Uniqueness and Convergence of Analytic Integrals in Harmonic and Spectral Analysis

本专题主要研究复分析与调和分析在数学物理中的应用.它由两部分组成。第一部分是专门研究希尔伯特变换，数学分析的经典对象之一。第二部分涉及特殊函数和微分算子谱问题的应用。尽管希尔伯特变换是复杂和真实的分析中研究最多的元素之一，但它远未被完全理解。这个建议的第一部分涉及一个长期存在的问题的有界性的两个权重希尔伯特变换和相关的主题。该项目的第二部分包含与著名的Beurling-Malliavin理论的推广和应用有关的问题。这个理论最初是在20世纪60年代早期发展起来的，用来解决区间上平方可和函数空间中指数函数的完备性问题，这是调和分析的典型问题之一。最近开发的Toeplitz算子方法允许一个扩展的经典理论，并将其应用到其他家庭的特殊功能。另一个重要的应用领域是微分算子的正、逆和混合谱问题，如薛定谔算子、克莱因弦算子和更一般的微分方程正则系统。 该项目中考虑的应用与二阶微分方程有关，例如薛定谔方程或弦方程，这些方程用于数学物理中，以模拟量子系统，波传播和各种其他物理现象的行为。这些问题的一个重要方面是通过查看光谱数据来分析系统的物理特性的能力。该项目的很大一部分致力于进一步开发这种光谱分析所需的数学工具。PI将积极让他的学生参与该项目，并将在德克萨斯农工大学的研究生课程中继续审查与该项目相关的最新成果。该项目取得的成果将在科学期刊上发表，并在研究会议上报告。

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.