Studies of Correspondence between Classical and Quantum Dynamical Systems

经典动力系统与量子动力系统对应关系的研究

• 批准号: 14540210
• 负责人: KUWABARA Ruishi
• 金额: $ 0.96万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540210/
• 关键词: spectral geometry; Schroedinger operator; quantization condition; semi-classical analysis; Fourier integral operator; magnetic field; gauge field; Schrodinger operator; spectrum; magnetic flow; semi-classsical analysis; periodic orbit; Fourier

The main result obtained in the research concerns with the classical-quantum correspondence for the mechanical system in a magnetic field. The result was presented at the international workshop ‘Spectral theory of differential operators and the inverse problems' held in the Research Institute for Mathematical Sciences in Kyoto University (October 28-November 1) in 2002. The title of the talk was ‘Quantum energies and classical orbits in a magnetic field'. Later the paper described the details of the results was published in the Proceedings of the Workshop from American Mathematical Society in 2004.The result of our research is the following. We remark first that a magnetic field on a manifold is regarded as the curvature of a connection given on a principal bundle, and the dynamical flow in the magnetic field can be analyzed as the geodesic flow on the bundle relative to so-called the Kaluza-Klein metric. On the basis of this formulation we considered the relationship between the class … More ical flow and the quantum system (the Schroedinger operator), as a result, we clarified that some classical orbit satisfying ‘quantization condition' corresponds to an approximate energy level in a semi-classical sense.This result is a generalization of the former result by Ralston and Guillemin for the geodesic flow to the case of magnetic flow. It also gives an interesting view to the trace formula. The key tool for the research was the theory of Fourier integral operators of Hermite type which is developed by Boutet de Monvel and Guillemin.Next we aimed to generalize the results for the magnetic system (the U(1)-gauge system) to the non-abelian gauge systems. We first reconstructed a geometric formulation (originally due to Guillemin, Uribe, Zelditch and so on) of the system in the frame-work of principal G-bundle, and obtained some results for the system in the non-abelian gauge field, which is a generalization of the results for the system in the magnetic field (the U(1)-gauge field). More precisely we clarified the relationship between so-called the Maslov quantization condition (and classical periodic orbits) and the asymptotic properties of quantum energy levels. The paper containing these results is being prepared. Less

研究的主要结果涉及磁场中力学系统的经典-量子对应。这一结果已于2002年10月28日至11月1日在京都大学数学科学研究所举行的“微分算符的谱理论和反问题”国际研讨会上公布。演讲的题目是《磁场中的量子能量和经典轨道》。随后，论文详细介绍了研究结果，并于2004年发表在美国数学学会的《研讨会论文集》上。我们首先指出，流形上的磁场被看作是主丛上给定的联络的曲率，磁场中的动态流可以被分析为相对于所谓的Kaluza-Klein度规在主丛上的测地线流。在这个公式的基础上，我们考虑了…类之间的关系更多的流和量子系统(薛定谔算符)，我们澄清了一些满足量子化条件的经典轨道对应于半经典意义下的近似能级，这是Ralston和Guillmin关于测地线流的结果在磁流情况下的推广。它还为迹公式提供了一个有趣的观点。研究的关键工具是Boutet de Monvel和Guilline发展的Hermite型傅立叶积分算符理论。接下来，我们的目的是将关于磁性系统(U(1)规范系统)的结果推广到非阿贝尔规范系统。我们首先在主G丛的框架下重建了系统的几何形式(最初由Guillemin，Uribe，Zelditch等人给出)，并得到了非阿贝尔规范场中系统的一些结果，推广了磁场(U(1)规范场)中的结果。更确切地说，我们阐明了所谓的Maslov量子化条件(和经典周期轨道)与量子能级的渐近性质之间的关系。包含这些结果的论文正在编写中。较少

项目成果

Eigenvalues associated with a periodic orbit of the magnetic flow

与磁流周期轨道相关的特征值

• 发表时间: 2004
• 期刊: Contemporary Mathematics 348
• 作者: Ruishi Kuwabara

Ruishi Kuwabara: "Eigenvalues associated with a periodic orbit of the magnetic flow"Contemporary Mathematics. (発表予定). (2003)

Ruishi Kuwabara：“与磁流周期轨道相关的特征值”（待发表）。