Jacobi matrices, Toda maps, and the absolutely continuous spectrum

本课题研究具有绝对连续谱的雅可比矩阵和更一般的差分算子及其相关的动力学性质。当人们对绝对连续谱感兴趣时，移位映射引起的动力系统是相关的，然后由Toda流实现的映射值得特别注意，因为它们保留了这种系统的结构性质。该项目将采用这种方法，并使Toda地图在绝对连续光谱的研究中发挥突出作用。本项目将研究的数学问题与纯数学的几个经典领域有关；然而，他们的主要动机来自量子力学理论。更具体地说，它们与量子粒子（比如电子）的长时间行为问题直接相关。对这些问题进行严格的数学研究，往往会对量子世界的基本机制和微妙效应有更清晰、更可靠的理解。

This project deals with Jacobi matrices and more general difference operators with some absolutely continuous spectrum and related dynamical properties of such operators. The dynamical systems that are induced by the shift map are relevant when one is interested in the absolutely continuous spectrum, and then maps implemented by Toda flows deserve special attention because they preserve the structural properties of such sytems. The project will pursue this approach and give a prominent role to Toda maps in the study of the absolutely continuous spectrum.The mathematical questions that will be studied in this project are related to several classical areas of pure mathematics; however, they derive their main motivation from the theory of quantum mechanics. More specifically, they are directly related to questions about the long time behavior of a quantum particle (say, an electron). The rigorous mathematical investigation of such issues often leads to a clearer and more secure understanding of basic mechanisms and subtle effects in the quantum world.