Mathematical Sciences: Well-Posed Inverse Problems

该项目的重点是确定 结构的微分算子的信息 运营商的频谱。 用非数学术语来说，这意味着 一个人试图重建一个对象（如 阻碍流动）或力场， 远程观察 这项工作将集中在 从量子反向散射数据重建。 若干问题 起来。 第一个问题是， 信息可用于重建，无论 结果电势（所寻求的对象）是唯一确定的 以及通过一些实际的 手段 大部分工作都是为了理解 三维薛定谔方程 该方程具有 在一个维度上被广泛研究。 在三维空间中， 进步的障碍要大得多，这只是 最近，该地区的数学研究表明， 中求进工作总 解以纯指数形式给出 一部分加一个术语，从知识的恢复， 问题的物理性质-散射振幅。 逆散射问题包括恢复 薛定谔方程的散射势部分 振幅 人们遇到的一个直接问题是， 逆问题是超定的;人们被迫描述 这些散射振幅可以由三个- 维度潜力 目前正在研究的一个程序是限制 散射振幅（通常定义在五维上 空间）到三维流形。 其中之一是 所谓的反向散射数据。 这种选择合理地导致了 在那些已知潜在的情况下取得良好进展 小了 目前的工作将继续沿着同样的思路进行。 一 主要目标是定义适当的函数类， 将给出一个可控的、清晰的后向散射图。 在 此外，将继续就有关问题开展工作， 解决逆问题所需的数据集，指定 逆映射的范围，并表明Frechet导数 地图是可逆的。

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.