Study of the mathematical foundation of energy level statistics.

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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.