Mathematical Sciences: Phase-space Analysis and Scattering Theory of Shcrodinger Type Hamiltonians

数学科学：相空间分析和薛定谔型哈密顿量的散射理论

• 批准号: 8905772
• 负责人: Avraham Soffer
• 金额: $ 6.29万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905772&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Phase; space; Analysis

Professor Soffer's project is concerned with the spectral and scattering theory of hamiltonian systems. It includes as a major part the study of N-body long range potential systems and quasiparticle dynamics such as spin waves. Related problems will also be tackled, including the spectral theory of time-dependent hamiltonians and multichannel nonlinear scattering. It is expected that this work will provide a rigorous basis for some important concepts in physics as well as new mathematical techniques of analyzing the asymptotic behavior of solutions of partial differential equations. The general idea here is the use of sophisticated mathematics to model the behavior of physical systems, especially those involving several interacting particles. (These might be electrons interacting by Coulomb forces, or at the opposite extreme, stars interacting gravitationally.) Such a system may variously be described by a system of partial differential equations, or by an operator (called the hamiltonian) on a space of functions that represent the possible physical states. There is in principle a quite straightforward relationship between the hamiltonian and the time evolution of the system. Working this out for a given class of hamiltonians in sufficient detail to make strong qualitative statements about what happens to the system in the long run is a major analytical task, however, and the focus of the mathematical research for this project.

索弗教授的项目涉及哈密尔顿系统的光谱和散射理论。它的主要部分包括 N 体长程势系统和准粒子动力学（例如自旋波）的研究。相关问题也将得到解决，包括时间相关的哈密顿量的谱理论和多通道非线性散射。预计这项工作将为物理学中的一些重要概念以及分析偏微分方程解的渐近行为的新数学技术提供严格的基础。 这里的总体思想是使用复杂的数学来模拟物理系统的行为，特别是那些涉及多个相互作用粒子的系统。 （这些可能是通过库仑力相互作用的电子，或者在相反的极端，通过引力相互作用的恒星。）这样的系统可以通过偏微分方程组或函数空间上的算子（称为哈密顿）来不同地描述代表可能的物理状态。原则上，哈密顿量和系统的时间演化之间存在非常直接的关系。然而，对给定类别的哈密顿量进行足够详细的计算，以便对系统长期发生的情况做出强有力的定性陈述，这是一项主要的分析任务，也是该项目数学研究的重点。