Study of the mathematical foundation of energy level statistics.

1.一维奇异势Schrodinger算子的定义及其在随机系统中的应用：1994年，H. P. McKean考虑了有限区间上的白色噪声势Schrodinger算子，并研究了其第一特征值的概率分布，但没有提到像白色噪声那样的奇异势算子的定义的基本问题。福岛、中尾和南已经给出了以连续函数的形式导数为势的薛定谔算子的公式。但每一种方法在应用于实际情况时都存在一定的技术困难。在我们的研究中，我们发现Savchuk和Shkalikov（1999）最近提出的简明公式对我们的目的是有效的。特别是，他们的概念“准导数“使我们能够证明斯特姆振荡定理需要填补差距在麦基恩的理论。这项工作是与K合作完成的。永井2号。Galton-Watson树的顶点数分布：如A. Khourunzhy（Adv. In Appl. Probab. Vol.33，No.1（2001）124-140），为了研究随机矩阵谱的涨落，需要计算随机树的顶点数。另一方面，随机树作为Galton-沃森过程（离散时间分支过程）的轨迹而获得。我们称这种树为高尔顿-沃森树。继续R. Otter（1949）中，我们得到了关于Galton-沃森树顶点数的一些新结果。

1. The definition of one-dimensional Schrodinger operators with singular potentials and its application to random systems : In 1994, H.P. McKean considered the Schrodinger operator with white noise potential on a finite interval, and investigated the probability distribution of its first eigenvalue, but did not mention the fundamental question of the definition of operators with singular potentials like white noise. The formulation of a Schrodinger operator which has as its potential the formal derivative of a continuous function had been already given by Fukushima, Nakao and Minami. But each of their methods had some technical dificulty when applied to the present situation. In our study, we found that the recent concise formulation due to Savchuk and Shkalikov (1999) is effective for our purpose. In particular, their notion of "quasi-derivative "enabled us to prove the Sturm's oscillation theorem needed in filling the gap in McKean's theory. This work was done in collaboration with K. Nagai.2. The distribution of the number of vertices of a Galton-Watson tree : As is shown e.g. the recent work by A. Khourunzhy (Adv. In Appl. Probab. Vol.33, No.l (2001) 124-140), one needs to count the number of vertices of random trees in order to study the fluctuation of the spectrum of random matrices. On the other hand, random trees are obtained as the trajectory of a Galton- Watson process (a discrete time branching process). We shall call this type of trees the Galton-Watson trees. Continuing the pioneering work of R. Otter (1949), we obtained some new results on the number of vertices of Galton- Watson trees.