Inner Functions, Spectra, and Scattering

本项目主要研究复谐分析在微分方程组谱问题中的应用。微分方程系统允许来自不同科学领域的研究人员用数学术语来模拟复杂的物理系统。微分系统的典型数学问题之一是谱逆问题；从系统的光谱信息中恢复系统的问题。在本课题中，我们将复变函数理论和谐波分析的新方法应用于反谱问题。我们的方法在逼近理论、预测理论、信号处理和数学物理中都有应用。本项目中的一些问题属于不确定性原理（UP）领域，涉及Mark Krein和Norbert Wiener等杰出数学家提出的经典问题。我们在这个项目中解决问题的方法是基于所谓的Toeplitz算子的使用。下一阶段Toeplitz方法在谐波分析和频谱理论中的应用是本项目的主要课题。在提出的Toeplitz方法的进一步应用中，扩展了著名的Gelfand-Levitan理论在微分算子和Krein正则系统与Riemann - ζ函数的渐近之间的连接的谱问题领域。项目的这一步的成功完成将创建一个基于Toeplitz阶新概念的谐波分析和频谱理论的大量问题的系统视图，该概念在PI最近的一篇论文中提出。对Toeplitz顺序的研究是对Nikolai Makarov（加州理工学院）和PI最近的论文中发展起来的所谓的Toeplitz方法的研究的延续。下一阶段Toeplitz方法的应用包含了调和分析和谱理论的几个经典开放问题，包括一般完备性问题，薛定谔和狄拉克算子的谱问题，以及所谓的Krein - de Branges理论的Toeplitz算子版本，该理论旨在连接复杂分析和谱分析。PI最近在美国和国际研究中心开设了几门关于Toeplitz方法的迷你课程，其中包括在法国图卢兹的一个研究项目中讲授的Toeplitz秩序课程。所有的课程都面向职业生涯初期的研究人员和研究生。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on applications of complex and harmonic analysis to spectral problems for systems of differential equations. Systems of differential equations allow researchers from various fields of science to model complicated physical systems in mathematical terms. One of the canonical mathematical problems concerning differential systems is the inverse spectral problem; the problem of recovering the system from its spectral information. In this project, we apply new methods of complex function theory and harmonic analysis to inverse spectral problems. Our methods have applications in approximation theory, prediction theory, signal processing and mathematical physics. Some of the problems in this project belong to the area of Uncertainty Principle (UP) and concern classical questions posed by such prominent mathematicians as Mark Krein and Norbert Wiener. Our approach to the problems in this project is based on the use of so-called Toeplitz operators. The next stage of the applications of the Toeplitz approach to harmonic analysis and spectral theory is the main topic of this project.Among the proposed further applications of the Toeplitz approach is an extension of the well-known Gelfand-Levitan theory in the area of spectral problems for differential operators and connections between Krein's canonical systems and the asymptotics of the Riemann zeta-function. Successful completion of this step of the project will create a systematic view of the large variety of problems of harmonic analysis and spectral theory based on the new notion of Toeplitz order, presented in one of PI's recent papers. The study of Toeplitz order is a continuation of the study of the so-called Toeplitz approach to the UP developed in recent papers of Nikolai Makarov (Caltech) and the PI. The next stage of the applications of the Toeplitz approach contains several classical open problems of harmonic analysis and spectral theory, including general completeness problems, spectral problems for Schroedinger and Dirac operators and a Toeplitz operator version of the so-called Krein - de Branges theory, which was designed to connect complex and spectral analysis. The PI recently gave several mini-courses on the Toeplitz approach in the US and at international research centers, including a course on the Toeplitz order presented at a research program in Toulouse, France. All of the courses were oriented towards early-career researchers and graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.