This project is concerned with some problems in approximation theory and with the scattering theory for partial differential equations. For classical orthogonal systems, such as polynomials on the unit circle, Krein systems and strings, the relation between the measure of orthogonality and parameters of the system will be studied. The problems of asymptotical analysis of polynomials will be addressed as well. The study will be focused on better understanding the scattering for multidimensional Schrodinger and Dirac operators. The goal here is to obtain analogs of the sharp one-dimensional results that were recently proved in the framework of Szego theory. The theory of hyperbolic Schrodinger pencils as well as some special stochastic differential equations will be studied and applied for that purpose.
The research of PI will be focused on some classical problems of approximation theory and the theory of partial differential equations. These two seemingly unrelated areas have recently enjoyed considerable progress and synthesis of ideas. The scattering theory for partial differential equations is widely used to describe the propagation of electromagnetic and acoustic waves in the medium. The cases when the media is rough or random are especially challenging and attracted considerable attention in both physics and mathematics. In this project, new analytical methods will be developed and applied to deepen our understanding of these physical processes. The work on the project will require applications of ideas from many other areas of mathematics: general spectral theory, probability, harmonic analysis and that will have an impact on these fields as well.
