Toeplitz approach to the Uncertainty Principle

这个项目是在复数和调和分析领域。它涉及调和分析中属于测不准原理的一组问题，测不准原理是量子力学中出现的同名原理的相对论。源于诺伯特·维纳在数学中的工作和沃纳·海森伯格在物理学中的工作，测不准原理领域仍然存在许多数学挑战，该领域在邻近领域有许多应用。与测不准原理相关的几个经典问题，几十年前由诺曼·莱文森、安德烈·科尔莫戈洛夫和诺伯特·维纳等著名数学家提出的问题，仍然悬而未决。其中的某些问题将在这个项目中进行研究。最近三十年出现的复数和调和分析的现代方法提出了新的方法来挑战测不准原理的经典挑战。最近，首席调查员解决了两个这样的问题，即所谓的缺口和类型问题。这些问题在逼近理论、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·马卡洛夫和首席研究员在最近的论文中提出了一种新的不确定原理领域的方法。新方法的主要组成部分之一是使用Toeplitz运算符，这解释了项目的标题。实际上，Toeplitz方法在调和分析和频谱理论中的下一阶段应用是本项目的重点。它包含几个经典的开放问题。要研究的课题包括：一般完备性问题，它将扩展前面提到的“类型”问题的结果；薛定谔和狄拉克算子的谱问题；以及所谓的Krein-de Brange理论的Toeplitz算符版本，该理论旨在将复数分析和谱分析联系起来。这项研究的成功完成将为不确定性原则领域中的大量问题创造一个新的和系统的观点。该项目的一部分将与首席研究员在德克萨斯农工大学的研究生合作完成。最近，美国和国际研究中心的首席研究人员开设了几门关于Toeplitz方法应用于不确定原理的小型课程。所有课程都面向年轻的研究人员和研究生。首席研究员尼古拉·马卡洛夫和米哈伊尔·索丁正在写一本书，这本书将包含对克莱因-德·布兰奇理论和其他与不确定性原理相关的主题的现代描述。

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.