Mathematical Sciences: Localization in Mathematical Models of Disordered Media

9706076 Stolz Dr. G.位于伯明翰的亚拉巴马大学的斯托尔兹将进行一项关于“本地化”在各个班级出现的研究项目 数学物理学中的随机算子。局部化，即存在具有密集纯点谱和指数衰减本征函数的能量区域，已被发现是共同的， 无序介质的大多数一维模型。Stolz博士的研究旨在更好地理解多维无序介质的数学模型。具体目标包括 (i)将现有的局部化结果扩展到对无序介质（如合金和非晶材料）给出比以前研究的模型更真实的描述的模型; (ii)基于随机点相互作用的三维模型的深入研究，其具有提供高能光谱特性的新见解的潜力，并且还可以用于研究随机表面和随机聚合物的物理性质。 无序介质在各种物理学科中被广泛研究，例如固态物理学，光学和电动力学，以及材料科学。它们包括各种类型的材料，例如合金、具有杂质的晶体（例如半导体）和无定形晶体。 固体随机算子提供了无序介质的数学模型。 这些算子可以通过使用数学光谱和散射理论的方法来研究。它们可以描述并预测 导电性和波在无序介质中的传播特性。例如，Stolz博士所研究的局部化的数学现象，描述了绝缘体（电绝缘体，分别是声音绝缘体，取决于底层的物理模型）的特征。 虽然这些模型已经获得了一些重要的结果，但研究更现实的模型并更B地从理论上理解潜在的问题将是未来研究的目标。 物理现象。

9706076 Stolz Dr. G. Stolz of the University of Alabama at Birmingham will conduct a research project on the appearance of "localization" in various classes of random operators which arise in mathematical physics. Localization, i.e. the existence of energy regions with dense pure point spectrum and exponentially decaying eigenfunctions, has been found to be common to most one dimensional models of disordered media. Dr. Stolz' research will aim at a better understanding of mathematical models of multidimensional disordered media. Specific goals include (i) the extension of existing results on localization to models which give a more realistic description of disordered media, such as alloys and amorphous materials, than the models studied previously; (ii) an intensified investigation of three-dimensional models based on random point-interactions, which have potential of providing new insights in high energy spectral characteristics and can also be used to study physical properties of random surfaces and random polymers. Disordered media are studied extensively in various physical disciplines such as solid state physics, optics, and electrodynamics, as well as in materials science. They comprise various types of materials, for example alloys, crystals with impurities (e.g. semiconductors), and amorphous solids. Random operators provide mathematical models of disordered media. These operators can be studied by using methods from mathematical spectral and scattering theory. They allow to describe and thus predict conductivity properties and the characteristics of wave propagation in disordered media. For example, the mathematical phenomenon of localization, as studied by Dr. Stolz, characterizes insulators (electrical insulators, respectively sound insulators, depending on the underlying physical model). While a number of important results have been obtained for these models, it will be the goal of future research to study more realistic models and to get a b etter theoretical understanding of the underlying phenomena from physics.