Boundary Behavior of Analytic Functions

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

The research will focus on several problems concerning the boundary behavior of analytic functions and their applications in functional analysis and mathematical physics. In analytic function theory the PI plans to study the interaction between singular and non-singular components of Cauchy integrals near the boundary of a complex domain. The results will then be applied to study the resolvent functions of linear operators in perturbation problems. This area has connections with mathematical models of solid state physics, which provides most of the motivation. Another part of the project is the study of the connections between the Beurling-Malliavin theory, inverse spectral problems for partial differential equations and properties of kernels of Toepliz operators. Analytic Function Theory is one of the classical yet rapidly developing parts of modern mathematics. The present stage of its development features many promising new applications in various parts of mathematics and physics. One of the cornerstones of Analytic Function Theory is the integral formula named after the famous French mathematician Cauchy. The importance of this formula is due to the fact that most analytic functions in complex domains can be defined through Cauchy integrals. The PI plans to study various properties of Cauchy integrals near the boundaries of their domains. The results will then be applied in several areas of mathematics as well as in mathematical physics. The applications in mathematical physics will concern, among other things, mathematical models of wave propagation. The general question that will be considered is how to recover full information about a disordered environment from the spectrum of the wave operator and partial information on the potential field.