Geometric Scattering Theory, and Propagation and Decay of Waves

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The principal investigator will study mathematical problems in the subject of particle-wave duality. This duality has been successfully investigated for a long time, especially in the setting of closed systems, but many of its important manifestations are poorly understood, especially in the settings of open and mixed systems which are the principal investigator's focus. In open systems the method of complex scaling studied by the principal investigator appears in numerical analysis as the method of perfectly matched layers, where it has been useful in the engineering of nanostructures and microelectromechanical systems. Mixed systems like the ones studied by the principal investigator are important in electromagnetics, communications, optics, quantum waveguides, and semiconductors.The main dynamical object studied by the principal investigator is the trapped set; this is the set of particle trajectories which remain in a bounded set for all time. The geometry of this set is of central importance in determining salient features of wave evolution, such as regularity, decay, and asymptotics. The principal investigator will study these features in the important but still comparatively mysterious situations of heavy trapping, rough coefficients, and noncompact trapped sets.