Probabilistic and statistical approach to spectra of quantum Hamiltonians.

1.考虑真实的对称矩阵X^&lt;（n）&gt;，n <greater than or equal>1，其元素是独立同分布的高斯随机变量.众所周知，X^&lt;（n）&gt;的本征值的经验分布，在适当的标度下，收敛于维格纳半圆定律，即n * *。最近，T.Chan、L.Rogers、Z.Shi和Y.Takahashi通过考虑X^&lt;（n）&gt;的一个虚构的时间演化X^&lt;（n）&gt;（t）给出了这个定理的另一种证明，这是一个可以追溯到F.Dyson（1962）的想法。他们研究了由X^&lt;（n）&gt;的特征值产生的n维扩散过程所满足的随机微分方程（XN）。然而，由于该方程系数的奇异性，为了证实其解的存在，他们被卷入了不必要的复杂分析中。另一方面，我们简化了他们的论证，注意到就我们所关心的半圆定律的推导而言， 关于我们 给出了经验分布的Stieltjes变换所满足的一个较简单的条件。我们还证明了，通过考虑X^&lt;（n）&gt;的预解式，可以很容易地导出这个矩阵。这些结果是通过与T. Hiratsuka先生的共同工作获得的。我们给出了一个数学基础的现象学方面的“水平统计”，这是一个共同的问题，在理论的无序系统，随机矩阵，量子混沌。这只能通过将量子哈密顿量的全部或部分谱视为驻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。我们阐明了以下几点：（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著名的Palm-Khinchin等式的推论。（2）当我们把点过程看作谱的模型时，观察能级间距分布等，就等于观察点过程的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