Mathematical Sciences: Research in Phenomena Occurring in Disordered Systems

克莱因教授的项目与数学模型有关 在无序的物理系统中发生的现象， 例如超流性和超导性。部分工作 将涉及分析马-哈尔佩林-李模型， 无序超流体;这是一个晶格模型， 包含随机项。克莱因教授还将继续他的 固体物理学中的安德森模型研究。 具体来说，他将研究随机的光谱特性， 薛定谔算子与相关的潜力，并将尝试 在各种情况下获得弱无序扩展。 大量的数学物理学都与 通过空间上的算子对物理系统建模， 表示系统可能状态的函数。在 一个自伴算子叫做哈密顿算子， 大致相当于能量，可以很容易地写下来， 物理假设的基础。一个数学上优雅的， 直接的关系，然后产生的时间演变， 从哈密顿量中推导出系统然而，这种关系是 在具体的术语中或多或少是无用的，除非一个人知道一些事情。 关于光谱（可能的能级）和相关的 哈密顿量的数学工具分析这些 通常是困难和微妙的。一个额外的复杂 当哈密顿量包含随机项时，因子进入，如 显然，为了模拟某些现象， 成功地，例如超流性和超导性。 克莱因教授的项目是扩大我们对 这些随机的模型。

Professor Klein's project has to do with mathematical models of phenomena that occur in disordered physical systems, for example superfluidity and superconductivity. Part of the work will involve analysis of the Ma-Halperin-Lee model for strongly disordered superfluids; this is a lattice model whose Hamiltonian contains a random term. Professor Klein will also continue his research on the Anderson model in solid state physics. Specifically, he will investigate spectral properties of random Schrodinger operators with dependent potentials, and will try to obtain weak disorder expansions in various situations. A great deal of mathematical physics is concerned with the modeling of physical systems by means of operators on spaces of functions representing the possible states of the system. In the general picture, a selfadjoint operator called the Hamiltonian, corresponding roughly to energy, can readily be written down on the basis of physical assumptions. A mathematically elegant and straightforward relationship then produces the time evolution of the system from the Hamiltonian. This relationship is however more or less useless in specfic terms unless one knows something about the spectrum (possible energy levels) and associated mathematical paraphernalia for the Hamiltonian. Analyzing these is generally difficult and subtle. An additional complicating factor enters in when the Hamiltonian contains random terms, as apparently it must in order to model certain kinds of phenomena successfully, for instance superfluidity and superconductivity. Professor Klein's project is to enlarge our understanding of these sorts of random models.