Probabilistic and statistical approach to spectra of quantum Hamiltonians.

1. 考虑实对称矩阵X^<(n)>, n<大于等于>1，其项是独立且同分布的高斯随机变量。众所周知，X^<(n)>的特征值的经验分布在适当的标度下收敛于Wigner半圆定律为n * *。最近，t.c an， L.Rogers, z.s shi和Y.Takahashi通过考虑X^<(n)>的虚拟时间演化X^<(n)>(t)给出了该定理的替代证明，这是一个可以追溯到F.Dyson（1962）的想法。他们研究了由X^<(n)>的特征值引起的n维扩散过程所满足的随机微分方程（SDE）。但是，由于该SDE系数的奇异性，它们被卷入不必要的复杂分析，以确认其解的存在。另一方面，我们简化了他们的论点，注意到，就我们所关心的半圆定律的推导而言，它足以观察到一个由经验分布的Stieltjes变换满足的更简单的SDE。我们还表明，通过考虑X^<(n)>的分辨率，可以很容易地推导出该SDE。这些结果是通过与平冢先生的共同工作获得的。我们为“水平统计”的现象学方面提供了数学基础，这是无序系统、随机矩阵和量子混沌理论中的一个常见问题。这只能通过把量子哈密顿谱的全部或一部分看作是一个驻定点过程的典型样本来实现，尽管物理学家似乎还没有清楚地认识到这一点。我们澄清了以下几点：(1)水平统计中出现的许多平均量之间的数学关系是点过程理论中众所周知的Palm-Khinchin等式的推论。(2)观测能级间距分布等，当我们将其视为光谱模型时，相当于观测点过程的Palm测度。少

1. Consider real symmetric matrices X^<(n)>, n<greater than or equal>1 , whose entries are independent and identically distributed Gaussian random varibles. It is well known that the empirical distribution of the eigenvalues of X^<(n)>, under a suitable scaling, converges to the Wigner semi-circle law as n * *. Recently, T.Chan, L.Rogers, Z.Shi, and Y.Takahashi gave alternative proofs to this theorem by considering a fictitious time evolution X^<(n)>(t) of X^<(n)>, which is an idea going back to F.Dyson (1962). They investigated the stochastic differential equation (SDE) satisfied by the n-dimensional diffusion process arising from the eigenvalues of X^<(n)>. However, because of singularity of the coefficients of that SDE, they were involved in an unnecessarily complicated analysis in order to confirm the existence of its solution. We, on the other hand, simplified their argument by noting that as far as we are concerned with the derivation of the semi- circle law, it suffices to obser … More ve a simpler SDE satisfied by the Stieltjes transform of the empirical distribution. We also showed that this SDE is easily derived by considering the resolvent of X^<(n)>. These results have been obtained through a joint work with Mr. T.Hiratsuka.2. We gave a mathematical foundation to the phenomenological side of "level statistics ", which is a common issue in the theories of disordered systems, random matrices, and quantum chaos. This is only possible by regarding the whole or a part of the spectrum of a quantum Hamiltonian as a typical sample of a stationary point process, though this does not seem to have been clearly recognized by physicists. We clarified the following points :(1)Many of mathematical relations between average quantities appearing in level statistics are corollaries of Palm-Khinchin equality, which is well known in point process theory.(2)Observing level spacing distribution etc. amounts to observing the Palm measure of the point process, when we regard it as the model of the spectrum. Less