Dynamics of few-body quantum systems and chaos

本课题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，化学反应的动力学过程和介观现象是最有趣的。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（阿诺德网）的性质对动力行为的影响。其次，研究了稳定流形和不稳定流形在高维相空间中的相交问题。第三，我们研究了系统的对称性如何导致经典系统和量子系统之间的差异。我们的研究结果如下。首先，在振动高激发分子的研究中，我们获得了利用阿诺德网利用外部激光场操纵分子系统的一些线索。特别是，我们发现分子内振动能量重分布（IVR）的非遍历性起着重要的作用。其次，通过推广李摄动，得到了一种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变形研究中使用的分子置换群方法，对于涉及混沌行为的大变形的研究是不够的。对于我们研究的未来发展，以下几个方向是最重要的。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。特别是，为了提高非线性共振与激光场结合产生的效应，有必要利用阿诺德网设计激光场的一般方法。其次，基于稳定流形和不稳定流形的半经典公式是分析量子-经典对应关系的必要条件。第三，我们需要一种方法来找出分子排列群的子群如何对应于阿诺德网。

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.