Mathematical Sciences: Multidimensional Inverse Scattering Problems

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This project consists of an investigation of the inverse scattering problems for the multidimensional Schrodinger equations when the potential is independent of energy and when the potential is proportional to energy. In the former case, the Wiener-Hopf factorization and the generalized Muskhelishvili- Vekua technique will be used to investigate the characterization of the scattering data, the relationship between bound states and partial indices, the inversion from partial data, and the stability of the inversion. In the latter case, the inverse scattering problem will be formulated as a Riemann-Hilbert problem with a discontinuity of almost periodic type and its solution will be obtained through a generalized Wiener-Hopf factorization technique. The solution of the inverse scattering problem with energy- independent potentials is equivalent to determining molecular, atomic, and nuclear forces in terms of the scattering data obtained in collision experiments, and the determination of these forces is one of the fundamental problems in physics. The inverse scattering problem with energy-dependent potentials is equivalent to the determination of the properties of a medium using only the measurements made on the boundary; such a problem has many important applications in wave propagation, seismology, nondestructive testing, oil exploration, atmospheric profile inversion, and other areas.