Mathematical Sciences: Spectral Properties of Random Media

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9707049 Hislop Localization at band-edge energies for Schroedinger operators is now reasonably well understood for simple models of disorder, like Anderson-type potentials. A part of this project is directed towards extending these results to other interesting systems for which disorder strongly affects wave propagation: acoustic, electromagnetic, and elastic waves. Another goal is to extend the types of disorder allowed to more realistic models such as the Poisson and random displacement models. This will require an improvement of the Wegner estimate obtained in previous works. These techniques will be useful in studying the question of correlated systems and high-energy localization for two-dimensional disordered system. The propagation of classical acoustic and electromagnetic waves and of a single electron in perfectly ordered media, like a crystal, is well understood. There are allowed and forbidden energies of propagation for the waves. At the allowed energies, the waves propagate out to infinity behaving almost as if they were free waves. The effects of randomly distributed impurities in solids, or randomly distributed defects in an otherwise perfect material, is profound: the particles and waves no longer propagate out to infinity. Rather, at many energies, the transport of energy will be completely suppressed. This is effect is referred to as localization. The goal of this research is to better understand the mechanisms of localization for classical and quantum mechanical systems for many models of disorder. These results have applications to many engineering devices such as the effect of random imperfections in wave guides, photonic crystals, and superlattices.