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Toeplitz Order and Spectral Problems

托普利兹阶和谱问题

基本信息

批准号：
1665264
负责人：
Alexei Poltoratski
金额：
$ 18.6万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665264&HistoricalAwards=false
关键词：
Toeplitz Order Spectral Problems

项目摘要

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.

这个项目与谐波分析中的不确定性原理（UP）有关。简单地说，这个原则说，一个函数及其傅里叶变换不为零的集合不能同时很小。从诺伯特·维纳在数学和维尔纳·海森堡在物理学的工作中，UP领域仍然存在许多数学挑战。它的问题在邻近领域有许多应用。几十年前，著名数学家诺曼·莱文森（Norman Levinson）、安德烈·柯尔莫哥洛夫（Andrei Kolmogorov）和诺伯特·维纳（Norbert Wiener）提出了UP的几个经典问题，这些问题至今仍未解决。本项目对其中的一些问题进行了研究。在过去的30年中出现的复杂和谐波分析的现代方法提出了新的方法来应对UP的经典挑战。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。在拟议的研究课题是概括的所谓差距和类型问题的调和分析，一个扩展的著名Gelfand-Levitan理论领域的谱问题的微分算子和连接之间的克莱因的典型系统的微分方程和渐近的黎曼zeta函数。成功完成项目的这一步将创建一个系统的观点，在UP领域的各种问题的基础上，在提案中讨论的Toeplitz秩序的新概念。Toeplitz顺序的研究是在Nikolai Makarov（Caltech）和主要研究者最近的论文中开发的所谓Toeplitz方法的研究的延续。下一阶段的应用程序的Toeplitz方法包含几个经典的开放问题的调和分析和频谱理论，包括一般的完整性问题，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。在其他应用程序中，该项目包括一个关于薛定谔算子和微分方程正则系统的谱问题设置中的不确定性量化问题。