Multiscattering and Absolutely Continuous Spectrum

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The main objective of this research is to study random anddeterministic differential operators with mixed spectrum containing a dense pure point spectrum (corresponding to localized states) and an absolutely continuous spectrum (corresponding to extended states). According to the classical Anderson conjecture, both types of spectra coexist for random Schrodinger operators with "small" disorder. This conjecture hasnot been proved yet in a rigorous fashion. The award will support research on a rigorous proof of this conjecture when the disorder is not necessarily small in magnitude, but instead sparsely distributed.The awardees will also study related non-linear evolution problems in suitable classes of sparse functions.The problems under consideration concern the behavior of quantumparticles or waves in a cloud of localized and randomly distributed obstacles. The interaction of a particle with each individual obstacle can bedescribed in terms of classical scattering theory, while the globalpicture is a subject of multiscattering analysis. There are twodifferent scenarios which are not necessarily mutually exclusive: either particles or waves are trapped by the system of obstacles,or they can propagate to infinity. This research will investigate which scenario holds under what circumstances, in the case of very thinly distributed obstacles. This theory is essential to an understanding of wave propagation in disordered media, such as crystals with randomly distributed microdefects. A concrete application is inquality control of silicon surfaces using lasers.