Quantum stochastic analysis - Transforms and spectral analysis

2014 Fiscal Year Final Research Report

• Project/Area Number: 23340027
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: OBATA Nobuaki 東北大学, 情報科学研究科, 教授 (10169360)
• Co-Investigator(Kenkyū-buntansha): FUKUIZUMI Reika 東北大学, 大学院情報科学研究科, 准教授 (00374182) ; HASEGAWA Takehisa 東北大学, 大学院情報科学研究科, 助教 (10528425) ; SEGAWA Etsuo 東北大学, 大学院情報科学研究科, 准教授 (30634547) ; KUBO Hideo 北海道大学, 大学院理学研究院, 教授 (50283346) ; HIAI Fumio 東北大学, 名誉教授 (30092571) ; SUZUKI Kanako 茨城大学, 理学部, 准教授 (10451519)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: 関数解析 / 量子確率論 / 無限次元解析 / 複雑ネットワーク / スペクトル解析 / フォック空間 / 量子ホワイトノイズ / 量子確率過程

Outline of Final Research Achievements

For the development of quantum stochastic analysis we focused on 'quantum white noise calculus' from analytic aspect and 'spectral analysis of complex networks' from algebraic aspect. We aimed at the establishment of the mathematical fundamentals and the paradigm for collaborating with other research fields for applications. By means of quantum white noise calculus, the Bogoliubov transform and the Girsanov transform are characterized by the white noise differential equations of new types. A quantum probabilistic method is applied to the spectral analysis of digraphs such as Manhattan product. The phase transition of various dynamics on networks is studied in detail with the help of numerical computation. New statistical properties of quantum walks on graphs such as localization are obtained by generalizing the existing method of spectral analysis.

Free Research Field

量子確率論