Level Statistics for Random Schrodinger Operators

2018 Fiscal Year Final Research Report

• Project/Area Number: 26400145
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Gakushuin University
• Principal Investigator: Nakano Fumihiko 学習院大学, 理学部, 教授 (10291246)
• Research Collaborator: Minami Nariyuki ; Ueki Naomasa ; Sadahiro Taizo
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 準位統計 / ランダム行列理論 / サインベータ過程 / ポアソン過程

Outline of Final Research Achievements

We studied the one dimensional random Schroedinger operators with decaying facto, and with decaying coupling constant, in the order of -alpha at infinity, and our results are as follows.
(i) the level statistics are : Clock, Sine_{\beta}, and Poisson processes, depending on \alpha is bigger than, equal to, and less than 1/2, (ii) the asymptotic expansion of the number of eigenvalues on a given interval, which shows different behavior depending on the value of alpha, (iii) as a related question, the level statistics of the Gaussian beta ensemble at high temperature is Poisson.

Free Research Field

ランダムシュレーディンガー作用素

Academic Significance and Societal Importance of the Research Achievements

１次元での様々なランダムシュレーディンガー作用素において、その準位統計を考え、それがポアソン過程以外になるものを得たこと、またランダム行列理論との関係を具体的に示したこと。またランダムシュレーディンガー作用素の研究手法のランダム行列理論への応用例。