Singularities of solutions to Schrodinger equations

• Project/Area Number: 17340033
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: NAKAMURA Shu The University of Tokyo, 大学院・数理科学研究科, 教授 (50183520)
• Co-Investigator(Kenkyū-buntansha): 谷島 賢二 学習院大学, 理学部, 教授 (80011758) ; 土居 伸一 大阪大学, 大学院・理学研究科, 教授 (00243006)
• Project Period (FY): 2005 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥17,160,000 (Direct Cost: ¥15,000,000、Indirect Cost: ¥2,160,000); Fiscal Year 2008: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2007: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2006: ¥3,600,000 (Direct Cost: ¥3,600,000); Fiscal Year 2005: ¥4,200,000 (Direct Cost: ¥4,200,000)
• Keywords: 偏微分方程式 / 量子力学 / 超局所解析 / シュレディンガー方程式 / 超局所的特異性 / 特異性の伝播 / 散乱理論 / 散乱多様体 / 調和振動子 / シュレディンガー作用素 / 半古典解析 / 解の特異性 / 平滑化作用 / 波面集合 / 偏微分方程式の解の特異性

Research Abstract

シュレディンガー方程式の解の特異性の伝播に関する研究を行い、特に、非束縛的な仮定の下で、解の超局所特異性を古典力学的散乱理論を用いて特徴づけることに成功した。この成果は、当初はユークリッド空間上の短距離型摂動の場合に証明され、さらに長距離型摂動、散乱多様体、摂動を持つ調和振動子、また解析的特異性の特徴付けへと拡張された。また、実解析的な平滑化作用に関する研究成果も得た。

Research Products

• [Journal Article] Wave front set for solutions to Schrodinger equations (2009)
  • Author(s): Nakamura, S.
  • Journal Title: J.Functional Analysis 256 Pages: 1299-1309
  • Peer Reviewed
• [Journal Article] Semiclassical singular-ity propagation property for Schrodinger equations (2009)
  • Author(s): Nakamura, S.
  • Journal Title: J. Math. Soc. Japan 61 Pages: 177-211
  • Peer Reviewed
• [Journal Article] Spectral extrema and Lifshitz tails for non monotonous alloy type models (2009)
  • Author(s): Klopp, F., Nakamura,S.
  • Journal Title: Commun. Math. Phys 287 Pages: 1133-1143
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to perturbed harmonic oscillators. Comm (2009)
  • Author(s): Mao, S., Nakamura, S.
  • Journal Title: Partial Differential Equations 34 Pages: 506-519
  • Peer Reviewed
• [Journal Article] Wave front set for solutions to Schrodinger equations (2009)
  • Author(s): S. Nakamura
  • Journal Title: J. Functional Analysis 256 Pages: 1299-1309
  • Peer Reviewed
• [Journal Article] Semiclassical singularity propagation property for Schrodinger equations (2009)
  • Author(s): S. Nakamura
  • Journal Title: J. Math. Soc. Japan 61 Pages: 177-211
  • Peer Reviewed
• [Journal Article] Spectral extrema and Lifshitz tails for nOn monotonous alloy type models (2009)
  • Author(s): S. Nakamura, F. Klopp
  • Journal Title: Commun. Math. Phys. 287 Pages: 1133-1143
  • Peer Reviewed
• [Journal Article] Singularities of solutions to Schrodinger equation on scattering manifold. (2009)
  • Author(s): k Ito, S. Nakamura
  • Journal Title: American J. Math. (印刷中)
  • Peer Reviewed
• [Journal Article] Wave front set for solutions, to perturbed harmonic oscillators. (2009)
  • Author(s): S. Mao, S. Nakamura
  • Journal Title: Comm. Partial Differential Equations (印刷中)
  • Peer Reviewed
• [Journal Article] Analytic singularities for long range Schrodinger equations (2008)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V.
  • Journal Title: Comptes Rendus Mathematique 346, 15-16 Pages: 849-852
  • Peer Reviewed
• [Journal Article] On Lp bounded-ness of wave operators for 4-dimensional Schrodinger operators with threshold singularities (2008)
  • Author(s): Jensen, A., Yajima, K.
  • Journal Title: Proc. Lond. Math. Soc 86 Pages: 136-162
  • Peer Reviewed
• [Journal Article] Analytic singularities for long range Schrodinger equations (2008)
  • Author(s): A. Martinez, S. Nakamura, V. Sordoni
  • Journal Title: Comptes Rendus Mathematique 346 Pages: 849-852
  • Peer Reviewed
• [Journal Article] Analytic Smoothing Effect for the Schrodinger Equation with Long-Range Perturbation, Comm (2006)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V.
  • Journal Title: Pure Appl. Math 59 Pages: 1330-1351
  • Peer Reviewed
• [Journal Article] Bounds on the spectral shift function and the density of states (2006)
  • Author(s): Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselic, I..
  • Journal Title: Commun. Math. Phys 262 Pages: 489-503
  • Peer Reviewed
• [Journal Article] The L^p boundedness of wave operators for Schrodinger operators with threshold singularities (2006)
  • Author(s): Yajima, K.
  • Journal Title: I.. The odd dimensional case. J. Math. Sci. Univ. Tokyo 13 Pages: 43-93
  • Peer Reviewed
• [Journal Article] The L^p bounded-ness of wave operators for Schrodinger operators with threshold singularities.II. Even dimensional case (2006)
  • Author(s): Finco, D., Yajima,K.
  • Journal Title: J. Math. Sci.Univ. Tokyo 13 Pages: 277-346
  • Peer Reviewed
• [Journal Article] Propagation of singularities for Schrodinger equations on the Euclidean space with a scattering metric (2006)
  • Author(s): Ito, K.
  • Journal Title: Comm. Partial Differential Equations 31 Pages: 1735-1777
  • Peer Reviewed
• [Journal Article] Analytic smoothing effect for the Schrodinger equation with long-range perturbation. (2006)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V.
  • Journal Title: Communicaitions in Pure and Applied Mathematics 59・9 Pages: 1330-1351
• [Journal Article] Bounds on the spectral shift function and the density of states. (2006)
  • Author(s): Hundertmark, D., Killip, R., Nakamura, S., Stollman, P., Veselic, I.
  • Journal Title: Communications in Mathematical Physics 261・2 Pages: 489-503
• [Journal Article] The $L_p$ boundedness of wave operators for Schrodinger operators with threshold singularities. II. Even dimentional case (2006)
  • Author(s): Finco, Domenico, Yajima, Kenji
  • Journal Title: J. Math. Sci. Univ. Tokyo 13・3 Pages: 277-346
• [Journal Article] The $Lsp p$ boundedness of wave operators for Schrodinger operators with threshold singularities. I. The odd dimensional case. (2006)
  • Author(s): Yajima, Kenji
  • Journal Title: J. Math. Sci. Univ. Tokyo 13・1 Pages: 43-93
• [Journal Article] $scr Hsp s$ solutions for nonlinear Schrodinger equations with potentials superquadratic at infinity. (2006)
  • Author(s): Zhang, Guoping, Yajima, Kenji, Liu, Fengshan
  • Journal Title: Physics Letters A 356・2 Pages: 95-98
• [Journal Article] Analytic Smoothing Effect for the Schr" odinger Equation with Long-Range Perturbation (2006)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V.
  • Journal Title: Conn.Pure Appl.Math.
• [Journal Article] Bounds on the spectral shift function and the density of states (2006)
  • Author(s): Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselic', I.
  • Journal Title: Commun.Math.Phys. 262.2 Pages: 489-503
• [Journal Article] $L^p$-boundedness of wave operators with threshold singularities I, Odd dimensional case (2006)
  • Author(s): Kenji Yajima
  • Journal Title: J.math.Ssci.Univ.Tokyo
• [Journal Article] Propagation of the homogeneous wave front set for Schrodinger equations (2005)
  • Author(s): Nakamura, S
  • Journal Title: Duke Math. J 126 Pages: 349-367
  • Peer Reviewed
• [Journal Article] Dispersive estimates for Schrodinger equations with threshold resonance and eigenvalue (2005)
  • Author(s): Yajima, K.
  • Journal Title: Comm. Math. Phys 259 Pages: 475-509
  • Peer Reviewed
• [Journal Article] Smoothness of solutions for Schrodinger equations with unbounded potentials (2005)
  • Author(s): Doi, S.
  • Journal Title: Publ. Res. Inst. Math. Sci 41 Pages: 175-221
  • Peer Reviewed
• [Journal Article] Propagation of the homogenous wave front set for Schr" odinger equations. (2005)
  • Author(s): Nakamura, S.
  • Journal Title: Duke Math.J. 126.2 Pages: 349-367
• [Journal Article] Dispersive estimates fro Schr" odinger equations with thershold resonance and eigenvalue (2005)
  • Author(s): Kenji Yajima
  • Journal Title: Commun.Math.Phys. 259.2 Pages: 475-509
• [Journal Article] Singularities of solutions to Schrodinger equation on scattering manifold
  • Author(s): Ito, K., Nakamura, S.
  • Journal Title: American J. Math (印刷中)
  • Peer Reviewed
• [Journal Article] Analytic wave front for solutions to Schrodinger equation
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V.
  • Journal Title: Advances in Math (印刷中)
  • Peer Reviewed
• [Presentation] Time-dependent scattering theory for Schrodinger operators on scattering manifolds (2008)
  • Author(s): 中村周
  • Organizer: BIRS Workshop : Mathematical Theory of Resonances. Banff
  • Place of Presentation: Canada
  • Year and Date: 2008-10-23
• [Presentation] Time dependent scattering theory for Schrodinger operators on scattering manifolds (2008)
  • Author(s): S. Nakamura
  • Organizer: "Mathematical Theory of Resonances"
  • Place of Presentation: BIRS,バンフ、カナダ
  • Year and Date: 2008-10-23
• [Presentation] Propagation of singularities for Schrodinger operators (2008)
  • Author(s): 中村周
  • Organizer: Second Symposium on Scattering and Spectral Theory, Serrambi
  • Place of Presentation: Brazil
  • Year and Date: 2008-08-21
• [Presentation] Propagation of singularities for Schrodinger operators (2008)
  • Author(s): S. Nakamura
  • Organizer: "Second Symposium on Scattering and Spectral Theory"
  • Place of Presentation: Serrambi,ブラジル
  • Year and Date: 2008-08-21
• [Presentation] Lifshitz tails for non monotonous alloy type model (2008)
  • Author(s): 中村周
  • Organizer: Oberwolfach Workshop : Disordered Systems : Random Schrodinger Operators and Random Matrices,Oberwolfach
  • Place of Presentation: Germany
  • Year and Date: 2008-03-28
• [Presentation] Remarks on scattering on scattering manifold (2008)
  • Author(s): 中村周
  • Organizer: 研究集会「スペクトル・散乱理論とその周辺」(京都大学・数理解析研究所)
  • Year and Date: 2008-01-16
• [Presentation] Remarks on scattering on scattering manifolds (2008)
  • Author(s): 中村周
  • Organizer: 研究集会「スペクトル理論とその周辺」
  • Place of Presentation: 京都大学数・理解析研究所
  • Year and Date: 2008-01-16
• [Presentation] Singularity of solutions to Schrodinger equation on scattering manifold (2007)
  • Author(s): 中村周
  • Organizer: Mathematical Physics Seminar at Inst.H. Poincare
  • Place of Presentation: Paris
  • Year and Date: 2007-07-31
• [Presentation] Singularity of solutions to Schrodinger equation on scattering manifold (2007)
  • Author(s): 中村周
  • Organizer: Seminar at Euler Institute
  • Place of Presentation: St.Petersburg
  • Year and Date: 2007-07-31
• [Presentation] Singularity of solutions on scattering manifold (2007)
  • Author(s): 中村周
  • Organizer: アンリ・ポアンカレ研究所 数理物理学セミナー
  • Place of Presentation: パリ Inst. H. Poincare
  • Year and Date: 2007-06-11
• [Presentation] Singularity of solutions to Schrodinger equation on scattering manifold (2007)
  • Author(s): 中村周
  • Organizer: Seminar at Univ. Bologna
  • Year and Date: 2007-06-04
• [Presentation] Topics on scattering theory for Schrodinger operators (2007)
  • Author(s): 中村周
  • Organizer: Seminar at Univ
  • Place of Presentation: Paris
  • Year and Date: 2007-05-29
• [Presentation] Topics in scattering theory for Schroedinger operators (2007)
  • Author(s): 中村周
  • Organizer: パリ大学第13校 セミナー
  • Place of Presentation: パリ大学第13校
  • Year and Date: 2007-05-29
• [Presentation] シュレディンガー方程式の解の特異性とその周辺 (2007)
  • Author(s): 中村周
  • Organizer: 日本数学会年会・企画特別講演
  • Place of Presentation: 埼玉大学
  • Year and Date: 2007-03-30
• [Presentation] Singularity of solutions to Schrodinger equations (2006)
  • Author(s): 中村周
  • Organizer: International Conference : Spectral Theory of Random Operators and Related Fields in Probability Theory
  • Place of Presentation: Kyoto Univ
  • Year and Date: 2006-12-11
• [Presentation] Singularity of solutions to Schrodinger equations (2006)
  • Author(s): 中村周
  • Organizer: Seminar at Dept.Math., The Australian National University
  • Place of Presentation: Canberra, Australia
  • Year and Date: 2006-10-30
• [Presentation] Singularity of solutions to Schrodinger equation with variable coefficients (2006)
  • Author(s): 中村周
  • Organizer: PASI(Pan-American Advanced Studies Institute)
  • Place of Presentation: Santiago, Chile
  • Year and Date: 2006-07-31
• [Presentation] Semiclassical singularity propagation property for Schrodinger equation with long-range perturbation (2006)
  • Author(s): 中村周
  • Organizer: BIRS Workshop : Schrodinger Evolution Equations, Banff
  • Place of Presentation: Canada
  • Year and Date: 2006-04-23
• [Book] 関数解析 (2006)
  • Author(s): 岡本 久, 中村 周
  • Total Pages: 286
  • Publisher: 岩波書店