171/jst/249
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle (2018)
  • Author(s): Setsuro Fujiie, Jens Wittsten
  • Journal Title: Discrete and Contijuous Dynamical Systems Volume: 38 Issue: 8 Pages: 3851-3873
  • DOI: 10.3934/dcds.2018167
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Resonances for homoclinic trapped sets (2018)
  • Author(s): Jean-Francois Bony, Setsuro Fujiie, Thierry Ramond, Maher Zerzeri
  • Journal Title: Asterisque Volume: 405 Pages: 1-314
  • DOI: 10.24033/ast.1055
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Some quantum effects by crossings of classical trajectories in matrix Schroedinger operators (2022)
  • Author(s): S. Fujiie
  • Organizer: 愛媛大学解析セミナー
  • Invited
• [Presentation] Eigenvalues and resonances created by energy-level crossings (2020)
  • Author(s): Setsuro Fujiie
  • Organizer: Schroedinger Operators and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Double well problem for a system of Schroedinger operators (2019)
  • Author(s): Setsuro Fujiie
  • Organizer: ハミルトン系とその周辺
  • Invited
• [Presentation] Eigenvalue splitting for a system of Schroedinger operators (2019)
  • Author(s): Setsuro Fujiie
  • Organizer: The 17TH Linear and Nonlinear Waves
  • Invited
• [Presentation] Eigenvalue splitting for a system of Schroedinger operator with an energy-level crossing (2019)
  • Author(s): Setsuro Fujiie
  • Organizer: Seminaire EDP et Physique mathematique
  • Int'l Joint Research / Invited
• [Presentation] Methode BKW exacte appliquee a un operateur de Dirac non-auto-adjoint (2019)
  • Author(s): Setsuro Fujiie
  • Organizer: Groupe de travail semi-classique
  • Invited
• [Presentation] Resonances for a system of Schrodinger operators associated with a energy level crossing (2018)
  • Author(s): Setsuro Fujiie
  • Organizer: International Conference Spectral Theory and Mathematical Physics
  • Int'l Joint Research / Invited
• [Presentation] Resonances for homoclinic strapped sets (2018)
  • Author(s): 藤家雪朗
  • Organizer: 微分方程式の総合的研究
• [Funded Workshop] Himeji Conference on Partial Differential Equations (2023)
• [Funded Workshop] Quantum Resonances and Related Topics (2019)
• [Funded Workshop] Himeji Conference on Partial Differential Equations (2019)