Wave Propagation and Scattering for Periodic and Randomly Perturbed Periodic Media.

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This project will provide a complete asymptotic analysis of the propagation of wave packets with narrow spectra through finite slabs of periodic layered media and periodic quantum graphs. The work will identify conditions under which small group velocity is accompanied by near transparency. This analysis will be complemented by a study of the effects due to compact and random perturbations of the media; in particular, the transition from models with absolutely continuous spectrum to complete localization will be investigated. Effective statistical estimates will be given for physical characteristics of the pulses in terms of the noise in the media.The project will furnish the mathematical background for a fundamental problem that has been very actively discussed in physical literature during the last several years: construction of efficient optical devices for slowing down of optical signals. These devices can find many applications, for example in synchronizing the work of very fast optical elements with much slower electronics.