Microlocal and semiclassical methods in spectral and scattering theory

• Project/Area Number: 15H03622
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Gakushuin University (2019); The University of Tokyo (2015-2018)
• Principal Investigator: 中村 周 学習院大学, 理学部, 教授 (50183520)
• Co-Investigator(Kenkyū-buntansha): 藤家 雪朗 立命館大学, 理工学部, 教授 (00238536) ; 足立 匡義 京都大学, 人間・環境学研究科, 教授 (30281158)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Granted (Fiscal Year 2019)
• Budget Amount: ¥16,640,000 (Direct Cost: ¥12,800,000、Indirect Cost: ¥3,840,000); Fiscal Year 2019: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2017: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2016: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2015: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
• Keywords: 関数解析学 / 偏微分方程式論 / スペクトル・散乱理論 / 半古典解析 / 超局所解析 / シュレディンガー方程式 / 散乱理論 / 半古典極限

Outline of Annual Research Achievements

研究代表者の中村は，長距離型摂動を持つ拡張されたシュレディンガー作用素に対する散乱行列がフーリエ積分作用素であること，またその応用として散乱行列のスペクトルが決定できるモデルについての研究を継続し，論文をほぼ完成させた．また，只野之英氏との共同研究で，離散シュレディンガー作用素が連続極限でシュレディンガー作用素にノルム・レゾルベント収束すること，したがってスペクトルが収束することを証明した．

研究分担者の藤家は，2018年度は，主に次の２つの課題を研究した．(1) A.Martinez, 渡部との共同研究で，エネルギー交差を持つ一次元シュレディンガー方程式系の量子共鳴の半古典極限における漸近分布について，エネルギー交差レベルより高いレベルにおいて，超局所解析の手法を用いて量子共鳴の漸近分布を明らかにした．(2) S.Kamvissisとの共同研究で，非線形シュレディンガー方程式のLax対として現れるZakharov-Shabat作用素の反射係数の漸近挙動，固有値の漸近分布を，完全WKB法を用いて明らかにした．

研究分担者の足立は，神戸大学の清瀬周氏との共同研究で，(1) 定電場と定磁場の両方が印加されている空間内の多体量子力学系に対するAvron-Herbst型の公式を導出し，それを応用して，１個の粒子のみ荷電しており，他の粒子は非荷電であるような系に対する漸近完全性を証明した; (2)時間周期的ポテンシャルをもつ２体Schrodinger作用素に付随するFloquet Hamiltonianに対する新たなconjugate operatorを提唱しMourre評価を得た．その研究の続きで，時間周期的ポテンシャルをもつ３体Schrodinger作用素に付随するFloquet Hamiltonianに対するconjugate operatorを構成し，Mourre評価を得た．

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

研究実績の概要にあるように，研究目的に合致した多数の研究成果が得られた．具体的には，例えば，中村の研究成果は，散乱理論の中核的な研究対象の一つである散乱行列の数学的構造に対する理解を深める結果である．藤家の研究成果は，古典力学的に複雑な構造を持つ力学系を量子化した系の半古典極限におけるスペクトル構造に新たな光をあてるものである．足立の研究成果は，外部電場，磁場等の下での電子の散乱理論の解析という，長い歴史を持つ困難な研究領域に確実な発展をもたらした．
数学の研究に関しては、研究成果が出版されるには時間がかかるが，多くの研 究成果はすでにプレプリント等の形で公表されている．また，研究目的の一つである若手研究者に対する研究助成も着実に行われており，上記の研究成果の多くは，若手研究者との共同研究成果であることも，それを反映している．

Strategy for Future Research Activity

前項までで述べたように，研究計画の主目的である量子力学のスペクトル理論，散乱理論を超局所解析的な手法，アイデアを利用して深める，という課題は着実に実行されている．したがって，当初の研究計画にもとづき，引き続き，研究代表者，研究分担者の自発的な研究活動によって，当該研究計画の目標の実現を目指す．具体的には，国際交流，国際会議での研究発表のための外国旅費の支出，国内における研究交流のための国内旅費の支出に予算の大部分を引き続き充てる．必要に応じて，それに付随する諸費用，謝金，研究のための計算機環境の整備，などに予算を支出する．

Research Products

• [Int'l Joint Research] ボルドー大学・数学研究所/パリ11大学（オルセイ）・数学研究所/パリ13大学・解析,幾何,応用研究所(フランス)
• [Int'l Joint Research] オーフス大学・数学科/オールボー大学・数学科(デンマーク)
• [Int'l Joint Research] マドリッド工科大学(スペイン)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Int'l Joint Research] Universidad Politecnica de Madrid(スペイン)
• [Int'l Joint Research] Univ. Paris-Sud/Universite Paris 13/Universite de Bordeaux(フランス)
• [Int'l Joint Research] Aarhus Univ./Aalborg Univ.(デンマーク)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Int'l Joint Research] オルセイ大学/ボルドー大学(フランス)
• [Int'l Joint Research] マドリッド工科大学(スペイン)
• [Int'l Joint Research] グラーツ大学(オーストリア)
• [Int'l Joint Research] ベイラー大学(アメリカ合衆国)
• [Int'l Joint Research] ロンドン大学キングス・カレッジ(連合王国)
• [Int'l Joint Research] ボルドー大学(フランス)
• [Int'l Joint Research] ボローニャ大学(イタリア)
• [Int'l Joint Research] マドリッド工科大学(スペイン)
• [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 Pages: 315-348
  • DOI: 10.4171/jst/249
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Semiclassical trace formula and spectral shift function for systems via a stationary approach (2019)
  • Author(s): M.Assal, M.Dimassi, S.Fujiie
  • Journal Title: Int. Math. Res. Not. Volume: 印刷中 Pages: 1227-1264
  • DOI: 10.1093/imrn/rnx149
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Remarks on the Avron-Herbst type formula for N-body quantum systems in constant electric and magnetic fields (2019)
  • Author(s): Tadayoshi Adachi and Amane Kiyose
  • Journal Title: New Horizons in Mathematical Physics Volume: 3 Pages: 1-12
  • DOI: 10.22606/nhmp.2019.31001
  • Peer Reviewed / Open Access
• [Journal Article] Spectral shift functions and Dirichlet-to-Neumann maps (2018)
  • Author(s): Behrndt Jussi, Gesztesy Fritz, Nakamura Shu
  • Journal Title: Mathematische Annalen Volume: 371 Pages: 1255-1300
  • DOI: 10.1007/s00208-017-1593-4
  • Peer Reviewed / Open Access / 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 Continuous Dynamical Systems Volume: 38 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Barrier-top resonances for non-globally analytic potentials (2018)
  • Author(s): J.-F.Bony, S.Fujiie, T.Ramond, M.Zerzeri
  • Journal Title: Journal of Spectral Theory Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle (2018)
  • Author(s): S.Fujiie, J.Wittsten
  • Journal Title: Discrete and Continuous Dynamical Systems Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Microlocal Resolvent Estimates, Revisited (2017)
  • Author(s): Nakamura, Shu
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 24 Pages: 239-257
  • Peer Reviewed
• [Journal Article] Improving the Lieb-Robinson bound for long-range interactions. (2017)
  • Author(s): Matsuta, T., Koma, T., Nakamura, S.
  • Journal Title: Ann. H. Poincare Volume: 18 Pages: 519-528
  • DOI: 10.1007/s00023-016-0526-1
  • Peer Reviewed
• [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
  • DOI: 10.1016/j.crma.2017.06.008
  • Peer Reviewed / Open Access / 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 Pages: 5880-5895
  • DOI: 10.1016/j.jde.2017.02.017
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Microlocal resolvent estimates, revisited (2017)
  • Author(s): Nakamura, S.
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 24 Pages: 239-257
  • Peer Reviewed
• [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 Pages: 9-21
  • Peer Reviewed / Int'l Joint Research
• [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 / Int'l Joint Research
• [Journal Article] Microlocal properties of scattering matrices (2016)
  • Author(s): Nakamura, S.
  • Journal Title: Comm. Partial Differential Equation Volume: 41 Pages: 894-912
  • DOI: 10.1080/03605302.2016.1167082
  • Peer Reviewed
• [Journal Article] Quantum scattering in a periodically pulsed magnetic field (2016)
  • Author(s): T. Adachi and M. Kawamoto
  • Journal Title: Annales Henri Poincare Volume: 17 Pages: 2409-2438
  • DOI: 10.1007/s00023-016-0457-x
  • Peer Reviewed
• [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 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 Pages: 1787-1814
  • DOI: 10.1080/03605302.2015.1053567
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Topics on Microlocal Scattering Theory for Discrete Schrodinger Operators (2019)
  • Author(s): 中村 周
  • Organizer: “Spectra of Random Operators and Related Topics”, 2019, January 9-11, Kyoto University 2019年1月9日
  • Int'l Joint Research / Invited
• [Presentation] 長距離型散乱の散乱行列の超局所的構造とスペクトルについて (2019)
  • Author(s): 中村 周
  • Organizer: 京都大学・作用素論セミナー 2019年1月25日
  • Invited
• [Presentation] 量子力学の数学的構造と古典力学 (2019)
  • Author(s): 中村 周
  • Organizer: 東京大学・数理科学研究科・談話会 2019年3月22日
  • Invited
• [Presentation] On the Mourre estimates for Floquet Hamiltonians (2019)
  • Author(s): 足立 匡義
  • Organizer: 解析学研究特別セミナー 2019年4月20日
  • Invited
• [Presentation] Schrodinger operators with long-range potentials (2018)
  • Author(s): 中村 周
  • Organizer: Workshop “Analytical and Numerical Methods in Quantum Transport”, Aalborg University 2018年5月28日
  • Int'l Joint Research / Invited
• [Presentation] 長距離型散乱の散乱行列について (2018)
  • Author(s): 中村 周
  • Organizer: 研究集会「2018 夏の作用素論シンポジウム」徳島 2018年9月1日
• [Presentation] Remarks on scattering matrices for long-range perturbation (2018)
  • Author(s): 中村 周
  • Organizer: 第28回「数理物理と微分方程式」函館 2018年11月2日
• [Presentation] Properties of scattering matrix with long-range perturbations (2018)
  • Author(s): 中村 周
  • Organizer: “International Conference Spectral Theory and Mathematical Physics 2018”, Pontificia Universidad Catolica de Chile 2018年12月3日
  • Int'l Joint Research / Invited
• [Presentation] Remarks on the Avron-Herbst type formula for N-body quantum systems in constant electric and magnetic fields (2018)
  • Author(s): 足立 匡義
  • Organizer: 神楽坂解析セミナー 2018年11月23日
  • Invited
• [Presentation] Quantum scattering in spatially homogeneous electromagnetic fields (2018)
  • Author(s): 足立 匡義
  • Organizer: 研究集会「微分方程式の総合的研究」 2018年12月22日
  • Invited
• [Presentation] Properties of scattering matrix with long-range perturbations (2018)
  • Author(s): Shu Nakamura
  • Organizer: Universidad Politecnica de Madrid, Spain
  • Int'l Joint Research / Invited
• [Presentation] On quantum scattering in time-dependent electromagnetic fields (2018)
  • Author(s): Tadayoshi Adachi
  • Organizer: Workshop on Spectral and Scattering Theory, 京都大学
  • Int'l Joint Research / Invited
• [Presentation] Remarks on long-range scattering (2017)
  • Author(s): Shu Nakamura
  • Organizer: Aarhus University, Denmark
  • Int'l Joint Research / Invited
• [Presentation] 定電磁場内の量子力学系に対する Avron-Herbst 型 公式について (2017)
  • Author(s): 足立匡義
  • Organizer: 第2回ひこね解析セミナー，大学サテライト・プラザ彦根
  • Invited
• [Presentation] 定電磁場内の量子力学系に対する Avron-Herbst 型 公式について (2017)
  • Author(s): 足立匡義
  • Organizer: 応用解析研究会，早稲田大学
  • Invited
• [Presentation] On quantum scattering in time-dependent electromagnetic fields (2017)
  • Author(s): Tadayoshi Adachi
  • Organizer: Tosio Kato Centennial Conference, University of Tokyo
  • Int'l Joint Research / Invited
• [Presentation] Resonances semiclassiques aux sommets (2017)
  • Author(s): Setsuro, Fujiie
  • Organizer: Seminaire Math-Physique, Institut Mathematique de Bourgogne
  • Int'l Joint Research / Invited
• [Presentation] Propagation of singularities at a non analytic hyperbolic fixed point and application to the quantization of resonances (2017)
  • Author(s): Setsuro, Fujiie
  • Organizer: Tosio Kato Centennial Conference, University of Tokyo
  • Int'l Joint Research / Invited
• [Presentation] Microlocal method for the semiclassical distribution of quantum resonances (2017)
  • Author(s): Setsuro, Fujiie
  • Organizer: nd Summer School: Various aspects of mathematical physics, Euler International Mathematical Institute
  • Int'l Joint Research / Invited
• [Presentation] 磯崎・北田modifiersについて (2016)
  • Author(s): 中村周
  • Organizer: 研究集会 第27回「数理物理と微分方程式」
  • Place of Presentation: かんぽの宿・富山（富山県・富山市）
  • Year and Date: 2016-11-25
• [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
  • Place of Presentation: 首都大学東京（東京都）
  • Year and Date: 2016-10-17
  • Invited
• [Presentation] 散乱理論における超局所的手法についての話題 (2016)
  • Author(s): 中村周
  • Organizer: 共同研究「微分方程式に対する散乱理論の展開」
  • Place of Presentation: 京都大学・数理解析研究所（京都府・京都市）
  • Year and Date: 2016-09-07
  • Int'l Joint Research / Invited
• [Presentation] Microlocal Scattering Theory for Discrete Schroedinger Operators and Related Topics (2016)
  • Author(s): Shu Nakamura
  • Organizer: 研究集会「保存則をもつ偏微分方程式に対する解の正則性, 特異性および長時間挙動の研究」
  • Place of Presentation: 京都大学・数理解析研究所（京都府・京都市）
  • Year and Date: 2016-06-06
  • Int'l Joint Research / Invited
• [Presentation] Microlocal Scattering Theory for Discrete Schroedinger Operators and Related Topics (2016)
  • Author(s): Shu Nakamura
  • Organizer: Workshop “Solid Math 2016”
  • Place of Presentation: オールボー（デンマーク）
  • Year and Date: 2016-05-26
  • Int'l Joint Research / Invited
• [Presentation] Quantization of resonances in the semiclassical limit (2016)
  • Author(s): 藤家雪朗
  • Organizer: Topics in Mathematic
  • Place of Presentation: Dipartimento di Matematica, Universita di Bologna
  • Year and Date: 2016-03-22
  • Int'l Joint Research / Invited
• [Presentation] Remarks on quantum scattering in a periodically pulsed magnetic field (2016)
  • Author(s): 足立匡義
  • Organizer: 解析セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 2016-02-23
• [Presentation] 時間周期的電磁場内での量子散乱について (2016)
  • Author(s): 足立匡義
  • Organizer: 大談話会（理学研究科数学教室・数理解析研究所）
  • Place of Presentation: 京都大学（京都府・京都市）
  • Invited
• [Presentation] 時間周期的電磁場内での量子散乱について (2016)
  • Author(s): 足立匡義
  • Organizer: 作用素論セミナー
  • Place of Presentation: 京都大学（京都府・京都市）
  • Invited
• [Presentation] Microlocal method for the semiclassical distribution of resonances Workshop on the Boltzmann Equation (2016)
  • Author(s): S. 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）
  • Int'l Joint Research / Invited
• [Presentation] Quantum scattering in a periodically pulsed magnetic field (2015)
  • Author(s): 足立匡義
  • Organizer: 研究集会「微分方程式の総合的研究」
  • Place of Presentation: 東京大学
  • Year and Date: 2015-12-20
  • Invited
• [Presentation] High energy asymptotics of the scattering matrices for Schroedinger and Dirac operators (2015)
  • Author(s): 中村周
  • Organizer: Conference “Semiclassical Analysis and Non-Self- adjoint Operators”
  • Place of Presentation: CIRM, Marseille, France
  • Year and Date: 2015-12-14
  • Int'l Joint Research / Invited
• [Presentation] Resonances near an energy-level crossing (2015)
  • Author(s): 藤家雪朗
  • Organizer: 第２回 解析学の耳袋
  • Place of Presentation: プラザヴェルデ（沼津）
  • Year and Date: 2015-10-29
  • Invited
• [Presentation] 時間周期的パルス磁場内での量子散乱 (2015)
  • Author(s): 足立匡義
  • Organizer: 神楽坂解析セミナー
  • Place of Presentation: 東京理科大学
  • Year and Date: 2015-10-23
  • Invited
• [Presentation] Resonances near an energy-level crossing (2015)
  • Author(s): 藤家雪朗
  • Organizer: 研究集会「Microlocal Analysis and Singular Perturbation Theory」
  • Place of Presentation: 京都大学数理解析研究所
  • 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): 藤家雪朗
  • 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
  • Year and Date: 2015-09-02
  • Invited
• [Presentation] High energy asymptotics of the scattering matrices for Schroedinger-type operators and Dirac operators (2015)
  • Author(s): 中村周
  • Organizer: Workshop: “Semiclassical Analysis, Spectral Theory and Resonances”
  • Place of Presentation: Erwin Schroedinger Institute, Vienna, Austria
  • Year and Date: 2015-08-24
  • Int'l Joint Research / Invited
• [Presentation] High energy asymptotics of scattering matrices (2015)
  • Author(s): 中村周
  • Organizer: Conference “Topics in Analysis and Mathematical Physics”
  • Place of Presentation: Aalborg University, Aalborg, Denmark
  • Year and Date: 2015-05-29
  • Int'l Joint Research / Invited
• [Presentation] Remarks on quantum scattering in a periodically pulsed magnetic field (2015)
  • Author(s): 足立匡義
  • Organizer: Conference “Topics in Analysis and Mathematical Physics”
  • Place of Presentation: Aalborg University, Aalborg, Denmark
  • Year and Date: 2015-05-29
  • Int'l Joint Research / Invited
• [Presentation] Microlocal analysis of scattering matrix, and related topics (2015)
  • Author(s): 中村周
  • Organizer: Cardiff University, Analysis Seminar
  • Place of Presentation: Cardiff University, Cardiff, UK
  • Year and Date: 2015-04-27
  • Int'l Joint Research / Invited
• [Presentation] Microlocal properties of scattering matrices and related topics (2015)
  • Author(s): 中村周
  • Organizer: King’s College Analysis Seminar
  • Place of Presentation: King’s College London, UK
  • Year and Date: 2015-04-23
  • Int'l Joint Research / Invited
• [Presentation] Microlocal properties of scattering matrices (2015)
  • Author(s): 中村周
  • Organizer: Workshop: “Almost- periodic and other ergodic problems”
  • Place of Presentation: Isaac Newton Inst. Math. Sci., Cambridge, UK
  • Year and Date: 2015-04-07
  • Int'l Joint Research / Invited
• [Funded Workshop] Mathematics of Schrodinger Equations and Related Topics, 2019 Jan 6-8, 土肥 (2019)