Spectral and scattering theory on geometric objects

• Project/Area Number: 25800073
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Kobe University (2014-2016); University of Tsukuba (2013)
• Principal Investigator: Ito Kenichi 神戸大学, 理学研究科, 准教授 (90512509)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 関数方程式 / 多様体上での解析 / 数理物理 / 偏微分方程式論 / 差分作用素 / シュレーディンガー方程式 / 閾値レゾナンス / リーマン多様体 / スペクトル幾何 / 関数方程式論

Outline of Final Research Achievements

I constructed a new general framework for the stationary scattering theory for the Schroedinger operator on a manifold with asymptotically Euclidean and/or hyperbolic funnel ends. I also succeeded in formulating the threshold resonances in a natural manner for the discrete Schroedinger operators on the discrete line and discrete half-line. Moreover, I obtained explicit expressions for the branching parts around thresholds for the resolvent of the discrete Laplacian on the square lattice.

Research Products

• [Int'l Joint Research] オーフス大学(デンマーク)
• [Journal Article] Rellich's theorem and N-body Schr\"odinger operators (2016)
  • Author(s): K. Ito, E. Skibsted
  • Journal Title: Reviews in Mathematical Physics Volume: 28 Issue: 05 Pages: 1650010-1650010
  • DOI: 10.1142/s0129055x16500100
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] A complete classification of threshold properties for one-dimensional discrete Schr\"odinger operators (2015)
  • Author(s): K. Ito, A. Jensen,
  • Journal Title: Reviews in Mathematical Physics Volume: 27 Issue: 01 Pages: 1550002-1550002
  • DOI: 10.1142/s0129055x15500026
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Absence of positive eigenvalues for hard-core $N$-body systems (2014)
  • Author(s): K. Ito, E. Skibsted
  • Journal Title: Annales Henri Poincar\'e Volume: 15 Issue: 12 Pages: 2379-2408
  • DOI: 10.1007/s00023-013-0309-x
  • Peer Reviewed
• [Journal Article] Microlocal properties of scattering matrices for Schrodinger equations on scattering manifolds (2013)
  • Author(s): Ito, K., Nakamura, S
  • Journal Title: Analysis and PDE Volume: 6(No. 2) Issue: 2 Pages: 257-286
  • DOI: 10.2140/apde.2013.6.257
  • Peer Reviewed
• [Journal Article] Absence of embedded eigenvalues for Riemannian Laplacians (2013)
  • Author(s): K. Ito, E. Skibsted
  • Journal Title: Advances in Mathematics Volume: 258 Pages: 945-962
  • DOI: 10.1016/j.aim.2013.08.023
  • NAID: 120007136801
  • Peer Reviewed
• [Presentation] Resolvent expansions for the Schr\"odinger operator on the discrete half-line (2017)
  • Author(s): K. Ito
  • Organizer: HMAセミナー・冬の研究会 2017
  • Place of Presentation: 広島大学
  • Year and Date: 2017-01-20
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends （4回連続講演） (2017)
  • Author(s): K. Ito
  • Organizer: Tokyo-Berkeley Mathematics Workshop "Partial Differential Equations and Mathematical Physics"
  • Place of Presentation: 東京大学
  • Year and Date: 2017-01-09
  • Int'l Joint Research / Invited
• [Presentation] Branching form of the resolvent at threshold for discrete Laplacians (2016)
  • Author(s): K. Ito
  • 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
  • Year and Date: 2016-11-26
• [Presentation] Branching form of the resolvent at threshold for discrete Laplacians (2016)
  • Author(s): K. Ito
  • Organizer: QMath13: Mathematical Results in Quantum Physics
  • Place of Presentation: Georgia Institute of Technology, USA
  • Year and Date: 2016-10-09
  • Int'l Joint Research
• [Presentation] Branching form of the resolvent at threshold for discrete Laplacians (2016)
  • Author(s): K. Ito
  • Organizer: Math/Phys Seminar
  • Place of Presentation: Aarhus University, Denmark
  • Year and Date: 2016-08-18
  • Int'l Joint Research / Invited
• [Presentation] Interpretation of complex resonances (and a remark on Rellich's theorem) (2016)
  • Author(s): K. Ito
  • Organizer: RIMS共同研究「線形及び非線形分散型方程式に関する最近の進展」
  • Place of Presentation: 京都大学
  • Year and Date: 2016-05-19
  • Invited
• [Presentation] Asymptotic behavior of eigenfunctions on manifold with ends (2015)
  • Author(s): 伊藤健一
  • Organizer: 微分トポロジーセミナー
  • Place of Presentation: 京都大学（京都府・京都市）
  • Year and Date: 2015-11-17
  • Invited
• [Presentation] A remark on Rellich's theorem (2015)
  • Author(s): 伊藤健一
  • Organizer: 波動セミナー
  • Place of Presentation: 北海道大学（北海道・札幌市）
  • Year and Date: 2015-11-09
  • Invited
• [Presentation] A remark on Rellich's theorem (2015)
  • Author(s): 伊藤健一
  • Organizer: 研究集会「第26回 数理物理と微分方程式」
  • Place of Presentation: ニューサンピア姫路ゆめさき（兵庫県・姫路市）
  • Year and Date: 2015-11-01
• [Presentation] A complete classification of threshold properties for one-dimensional discrete Schr\"odinger operators (2015)
  • Author(s): Kenichi Ito
  • Organizer: Young Researchers Symposium
  • Place of Presentation: Santiago (Chile)
  • Year and Date: 2015-07-24
  • Int'l Joint Research
• [Presentation] Complete classification of threshold properties for one-dimensional discrete Schr\"odinger operators (2015)
  • Author(s): Kenichi Ito
  • Organizer: Topics in Analysis and Mathematical Physics, A conference in honor of Arne Jensen's 65th birthday
  • Place of Presentation: Aalborg (Denmark)
  • Year and Date: 2015-05-30
  • Int'l Joint Research / Invited
• [Presentation] Stationary scattering theory on manifold with ends (LAP and RC) (2015)
  • Author(s): 伊藤健一
  • Organizer: 偏微分方程式姫路研究集会
  • Place of Presentation: イーグレ姫路
  • Year and Date: 2015-03-07
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2015)
  • Author(s): 伊藤健一
  • Organizer: 第7回名古屋微分方程式研究集会 (The 7th Nagoya Workshop on Differential Equations)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2015-03-04
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-12-13
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 第3回 信州関数解析シンポジウム
  • Place of Presentation: 信州大学
  • Year and Date: 2014-12-02
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 東京大学火曜解析セミナー
  • Place of Presentation: 東京大学
  • Year and Date: 2014-11-25
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 研究集会「第25回 数理物理と微分方程式」
  • Place of Presentation: 神奈川県足柄下郡箱根町強羅 四季の湯強羅静雲荘
  • Year and Date: 2014-11-01
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 第20回南大阪応用数学セミナー
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2014-10-25
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: RIMS研究集会「スペクトル理論とその周辺(Spectral and Scattering Theory and Related Topics)」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-10-17
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: NLPDEセミナー
  • Place of Presentation: 京都大学
  • Year and Date: 2014-10-10
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: RIMS研究集会「偏微分方程式に付随する確率論的問題」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-09-19
  • Invited
• [Presentation] Stationary scattering theory on manifold with ends （4回連続講演） (2014)
  • Author(s): 伊藤健一
  • Organizer: RIMS共同研究「多様体・格子・グラフ上のシュレディンガー方程式のスペクトル・散乱理論」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-09-10 – 2014-09-12
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): Kenichi Ito
  • Organizer: Analysis Seminar
  • Place of Presentation: Aarhus University
  • Year and Date: 2014-08-20
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 偏微分方程式セミナー
  • Place of Presentation: 北海道大学
  • Year and Date: 2014-07-28
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 愛媛大学解析セミナー
  • Place of Presentation: 愛媛大学
  • Year and Date: 2014-06-21
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: RIMS共同研究「線形および非線形分散型方程式に関する最近の進展」
  • Place of Presentation: 京都大学
  • Year and Date: 2014-05-21
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 大阪大学数学教室 微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2014-05-16
  • Invited
• [Presentation] Absence of $B_0^*$-eigenfunctions for the Schr\"odinger operator on manifold with ends (2014)
  • Author(s): 伊藤健一
  • Organizer: 神戸大学理学研究科数学教室談話会
  • Place of Presentation: 神戸大学
  • Year and Date: 2014-04-16
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2014)
  • Author(s): 伊藤健一
  • Organizer: 研究集会「スペクトル・散乱 鹿児島シンポジウム」
  • Place of Presentation: 鹿児島大学（鹿児島県）
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2014)
  • Author(s): 伊藤健一
  • Organizer: 東京大学火曜解析セミナー
  • Place of Presentation: 東京大学（東京都）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2014)
  • Author(s): 伊藤健一
  • Organizer: 解析セミナー
  • Place of Presentation: 神戸大学（兵庫県）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2014)
  • Author(s): 伊藤健一
  • Organizer: 筑波大学解析セミナー
  • Place of Presentation: 筑波大学（茨城県）
• [Presentation] Classification of threshold properties of one-dimensional discrete Schr\"odinger operators (2014)
  • Author(s): 伊藤健一
  • Organizer: 日本数学会2014年度年会 函数解析学分科会特別講演
  • Place of Presentation: 学習院大学（東京都）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2013)
  • Author(s): 伊藤健一
  • Organizer: RIMS共同研究「線形および非線形分散型方程式の研究」
  • Place of Presentation: 京都大学（京都府）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2013)
  • Author(s): 伊藤健一
  • Organizer: 応用解析セミナー
  • Place of Presentation: 東北大学（宮城県）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2013)
  • Author(s): 伊藤健一
  • Organizer: 研究集会「Linear and Nonlinear Waves, No.11」
  • Place of Presentation: ピアザ淡海滋賀県立県民交流センター（滋賀県）
  • Invited
• [Presentation] Threshold properties of one-dimensional discrete Schr\"odinger operators (2013)
  • Author(s): 伊藤健一
  • Organizer: 研究集会「第24回 数理物理と微分方程式」
  • Place of Presentation: 弘前市 星と森のロマントピア（青森県）