Inverse Scattering Problems and Applications

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This research is to study various aspects of inverse scattering problems arising in wave propagation and in quantum mechanics and their applications. These include wave propagation in a nonhomogeneous and absorptive medium and the determination ofthe nonhomogeneities and absorptivity of the medium, X-ray reflectometry and the determination of material properties ofstratified thin films by probing them with X-rays, focusing of waves at target locations and the determination medium propertiesby using wave focusing, developing exact quadratures to analytically continue a reflection coefficient from an interval to larger domains and their numerical implementation, and solving inversescattering problems in order to recover functions with slower decay conditions at infinity.The inverse scattering problems investigated have important applications in manyareas such as materials science, nondestructive testing, acoustic imaging, and remote sensing. The principal aim is to determineproperties of a target in a remote fashion: send a wave onto thetarget, analyze the scattered wave, and infer the properties of thetarget from the scattering data. In addition to its practical importance in physical sciences, engineering, and other applied areas, the research will contribute mathematical techniques to various areas of mathematics and it will also help to train some graduate and undergraduate students in applied mathematics.