Inverse Scattering Problems and Applications

本研究旨在探讨波传播与量子力学中逆散射问题的各个方面及其应用。 其中包括波在非均匀吸收性介质中的传播以及介质的非均匀性和吸收率的测定，X射线反射测量法以及用X射线探测分层薄膜来测定分层薄膜的材料性质，波在目标位置的聚焦以及用波聚焦来测定介质性质，开发精确求积以将反射系数从间隔解析地延续到更大的域及其数值实现，逆散射问题在材料科学、无损检测、声成像、光学成像等领域有着重要的应用，和遥感。 其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，并从散射数据中推断目标的属性。 除了其在物理科学，工程和其他应用领域的实际重要性，研究将有助于数学技术的各个领域的数学，它也将有助于培养一些研究生和本科生在应用数学。

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.