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

索弗教授的课题涉及哈密顿系统的光谱和散射理论。它包括作为主要部分的n体远程势系统和准粒子动力学，如自旋波的研究。相关的问题也将被处理，包括随时间哈密顿量的光谱理论和多通道非线性散射。期望这项工作将为一些重要的物理概念提供严格的基础，并为分析偏微分方程解的渐近行为提供新的数学技术。这里的一般思想是使用复杂的数学来模拟物理系统的行为，特别是那些涉及几个相互作用的粒子的系统。（这些可能是电子与库仑力的相互作用，或者在相反的极端，恒星与引力的相互作用。）这样一个系统可以用偏微分方程系统来描述，也可以用代表可能物理状态的函数空间上的一个算子（称为哈密顿算子）来描述。原则上，在哈密顿量和系统的时间演化之间有相当直接的关系。然而，对于给定的一类哈密顿量，用足够的细节来对系统在长期运行中发生的情况做出强有力的定性陈述是一项主要的分析任务，也是本项目数学研究的重点。

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.