Spectra of Elliptic Operators on Manifolds and Classical Mechanics

本课题的目的是研究黎曼流形上的经典力学性质与薛定谔算子谱之间的关系，特别是磁场中的力学性质。将磁场视为流形上的一个封闭的二次型，将带电粒子在磁场中的运动表示为具有被二次型扭曲的辛结构的哈密顿系统的流.另一方面，关联的量子系统或薛定谔算符是复线丛上的拉普拉斯算子，复线丛自然地由流形上的积分闭二形式（磁场）定义。在此背景下，我们得到了以下结果：1.我们考虑了磁通哈密顿系统不变环面的量子化条件，并借助于傅里叶积分算符理论阐明了由量子化条件确定的（半经典）能级给出了量子系统真能量的“好”近似.此外，我们还考虑了磁通稳定周期轨道的“量子化条件”，并同样阐明了适当周期轨道的经典“能量”给出了相关量子系统能量的近似值。本文的研究工具是Hermite型Fourier积分算子理论，它是对应于迷向子流形的算子。

The purpose of the research project is to investigate the relationships between the properties of classical mechanics and the spectrum of the associated Schrodinger operator on the Riemannian manifolds.Particularly, we have payed attention to the mechanics in a magnetic field on the Riemannian manifold. A magnetic field is regarded as a closed two-form on the manifold, and the motion of a charged particle in the magnetic field is formulated as the flow of the Hamiltonian system with the symplectic structure twisted by the two-form. On the other hand, the associated quantum system or the Schrodinger operator is the Laplacian on the complex line bundle naturally defined by the integral closed two-form (the magnetic field) on the manifold. In this context, we have obtained the following results :1. We have considered the quantization condition for the invariant torus of the Hamiltonian system of magnetic flow, and have clarified by virtue of the theory of Fourier integral operators that the (semi-classical) energy levels determined by the quantization condition give a "good" approximation of the true energies of the quantum system.2. We have, moreover, considered the "quantization condition" for the stable periodic orbits of the magnetic flow, and have similarly clarified that the classical "energy" of the a suitable periodic orbit gives an approximation of the energy of the associated quantum system. The tool used in this research if the theory of Fourier integral operators of Hermite type which are the operator corresponding to the isotropic submanifold.