9401491 Simon Professor Simon will continue his research looking into the spectral theory of Schrodinger operators and research concerning inverse spectral and inverse scattering problems. Both sets of research exploit detailed studies of the Krein spectral shift. Among areas that will be explored are: extending the rank one theory to include trace class perturbations, and the inverse problem for unbounded potentials, which should be determined in some way by the eigenvalues and Dirichlet eigenvalues. In particular, an attempt will be made to describe the isospectral manifold for this problem. Professor Simon will study a number of problems in the mathematical theory connected with quantum mechanics. One aspect of his work involves the study of long-time behavior in the model of electrons in solid and random media, whose properties are at the forefront of innovative materials design. Simon also expects to study problems in inverse spectral and scattering theory: the general issue of determining a microscopic structure from data that can be readily observed. Inverse theory is central to areas as varied as radar detection of submarines and the interpretation of medical tomography. ***
