Spectra of Elliptic Operators on Manifolds and Classical Mechanics

流形和经典力学上的椭圆算子谱

基本信息

批准号：
11640205
负责人：
KUWABARA Ruishi
金额：
$ 0.32万
依托单位：
The University of Tokushima
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

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.

该研究项目的目的是研究经典力学的性质与黎曼流形上相关薛定谔算子的谱之间的关系。特别是，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元形式，带电粒子在磁场中的运动被公式化为具有被二元形式扭曲的辛结构的哈密顿系统的流动。另一方面，相关的量子系统或薛定谔算子是由流形上的积分封闭二元形式（磁场）自然定义的复线束上的拉普拉斯算子。在此背景下，我们取得了以下成果： 1．我们考虑了磁流哈密顿系统不变环面的量子化条件，并借助傅立叶积分算子理论阐明了由量子化条件确定的（半经典）能级给出了“良好”的近似量子系统的真实能量。2．此外，我们还考虑了磁流稳定周期轨道的“量子化条件”，并类似地阐明了合适周期轨道的经典“能量”给出了相关量子系统能量的近似值。本研究使用的工具是 Hermite 型傅立叶积分算子理论，该算子是对应于各向同性子流形的算子。