Toeplitz Order and Spectral Problems

本课题涉及谐波分析中的不确定原理（UP）领域。简单地说，这个原理说明一个函数及其傅里叶变换不为零的集合不可能同时很小。由于数学上的诺伯特·维纳和物理上的维尔纳·海森堡的工作，UP领域仍然面临许多数学上的挑战。它的问题在邻近的领域有许多应用。几十年前由Norman Levinson、Andrei Kolmogorov和Norbert Wiener等杰出数学家提出的几个经典问题至今仍未解决。本课题对其中的一些问题进行了研究。在过去30年里出现的现代复谐分析方法为UP的经典挑战提出了新的方法。这些问题在近似理论、预测理论、微分算子谱理论和数学物理中都有重要的应用。提出的研究课题包括谐波分析中所谓的间隙和类型问题的推广，微分算子谱问题领域中著名的Gelfand-Levitan理论的扩展，以及Krein微分方程正则系统与黎曼ζ函数渐近性之间的联系。该项目的这一步骤的成功完成将根据提案中讨论的Toeplitz秩序的新概念，对UP领域的各种各样的问题产生系统的看法。对Toeplitz顺序的研究是对所谓的Toeplitz方法的研究的延续，该方法是由Nikolai Makarov（加州理工学院）和首席研究员在最近的论文中提出的。下一阶段Toeplitz方法的应用包含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，该理论旨在连接复分析和谱分析。在其他应用中，该项目包括薛定谔算子和正则微分方程系统的谱问题设置中的不确定性量化问题。

This project is related to the area of the Uncertainty Principle (UP) in Harmonic Analysis. Briefly, this principle says the sets where a function and its Fourier transform are non-zero cannot be simultaneously small. Stemming from the work of Norbert Wiener in mathematics and Werner Heisenberg in physics, the area of UP still presents many mathematical challenges. Its problems have a number of applications in adjacent fields. Several classical problems of UP, posed decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov and Norbert Wiener, remain open. Some of such problems are studied in this project. Modern methods of Complex and Harmonic Analysis that appeared in the last 30 years suggest new approaches to the classical challenges of UP. These problems have a number of important applications in Approximation Theory, Prediction Theory, Spectral Theory of differential operators and Mathematical Physics. Among the proposed topics of research are generalizations of the so-called Gap and Type problems in Harmonic Analysis, 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 of differential equations and 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 in the area of UP based on the new notion of Toeplitz Order discussed in the proposal. The study of Toeplitz Order is a continuation of the study of the so-called Toeplitz approach to UP developed in recent papers of Nikolai Makarov (Caltech) and the principal investigator. 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. Among other applications, the project includes a problem on Uncertainty Quantification in the settings of spectral problems for Schroedinger opearators and canonical systems of differential equations.