Sacks The investigator studies the one-dimensional inverse scattering problem of quantum mechanics, in which the solution to be found is a potential in the Schrodinger equation, and develops effective numerical methods for computing the solution. A variety of inverse scattering problems are known to be uniquely solvable when a complex valued function, known as the reflection coefficient, is given as data. In many interesting applications, however, the reflection coefficient cannot be measured, but instead only its amplitude can be determined experimentally. The phase problem in inverse scattering consists then in solving the inverse scattering problem despite the lack of phase information contained in the reflection coefficient. In general these missing data introduce genuine non-uniqueness into the problem, and one must compensate by making use of other kinds of information. For example, one might restrict the class of admissible solutions in some way consistent with the physics of the situation, or alternatively one might have available some other kinds of data related to the usual scattering data. Aside from uniqueness questions, there is also a need to develop accurate and reliable computational techniques. The model problem --- the one-dimensional inverse scattering problem --- arises in neutr on and x-ray reflection studies of surface structure. Related problems in optics are also studied. It is natural in such applications to require the potential to vanish on a half line, and this substantially, although not completely, removes the ambiguity due to the missing phase information. A main goal of this project is to identify various types of further supplementary information whose specification allows for unique recovery of the potential. Also, numerical methods are developed for computation of the potential in each case. An important aspect of the numerical approach developed here is that ultimately it is only necessary to solve an optimization problem over a relatively small dimensional parameter space. In a number of areas of science and engineering it is of interest to characterize the chemical structure of surfaces and interfaces on very small length scales. Aside from its intrinsic theoretical interest, this kind of capability has technological applications, especially in materials science and biology. One experimental technique which has received considerable attention in recent years is the use of so-called reflectivity data: a carefully prepared beam of x-rays or neutrons is aimed at the surface whose material properties are sought, and the reflected beam is carefully measured. Due to the interaction of the beam with the surface, the reflected beam encodes a great deal of information about the surface, and one is thus confronted with the problem of properly interpreting such data. The main goal of this project is to investigate certain analytical and numerical methods for making such inferences. The reflected beam may be characterized by amplitude and phase components, but conventional measuring devices are sensitive only to the amplitude component. It is well understood that considerable ambiguity in the interpretation of reflectivity data is associated with the absence of the phase component. To compensate for this one hopes to use various kinds of supplementary information which may be available. In this project the investigator seeks to elucidate from a mathematical point of view how it may be possible to incorporate such extra information into automatic data processing techniques.
