Semiclassical foundations of single particle and collective phenomena in quantum chaos

近年来，由于经典混沌系统周期轨道相关定量估计技术的发展，半经典理论得到了长足的发展。项目的第一部分侧重于将半经典理论应用于一类单粒子混沌系统，该系统具有两种运动类型，彼此被动力势垒分开。这类系统的谱统计是非常有趣的，因为它是异常的，偏离了随机矩阵理论（RMT）的标准系综。项目的第二部分处理混沌系统中的特征函数分布问题。在遍历系统中，几乎所有的特征函数在相空间中都是均匀分布的，但仍然可能存在特殊的局域状态序列（疤痕）。主要目标是通过证明相应的半经典测度的度量熵的下界来理解混沌系统中这种局部化的可能“强度”的限制。该项目的第三部分致力于多粒子系统。这里出现的主要新现象是（典型的）混沌单粒子和（典型的）规则集体动力学的共存。我的长期目标是发展一个半经典理论的集体谱激励，适当地捕捉两种动力现象。

In recent years the semiclassical theory has experienced considerable progress due to the development of the techniques for quantitative estimations of periodic orbit correlations in classically chaotic systems. The first part of the project focuses on the application of the semiclassical theory to a class of single particle chaotic systems with two types of motion separated from each other by a dynamical barrier. The spectral statistics of such systems is of great interest, since it is anomalous and deviates from standard ensembles of Random Matrix Theory (RMT).The second part of the project deals with the problem of eigenfunction distributions in chaotic systems. In ergodic systems almost all eigenfunctions are equidistributed in phase space, but exceptional sequences of localised states (scars) might still exist. The main objective is to understand the restriction on the possible “strength” of such a localisation in chaotic systems by proving lower bounds on the metric entropies of the corresponding semiclassical measures.The third part of the project is devoted to many-particle systems. The main new phenomenon appearing here is coexistence of (typically) chaotic single particle and (typically) regular collective dynamics. My long term goal is to develop a semiclassical theory for collective spectral excitations which properly catches both dynamical phenomena.