Geometric methods for the Schrodinger equation

• Project/Area Number: 17K05325
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: The University of Tokyo (2018-2022); Kobe University (2017)
• Principal Investigator: ITO Kenichi 東京大学, 大学院数理科学研究科, 准教授 (90512509)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: シュレーディンガー作用素 / スペクトル理論 / 散乱理論 / 差分作用素 / 数理物理学 / 関数解析学 / 量子力学 / 偏微分方程式論 / シュレーディンガー方程式 / 数理物理 / 閾値レゾナンス / 偏微分方程式 / 多体問題 / 関数方程式論 / 解析学 / 幾何学

Outline of Final Research Achievements

In this research project I applied a commutator method that I had invented to the Schrodinger operators with N-body, Stark or long-range potentials, and obtained a series of results including the stationary scattering theory. In addition, I completely classified threshold resonances of the Schrodinger operator on a graph with rays. Moreover, I obtained asymptotic expansions of the resolvent around thresholds for the Laplacian on the 2-dimensional square lattice.

Academic Significance and Societal Importance of the Research Achievements

Schrodinger作用素のスペクトル・散乱理論に有効な新しい交換子スキームの導入に成功した．このスキームは高度な関数解析学的手法や擬微分作用素の理論を必要とせず，古典的な交換子計算のみで完結する，簡素なものである．そのため，広範な関連研究領域においてさらなる応用や発展が見込まれる．また，2次元格子状の格子Green関数に対しても新奇性の高い知見が得られた．これは今後の関連研究への足掛かりとして有用なものと期待される．

Research Products

• [Int'l Joint Research] Aarhus University/Aalborg University(デンマーク)
• [Int'l Joint Research] Aarhus University/Aalborg University(デンマーク)
• [Int'l Joint Research] Aarhus University/Aalborg University(デンマーク)
• [Int'l Joint Research] Aarhus University/Aalborg University(デンマーク)
• [Int'l Joint Research] Aarhus Unversity/Aalborg University(デンマーク)
• [Int'l Joint Research] Aarhus University/Aalborg University(Denmark)
• [Journal Article] Stationary scattering theory for one-body Stark operators, I (2022)
  • Author(s): T. Adachi, K. Itakura, K. Ito, E. Skibsted
  • Journal Title: Pure and Applied Functional Analysis Volume: 7
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Stationary Scattering Theory for One-body Stark Operators, II (2022)
  • Author(s): Kenichi Ito, Erik Skibsted
  • Journal Title: Annales Henri Poincare Volume: 23 Issue: 2 Pages: 513-548
  • DOI: 10.1007/s00023-021-01101-9
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions (2021)
  • Author(s): Kenichi Ito, Arne Jensen
  • Journal Title: Integral Equations and Operator Theory Volume: 93 Issue: 3 Pages: 32-32
  • DOI: 10.1007/s00020-021-02648-2
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Stationary scattering theory on manifolds (2021)
  • Author(s): Kenichi Ito, Erik Skibsted
  • Journal Title: Annales de l'Institut Fourier Volume: 71 Issue: 3 Pages: 1065-1119
  • DOI: 10.5802/aif.3417
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] New methods in spectral theory of N-body Schrodinger operators (2021)
  • Author(s): Adachi T.、Itakura K.、Ito K.、Skibsted E.
  • Journal Title: Reviews in Mathematical Physics Volume: 33 Issue: 05 Pages: 2150015-2150015
  • DOI: 10.1142/s0129055x2150015x
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Spectral theory for 1-body Stark operators (2020)
  • Author(s): Adachi T., Itakura K., Ito K., Skibsted E.
  • Journal Title: Journal of Differential Equations Volume: 268 Issue: 9 Pages: 5179-5206
  • DOI: 10.1016/j.jde.2019.11.006
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Radiation condition bounds on manifolds with ends (2020)
  • Author(s): Ito K., Skibsted E.
  • Journal Title: Journal of Functional Analysis Volume: 278 Issue: 9 Pages: 108449-108449
  • DOI: 10.1016/j.jfa.2019.108449
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacians (2019)
  • Author(s): Ito Kenichi, Jensen Arne
  • Journal Title: Journal of Functional Analysis Volume: 277 Issue: 4 Pages: 965-993
  • DOI: 10.1016/j.jfa.2019.05.018
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Time-dependent scattering theory on manifolds (2019)
  • Author(s): Ito K., Skibsted E.
  • Journal Title: Journal of Functional Analysis Volume: 277 Issue: 5 Pages: 1423-1468
  • DOI: 10.1016/j.jfa.2019.05.016
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Resolvent expansion for the Schrodinger operator on a graph with infinite rays (2018)
  • Author(s): Kenichi Ito, Arne Jensen
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 464 Issue: 1 Pages: 616-661
  • DOI: 10.1016/j.jmaa.2018.04.022
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Resolvent expansions for the Schr\"odinger operator on the discrete half-line (2017)
  • Author(s): K. Ito, A. Jensen
  • Journal Title: Journal of Mathematical Physics Volume: 58 Issue: 5
  • DOI: 10.1063/1.4982957
  • NAID: 120006318804
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Generalized Fourier transform for $C^2$ potentials (2023)
  • Author(s): Kenichi Ito
  • Organizer: Mathematics seminar (Aarhus University)
  • Int'l Joint Research / Invited
• [Presentation] Pseudodifferential expression for the S-matrix of a perturbed Stark Hamiltonian (2022)
  • Author(s): Kenichi Ito
  • Organizer: Mathematics seminar (Aarhus University)
  • Int'l Joint Research / Invited
• [Presentation] Strong radiation bounds for long-range perturbations (2022)
  • Author(s): 伊藤健一
  • Organizer: 愛媛解析セミナー（愛媛大学）
  • Invited
• [Presentation] Pseudodifferential expression for the S-matrix of perturbed Stark Hamiltonian (2021)
  • Author(s): 伊藤健一
  • Organizer: 第174回神楽坂解析セミナー（東京理科大学）
  • Invited
• [Presentation] Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions (2021)
  • Author(s): Kenichi Ito
  • Organizer: Effective models, critical phenomena and spectral methods in Quantum Transport
  • Int'l Joint Research / Invited
• [Presentation] Hypergeometric expression for the fundamental solution to the 2-dimensional discrete Laplacian（2次元離散Laplace作用素の基本解に対する超幾何表示） (2021)
  • Author(s): 伊藤健一
  • Organizer: 数学域談話会（筑波大学）
  • Invited
• [Presentation] Pseudodifferential expression for the S-matrix of perturbed Stark Hamiltonian (2021)
  • Author(s): 伊藤健一
  • Organizer: 信州微分方程式セミナー（信州大学）
  • Invited
• [Presentation] Hypergeometric expression for resolvent of the discrete Laplacian in low dimension (2020)
  • Author(s): Kenichi Ito
  • Organizer: 日本数学会2020年度年会
• [Presentation] Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions (2020)
  • Author(s): Kenichi Ito
  • Organizer: 微分方程式の総合的研究
  • Invited
• [Presentation] Commutator method for the Stark Hamiltonian (2019)
  • Author(s): Kenichi Ito
  • Organizer: 早稲田大学 応用解析研究会
  • Invited
• [Presentation] Zeroth order conjugate operator in N-body Schrodinger operators (2019)
  • Author(s): Kenichi Ito
  • Organizer: RIMS共同研究（公開型）「量子場の数理とその周辺」
  • Invited
• [Presentation] Commutator method for the Stark Hamiltonian (2019)
  • Author(s): Kenichi Ito
  • Organizer: 信州大学偏微分方程式研究集会
  • Invited
• [Presentation] Commutator method for the Stark Hamiltonian, (2019)
  • Author(s): Kenichi Ito
  • Organizer: QMath14: Mathematical Results in Quantum Physics
  • Int'l Joint Research
• [Presentation] Hypergeometric expressions for resolvents of the discrete Laplacians in low dimensions (2019)
  • Author(s): Kenichi Ito
  • Organizer: The 17th Linear and Nonlinear Waves
  • Invited
• [Presentation] Hypergeometric expressions for resolvents of the discrete Laplacians in low dimensions (2019)
  • Author(s): Kenichi Ito
  • Organizer: 研究集会「第29回数理物理と微分方程式」
• [Presentation] Rellich's theorem for the Stark Hamiltonian (2019)
  • Author(s): Kenichi Ito
  • Organizer: シュレーディンガー方程式の数理とその周辺
  • Int'l Joint Research / Invited
• [Presentation] Commutator methods for the Stark Hamiltonian (2019)
  • Author(s): Kenichi Ito
  • Organizer: Workshop on "Spectral theory & semiclassical analysis"
  • Int'l Joint Research / Invited
• [Presentation] Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends (2018)
  • Author(s): Kenichi Ito
  • Organizer: The Role of Metrics in the Theory of Partial Differential Equations, The 11th MSJ-SI
  • Int'l Joint Research / Invited
• [Presentation] New methods in spectral theory of N-body Schrodinger operators (2018)
  • Author(s): Kenichi Ito
  • Organizer: XIX International Congress on Mathematical Physics
  • Int'l Joint Research
• [Presentation] Zeroth order conjugate operator in N-body Schrodinger operators (2018)
  • Author(s): Kenichi Ito
  • Organizer: 2018夏の作用素論シンポジウム
• [Presentation] Explicit formula for singular part of the resolvent at threshold for multi-dimensional discrete Laplacian (2回講演) (2018)
  • Author(s): Kenichi Ito
  • Organizer: RIMS共同研究（グループ型）「スペクトル解析におけるスケール不変構造と摂動論の新展開 (Scaling invariant structures for spectral analysis and perturbation theory)」
  • Int'l Joint Research / Invited
• [Presentation] Zeroth order conjugate operator in N-body Schrodinger operators (2018)
  • Author(s): Kenichi Ito
  • Organizer: Math/Phys Seminar, Aarhus University
  • Int'l Joint Research / Invited
• [Presentation] 一般化固有関数の漸近挙動と散乱理論 (2018)
  • Author(s): Kenichi Ito
  • Organizer: 談話会, 東京大学大学院数理科学研究科
  • Invited
• [Presentation] Zeroth order conjugate operator in N-body Schrodinger operators (2018)
  • Author(s): Kenichi Ito
  • Organizer: 研究集会「第28回 数理物理と微分方程式」
• [Presentation] Zeroth order conjugate operator in N-body Schrodinger operators (2018)
  • Author(s): Kenichi Ito
  • Organizer: International Conference Spectral Theory and Mathematical Physics
  • Int'l Joint Research / Invited
• [Presentation] Resolvent expansion for the Schr\"odinger operator on a graph with rays I, II （2回連続講演） (2018)
  • Author(s): K. Ito
  • Organizer: スペクトル・散乱 葛飾シンポジウム
  • Invited
• [Presentation] Resolvent expansion for the Schr\"odinger operator on a graph with rays (2018)
  • Author(s): K. Ito
  • Organizer: One day meeting in mathematical quantum mechanics
  • Int'l Joint Research / Invited
• [Presentation] Wave front set for the damped wave equations (2018)
  • Author(s): K. Ito
  • Organizer: 神戸大学解析セミナー
  • Invited
• [Presentation] Wave front set for the damped wave equations (2018)
  • Author(s): K. Ito
  • Organizer: Saga Workshop on Partial Differential Equations
  • Invited
• [Presentation] Resolvent expansions for the Schr\"odinger operator on the discrete half-line (2017)
  • Author(s): K. Ito
  • Organizer: RIMS合宿型セミナー「Workshop on linear and nonlinear dispersive equations and related topics」
  • Int'l Joint Research / Invited
• [Presentation] Resolvent expansions for the Schr\"odinger operator on the discrete half-line (2017)
  • Author(s): K. Ito
  • Organizer: 第148回神楽坂解析セミナー
  • Invited
• [Presentation] Resolvent expansions for the Schr\"odinger operator on the discrete half-line (2017)
  • Author(s): K. Ito
  • Organizer: 2017夏の作用素論シンポジウム
• [Presentation] Resolvent expansions for the Schr\"odinger operator on a graph with rays (2017)
  • Author(s): K. Ito
  • Organizer: Tosio Kato Centennial Conference
  • Int'l Joint Research / Invited
• [Presentation] Resolvent expansion for the Schr\"odinger operator on a graph with rays (2017)
  • Author(s): K. Ito
  • Organizer: 九州関数方程式セミナー
  • Invited
• [Remarks] Homepage of Kenichi Ito
  • URL: https://www.ms.u-tokyo.ac.jp/~ito/
• [Remarks] Homepage of Kenichi Ito
  • URL: http://www.ms.u-tokyo.ac.jp/~ito/
• [Funded Workshop] Schrodinger Operators and Related Topics (2020)