Toeplitz approach to the Uncertainty Principle

本项目属于复谐分析领域。它关注的是调和分析中的一系列问题，这些问题属于测不准原理的范畴，测不准原理是量子力学中出现的同名原理的一个亲戚。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，测不准原理领域仍然存在许多数学上的挑战，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，几十年前由Norman Levinson， Andrei Kolmogorov和Norbert Wiener等杰出数学家提出的问题，仍然没有解决。其中一些问题将在本项目中进行研究。近三十年来出现的现代复谐分析方法提出了解决测不准原理经典挑战的新方法。两个这样的问题，即所谓的差距和类型问题，最近已经由首席研究员解决了。这些问题在近似理论、预测理论、微分算子的谱理论和数学物理中有许多重要的应用。尼古拉·马卡罗夫（Nikolai Makarov）及其首席研究员在最近的论文中提出了不确定性原理领域的一种新方法。新方法的主要组成部分之一是使用Toeplitz操作器，这也解释了该项目的名称。事实上，Toeplitz方法在谐波分析和频谱理论中的下一阶段应用是本项目的重点。它包含了几个经典的开放问题。要进行的研究课题包括：一般完备性问题，它将扩展前面提到的“类型”问题的结果；薛定谔算子和狄拉克算子的谱问题；以及所谓的克林-德布朗日理论的Toeplitz算子版本，该理论旨在将复杂分析和光谱分析联系起来。这项研究的成功完成将为不确定性原理领域的大量问题创造一个新的和系统的观点。这个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员开设了几门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，首席研究员正在撰写一本书，其中将包含克林-德·布朗日理论的现代描述以及与不确定性原理相关的其他主题。

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.