Research in Approximation and Scattering Theory

本课题研究偏微分方程解的近似理论和散射理论中的一些问题。对于经典的正交系，如单位圆上的多项式、Krein系和弦，将研究正交性度量与系统参数之间的关系。此外，还将讨论多项式的渐近分析问题。这项研究将集中于更好地理解多维薛定谔和狄拉克算符的散射。这里的目标是获得与最近在Szego理论框架中证明的尖锐的一维结果类似的结果。为此，我们将研究和应用双曲薛定谔铅笔理论以及一些特殊的随机微分方程。PI的研究将集中在逼近理论和偏微分方程组理论的一些经典问题上。这两个看似无关的领域最近取得了长足的进步和思想的综合。偏微分方程组的散射理论被广泛用于描述电磁波和声波在介质中的传播。当媒体粗糙或随机时，情况尤其具有挑战性，在物理和数学领域都引起了相当大的关注。在这个项目中，将开发和应用新的分析方法来加深我们对这些物理过程的理解。该项目的工作将需要应用许多其他数学领域的思想：一般谱理论、概率、调和分析，这也将对这些领域产生影响。

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.