Inverse Scattering Problems

9803219 Aktosun它被提议研究量子力学和波传播中出现的某些逆散射问题。 这类问题由频域中的薛定谔方程、其变体、波动方程和电报方程所支配。 利用与黎曼-希尔伯特问题、算子因子分解和微分算子谱分析相关的技术，目的是根据一组合适的散射数据恢复相关微分方程中出现的系数。除了在物理学、工程学和其他应用领域中的实际重要性外，这些反问题还为非线性微分方程、积分方程、边值问题和应用泛函分析等数学领域提供了数学方法。地震学、地球物理学和遥感学。 在材料科学中，逆散射问题的解决方案相当于通过分析从材料散射的中子束来确定材料的表面和亚表面结构，例如，帮助理解聚合物成分的结合。 在量子力学中，它对应于通过分析粒子的碰撞束来确定将分子、原子和亚原子粒子保持在一起的力。 在遥感中，它通过分析从未知目标散射的声波或电磁波来帮助描绘这些目标。

9803219AktosunIt is proposed to study certain inverse scattering problems arising in quantum mechanics and wave propagation. Such problems are governed by the Schroedinger equation, its variants, the wave equation and the telegraphy equation in the frequency domain. Using techniques related to Riemann-Hilbert problems, operator factorizations, and spectral analysis of differential operators, the aim is to recover coefficients appearing in the relevant differential equations in terms of a suitable set of scattering data. In addition to their practical importance in physics, engineering, and other applied fields, these inverse problems contribute mathematical techniques to various areas of mathematics such as nonlinear differential equations, integral equations, boundary value problems, and applied functional analysis.The inverse scattering problems under study are motivated by their important applications in many areas such as materials science, nondestructive testing, acoustic imaging, seismology, geophysics, and remote sensing. In materials science the solution of the inverse scattering problem amounts to determining surface and subsurface structure of materials by analyzing neutron beams scattered off the materials, for example, helping to understand bonding of polymer components. In quantum mechanics it corresponds to the determination of forces holding molecules, atoms, and subatomic particles together by analyzing colliding beams of particles. In remote sensing it helps to profile unknown targets by analyzing acoustic or electromagnetic waves scattered off such targets.