Inverse Scattering Problems

[8032 . 19]提出研究量子力学和波传播中出现的某些逆散射问题。这类问题是由薛定谔方程、它的变体、波动方程和频域的电报方程控制的。利用与黎曼-希尔伯特问题、算子分解和微分算子的光谱分析相关的技术，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、工程和其他应用领域的实际重要性外，这些反问题还为各种数学领域（如非线性微分方程、积分方程、边值问题和应用泛函分析）贡献了数学技术。反散射问题在材料科学、无损检测、声成像、地震学、地球物理和遥感等领域的重要应用推动了反散射问题的研究。在材料科学中，反散射问题的解决相当于通过分析从材料中散射出来的中子束来确定材料的表面和地下结构，例如，帮助理解聚合物组分的结合。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、原子和亚原子粒子结合在一起的力。在遥感中，它通过分析从这些目标上散射出来的声波或电磁波来帮助探查未知目标的轮廓。

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.