Mathematical Analysis of Quantum Physics

• Project/Area Number: 18340041
• 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
• Principal Investigator: YAJIMA Kenji (KENJI Yajima) Gakushuin University, 理学部, 教授 (80011758)
• Co-Investigator(Kenkyū-buntansha): FUJIWARA Daisuke 学習院大学, 理学部, 教授 (10011561) ; NAKAMURA Shu 東京大学, 大学院・数理科学研究科, 教授 (50183520) ; MIZUTANI Akira 学習院大学, 理学部, 教授 (80011716) ; WATANABE Kazuo 学習院大学, 理学部, 助教 (90260851) ; SHIMOMURA Akihiro 首都大学東京, 大学院・理工学研究科, 准教授 (00365066)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥10,000,000 (Direct Cost: ¥8,200,000、Indirect Cost: ¥1,800,000); Fiscal Year 2009: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2008: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2007: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2006: ¥2,200,000 (Direct Cost: ¥2,200,000)
• Keywords: 関数方程式 / 関数解析 / 実解析 / 量子力学 / シュレーディンガー作用素 / シュレーディンガー方程式 / 線形偏微分方程式 / 散乱理論 / 特異性伝播 / ファインマン経路積分 / スペクトル理論 / 数理物理学 / 関数解析学 / 偏微分方程式 / 基本解 / 経路積分 / 量子物理学 / 特異性伝播問題 / 非線形偏微分方程式 / 分散型評価 / 非線形方程式

Research Abstract

We studied mathematical problems on Schroedinger equations which are fundamental equation for describing the dynamics of quantum particles and obtained following results : (1) New results are obtained on the behavior at infinity of the fundamental solutions of the initial value problems ; (2) new methods for studying the propagation of singularities of solutions are found ; (3) unsolved problems on wave operators of scattering have been solved ; (4) new spectral properties of random Schroedinger operators are found.

Research Products

• [Journal Article] Spatial growth of fundamental solutions for certain perturbations of the harmonic oscillator (2010)
  • Author(s): A. Jensen, K. Yajima
  • Journal Title: Review in Mathematical Physics 22 Pages: 193-206
  • Peer Reviewed
• [Journal Article] Spatial growth of fundamental solutions for certain perturbations of the harmonic oscillator (2010)
  • Author(s): A.Jensen, K.Yajima
  • Journal Title: Reviews in Mathematical Physics 22 Pages: 193-206
  • Peer Reviewed
• [Journal Article] Semiclassical singularity propagation property for Schroedinger equations. (2009)
  • Author(s): S. Nakamura
  • Journal Title: Journal of the Mathematical Society of Japan 61 Pages: 177-211
  • Peer Reviewed
• [Journal Article] Spectral extrema and Lifshitz tails for non monotonous alloy type models. (2009)
  • Author(s): F. Klopp, S. Nakamura
  • Journal Title: Commun. in Mathematical Physics 287 Pages: 1133-1143
  • Peer Reviewed
• [Journal Article] Analytic wave front for solutions to Schroedinger equation (2009)
  • Author(s): A. Martnez, S. Nakamura, B.V. Sordoni
  • Journal Title: Advances in Mathematics 222 Pages: 1277-1307
  • Peer Reviewed
• [Journal Article] Singularities of solutions to Schroedinger equation on scattering manifold. (2009)
  • Author(s): K. Ito, S. Nakamura
  • Journal Title: American Journal of Mathematics 131 Pages: 1835-1865
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to Schroedeinger equations (2009)
  • Author(s): S. Nakamura
  • Journal Title: Journal of Functional analysis 256 Pages: 1299-1309
  • Peer Reviewed
• [Journal Article] Semi-classical singularity propagation property for Schrodinger equations. (2009)
  • Author(s): S.Nakamura
  • Journal Title: Journal of the Mathematical Society of Japan 61 Pages: 177-211
  • Peer Reviewed
• [Journal Article] Spectral extrema and Lifshitz tails for non monotonous alloy type models. (2009)
  • Author(s): F.Klopp, S.Nakamura
  • Journal Title: Communication in Mathematical Physics 287 Pages: 1133-1143
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to perturbed harmonic oscillators. (2009)
  • Author(s): S.Mao, S.Nakamuara
  • Journal Title: Communications in Partial Differential Equations 34 Pages: 506-519
  • Peer Reviewed
• [Journal Article] Analytic wave front for solutions to Schrodinger equation (2009)
  • Author(s): A.Martnez, S.Nakamura, B.V.Sordoni
  • Journal Title: Advancess in Mathematics 222 Pages: 1277-1307
  • Peer Reviewed
• [Journal Article] Singularities of solutions to Schrodinger equation on scattering manifold. (2009)
  • Author(s): K.Ito, S.Nakamuar
  • Journal Title: American Journal of Mathematics 131 Pages: 1835-1865
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to Schroedinger equations (2009)
  • Author(s): Shu Nakamura
  • Journal Title: J. Funct. Anal Vol.256 Pages: 1299-1309
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to Schroedeinger equations (2009)
  • Author(s): S.Nakamura
  • Journal Title: Journal of Functional analysis 256 Pages: 1299-1309
  • Peer Reviewed
• [Journal Article] On L^p boundedness of wave operators for 4-dimensional Schroedinger operators with threshold singularities (2008)
  • Author(s): A. Jensen, K. Yajima
  • Journal Title: Proceedings of London Mathematical Society 96(3) Pages: 136-162
  • Peer Reviewed
• [Journal Article] Analytic singularities for long range Schroedinger equations (2008)
  • Author(s): A. Martinez, S. Nakamura, V. Sordoni
  • Journal Title: Comptes Randus Mathematiques 346 Pages: 859-852
  • Peer Reviewed
• [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semi-classical approximations (2008)
  • Author(s): N. Kumano-go, D. Fujiwara
  • Journal Title: Bulletin des Sciences Mathematiques 132 Pages: 313-357
  • Peer Reviewed
• [Journal Article] Phase space Feynman path integral via piecewise bicharacteristic paths and their semi-classical approximations (2008)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: Bull. Sci. Math vol.132 Pages: 313-357
  • Peer Reviewed
• [Journal Article] On L^p boundedness of wave operators for 4-dimensional Schroedinger operators with threshold singularities (2008)
  • Author(s): A.Jensen, K.Yajima
  • Journal Title: Proceedings of London Mathematical Society(3) 96 Pages: 136-162
  • Peer Reviewed
• [Journal Article] Analytic singularities for long range Schroedinger equations (2008)
  • Author(s): A.Martinez, S.Nakamura, V.Sordoni
  • Journal Title: Comptes Randus Mathematiques 346 Pages: 859-852
  • Peer Reviewed
• [Journal Article] On L^p boundeness of wave operators for four dimensional Schrodinger operators with threshold singularities (2008)
  • Author(s): Arne Jensen
  • Journal Title: Proceeding of the London Mathematical Society 96 Pages: 136-162
  • Peer Reviewed
• [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations, (2008)
  • Author(s): Naoto Kumano-go
  • Journal Title: Bulletin des Sciences Mathematiques (In press)
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions to Schroedinger equations with a subcritical dissipative nonlinearity (2007)
  • Author(s): N. Kita, A. Shimomura
  • Journal Title: Journal of Differential Equations 242
  • Peer Reviewed
• [Journal Article] Scattering theory for the Schroedinger-improved Boussinesq system in two space dimensions (2007)
  • Author(s): A. Shimomura
  • Journal Title: Asymptotic Analysis 51 Pages: 167-187
  • Peer Reviewed
• [Journal Article] Non-selfadjoint perturbation of Schroedinger and wave equations (2007)
  • Author(s): M. Kadowaki, K. Watanabe
  • Journal Title: Advanced Studies and Pure Mathematics 47 Pages: 137-157
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions to Schrodinger equations with a subcritical dissipative nonlinearity (2007)
  • Author(s): Naoyasu Kita
  • Journal Title: Journal of Differential Equations 242 Pages: 192-210
  • Peer Reviewed
• [Journal Article] Scattering theory for the Schrodinger-improved Boussinesq system in two space dimensions (2007)
  • Author(s): Akihiro Shimomura
  • Journal Title: Asymptotic Analysis 51 Pages: 167-187
  • Peer Reviewed
• [Journal Article] Global existence and asymptotic behavior of solution to the fourth order nonlinear Schrodinger type equation (2007)
  • Author(s): Jun Segata
  • Journal Title: Communications in Applied Analysis Vol.11 11 Pages: 169-188
  • Peer Reviewed
• [Journal Article] Non-selfadjoint perturbation of Schr\"odinger and wave equations, pp. (2007)
  • Author(s): M.Kadowaki
  • Journal Title: Advanced Studies and Pure Math. 47 Pages: 137-157
  • Peer Reviewed
• [Journal Article] Some results on spectral analysis of non-selfadjoint perturbati on for Schrodinger and wave equations, (2007)
  • Author(s): M.Kadowaki
  • Journal Title: Proceedings of 6'th ISAAC congress, To appear
• [Journal Article] The L^p boundedness of wave operators with threshold singularities I, Odd dimensional case (2006)
  • Author(s): K. Yajima
  • Journal Title: Journal of Mathematical Sciences, University of Tokyo 13 Pages: 43-93
  • Peer Reviewed
• [Journal Article] The L^p boundedness of wave operators with threshold singularities II, Even dimensional case (2006)
  • Author(s): D. Finco, K. Yajima
  • Journal Title: Journal of Mathematical Sciences, University of Tokyo 13 Pages: 277-346
  • Peer Reviewed
• [Journal Article] Analytic smoothing effect for the Schroedinger equations with long range perturbation (2006)
  • Author(s): A. Martinez, S. Nakamura, V. Sordoni
  • Journal Title: Communications in Pure and Applied Mathematics 59 Pages: 1330-1361
  • Peer Reviewed
• [Journal Article] Bounds on spectral shift functions and the density of states (2006)
  • Author(s): D. Hundredmark, R. Killip, S. Nakamura, P. Stollmann, I. Veselic
  • Journal Title: Communications in Mathematical Physic 262
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of solutions of Schroedinger equations with dissipative nonlinearities (2006)
  • Author(s): A. Shimomura
  • Journal Title: Commun. in Partial Differential Equations 31 Pages: 1407-1423
  • Peer Reviewed
• [Journal Article] Nonexistence of scattering states for some quadratic nonlinear Schroedinger equation in two space dimensions (2006)
  • Author(s): A. Shimomura, Y. Tsutsumi
  • Journal Title: Differential and Integral Equation 19 Pages: 1047-1060
  • Peer Reviewed
• [Journal Article] An improved remainder estimates of stationary phase method for some oscillatory integrals over a space of large dimensions (2006)
  • Author(s): D. Fujiwara, N. Kumano-go
  • Journal Title: Funkcilaj Ekvacioj 49 Pages: 59-86
  • NAID: 130000140660
  • Peer Reviewed
• [Journal Article] The second term of semi-classical expansion for Feynman path integrals with integrand of polynomial growth (2006)
  • Author(s): D. Fujiwara, N. Kumano-g
  • Journal Title: Journal of the Mathematical Society of Japan 58 Pages: 847-867
  • Peer Reviewed
• [Journal Article] The L^p boundednes of wave operators for Schroedinger operators with threshold singularities I, Odd dimensional case (2006)
  • Author(s): Kenji Yajima
  • Journal Title: J. Math. Sci vol. 13 Pages: 43-93
  • Peer Reviewed
• [Journal Article] The L^p boundednes of wave operators for Schroedinger operators with threshold singularities II, Even dimensional case (2006)
  • Author(s): Domenico Finco, Kenji Yajima
  • Journal Title: J. Math. Sci Univ. Tokyo vol. 13 Pages: 277-356
  • Peer Reviewed
• [Journal Article] Analytic smoothing effect for Schroedinger equations with long range perturbation (2006)
  • Author(s): Andre Martinez, Shu Nakamura, VaniaSordoni
  • Journal Title: Comm. Pure. Appl. Math vol. 59 Pages: 1330-1361
  • Peer Reviewed
• [Journal Article] the L^p boundedness of wave operators for Schroedinger operators with threshold singularities I, Odd dimensional case (2006)
  • Author(s): Yajima, K.
  • Journal Title: Journal of Mathematical Sciences, University of Tokyo 13/1 Pages: 43-93
• [Journal Article] The L^p boundedness of wave operators for Schroedinger operators with threshold singularities II, Even dimensional case (2006)
  • Author(s): Fince, D., Yajima, K.
  • Journal Title: Journal of Mathematical Sciences, University of Tokyo 13/2 Pages: 277-346
• [Journal Article] Analytic smoothing effect for the Schroedinger equations with long range perturbation (2006)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, W.
  • Journal Title: Communications on Pure and applied Mathematics 59/9 Pages: 1330-1361
• [Journal Article] Bounds on spectral shift functions and the density of states (2006)
  • Author(s): Hundredmark, D., Killip, R., Nakamura, S., Stollann, P, Veselic, I.
  • Journal Title: Communications in Mathematical Physics 262 Pages: 489-503
• [Journal Article] asymptotic behavior of solutions of Schroedinger equations with dissipative nonlineraties (2006)
  • Author(s): Shimomura, A.
  • Journal Title: Communications in Partial Differential Equations 31 Pages: 1407-1423
• [Journal Article] Nonexistence of scattering states for some quadratic nonlinera Schreodinger equation in tow space dimensions (2006)
  • Author(s): Shimomura, A., Tsutsumi, Y.
  • Journal Title: Differential and Integral equations 19 Pages: 1047-1060
• [Journal Article] an improved remainder estimate of stationary phase method for some oscillatory integrals over a space of large dimension. (2006)
  • Author(s): Fujiwara, D., Kumanogo, N.
  • Journal Title: Funkcialaj Ekvacioj 49/1 Pages: 59-86
  • NAID: 130000140660
• [Journal Article] The second term of semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth. (2006)
  • Author(s): Fujiwara, D., Kumanogo, N.
  • Journal Title: Journal of the Mathematical Society of Japan 58/3 Pages: 847-867
  • NAID: 10018381149
• [Journal Article] Interface vanishing for solutions to Maxwell and Stokes systems, (2006)
  • Author(s): Kobayashi, T., Suzuki, T., Watanabe, K
  • Journal Title: Journalof Mathematical Fluid Mechanics 8/3 Pages: 382-397
• [Journal Article] Semiclassical singularity propagation property for Schrodinger equations
  • Author(s): Shu Nakamura
  • Journal Title: Journal of the Mathematical society of Japan (to appear)
  • Peer Reviewed
• [Presentation] Remarks on scattering theory for Schr?dinger operators on scattering manifolds (2010)
  • Author(s): S. Nakamura
  • Organizer: Spectral Problems for Quantum Hamiltonians
  • Place of Presentation: Centre Interfacultaire Bernoulli, EPFL, Lausanne
  • Year and Date: 2010-02-22
• [Presentation] Remarks on scattering theory for Schrodinger operators on scattering manifolds (2010)
  • Author(s): S.Nakamura
  • Organizer: Spectral Problems for Quantum Hamiltonians
  • Place of Presentation: Centre Interfacultaire Bernoulli, EPFL, Lausanne,
  • Year and Date: 2010-02-22
• [Presentation] Remarks on Fundamental Solutions to Schroedinger Equation with Variable Coefficients (2010)
  • Author(s): S. Nakamura
  • Organizer: Spectral and Dynamical Properties of Quantum Hamiltonians
  • Place of Presentation: Centre Interfacultaire Bernoulli, EPFL, Laussane
  • Year and Date: 2010-02-16
• [Presentation] Remarks on Fundamental Solutions to Schrodinger Equation with Variable Coefficients (2010)
  • Author(s): S.Nakamura
  • Organizer: Spectral and Dynamical Properties of Quantum Hamiltonians,
  • Place of Presentation: Centre Interfacultaire Bernoulli, EPFL, Laussane
  • Year and Date: 2010-02-16
• [Presentation] 大次元空間上での停留位相法の剰余項評価とその経路積分への応用 (2010)
  • Author(s): 藤原大輔
  • Organizer: Encounter with Mathematics
  • Place of Presentation: 中央大学理工学部
  • Year and Date: 2010-01-08
• [Presentation] Lifshitz tails for Schroedinger operators with non-sign definite random potentials (2009)
  • Author(s): S. Nakamura
  • Organizer: Modeling and Understanding Random Hamiltonians: Beyond Monotonicity, Linearity and Independence
  • Place of Presentation: Oberwolfach Mathematics Institute
  • Year and Date: 2009-12-08
• [Presentation] Lifshitz tails for Schrodinger operators with non-sign definite random potentials", (2009)
  • Author(s): S.Nakamura
  • Organizer: "Modeling and Understanding Random Hamiltonians : Beyond Monotonicity, Linearity and Independence
  • Place of Presentation: Oberwolfach Mathematics Institute
  • Year and Date: 2009-12-08
• [Presentation] 調和振動子のある種の摂動の基本解 (2009)
  • Author(s): 谷島賢二
  • Organizer: 京都大学数理解析研究所研究集会「スペクトル・散乱理論とその周辺」
  • Place of Presentation: 京都大学人間環境学部
  • Year and Date: 2009-12-04
• [Presentation] On fundamental solution of Schroedinger equations研究集会 (2009)
  • Author(s): 谷島賢二
  • Organizer: Stochastic Problems and Nonlinear PDE
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2009-12-01
• [Presentation] On fundamental solution of Schroedinger equations (2009)
  • Author(s): 谷島賢二
  • Organizer: 研究集会[Stochastic Problems and Nonlinear PDE]
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2009-12-01
• [Presentation] Remarks on scatering theory on scattering manifolds Colloque Franco-Tunisien (2009)
  • Author(s): S. Nakamura
  • Organizer: d'equation aux derives pariellles
  • Place of Presentation: Hammamet, Tunisia
  • Year and Date: 2009-09-28
• [Presentation] Remarks on scatering theory on scattering manifolds (2009)
  • Author(s): S.Nakamura
  • Organizer: Colloque Franco-Tunisien, dequation aux derivees pariellles,
  • Place of Presentation: Hammamet, Tunisia,
  • Year and Date: 2009-09-28
• [Presentation] シュレーディンガー方程式の分散型評価 (2009)
  • Author(s): 谷島賢二
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 大阪大学豊中キャンパス
  • Year and Date: 2009-09-25
• [Presentation] シュレーディンガー方程式の分散型評価 (2009)
  • Author(s): 谷島賢二
  • Organizer: 日本数学会秋季総合分科会「秋季賞受賞講演」
  • Place of Presentation: 大阪大学豊中キャンパス
  • Year and Date: 2009-09-25
• [Presentation] シュレディンガー発展方程式の基本解について (2009)
  • Author(s): 中村周
  • Place of Presentation: 大阪電気通信大学
  • Year and Date: 2009-09-21
• [Presentation] 調和振動子の摂動の基本解の非有界性 (2009)
  • Author(s): 谷島賢二
  • Organizer: 研究集会第19回「微分方程式と数理物理」
  • Place of Presentation: ウエルハートピア熱海(熱海)
  • Year and Date: 2009-03-16
• [Presentation] Feynman経路積分の伊藤先生による定式化について (2008)
  • Author(s): 藤原大輔
  • Organizer: 伊藤清先生文化勲章祝賀講演会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-12-14
• [Presentation] Feynman 経路積分の伊藤先生による定式化について (2008)
  • Author(s): 藤原大輔
  • Organizer: 伊藤清先生文化勲章祝賀講演会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-12-14
• [Presentation] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth (2008)
  • Author(s): 藤原大輔
  • Organizer: Spectral Theory and Partial Differential equations
  • Place of Presentation: Copenhagen大学
  • Year and Date: 2008-11-21
• [Presentation] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth (2008)
  • Author(s): Daisuke Fujiwara
  • Place of Presentation: Copenhagen Univ. Denmark
  • Year and Date: 2008-11-21
• [Presentation] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth (2008)
  • Author(s): 藤原大輔
  • Organizer: Spectral Theory and Partial Differential equations
  • Place of Presentation: Copenhagen 大学
  • Year and Date: 2008-11-21
• [Presentation] Review of the fundamental solution to Schroedinger Equation (2008)
  • Author(s): Kenji Yajima
  • Organizer: Mathematical Horizon of Quantum Physics
  • Place of Presentation: Singapore国立大学
  • Year and Date: 2008-09-09
• [Presentation] Revies of the fundamental solution to Schroedinger equation (2008)
  • Author(s): Kenji Yajima
  • Organizer: Mathematical Horizon of quantum physics
  • Place of Presentation: National Univ. Singapore, Singapore
  • Year and Date: 2008-09-09
• [Presentation] Review of the fundamental solution to Schroedinger Equation (2008)
  • Author(s): 谷島賢二
  • Organizer: Mathematical Horizon of Quantum Physics
  • Place of Presentation: Singapore 国立大学(Singapore)
  • Year and Date: 2008-09-09
• [Presentation] On fundamental solution of Schroedinger equations (2008)
  • Author(s): Kenji Yajima
  • Organizer: Meccanica, A conference in honor of S. Graffi's 65-th birthday
  • Place of Presentation: Bologna大学
  • Year and Date: 2008-08-27
• [Presentation] On fundamental solution of Schroedinger equations, Meccanica (2008)
  • Author(s): Kenji Yajima
  • Organizer: A conference in honor of S. Graffi's 65-th birthday
  • Place of Presentation: Univ. Bologna, Italy
  • Year and Date: 2008-08-27
• [Presentation] On fundamental solution of Schroedinger equations (2008)
  • Author(s): 谷島賢二
  • Organizer: Meccanica, A conference in honor of S.Graffi's 65-th birthday
  • Place of Presentation: Bologna 大学(Italy)
  • Year and Date: 2008-08-27
• [Presentation] The L^p boundedness of wave operators for Schroedinger Operators (2008)
  • Author(s): Kenji Yajima
  • Organizer: 35-th Journees equations aux derives partielles
  • Place of Presentation: Evian
  • Year and Date: 2008-06-03
• [Presentation] The L^p boundedness of wave operators for Schroedinger operators (2008)
  • Author(s): Kenji Yajima
  • Organizer: 35-th Journees equations aux derives partielles
  • Place of Presentation: Evian, France
  • Year and Date: 2008-06-03
• [Presentation] The L^p boundedness of wave operators for Schroedinger Operators (2008)
  • Author(s): 谷島賢二
  • Organizer: 35-th Journees equations aux derives partielles
  • Place of Presentation: Evian(France)
  • Year and Date: 2008-06-03
• [Presentation] Unboundedness of the fundamental solution for some perturbations of harmonic oscillators (2008)
  • Author(s): Kenji Yajima
  • Organizer: レンヌ大学解析セミナー
  • Place of Presentation: Rennes University
  • Year and Date: 2008-05-25
• [Presentation] Unboundedness of the fundamental solution for some perturbations of harmonic oscillators (2008)
  • Author(s): 谷島賢二
  • Organizer: レンヌ大学解析セミナー
  • Place of Presentation: Rennes University(France)
  • Year and Date: 2008-05-25
• [Presentation] Topics of Schroedinger operators, SISSA (Trieste Italy) (2007)
  • Author(s): K. Yajima
  • Organizer: Mathematical Physics Seminar
  • Place of Presentation: SISSA, Trieste
  • Year and Date: 2007-05-06
• [Presentation] Topics of Schroedinger operators (2007)
  • Author(s): 谷島 賢二
  • Organizer: SISSA(Trieste Italy)Seminar
  • Place of Presentation: SISSA, Trieste
  • Year and Date: 2007-05-06
• [Presentation] Lectures on topics of Schroedinger operators Lecture Series of Instituto d'Alta Matematica (2007)
  • Author(s): K. Yajima
  • Place of Presentation: University Rome I
  • Year and Date: 2007-05-04
• [Presentation] Lectures on topics of Schroedinger operators (2007)
  • Author(s): Kenji Yajima
  • Organizer: Lecture series of Instituto d'Alta Matematica
  • Place of Presentation: Univ. Rome I, La Sapienza, Italy
  • Year and Date: 2007-05-04
• [Presentation] Lectures on topics of Schroedinger operators (2007)
  • Author(s): 谷島 賢二
  • Organizer: Lecture Series of Instituto d'Alta Matematica
  • Place of Presentation: University Rome I
  • Year and Date: 2007-05-04
• [Presentation] Existence of solution of Schroedinger operators (2007)
  • Author(s): K. Yajima
  • Organizer: Mathematical Physics Seminar
  • Place of Presentation: University of Bologna
  • Year and Date: 2007-04-13
• [Book] 応用解析ハンドブックのうち「Schroedinger方程式」の項(577から629ページ) (2010)
  • Author(s): 増田久弥 編集
  • Total Pages: 629
  • Publisher: シュプリンガー・ジャパン
• [Book] 応用解析ハンドブックのうち「Schroedinger方程式」の項 (2009)
  • Author(s): 谷島賢二
  • Publisher: シュプリンガー・ジャパン