Toeplitz approach to the Uncertainty Principle

这个项目是在复杂和谐波分析领域。它关注的是调和分析中属于测不准原理标题下的一系列问题，测不准原理是量子力学中出现的同名原理的相对原理。源于数学中的诺伯特·维纳和物理学中的维尔纳·海森堡的工作，不确定性原理领域仍然存在许多数学挑战，并且该领域在相邻领域有许多应用。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·柯尔莫戈罗夫和诺伯特·维纳等著名数学家提出的问题，仍然没有解决。本项目将研究其中的某些问题。在过去的三十年中出现的现代复分析和调和分析方法提出了新的方法来应对不确定性原理的经典挑战。两个这样的问题，所谓的差距和类型的问题，最近已经解决了首席研究员。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。 在不确定性原理领域的一种新方法是在尼古拉·马卡罗夫和首席研究员最近的论文中开发的。新方法的主要成分之一是使用Toeplitz运算符，这解释了该项目的名称。事实上，Toeplitz方法在谐波分析和谱理论中的应用的下一阶段是这个项目的重点。它包含几个经典的开放问题。在研究的主题将进行以下：一般的完整性问题，这将扩大的结果“类型”的问题前面提到的;频谱问题的薛定谔和狄拉克运营商;和Toeplitz运营商版本的所谓克莱因德布兰日理论，这是旨在连接复杂和频谱分析。这项研究的成功完成将为不确定性原理领域的各种问题创造一个新的和系统的观点。该项目的一部分将与主要研究者在德克萨斯州A M大学的研究生合作完成。美国和国际研究中心的首席研究员最近举办了几个关于不确定性原理的Toeplitz方法的迷你课程。所有的课程都面向年轻的研究人员和研究生。与尼古拉·马卡罗夫和米哈伊尔·索丁一起，首席研究员正在写一本书，其中将包含克莱因-德布兰日理论的现代描述和与不确定性原理有关的其他主题。

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.