Research on Spectra of Random Schrodinger Operators

该项目的目的是举办国际研讨会，促进日本对随机算子的研究。主工作坊于2006年12月11日(星期一)至15日(星期五)在京都大学吉田南校区举行，为期5天。题目是《概率论中的随机算子谱理论及相关领域》，与会人数约50人，其中包括7位来自国外的应邀嘉宾，我们可以有机会进行有意义的交流和研究。另一个较小的讲习班于2007年11月14日(星期三)至16日(星期五)在京都大学人类与环境研究研究生院举行，为期3天。题目是《随机算子及相关领域的光谱》，参与者约30人，其中包括1名来自国外的嘉宾，我们可以就研究进行密切的交流。在这些讲习班上讨论的主题非常丰富。这些主要是关于随机薛定谔算子和随机矩阵的。此外，还讨论了有关金融和政治学的各种专题。关于随机算子的主题主要是从数学的角度出发的。此外，还从物理角度讨论了主题。在这些研讨会上，Ueki讲授了Hislop-Klopp理论的一种改进，即具有随机向量势的薛定谔算子的Wegner估计及其在Anderson局部化证明中的应用。南美就安德森模型的能级统计猜想进行了演讲。Nakamura就薛定谔方程波前集的半经典刻画结果作了一次演讲。Kotani用佐藤关于无限维Grassmanian流形上动力系统的理论，给出了关于关于时空变量具有平稳性的KdV流的构造的讲座。

The aim of this project is to held international workshops to promote the research on random operators in Japan. The main workshop was held for 5 days from 2006 December 11 (Monday) to 15 (Friday) at Kyoto University, Yoshida South Campus. The title was "Spectral Theory of Random Operators and Related Fields in Probability Theory", the amount of the participants were about 50 including 7 guests invited from abroad, and we could have meaningful occasions for exchanging ideas for researches. Another smaller workshop was held for 3 days from 2007 November 14 (Wednesday) to 16 (Friday) at Kyoto University, Graduate School of Human and Environmental Studies. The title was "Spectra of Random Operators and Related Fields", the amount of the participants was about 30 including 1 guest from abroad and we could have close communications on researches. The topics treated in these workshops were very substantial. Those were mainly on random Schrodinger operators and random matrices. Moreover various topics on financial and political sciences were also treated. The topics on random operators were mainly from mathematical viewpoints. Moreover the topics from physical viewpoints were also treated. In these workshops, Ueki gave a lecture on a refinement of Hislop-Klopp theory on the Wegner estimate for Schrodinger operators with random vector potentials and its applications to the proof of the Anderson localization. Minami gave lectures on a conjecture on energy level statistics for the Anderson model. Nakamura gave a lecture on the results on the semi-classical characterization of the wave front set of Schrodinger equations. Kotani gave a lecture on the construction of the KdV flow having stationarity with respect to the space-time variables by Sato's theory on the dynamical system on the infinite dimensional Grassmanian manifolds.