Toeplitz approach to the Uncertainty Principle

该项目属于复数和调和分析领域。它涉及调和分析中属于不确定性原理标题的一系列问题，该原理是量子力学中出现的同名原理的亲戚。源于诺伯特·维纳（Norbert Wiener）在数学方面和维尔纳·海森堡（Werner Heisenberg）在物理学方面的工作，不确定性原理领域仍然面临着许多数学挑战，并且该领域在邻近领域有许多应用。与不确定性原理相关的几个经典问题，即诺曼·莱文森、安德烈·柯尔莫哥洛夫和诺伯特·维纳等著名数学家几十年前提出的问题，仍然悬而未决。本项目将研究其中某些问题。过去三十年出现的现代复调和调和分析方法提出了应对不确定性原理的经典挑战的新方法。首席研究员最近解决了两个这样的问题，即所谓的间隙和类型问题。这些问题在逼近论、预测理论、微分算子谱理论和数学物理中有许多重要的应用。 尼古拉·马卡罗夫和首席研究员最近的论文中提出了不确定性原理领域的一种新方法。新方法的主要成分之一是使用 Toeplitz 算子，这解释了该项目的标题。事实上，托普利茨方法在谐波分析和谱理论中的下一阶段应用是该项目的重点。它包含几个经典的开放问题。待研究的主题包括：一般完整性问题，这将扩展前面提到的“类型”问题的结果；薛定谔和狄拉克算子的谱问题；所谓的 Krein-de Branges 理论的 Toeplitz 算子版本，旨在连接复数分析和谱分析。这项研究的成功完成将为不确定性原理领域的各种问题建立一个新的、系统的观点。该项目的一部分将与德克萨斯农工大学首席研究员的研究生合作完成。美国和国际研究中心的首席研究员最近开设了几门关于托普利茨不确定性原理方法的迷你课程。所有课程均面向年轻研究人员和研究生。首席研究员正在与 Nikolai Makarov 和 Mikhail Sodin 一起撰写一本书，其中将包含对 Krein-de Branges 理论以及与不确定性原理相关的其他主题的现代解释。

This project is in the area of complex and harmonic analysis. It is concerned with the set of problems in harmonic analysis that fall under the heading of the Uncertainty Principle, a relative of the principle of the same name that arises in quantum mechanics. Stemming from the work of Norbert Wiener in mathematics and Werner Heisenberg in physics, the area of the Uncertainty Principle still presents many mathematical challenges, and the field has numerous applications to adjacent fields. Several classical problems associated with the Uncertainty Principle, problems posed decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov, and Norbert Wiener, remain open. Certain of those problems will be studied in this project. Modern methods of complex and harmonic analysis that appeared in the last thirty years suggest new approaches to the classical challenges of the Uncertainty Principle. Two such problems, the so-called gap and type problems, have recently been solved by the principal investigator. These problems have a number of important applications in approximation theory, prediction theory, spectral theory of differential operators, and mathematical physics. A new approach in the area of the Uncertainty Principle was developed in recent papers of Nikolai Makarov and the principal investigator. One of the main ingredients of the new approach is the use of Toeplitz operators, which explains the title of the project. Indeed, the next stage of the application of the Toeplitz approach in harmonic analysis and spectral theory is the focal point of this project. It contains several classical open problems. Among the topics of research to be pursued are the following: general completeness problems, which will expand the results on the "type" problem mentioned earlier; 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. Successful completion of this research will create a new and systematic view of the large variety of problems in the area of the Uncertainty Principle. A part of this project will be done in collaboration with the principal investigator's graduate students at Texas A&M University. Several mini-courses on the Toeplitz approach to the Uncertainty Principle were recently given by the principal investigator at U.S. and international research centers. All of the courses are oriented towards young researchers and graduate students. Together with Nikolai Makarov and Mikhail Sodin, the principal investigator is working on a book that will contain a modern account of the Krein-de Branges theory and other topics related to the Uncertainty Principle.