Dynamics of few-body quantum systems and chaos

本项目的目的是从经典量子的角度研究少体量子系统的动力学行为。特别是，化学反应和介观现象的动力学过程是最有趣的。在我们的研究中，我们注意以下三点。首先，我们研究了非线性共振网络（Arnold网）的性质如何影响动力学行为。其次，我们研究了高维相空间中稳定流形和不稳定流形的相交问题。第三，我们研究了系统的对称性是如何导致经典系统与量子系统之间的差异的。首先，在研究振动高激发态分子的过程中，我们获得了利用Arnold网在外场作用下操纵分子系统的一些线索。特别是，我们发现，非遍历的分子内振动能量再分配（IVR）起着重要的作用。其次，我们通过推广李微扰得到了一个计算稳定流形和不稳定流形的方法。第三，我们发现传统的分子置换群方法在研究大变形的混沌行为时存在不足，未来的研究方向包括以下几个方面.首先，我们需要一种方法来设计激光场，以提高反应过程的效率。特别地，为了增加由非线性共振和激光场的组合产生的效果，利用Arnold网设计激光场的一般方法是必要的。其次，基于稳定流形和不稳定流形的半经典表述对于分析量子-经典对应是必要的。第三，我们需要一种方法来找到分子置换群的子群如何对应于Arnold网。

The purpose of this project is to study dynamical behavior of few-body quantum systems from classical-quantum point of view. In particular, dynamical processes of chemical reactions and mesoscopic phenomena are among the most interesting. In our study, we pay attention to the following three points. Firstly, we study how the properties of the networks of nonlinear resonances (Arnold webs) affect dynamical behavior. Secondly, we investigate how stable manifolds and unstable manifolds intersect in high-dimensional phase space. Thirdly, we study how symmetry of the systems lead to the difference between classical and quantum systems.The results of our study show the following. First, in the study of vibrationally highly excited molecules, we obtain some clues in utilizing the Arnold web in manipulating molecular systems by external laser fields. In particular, we find that non-ergodicity of the intramolecular vibrational-energy redistribution (IVR) plays an important role. Second, we obtain a method to compute stable and unstable manifolds by generalizing the Lie perturbation. Third, we find that the method of the molecular permutation group, which is used in the conventional study of the deformation of molecules, is insufficient for the investigation of large deformations involving chaotic behavior.For future development of our study, the following directions are among the most important. First, we need a method to design laser fields to increase the efficiency of reaction processes. In particular, in order to increase the effects which result from the combination of nonlinear resonances and laser fields, general methods to design laser fields utilizing the Arnold web are necessary. Second, semiclassical formulation based on stable and unstable manifolds is necessary to analyze quantum-classical correspondence. Third, we need a method to find how subgroups of the molecular permutation groups correspond to the Arnold web.