Mathematical problems related to quantum chaos

量子混沌或经典-半经典-量子对应是围绕动力系统混沌问题的主要主题之一。本研究的目的是探讨以下三个方向的数学问题。1)离散模拟公式和2)跟踪公式。Y.Takahashi（与Y.Baba和I.Kubo）澄清了Li-Yorke混沌概念所包含的含义之一。T.Shirai得到了一个显著的结果：Gutzwiller公式的图形版本确实完全成立。许多粒子系统的流体动力极限及极限方程的分析。在这个方向上有几个结果。K.Ohkitani计算了发达湍流的Lyapunov指数（与Yamada）， H.Okamoto和Sakajo研究了涡旋片的卷起现象，K.Murota讨论了土体剪切的分岔。同样，由于H.Kawai和Y.Takei引起的奇异摄动的确切理论也在这个方向上提出。高桥先生组织了一个关于随机分析中的渐近理论的会议，包括关于振荡积分理论的最新发展的讲座。他还对16年前的结果给出了一个新的更简单的证明（与K.Hara），并试图从点过程及其缩放极限的角度描述随机矩阵理论的结果（部分，在Oberwolfach研究所公布）。同时，受Johansson（1995）在有限维（与t.g iyonobu）推广Szego的结果的启发，他开始了与Cauch奇异积分相关的二次型的随机分析研究。

Quantum chaos or the classical-semiiclassical-quantum correspondence is one of the main topics around the problem of the chaos of dynamical systems. The purpose of the research was to investigate the mathematical problems in the following three directions.1) Discrete analogue and 2) Trace formulas. One of the implications carried by the concept of Li-Yorke's chaos is clarified by Y.Takahashi (with Y.Baba and I.Kubo). T.Shirai obtained a remarkable result : a graph version of Gutzwiller's formula does hold exactly.3) Hydrodynamical limit of many particle systems and analysis of the limit equations. There are several results in this direction. K.Ohkitani computed Lyapunov exponents for the developped turbulence (with Yamada), H.Okamoto and Sakajo studied the roll-up phenomenon of vortex sheet, K.Murota discussed the bifurcation of soil shear. Also, the exact theory of singular perturbation due to H.Kawai and Y.Takei is suggesive in this direction. Y.Takahashi organized a meeting on asymptotic theory in stochastic analysis, including lectures concerning the recent development of oscillatory integral theory. He also gave a new simpler proof to the result of 16 years ago (with K.Hara) and tried to describe the results in random matrix theory from the view-point of point process and its scaling limit (partially, announced at Oberwolfach institute). Also, he started the stochastic-analytical study of the quadratic form associated with the Cauch singular integral inspired by a result of Johansson (1995) extending Szego's result in finite dimension (with T.Chiyonobu).