Study of spectral statistics for random operators in the framework of stationary point process theory

(1)引入了“渐近遍历性”的概念，作为自伴随算子的离散谱在一定尺度下与一个典型的平稳点过程的相似度的数学表述。我们还证明了一维薛定谔算子谱的渐近遍历性，并得到了离散Anderson模型的部分结果。(2)考虑半线上的一维薛定谔算子H，其势项由白噪声和趋于正无穷的均匀电场组成。尽管白噪声项具有奇异性，但我们证明了H可以作为自伴随算子实现，并且具有纯离散谱，其概率为1。

(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.