Spectra of Elliptic Operators on Manifolds and Classical Mechanics

本研究的目的是研究黎曼流形上经典力学的性质与伴随薛定谔算子的谱之间的关系，特别地，我们关注了黎曼流形上磁场中的力学。将磁场视为流形上的闭合二态，将带电粒子在磁场中的运动表示为辛结构被二态扭转的哈密顿系统的流动。另一方面，相关联的量子系统或薛定谔算符是复线丛上的拉普拉斯算符，该复线丛自然地由流形上的积分闭合二态(磁场)定义。在此背景下，我们得到了以下结果：1.我们考虑了磁流哈密顿系统不变环面的量子化条件，并利用傅里叶积分算符理论阐明了由量子化条件决定的(半经典)能级对量子系统的真实能量给出了一个“良好”的近似。此外，我们还考虑了稳定的磁流周期轨道的“量子化条件”，并同样阐明了一个合适的周期轨道的经典“能量”给出了相关量子系统能量的近似。本研究中使用的工具是Hermite型傅立叶积分算子理论，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.