2013 Fiscal Year Final Research Report
Study of spectral statistics for random operators in the framework of stationary point process theory
| Project/Area Number |
22540205
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Keio University |
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Principal Investigator |
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| Co-Investigator(Renkei-kenkyūsha) |
KOTANI Shinichi 関西学院大学, 理工学部, 教授 (10025463)
UEKI Naomasa 京都大学, 大学院人間・環境学研究科, 教授 (80211069)
NAKANO Fumihiko 学習院大学, 理学部, 教授 (10291246)
NAGAO Taro 名古屋大学, 大学院多元数理科学研究科, 教授 (10263196)
MAKINO Hironori 東海大学, 情報理工学部, 准教授 (40338786)
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| Project Period (FY) |
2010-04-01 – 2014-03-31
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| Keywords | スペクトル統計 / ランダム作用素 / 点過程 / 自己共役性 / 漸近エルゴード性 / 確率エアリー作用素 |
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| Research Abstract |
(1) We introduced the concept of "asymptotic ergodicity" as a mathematical formulation of the similarity, under some scaling, of discrete spectrum of a self adjoint operator with a typical realization of a stationary point process. We also showed asymptotic ergodicity forthe spectrum of a certain one dimensional Schroedinger operator, and obtained a partial result for discrete Anderson models.
(2) We considered a one dimensional Schroedinger operator H on the half line whose potential term consists of white noise plus a uniform electric field tending to plus infinity. Despite of the singularity of white noise term, we showed that H can be realized as a self-adjoint operator and has purely discrete spectrum, with probability one.
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