Inverse Scattering Problems

Aktosun DMS-9501053提出通过研究薛定谔方程来研究量子力学和波传播中的逆散射问题，当势与能量无关时。反问题包括在已知散射算符的情况下确定势。通过将其描述为动量空间中的Riemann-Hilbert问题，我们建议研究Riemann-Hilbert问题的可解性、散射数据的特征、束缚态的作用以及逆问题的稳定性。能量无关势的逆散射问题的解相当于根据碰撞实验获得的散射数据确定分子力、原子力和核力，这些力的确定不仅是量子力学中的重要问题，也是物理学中的基本问题之一。波在非均匀介质中传播时会产生含能量势的逆散射问题，它等价于通过分析介质中散射的波来确定介质的性质。这一问题在声学、地震学、无损检测、石油勘探、大气剖面反演以及其他使用介质外测量来预测介质性质的领域有许多重要的应用。

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.