Uniqueness and Convergence of Analytic Integrals in Harmonic and Spectral Analysis

这个项目涉及复数和调和分析的问题及其在数学物理中的应用。它由两部分组成。第一部分是关于希尔伯特变换的研究，它是数学分析的经典对象之一。第二部分涉及微分算子在特殊函数和谱问题上的应用。尽管希尔伯特变换是复杂和真实分析中研究最多的元素之一，但人们对它的理解还远远不够。这项建议的第一部分涉及一个长期存在的双权希尔伯特变换的有界性问题和相关主题。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。这一理论最早是在20世纪60年代初发展起来的，S是为了解决指数函数在区间上的平方可和函数空间中的完备性问题而发展起来的，这是调和分析的典型问题之一。最近发展起来的Toeplitz算子方法使人们可以扩展经典理论，并将其应用于其他特殊函数族。另一组重要的应用在于微分算子的正、逆和混合谱问题，例如薛定谔算子、Krein弦算子和更一般的标准微分方程组。这个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，它们在数学物理中被用来模拟量子系统的行为、波的传播和各种其他物理现象。这类问题的一个重要方面是通过查看光谱数据来分析系统物理特性的能力。该项目的很大一部分用于进一步开发这种光谱分析所需的数学工具。PI将积极地让他的学生参与这个项目，并将继续在他在德克萨斯农工大学的研究生课程中审查与这个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.