The problem of determining a quantum mechanical potential from its scattering amplitude goes back to the beginning of quantum mechanics. One aspect of the problem which has received little attention is finding data sets for which the problem is well-posed, i.e., that the mapping from the potential to the scattering data set is continuously invertible. This work focuses on one such set, the so-called backscattering data. It is the data which occurs from the restriction of the scattering amplitude to the direction opposite to the direction of the incident plane wave. In dimensions three and higher, the relationship of the backscattering and potential is well documented. In two dimensions or in cases where the data is restricted to half-spaces, there are obstacles which will be analyzed during the course of this project. Other work on inverse scattering will focus on the wave equation with variable speed of sound and the Schrodinger equation with both electric and magnetic potential. A second line of investigation concerns hyperbolic partial differential equations defined in domains with wedgelike boundaries and the propagation of singularities of solutions of initial-boundary value problems for second order equations in the presence of boundary wedges. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The inverse problems described in this proposal play a central role in these studies. Their object is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.
