Semi-classical analysis of the Schroedinger equations

• Project/Area Number: 15K04971
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Ritsumeikan University
• Principal Investigator: FUJIIE Setsuro 立命館大学, 理工学部, 教授 (00238536)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 準古典解析 / 超局所解析 / シュレディンガー作用素 / 量子共鳴 / エネルギー交差 / Schroedinger作用素 / 準古典極限 / 共鳴 / 固有値 / スペクトルシフト関数 / WKB法 / 共鳴極 / WKB解

Outline of Final Research Achievements

The main results of the research during this period are the following three. (i) We established a propagation of singularity theorem of the quantum system at a hyperbolic fixed point of the corresponding classical mechanics. We proved that if the semiclassical wave front set is empty on the incoming stable manifold associated to the fixed point, then it is also empty on the outgoing stable manifold. (ii) We prove a theorem that the spectral shift function of the Schroedinger operator with matrix-valued potential has a complete asymptotic expansion if its symbol has a scalar escape function. (iii) We clarified the semiclassical asymptotic distribution of resonances of the Schroedinger operator with matrix-valued potential for a model of one-dimensional 2 by 2 system with an energy level crossing.

Research Products

• [Int'l Joint Research] Universite Paris Sud/University Paris Nord/University Bordeaux(France)
• [Int'l Joint Research] University di Bologna(Italy)
• [Int'l Joint Research] University of Lund(Sweden)
• [Int'l Joint Research] ボルドー大学/パリ南大学/パリ北大学(フランス)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Int'l Joint Research] クレタ大学(ギリシャ)
• [Int'l Joint Research] ルント大学(スウェーデン)
• [Int'l Joint Research] パリ１１大学/パリ１３大学/ボルドー１大学(フランス)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Journal Article] Barrier-top resonances for non-globally analytic potentials (2017)
  • Author(s): J.F.Bony, S.Fujiie, T.Ramond, M.Zerzeri
  • Journal Title: Journal of Spectral Theory Volume: ー
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Propagation des singularites et resonances (2017)
  • Author(s): J.F.Bony, S.Fujiie, T.Ramond, M.Zerzeri
  • Journal Title: Comptes Rendus Mathematique Volume: 355 Pages: 887-891
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach (2017)
  • Author(s): M.Assal, M.Dimassi, S.Fujiie
  • Journal Title: International Mathematics Research Notes Volume: ー
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Molecular predissociation resonances near an energy-level crossing II: Vector field interaction (2017)
  • Author(s): S.Fujiie, A.Martinez, T.Watanabe
  • Journal Title: Journal of Differential Equations Volume: 262 Pages: 5880-5895
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle (2017)
  • Author(s): S.Fujiie, J.Wittsten
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: ー
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Molecular predissociation resonances near an energy-level crossing II: Vector field interaction (2017)
  • Author(s): Setsuro Fujiie, Andre Martinez, Takuya Watanabe
  • Journal Title: Journal of Differential Equations Volume: 262-12 Issue: 12 Pages: 5880-5895
  • DOI: 10.1016/j.jde.2017.02.017
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Quantization condition of resonances at energy-level crossing (2017)
  • Author(s): Setsuro Fujiie, Andre Martinez, Takuya Watanabe
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B61
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Quantization condition for multi-barrier resonances (2017)
  • Author(s): Jean-Francois Bony, Setsuro Fujiie, Thierry Ramond, Maher Zerzeri
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 印刷中
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Molecular predissociation resonances near an energy-level crossing I: Elliptic interaction (2016)
  • Author(s): Setsuro FUJIIE, Andre Martinez and Takuya WATANABE
  • Journal Title: Journal of Differential Equations Volume: 260 Issue: 5 Pages: 4051-4085
  • DOI: 10.1016/j.jde.2015.11.015
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] A time-independent approach for the study of the spectral shift function and an application to Stark Hamiltonians (2015)
  • Author(s): Mouez DIMASSI and Setsuro FUJIIE
  • Journal Title: Communications in Partial Differential Equations Volume: 40 Issue: 10 Pages: 1787-1814
  • DOI: 10.1080/03605302.2015.1053567
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Resonances semiclassiques aux sommets (2017)
  • Author(s): S.Fujiie
  • Organizer: Seminaire Math-Physique, Institut de Mathematique de Bourgogne
  • Invited
• [Presentation] Propagation of singularities at a hyperbolic fixed point and application to the quantization of resonances (2017)
  • Author(s): S.Fujiie
  • Organizer: Tosio Kato Centennial Conference
  • Int'l Joint Research / Invited
• [Presentation] Microlocal method for the semiclassical distribution of quantum resonances (2017)
  • Author(s): Setsuro Fujiie
  • Organizer: 2nd Summer School: Various aspects of mathematical physics
  • Int'l Joint Research / Invited
• [Presentation] 共鳴極の量子化則 - 超局所的見地から (2016)
  • Author(s): 藤家雪朗
  • Organizer: 首都大学東京理工学研究科大学院集中講義（物理学特別講義 I）
  • Place of Presentation: 首都大学東京（東京都・八王子市）
  • Year and Date: 2016-10-17
  • Invited
• [Presentation] Microlocal method for the semiclassical distribution of resonances (2016)
  • Author(s): Setsuro Fujiie
  • Organizer: Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics
  • Place of Presentation: 京都大学（京都府・京都市）
  • Year and Date: 2016-05-27
  • Int'l Joint Research / Invited
• [Presentation] Quantization of resonances in the semiclassical limit (2016)
  • Author(s): Setsuro Fujiie
  • Organizer: Topics in Matematica, Universita di Bologna
  • Place of Presentation: Universita di Bologna (Italia)
  • Year and Date: 2016-03-22
  • Invited
• [Presentation] Resonances near an energy-level crossing (2015)
  • Author(s): Setsuro Fujiie
  • Organizer: 第２回 解析学の耳袋
  • Place of Presentation: プラザヴェルデ（静岡県沼津市）
  • Year and Date: 2015-10-29
  • Invited
• [Presentation] Resonances near an energy-level crossing (2015)
  • Author(s): Setsuro Fujiie
  • Organizer: Microlocal Analysis and Singular Perturbation Theory
  • Place of Presentation: RIMS Kyoto University （京都府京都市）
  • Year and Date: 2015-10-06
  • Int'l Joint Research / Invited
• [Presentation] 捕捉された古典軌道と量子共鳴極の漸近分布 (2015)
  • Author(s): Setsuro Fujiie
  • Organizer: 研究集会「共鳴極と測地線の幾何の再解釈と新展開」
  • Place of Presentation: 京都大学数理解析研究所 （京都府京都市）
  • Year and Date: 2015-09-02
  • Invited
• [Presentation] WKB法と固有値・レゾナンスの漸近分布 (2015)
  • Author(s): Setsuro Fujiie
  • Organizer: 東京大学数理科学研究科数理科学特別講義VI
  • Place of Presentation: 東京大学数理科学研究科（東京都目黒区）
  • Year and Date: 2015-07-13
  • Invited
• [Book] `数学'論説 ホモクリニック軌道が生成する共鳴の準古典漸近分布 ―超局所的視点からのアプローチ― (2017)
  • Author(s): 藤家雪朗
  • Publisher: 日本数学会
• [Funded Workshop] Himeji Conference on Partial Differential Equations (2018)
• [Funded Workshop] Himeji Conference on Partial Differential Equations (2017)
  • Place of Presentation: イーグレ姫路（兵庫県・姫路市）
  • Year and Date: 2017-03-06
• [Funded Workshop] Lectures on Semi-Classical Analysis (2016)
  • Place of Presentation: 立命館大学（滋賀県・草津市）
  • Year and Date: 2016-07-03
• [Funded Workshop] Himeji Conference on Partial Differential Equations (2016)
  • Place of Presentation: イーグレ姫路（兵庫県姫路市）
  • Year and Date: 2016-03-02