Inverse Scattering Problems and Applications

本研究旨在研究波传播和量子力学中反散射问题的各个方面及其应用。这些包括波在非均匀和吸收介质中的传播和介质的非均匀性和吸收率的测定，x射线反射法和用x射线探测分层薄膜的材料特性的测定，波在目标位置的聚焦和用波聚焦测定介质特性。发展精确正交以解析地将反射系数从一个区间延续到更大的域及其数值实现，并解决反散射问题，以便在无穷远处恢复具有较慢衰减条件的函数。所研究的逆散射问题在材料科学、无损检测、声成像和遥感等领域有着重要的应用。主要目的是以远程方式确定目标的属性：向目标发送波，分析散射波，并从散射数据推断目标的属性。除了在物理科学、工程和其他应用领域具有实际意义外，这项研究还将为数学的各个领域贡献数学技术，并有助于培养一些应用数学的研究生和本科生。

This research is to study various aspects of inverse scattering problems arising in wave propagation and in quantum mechanics and their applications. These include wave propagation in a nonhomogeneous and absorptive medium and the determination ofthe nonhomogeneities and absorptivity of the medium, X-ray reflectometry and the determination of material properties ofstratified thin films by probing them with X-rays, focusing of waves at target locations and the determination medium propertiesby using wave focusing, developing exact quadratures to analytically continue a reflection coefficient from an interval to larger domains and their numerical implementation, and solving inversescattering problems in order to recover functions with slower decay conditions at infinity.The inverse scattering problems investigated have important applications in manyareas such as materials science, nondestructive testing, acoustic imaging, and remote sensing. The principal aim is to determineproperties of a target in a remote fashion: send a wave onto thetarget, analyze the scattered wave, and infer the properties of thetarget from the scattering data. In addition to its practical importance in physical sciences, engineering, and other applied areas, the research will contribute mathematical techniques to various areas of mathematics and it will also help to train some graduate and undergraduate students in applied mathematics.