Analysis of differential equations on graphs.

2013 Fiscal Year Final Research Report

• Project/Area Number: 23540181
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: TRUSHIN Igor 東北大学, 国際教育院, 准教授 (80600337)
• Co-Investigator(Kenkyū-buntansha): KUBO Hideo 北海道大学, 理学 (系)研究科(研究院), 教授 (50283346) ; MOCHIZUKI Hiyoshi 首都大学東京, 理工学研究科, 名誉教授 (80026773)
• Project Period (FY): 2011 – 2013
• Keywords: 関数方程式 / 散乱理論

Research Abstract

In this project we consider Schrodinger operators on noncompact graphs which consist of some infinite rays and compact part attached. Spectral and scattering problems on graphs arise as simplified models in mathematics, physics, chemistry and engineering when one considers the propagation of waves of different natures in thin, tube-like domains. We study scattering direct and inverse problems which are important in applied physics. (1)We treat an inverse scattering problem on a graph with an infinite ray and a loop joined at one point. Reconstruction procedure is presented.(2)We consider Schrodinger operators on noncompact star-shaped graphs including some finite rays. We show that our spectral representation formula provides the time dependent formulation of the scattering theory. The scattering operator is constructed in the configuration space, and then is related to the scattering matrix in the momentum space. Corresponding inverse scattering problem is investigated.

Research Products

• [Journal Article] A stationary approach to the scattering on noncompact graphs containing finite rays (2014)
  • Author(s): トルシン イゴール, 望月 清
  • Journal Title: Current Trends in Analysis and its Applications Volume: (in press)
  • Peer Reviewed
• [Journal Article] Spectral problems and scattering on noncompact graphs containing finite rays (2014)
  • Author(s): トルシン イゴール, 望月 清
  • Journal Title: J.Inverse Ill-Posed Probl Volume: (in press)
  • Peer Reviewed
• [Journal Article] On the scattering on a loop-shaped graph (2012)
  • Author(s): トルシン イゴール, 望月 清
  • Journal Title: Progress in Mathematics Volume: Vol.301 Pages: 227-245
  • DOI: 10.1007/978-3-0348-0454-7_12
  • Peer Reviewed
• [Presentation] Scattering on Graphs (2013)
  • Author(s): トルシン イゴール, 望月 清
  • Organizer: Seminar on Functional Analysis and Global Analysis
  • Place of Presentation: 東京理科大学
  • Year and Date: 2013-11-20
  • Invited
• [Presentation] On inverse scattering on non-compact graphs (2013)
  • Author(s): トルシン イゴール, 望月清
  • Organizer: 9th International ISAAC Congress
  • Place of Presentation: Crakow, Poland
  • Year and Date: 2013-08-09
  • Invited
• [Presentation] Stationary approach to the time dependent scattering theory on star graphs (2013)
  • Author(s): 望月 清, トルシン イゴール
  • Organizer: 信大関数解析研究会
  • Place of Presentation: 信州大学
  • Year and Date: 2013-03-14
  • Invited
• [Presentation] Spectral representations and scattering for Schro"dinger operators on noncompact star-shaped graphs (2012)
  • Author(s): 望月 清, トルシン イゴール
  • Organizer: Taiwan-Japan joint conference on PDE and analysis
  • Place of Presentation: Taipei, Taiwan
  • Year and Date: 2012-12-27
• [Presentation] Scattering on graphs (2012)
  • Author(s): トルシン イゴール, 望月清
  • Organizer: Inverse and Ill-Posed Problems of Mathematical Physics
  • Place of Presentation: Novosibirsk, Russia
  • Year and Date: 2012-08-11
  • Invited
• [Presentation] On inverse scattering on graphs (2012)
  • Author(s): トルシン イゴール, 望月 清
  • Organizer: Inverse and Ill-Posed Problems of Mathematical Physics
  • Place of Presentation: Novosibirsk, Russia
  • Year and Date: 2012-08-09
  • Invited
• [Presentation] Spectral and scattering problems on noncompact star graphs including finite rays (2012)
  • Author(s): 望月 清, トルシン イゴ ー ル
  • Organizer: 東海大発展方程式シンポジウム
  • Place of Presentation: 東海大学
  • Year and Date: 2012-03-09
• [Presentation] On inverse scattering on graphs (2011)
  • Author(s): トルシン イゴール, 望月清
  • Organizer: 太閤山数理科学研究会
  • Place of Presentation: 富山大学
  • Year and Date: 2011-11-23
  • Invited
• [Presentation] Scattering problems on graphs (2011)
  • Author(s): トルシン イゴール, 望月清
  • Organizer: 仙台偏微分方程式研究会
  • Place of Presentation: 東北大学
  • Year and Date: 2011-10-08
  • Invited
• [Presentation] 星型グラグ上の散乱問題 (2011)
  • Author(s): 望月 清, トルシン イゴール
  • Organizer: 夏の作用素論シンポジウム
  • Place of Presentation: 熊本市国際交流会館
  • Year and Date: 2011-09-05
• [Presentation] On inverse scattering on graphs (2011)
  • Author(s): トルシン イゴール, 望月 清
  • Organizer: 8 th International ISAAC Congress
  • Place of Presentation: People's Friendship Univ. Moscow, Russia
  • Year and Date: 2011-08-24
  • Invited
• [Remarks] 望月清 , トルシンイゴールSpectral representations and scattering for Schro"dinger operators on star graphs, 2011.10.12, 筑波大学解析セミナー[大学等のセミナー発表]
• [Remarks] 第9回青葉山勉強会「半古典近似:理論と数値計算」2013.4.19, 東北大学情報科学研究科 3 講演, 参加者 15 名[研究会の主催]
• [Remarks] 第8回青葉山勉強会「数理物理と解析学」, 2013.3.25, 東北大学情報科学研究科 3 講演, 参加者 15 名[研究会の主催]
• [Remarks] 第7回青葉山勉強会, 2013.2.7, 東北大学情報科学研究科 3 講演, 参加者15 名[研究会の主催]
• [Remarks] 仙台偏微分方程式研究会, 2011.10.8-10, 東北大学情報科学研究科 15 講演, 参加者 43 名[研究会の主催]