Scattering theoretic methods in the mathematics of disordered media

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

Research will be done to establish the phenomenon of localization for physically relevant models of disordered media where existing methods have widely failed in the most relevant case of dimension three. Particular models to be investigated are the Bernoulli-Anderson model, the Poisson model, and the Random Displacement model. The main obstacle to overcome is the lack of monotonicity properties for these models, which causes significant differences to the more successfully studied Anderson model. In previous work, the PI and his collaborators have shown that methods from scattering theory can be used to establish localization for one-dimensional non-monotonic models. These methods will lead to further results in one dimension and provide new tools in higher dimension such as using the total scattering cross section of single sites to identify energies where extended states exist. Some related goals are to establish Wegner estimates for the eigenvalues of finite box hamiltonians and Lifshitz tail asymptotics for the integrated density of states in the above models, as well as to understand the effects of correlations arising from long range single site scatterers. The mathematical theory of disordered media aims at understanding the spectral and scattering theoretic properties of irregular solids such as, for example, crystals with impurities, alloys, materials with lattice deviations and amorphous media. Different models of random operators are used to describe the various types of disorder. This provides a mathematical framework to decide on conductivity properties: If localization of states can be shown, then the solid is electrically insulating, while the existence of extended states characterizes a conducting material. The PI's research aims at establishing these properties for models of high physical relevance, which have so far resisted a rigorous mathematical treatment. For example, localization properties are not yet understood for realistic models of alloys. Ideas from statistical physics and solid state physics will be combined with methods from scattering theory, spectral theory and harmonic analysis to make progress.