Research in Approximation and Scattering Theory

本项目涉及偏微分方程的近似理论和散射理论中的一些问题。对于经典的正交系统，如单位圆上的多项式、Krein系统和弦，我们将研究正交性测度与系统参数之间的关系。多项式的渐近分析问题也将得到解决。该研究将集中在更好地理解多维薛定谔和狄拉克算子的散射。这里的目标是获得类似的尖锐的一维的结果，最近被证明在Szego理论的框架。双曲薛定谔束理论以及一些特殊的随机微分方程将被研究和应用于这一目的。PI的研究将集中在逼近理论和偏微分方程理论的一些经典问题上，这两个看似不相关的领域最近取得了相当大的进展和思想的综合。偏微分方程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。当介质是粗糙或随机的情况下，特别具有挑战性，并在物理和数学中引起了相当大的关注。在这个项目中，新的分析方法将被开发和应用，以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的想法从许多其他领域的数学：一般谱理论，概率，谐波分析，并将产生影响，这些领域以及。

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.