Mathematical Sciences: Well-Posed Inverse Problems

这个项目的重点之一是根据关于算子的谱的信息来确定微分算子的结构。在非数学术语中，这意味着一个人正试图从远程观测获得的数据中重建一个物体(如流动的障碍物)或力场。这项特别的工作将集中在从量子后向散射数据重建。出现了几个问题。首先，存在是否有足够的信息可用于重建的问题，所产生的势(所寻求的对象)是否唯一确定的问题，以及通过一些实际手段重建势的问题。大部分工作是理解三维薛定谔方程的继续努力。这个方程已经在一维上得到了广泛的研究。在三个方面，取得进展的障碍要大得多，直到最近，该领域的数学研究才显示出任何进展。解的形式是纯指数部分加上一个从问题的物理性质--散射幅度的知识中恢复出来的项。逆散射问题包括从散射幅度恢复薛定谔方程的势能部分。人们遇到的一个直接问题是反问题被过度确定；人们被迫表征那些可能由三维势引起的散射幅度。目前正在研究的一个程序是将散射幅度(通常在五维空间上定义)限制在三维流形上。其中之一就是所谓的后向散射数据。在已知潜力很小的情况下，这一选择取得了相当好的进展。目前的工作将继续沿着同样的思路进行。一个主要的目标是定义适当的函数类，它将给出受控的、定义良好的后向散射图。此外，还将继续研究寻找解决逆问题所需的最小数据集、指定逆映射的范围以及证明该映射的Frechet导数是可逆的相关问题。

The focus of this project is one of determining the structure of differential operators from information about the spectrum of the operator. In non-mathematical terms, this means that one is attempting to reconstruct an object (such as an obstruction to a flow) or a force field from data obtained by remote observations. This particular work will concentrate on reconstruction from quantum backscattering data. Several issues arise. First, there is the question of whether or not sufficient information is available for the reconstruction, whether the resulting potential (the object sought) is uniquely determined and the problem of reconstructing the potential by some practical means. Much of the work is a continuation of efforts to understand the three-dimensional Schroedinger equation. This equation has been studied extensively in one dimension. In three dimensions, the obstacles to progress are much greater, and it is only recently that mathematical research in the area has shown any progress. Solutions are given in terms of a pure exponential part plus a term which is recovered from knowledge of the physical properties of the problem - the scattering amplitude. The inverse scattering problem consists of recovering the potential part of the Schroedinger equation from the scattering amplitude. An immediate problem one encounters is that the inverse problem is over-determined; one is forced to characterize those scattering amplitudes which can arise from three- dimensional potentials. One procedure currently under investigation is to restrict the scattering amplitude (normally defined on five-dimensional space) to three-dimensional manifolds. One of these is the so-called backscattering data. This choice has led to reasonably good progress in those cases where the potential is known to be small. The present work will continue along the same vein. A primary objective is to define the proper function classes which will give a controlled, well-defined backscattering map. In addtion, work will continue on related issues of finding minimal data sets necessary to solve the inverse problem, specifying the range of the inverse map and showing that the Frechet derivative of the map is invertible.