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

索弗教授的项目涉及哈密顿系统的谱和散射理论。它主要包括N体长程势系统和准粒子动力学(如自旋波)的研究。相关问题也将被解决，包括含时哈密顿量的谱理论和多通道非线性散射。期望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的数学方法提供严格的基础。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，特别是那些涉及几个相互作用的粒子的系统。(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，恒星通过引力相互作用。)这样的系统可以由偏微分方程组或由表示可能的物理状态的函数空间上的算子(称为哈密顿量)来描述。在哈密顿量和系统的时间演化之间，原则上有一个非常直接的关系。然而，为给定的哈密顿学派计算出足够详细的这一点，以对系统长期发生的事情做出强有力的定性陈述，是一项主要的分析任务，也是这个项目的数学研究的重点。

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.