Probabilistic and statistical approach to spectra of quantum Hamiltonians.

1.考虑实对称矩阵X^&lt；(N)&gt；，n&lt；大于或等于&gt；1，其项是独立且同分布的高斯随机变量。众所周知，在适当的标度下，X^&lt；(N)&gt；的本征值的经验分布收敛于Wigner半圆律，即n**。最近，T.Chan，L.Rogers，Z.Shi.和Y.Takahashi通过考虑X^&lt；(N)&gt；(T)的虚构时间演化X^&lt；(N)&gt；(T)给出了这个定理的另一种证明，这是F.Dyson(1962)的一个想法。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(SDE)。然而，由于SDE系数的奇异性，他们为了确认其解的存在而进行了不必要的复杂分析。另一方面，我们简化了他们的论点，指出就我们所关心的半圆律的推导而言，只要观察…就足够了更有经验分布的Stieltjes变换所满足的更简单的SDE。我们还证明了通过考虑X^&lt；(N)&gt；的预解式可以很容易地得到这个SDE。这些成果是通过与T.Hiratsuka先生共同努力取得的。我们给出了“能级统计”的现象学方面的数学基础，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。这只能通过将量子哈密顿量的全部或部分光谱视为驻点过程的典型样本来实现，尽管物理学家似乎并没有明确认识到这一点。我们阐明了以下几点：(1)能级统计中出现的平均值之间的许多数学关系是Palm-Khinin等式的推论，这在点过程理论中是众所周知的。(2)当我们把点过程的Palm度量作为光谱模型时，观察能级间距分布等就等于观察点过程的Palm度量。较少

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