Mathematical Sciences: Inverse Scattering Problems

摘要Ralston/Ekin 9622310在这个建议中，我们提出了几个系数逆问题，我们计划用我们在文中介绍的方法来解决这些问题(Commun.数学课。Phys.173(1995))。在大多数情况下，我们希望从固定能量的散射数据(或等效的Dirichlet到Neumann映射)中恢复未知系数。通常假设系数以指数方式接近极限值，但在紧致集之外不一定是常量。我们计划研究的方程是具有外部规范势的薛定谔方程，指数趋于欧几里得度规的度规的Laplace-Beltrami方程，麦克斯韦方程，以及二维空间中具有磁势和电势的薛定谔方程。当人们希望从远程观测获得的数据中恢复物体的物理性质时，出现了逆散射问题。从广义上讲，这些问题包括无损检测和医学成像方法。我们打算研究的问题包括从发出的信号中恢复介质的电学性质，以及通过观察量子粒子的轨迹偏转来重建作用在量子粒子上的力。

Abstract Ralston/Eskin 9622310 In this proposal we present several inverse coefficient problems which we plan to attack using the methods introduced in our paper (Commun. Math. Phys.173 (1995)). In most cases we wish to recover the unknown coefficients from scattering data (or equivalently the Dirichlet to Neumann map) at a fixed energy. The coefficients are usually assumed to approach limiting values exponentially, but are not necessarily constant outside compact sets. The equations we plan to study are the Schroedinger equation with an external gauge potential, the Laplace-Beltrami equation for a metric which tends exponentially to the Euclidean metric, Maxwell's equations, and the Schroedinger equation with magnetic and electric potentials in two space dimensions. Inverse scattering problems arise when one wishes to recover physical properties of an object from data obtained by remote observations. Interpreted broadly these problems include methods of nondestructive testing and medical imaging. The problems we propose to study include recovering the electrical properties of a medium from signals sent through it, and the reconstruction of the forces acting on a quantum particle from the observation of the deflection of its trajectories.