Mathematical Sciences: Research in Phenomena Occurring in Disordered Systems

克莱因教授的项目涉及无序物理系统中发生的现象的数学模型，例如超流性和超导电性。部分工作将涉及分析强无序超流体的Ma-Halperin-Lee模型；这是一个哈密顿量包含随机项的晶格模型。克莱因教授还将继续他对固体物理学中安德森模型的研究。具体地说，他将研究具有相依势的随机薛定谔算符的谱性质，并试图在各种情况下获得弱无序展开。大量的数学物理涉及通过表示系统可能状态的函数空间上的算子来对物理系统进行建模。在一般情况下，一个称为哈密顿算符的自伴算符，大致对应于能量，可以很容易地在物理假设的基础上写下来。然后，一个数学上优雅而直接的关系从哈密顿量得出系统的时间演化。然而，这种关系在具体意义上或多或少是无用的，除非一个人知道一些关于哈密顿量的谱(可能的能级)和相关的数学工具。分析这些通常是困难和微妙的。当哈密顿量包含随机项时，另一个复杂的因素就会出现，显然它必须这样做才能成功地模拟某些类型的现象，例如超流和超导电性。克莱因教授的项目是扩大我们对这类随机模型的理解。

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.