Numerical verification for spectral distribution of differential operators and its applications

2008 Fiscal Year Final Research Report

• Project/Area Number: 18740055
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyushu University
• Principal Investigator: NAGATOU Kaori Kyushu University, 大学院・数理学研究院, 准教授 (40326426)
• Project Period (FY): 2006 – 2008
• Keywords: シュレディンガー作用素 / 本質的スペクトル / スペクトルギャップ / 数値的検証法

Research Abstract

バンド構造の本質的スペクトルを持つ1次元シュレディンガー作用素の固有値非存在範囲を精度保証付きで求める手法を開発し,検証数値例を与えた。本質的スペクトルのギャップにおける離散スペクトル(固有値)の存在・非存在は,半導体理論とも密接に関連する重要な問題である。本研究では,線形常微分方程式の基本解を計算機援用解析により厳密に求める手法をもとに,固有値が存在しない範囲を数学的に厳密に保証する方法を提案した。

Research Products

• [Journal Article] Validated computations for fundamental solutions of linear ordinary differential equations (2008)
  • Author(s): K. Nagatou
  • Journal Title: International Series of Numerical Mathematics Vol. 157 Pages: 43-50
  • Peer Reviewed
• [Journal Article] A computational approach to constructive a priori and a posteriori error estimates for Bi-Harmonic problems (2008)
  • Author(s): M. T. Nakao, K. Hashimoto and K. Nagatou
  • Journal Title: GAKUTO International Series, Mathematical Sciences and Applications 28 Pages: 139-148
  • Peer Reviewed
• [Journal Article] Numerical verification of stationary solutions for Navier-Stokes problems (2007)
  • Author(s): K. Nagatou, K. Hashimoto and M. T. Nakao
  • Journal Title: Journal of Computational and Applied Mathematics 199 Pages: 445-451
  • Peer Reviewed
• [Journal Article] A computational approach to constructive apriori error estimate for finite element approximations of bi-harmonic problems in nonconvex polygonal domains (2006)
  • Author(s): K. Hashimoto, K. Nagatou, M.T. Nakao
  • Journal Title: Information 9[4] Pages: 573-580
  • Peer Reviewed
• [Journal Article] Numerical verification method for infinite dimensional eigenvalue problems
  • Author(s): K. Nagatou
  • Journal Title: to appear in Japan Journal of Industrial and Applied Mathematics 26-2/3
  • Peer Reviewed
• [Presentation] Eigenvalue Excluding on 3-D Photonic Crystals (2009)
  • Author(s): Kaori Nagatou
  • Organizer: International Workshop on Numerical Verification and its Applications 2009
  • Place of Presentation: Miyako Island
  • Year and Date: 20090322-29
• [Presentation] Spectral Problem on 3-D Maxwell' s Equations (2009)
  • Author(s): Kaori Nagatou
  • Organizer: International workshop on verified computations and related topics
  • Place of Presentation: University of Karlsruhe (TH), Germany
  • Year and Date: 20090307-10
• [Presentation] Computer assisted proofs for differential equations (2009)
  • Author(s): Kaori Nagatou
  • Organizer: The 26th Kyushu Symposium on Partial Differential Equations
  • Place of Presentation: Kyushu University (Hakozaki Campus) International Hall
  • Year and Date: 20090126-28
• [Presentation] Validated computations for elliptic systems of FitzHugh-Nagumo type (2008)
  • Author(s): Kaori Nagatou
  • Organizer: 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2008)
  • Place of Presentation: University of Texas at El Paso, USA
  • Year and Date: 20080929-1003
• [Presentation] Eigenvalue excluding on 1-D Schroedinger operators (2008)
  • Author(s): Kaori Nagatou
  • Organizer: The 6th Gregynog Workshop on Computation and Analytic Problems in Spectral Theory, University of Wales, Gregynog Hall
  • Place of Presentation: Newtown(Powys), United Kingdom
  • Year and Date: 20080712-18
• [Presentation] Computer Assisted Proofs for Partial Differential Equations (2008)
  • Author(s): Kaori Nagatou
  • Organizer: Numbers, Functions, Equations' 08
  • Place of Presentation: Noszvaj, Hungary
  • Year and Date: 20080615-21
• [Presentation] Non-existence proof of point spectrum in a spectral gap (2008)
  • Author(s): Kaori Nagatou
  • Organizer: 2008 International Workshop on Numerical Verification and its Applications
  • Place of Presentation: Okinawa
  • Year and Date: 20080301-07
• [Presentation] Numerical verification for a spectral problem on 1-D Schroedinger operators (2008)
  • Author(s): Kaori Nagatou
  • Organizer: Spectral and Scattering Theory and Related Topics, RIMS
  • Place of Presentation: Kyoto University
  • Year and Date: 20080215-17
• [Presentation] 固有値問題に対する数値的検証法とその応用 (2008)
  • Author(s): 長藤かおり
  • Organizer: 日本応用数理学会2008年度年会・オーガナイズドセッション特別講演
  • Place of Presentation: 東京大学柏キャンパス
  • Year and Date: 2008-09-17
• [Presentation] 非線形微分方程式系の周期境界値問題に対する数値的検証法 (2008)
  • Author(s): 長藤かおり
  • Organizer: 日本数学会2009年度年会
  • Place of Presentation: 東京大学
  • Year and Date: 2008-03-29
• [Presentation] 1次元シュレディンガー作用素のスペクトルに対する数値的検証法 (2008)
  • Author(s): 長藤かおり
  • Organizer: 日本数学会2008年度年会特別講演
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-26
• [Presentation] Numerical verification method for a spectral problem on 1-D Schroedinger operators (2008)
  • Author(s): Kaori Nagatou
  • Organizer: Seminario Modellistica Differenziale Numerica
  • Place of Presentation: Universita' Di Roma "La Sapienza"
  • Year and Date: 2008-02-19
• [Presentation] Eigenvalue problems on 1-D Schroedinger operators (2007)
  • Author(s): Kaori Nagatou
  • Organizer: Conference on Inequalities and Applications' 07, De La Motte Castle
  • Place of Presentation: Noszvaj, Hungary
  • Year and Date: 20070909-15
• [Presentation] Eigenvalue excluding in essential spectrum gaps (2007)
  • Author(s): Kaori Nagatou
  • Organizer: International Workshop on Numerical Verification and its Applications
  • Place of Presentation: Waseda, Japan.
  • Year and Date: 20070226-0303
• [Presentation] シュレディンガー作用素のスペクトル・ギャップにおける固有値非存在検証 (2007)
  • Author(s): 長藤かおり
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] Validated computations for fundamental solutions of linear ordinary differential operators (2007)
  • Author(s): Kaori Nagatou
  • Organizer: NOLTA 2007 Special Session, Simon Fraser University at Harbour Centre
  • Place of Presentation: Vancouver, Canada
  • Year and Date: 2007-09-18
• [Presentation] Validated computations for a Schroedinger equation with essential spectrum (2007)
  • Author(s): Kaori Nagatou
  • Organizer: Mathematics colloquium
  • Place of Presentation: University of Debrecen, Hungary
  • Year and Date: 2007-02-15
• [Presentation] Validated computation for infinite dimensional eigenvalue problems (2006)
  • Author(s): Kaori Nagatou
  • Organizer: 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN2006)
  • Place of Presentation: Duisburg, Germany
  • Year and Date: 20060926-29
• [Presentation] Verified numerical computations for infinite dimensional eigenvalue problems (2006)
  • Author(s): Kaori Nagaotu
  • Organizer: Mathematics colloquium, University of Karlsruhe
  • Place of Presentation: Germany
  • Year and Date: 2006-09-19