Mathematical Sciences: Localization in Mathematical Models of Disordered Media

9706076伯明翰阿拉巴马大学的G·斯托尔茨博士将进行一项研究项目，研究数学物理中出现的各种随机算子中的“局部化”现象。局域化，即具有密集纯点谱和指数衰减本征函数的能区的存在，已被发现在大多数无序介质的一维模型中是共同的。斯托尔茨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)将现有的局域化结果扩展到模型，这些模型比以前研究的模型更真实地描述了无序介质，如合金和非晶态材料；(Ii)基于随机点相互作用的三维模型的强化研究，这可能为高能谱特性提供新的见解，也可以用于研究随机表面和随机聚合物的物理性质。无序介质在固体物理、光学、电动力学以及材料科学等不同的物理学科中得到了广泛的研究。它们包括各种类型的材料，例如合金、含杂质的晶体(例如半导体)和无定形固体。随机算符提供了无序介质的数学模型。这些算符可以用数学谱理论和散射理论的方法来研究。它们可以描述并预测无序介质中的导电性和波的传播特性。例如，正如斯托尔茨博士研究的那样，局部化的数学现象表征了绝缘体(电绝缘体，分别是声音绝缘体，取决于潜在的物理模型)。虽然这些模型已经取得了一些重要的结果，但未来的研究目标将是研究更现实的模型，并从物理上更好地从理论上理解潜在的现象。

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.