Scattering theory for linear and nonlinear Schrodinger equations

• Project/Area Number: 20540169
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: ADACHI Tadayoshi Kobe University, 大学院・理学研究科, 教授 (30281158)
• Co-Investigator(Kenkyū-buntansha): TAKAOKA Hideo 北海道大学, 大学院・理学研究院, 教授 (10322794)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 多体問題 / スペクトル理論 / 散乱理論 / 漸近完全性 / 電磁場 / 時間大域的適切性 / 波動作用素 / 逆問題 / 適切性 / 電荷移動モデル / 電荷輸送モデル

Research Abstract

The achievements of our research are as follows: 1. We improved some results obtained by several previous works on the asymptotic completeness for quantum systems in time-dependent homogeneous electric fields. 2. We obtained a new result on a certain problem of the inverse scattering for quantum systems in homogeneous electric fields asymptotically zero in time. 3. We obtained new results on the problems of the well-posedness and the scattering theory for various nonlinear dispersive equations, especially in critical cases.

Research Products

• [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition (2010)
  • Author(s): K.Nakanishi, H.Takaoka, Y.Tsutsumi
  • Journal Title: Discrete and Continuous Dynamical Systems. Series A 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation (2010)
  • Author(s): J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao
  • Journal Title: Inventiones Mathematicae 181 Pages: 39-113
  • Peer Reviewed
• [Journal Article] Scattering theory for two-body quantum systems with singular potentials in a time-periodic electric field (2010)
  • Author(s): T.Adachi, T.Kimura, Y.Shimizu
  • Journal Title: Journal of Mathematical Physics 51
  • Peer Reviewed
• [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition (2010)
  • Author(s): K.Nakanishi, H.Takaoka, Y.Tsutsumi
  • Journal Title: Discrete and Continuous Dynamical Systems. Series A Volume: 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation (2010)
  • Author(s): J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao
  • Journal Title: Inventiones Mathematicae Volume: 181 Pages: 39-113
  • Peer Reviewed
• [Journal Article] Stability in H^<1/2> of the sum of K solitons for the Benjamin-Ono equation (2009)
  • Author(s): S.Gustafson, H.Takaoka, T.-P.Tsai
  • Journal Title: Journal of Mathematical Physics 50
  • Peer Reviewed
• [Journal Article] Bilinear Strichartz estimates and applications to the cubic nonlinear Schrodinger equation in two space dimensions (2008)
  • Author(s): H.Takaoka
  • Journal Title: Hokkaido Mathematical Journal 37 Pages: 861-870
  • Peer Reviewed
• [Journal Article] Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R^3 (2008)
  • Author(s): J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao
  • Journal Title: Annales of Mathematics. Second Series 167 Pages: 767-865
  • Peer Reviewed
• [Journal Article] Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on R^2 (2008)
  • Author(s): J.Colliander, M.Keel, G.Staffilani, H.Takaoka, T.Tao
  • Journal Title: Discrete and Continuous Dynamical Systems. Series A 21 Pages: 665-686
  • Peer Reviewed
• [Journal Article] Bilinear Strichartz estimates and applications to the cubic nonlinear Schrodinger equation in two space dimensions (2008)
  • Author(s): H. Takaoka
  • Journal Title: Hokkaido Math. J. 37 Pages: 861-870
  • Peer Reviewed
• [Journal Article] Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R^3 (2008)
  • Author(s): J. Colliander, M. Keel, G Staffilani, H. Takaoka, T. Tao
  • Journal Title: Ann. of Math. 167 Pages: 767-865
  • Peer Reviewed
• [Journal Article] Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on R^2 (2008)
  • Author(s): J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao
  • Journal Title: Discrete Contin. Dyn. Syst. 21 Pages: 665-686
  • Peer Reviewed
• [Journal Article] On multidimensional inverse scattering in an external electric field asymptotically zero in time
  • Author(s): T.Adachi, T.Kamada, M.Kazuno, K.Toratani
  • Journal Title: to appear in Inverse Problems.
  • Peer Reviewed
• [Journal Article] Scattering in an external electric field asymptotically constant in time
  • Author(s): T.Adachi, A.Ishida
  • Journal Title: to appear in Journal of Mathematical Physics.
  • Peer Reviewed
• [Presentation] 時間周期的に変動する電場中の量子力学系に対する散乱理論 (2009)
  • Author(s): 足立匡義
  • Organizer: 日本数学会函数方程式論分科会特別講演
  • Place of Presentation: 大阪大学.
  • Year and Date: 2009-09-25
• [Presentation] 時間周期的に変動する電場中の量子力学系に対する散乱理論 (2009)
  • Author(s): 足立匡義
  • Organizer: 日本数学会函数方程式論分科会特別講演
  • Place of Presentation: 大阪大学(大阪府)
  • Year and Date: 2009-09-25
• [Presentation] M0orawetz評価式と非線形シュレディンガー方程式の散乱理論 (2008)
  • Author(s): 高岡秀夫
  • Organizer: 日本数学会北海道支部講演会
  • Place of Presentation: 北海道大学.
  • Year and Date: 2008-11-28
• [Presentation] Morawetz評価式と非線形シュレデインガー方程式の散乱理論 (2008)
  • Author(s): 高岡 秀夫
  • Organizer: 日本数学会北海道支部講演会
  • Place of Presentation: 北海道大学
  • Year and Date: 2008-11-28
• [Remarks] ホームページ
  • URL: http://www.math.kobe-u.ac.jp/~adachi/