Scattering theoretic methods in the mathematics of disordered media

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Research will be done to establish the phenomenon of localizationfor physically relevant models of disordered media where existingmethods have widely failed in the most relevant case of dimensionthree. Particular models to be investigated are theBernoulli-Anderson model, the Poisson model, and the RandomDisplacement model. The main obstacle to overcome is the lack ofmonotonicity properties for these models, which causes significantdifferences to the more successfully studied Anderson model. Inprevious work, the PI and his collaborators have shown thatmethods from scattering theory can be used to establishlocalization for one-dimensional non-monotonic models. Thesemethods will lead to further results in one dimension and providenew tools in higher dimension such as using the total scatteringcross section of single sites to identify energies where extendedstates exist. Some related goals are to establish Wegner estimatesfor the eigenvalues of finite box hamiltonians and Lifshitz tailasymptotics for the integrated density of states in the abovemodels, as well as to understand the effects of correlationsarising from long range single site scatterers.The mathematical theory of disordered media aims at understandingthe spectral and scattering theoretic properties of irregularsolids such as, for example, crystals with impurities, alloys,materials with lattice deviations and amorphous media. Differentmodels of random operators are used to describe the various typesof disorder. This provides a mathematical framework to decide onconductivity properties: If localization of states can beshown, then the solid is electrically insulating, while theexistence of extended states characterizes a conducting material.The PI's research aims at establishing these properties for modelsof high physical relevance, which have so far resisted a rigorousmathematical treatment. For example, localization properties arenot yet understood for realistic models of alloys. Ideas fromstatistical physics and solid state physics will be combined withmethods from scattering theory, spectral theory and harmonicanalysis to make progress.