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.
