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

Soffer教授的项目是关于 和哈密顿系统的散射理论。它包括作为一个 主要研究N体长程势系统， 准粒子动力学，如自旋波。相关问题将 也可以解决，包括时间依赖的光谱理论 哈密顿量和多通道非线性散射。是 预计这项工作将提供一个严格的基础， 物理学中的重要概念以及新的数学 分析解的渐近性态的技巧 偏微分方程 这里的总体思路是使用复杂的 数学建模物理系统的行为，特别是 涉及多个相互作用的粒子。（这些可能是 电子通过库仑力相互作用，或者相反 极端，恒星之间的引力相互作用。）这样的系统可以 可以用偏微分系统来描述 方程，或由空间上的算子（称为哈密顿算子） 表示可能的物理状态的函数。那里 原则上， 哈密顿量和系统的时间演化。查这个 对于给定的一类Hamilton，足够详细， 做出强有力的定性陈述 然而，从长远来看，系统是一项重大的分析任务， 这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.