Uniqueness and Convergence of Analytic Integrals in Harmonic and Spectral Analysis

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This project concerns problems in Complex and Harmonic Analysis with applications to Mathematical Physics. It consists of two parts. The first part is devoted to the study of the Hilbert transform, one of the classical objects of mathematical analysis. The second part involves applications to special functions and spectral problems for differential operators. Despite being one of the most studied elements of complex and real analysis, the Hilbert transform is far from being completely understood. The first part of this proposal deals with a long standing problem of boundedness of the two-weight Hilbert transform and related topics. The second part of the project contains problems related to generalizations and applications of the celebrated Beurling-Malliavin theory. This theory was originally developed in the early 1960's to solve the problem of completeness of exponential functions in the space of square-summable functions on an interval, one of the canonical problems of Harmonic Analysis. The recently developed Toeplitz operator approach allows one to extend the classical theory and apply it to other families of special functions. Another important set of applications lies in the area of direct, inverse and mixed spectral problems for differential operators, such as the Schroedinger operator, Krein's string operator and more general canonical systems of differential equations. The applications considered in this project are related to second-order differential equations, such as the Schroedinger equation or the string equation, that are used in mathematical physics to model the behavior of quantum systems, wave propagation and various other physical phenomena. One of the important aspects of such problems is the ability to analyze physical characteristics of the system by looking at spectral data. A large part of this project is devoted to the further development of the mathematical tools necessary for such spectral analysis. The PI will actively involve his students in this project and will continue to review recent results related to this project in his graduate courses at Texas A&M University. The results obtained in this project will be published in scientific journals and reported at research conferences.