Numerical verification for spectral distribution of differential operators and its applications

• 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
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,400,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
• 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 / Photonic Crvstal / バンド・ギャップ / 周期境界値問題 / 複素固有値 / シュレデンガー作用素 / 本的スペクトル

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] A computational approach to constructive a priori and a posteriori error estimates for Bi-Harmonic problems (2008)
  • Author(s): M.T. Nakao
  • Journal Title: GAKUTO International Series, Mathematical Sciences and Applications 28 Pages: 139-148
  • Peer Reviewed
• [Journal Article] Validated computations for fundamental solutions of linear ordinary differential equations (2008)
  • Author(s): K. Nagatou
  • Journal Title: International Series of Numerical Mathematics 157 Pages: 43-50
  • Peer Reviewed
• [Journal Article] Numerical verification method for spectral problems (2008)
  • Author(s): 長藤かおり
  • Journal Title: 京都大学数理解析研究所講究録 1607 Pages: 13-31
• [Journal Article] Validated computation for infinite dimensional eigenvalue problems (2008)
  • Author(s): K., Nagatou
  • Journal Title: 12th GAMM-IMACS International Symposion on Scientific Computing, Computer Arthmetic and Validated Numerics, IEEE Computer Society(Order Number E2821) (in press)
  • Peer Reviewed
• [Journal Article] Validated computation for fundamental solutions of linear ordinary differential equations (2008)
  • Author(s): K., Nagatou
  • Journal Title: Inequalities and Applications(to appear) (in press)
  • 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
  • NAID: 120000981497
  • Peer Reviewed
• [Journal Article] Numerical verification of stationary solutions for Navier-Stokes problems (2007)
  • Author(s): K.Nagatou, K.Hashimoto, M.T.Nakao
  • Journal Title: Journal of Computational and Applied Mathematics 199 Pages: 445-451
  • NAID: 120000981497
• [Journal Article] A computational approach to constructive a priori and a posteriori error estimates for Bi-Harmonic problems (2007)
  • Author(s): M.T.Nakao, K.Hashimoto, K.Nagatou
  • Journal Title: GAKUTO International Series, Mathematical Sciences and Applications (to appear)(印刷中)
• [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] A computational approach to constructive a priori 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
  • NAID: 120000981499
• [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
  • NAID: 10028169340
  • 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
• [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
• [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
• [Presentation] Computer assisted proofs for differential equations (2009)
  • Author(s): 長藤かおり
  • Organizer: The 26th Kyushu Symposium on Partial Differential Equatio
  • Place of Presentation: Kyushu University (Hakozaki Campus) International Hall
• [Presentation] Spectral Problem on 3-D Maxwell's Equations (2009)
  • Author(s): 長藤かおり
  • Organizer: International workshop on verified computations and related topics
  • Place of Presentation: University of Karlsruhe (TH), Germany
• [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] Numerical verification for a spectral problem on 1-Dschroedinger operators (2008)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: Seminario Modellistica Differenziale Numerica
  • Place of Presentation: Universita Di Roma La Sapienza
  • Year and Date: 2008-02-19
• [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
• [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
• [Presentation] Computer Assisted Proofs for Partial Differential Equations (2008)
  • Author(s): Kaori Nagatou
  • Organizer: Numbers, Functions, Equations' 08
  • Place of Presentation: Noszvaj, Hungary
• [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
• [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
• [Presentation] Computer Assisted Proofs for Partial Differential Equations (2008)
  • Author(s): 長藤かおり
  • Organizer: Numbers, Functions, Equations '08
  • Place of Presentation: Noszvaj, Hungary
• [Presentation] Eigenvalue excluding on 1-D Schroedinger operators (2008)
  • Author(s): 長藤かおり
  • Organizer: The 6th Gregynog Workshop on Computation and Analytic Problems in Spectral Theory
  • Place of Presentation: University of Wales, Gregynog Hall, Newtown, United Kingdom
• [Presentation] Validated computations for elliptic systems of FitzHugh-Nagumo type (2008)
  • Author(s): 長藤かおり
  • Organizer: 13th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2008)
  • Place of Presentation: the University of Texas at El Paso, USA
• [Presentation] 3次元Maxwell方程式に関するスペクトル問題について (2008)
  • Author(s): 長藤かおり
  • Organizer: 研究集会「流れ問題のための高品質数値解法と計算機援用解析学」
  • Place of Presentation: KKRホテル金沢(金沢共済会館)
• [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): 森藤孝志
  • Organizer: 応用数学合同研究集会
  • Place of Presentation: 龍谷大学瀬田キャンパス
• [Presentation] Numerical verification for a spectral problem on 1-Dschroedinger operators (2008)
  • Author(s): K, Nagatou
  • Organizer: Spetral and Scattering Theory and Related Topics
  • Place of Presentation: RIMS, Kyoto University
• [Presentation] Non-existence proof of point spectrum in a spectral gap (2008)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: 2008 International Workshop on Numerical Verification and its Applications
  • Place of Presentation: 東京第一ホテル(沖縄)
• [Presentation] Numerical verification method for PDEs (2008)
  • Author(s): K, Nagatou
  • Organizer: 2008 JSIAM-JSST 連合発表会, 精度保証付き数値計算チュートリアル
  • Place of Presentation: 首都大学東京
• [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 fundamental solutions of linear ordinary differential operators (2007)
  • Author(s): K, Nagatou
  • Organizer: NOLTA 2007 Special Session
  • Place of Presentation: Simon Fraser University at Harbour Centre, Vancouver, Canada
  • Year and Date: 2007-09-18
• [Presentation] Validated computations for a Schroedinger operator with band-gap essential spectrum (2007)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: NOLTA 2007 Mini-workshop
  • Place of Presentation: Simon Fraser University at Harbour Centre, Vancouver, Canada
  • Year and Date: 2007-09-17
• [Presentation] Numerical verification method for a Schroedinger equation with essential spectrum (2007)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: DIFFERENCIALEGYENLETEK SZEMINARIUM
  • Place of Presentation: Universitys of Szeged, Hungary
  • Year and Date: 2007-09-06
• [Presentation] スペクトル・ギャップにおける離散スペクトルの非存在検証について (2007)
  • Author(s): 長藤 かおり M, Plum, 中尾充宏
  • Organizer: 数値解析ミニシンポジウム
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2007-05-29
• [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] 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
• [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.
• [Presentation] Eigenvalue problems on 1-D Schroedingeroperators (2007)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: Conference on Inequalities and Applications 07
  • Place of Presentation: De La Motte Castle, Noszvaj, Hungary
• [Presentation] Validated computations for spectral problems on 1-D Schroedinger operators (2007)
  • Author(s): K, Nagatou・M. Plum・M, T, Nakao
  • Organizer: 「精度保証付き数値計算とその周辺」ワークショップ
  • Place of Presentation: 岐阜大学
• [Presentation] 1次元シュレディンガー作用素に対するスペクトル問題 (2007)
  • Author(s): 長藤 かおり, M, Plum, 中尾充宏
  • Organizer: 研究集会「流れ問題のだめの高品質数値解法と精度保証計算」,(主催者:田端正久)
  • Place of Presentation: 島根県民会館
• [Presentation] 非負行列のペロン・フロベニウス固有値の精度保証について (2007)
  • Author(s): 長藤かおり, 石井豊
  • Organizer: 応用数学合同研究集会
  • Place of Presentation: 龍谷大学瀬田キャンパス
• [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
• [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