Mathematical Sciences: Well-Posed Inverse Problems

这个项目的重点是从算子的频谱信息中确定微分算子的结构。在非数学术语中，这意味着人们试图从远程观测获得的数据中重建一个物体（如对流的障碍物）或力场。这项特别的工作将集中在量子后向散射数据的重建上。出现了几个问题。首先，是否有足够的信息可用于重建，所得到的势（所寻求的对象）是否唯一确定，以及通过一些实际手段重建势的问题。大部分的工作是对三维薛定谔方程的理解的延续。这个方程在一维上得到了广泛的研究。在三维空间中，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。解是用一个纯指数部分加上一个从问题的物理性质中恢复出来的项——散射振幅来给出的。逆散射问题包括从散射振幅中恢复薛定谔方程的势部分。人们遇到的一个直接问题是逆问题是过度确定的；人们不得不描述那些由三维势产生的散射振幅。目前正在研究的一种方法是将散射振幅（通常在五维空间上定义）限制为三维流形。其中之一就是所谓的后向散射数据。在那些已知潜力很小的情况下，这种选择导致了相当好的进展。目前的工作将沿着同样的脉络继续下去。主要目标是定义适当的函数类，这些函数类将给出一个受控的、定义良好的后向散射图。此外，我们还将继续研究寻找求解逆问题所需的最小数据集，指定逆映射的范围，并证明映射的Frechet导数是可逆的。

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.