Weikard 9401816 This award supports mathematical research on problems in the theory of differential equations, especially relating to the spectral theory of second order equations. The subject has a long, distinguished, history in which significant emphasis was placed on the study of the spectrum of Schrodinger operators during much of the current century. Fundamental advances were made following discoveries in the 1960's of the relationship of finite gap potentials, Weierstrass elliptic functions and inverse problems. The renewed interest in finite gap potentials was intensified when it was shown how potentials can have a finite number of bands if and only if they satisfy some equation in the Korteweg-deVriess hierarchy. Until very recently, all work focused on real-valued potentials. The isospectral problem and Korteweg-deVriess flow arising from complex-valued initial data was studied in the 1980's, marking the starting point for this research. A new approach has been opened based on a classical theorem of Picard concerning differential equations with elliptic coefficients all of whose solutions are meromorphic. This approach promises to be a powerful tool for tackling some of the open problems in the field, such as practical computation of band edges (by reduction to a linear algebraic eigenvalue problem) and the classification of isospectral manifolds of elliptic finite- gap potentials. Moreover it sheds light on an unexpected relationship between spectral properties in the algebraic as well as in the functional analytic sense of differential operators and global analytic properties of solutions of the associated differential equations. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***
