Aktosun DMS-9501053 It is proposed to investigate the inverse scattering problems arising in quantum mechanics and wave propagation by studying the Schroedinger equation when the potential is independent of energy and when the potential is proportional to energy. The inverse problem consists of the determination of the potential when the scattering operator is known. By formulating this as a Riemann-Hilbert problem in the momentum space, it is proposed to study the solvability of the Riemann-Hilbert problem, the characterization of the scattering data, the role of the bound states, and the stability of the inversion. The solution of the inverse scattering problem with energy-independent potentials corresponds to the determination of molecular, atomic, and nuclear forces in terms of the scattering data obtained in collision experiments; the determination of these forces is not only important in quantum mechanics but it is also one of the fundamental problems in physics. The inverse scattering problem with energy-dependent potentials arises in wave propagation in a nonhomogeneous medium, and it is equivalent to determining the properties of the medium by analyzing the waves scattered from the medium. This problem has many important applications in acoustics, seismology, nondestructive testing, oil exploration, atmospheric profile inversion, and other areas where the measurements taken outside the medium are used to predict the properties of the medium.
