Selected problems in perturbation theory, Schur multipliers, and Hankel and Toeplitz Operators in Noncommutative Analysis

非交换分析中的微扰理论、Schur 乘子以及 Hankel 和 Toeplitz 算子的精选问题

• 批准号: 1300924
• 负责人: Vladimir Peller
• 金额: $ 33万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300924&HistoricalAwards=false
• 关键词: Selected; problems; perturbation; theory; Schur

This project is devoted to the study of various problems in noncommutative analysis. In particular, it is concentrated on problems in perturbation theory. One of the most important problems of the proposal is the problem to estimate functions of perturbed noncommuting pairs of self-adjoint operators. This problem is very important in many applications, in particular, in quantum mechanics. Other important problems in perturbation theory deal with trace formulas for functions of perturbed operators, functional calculus for almost commuting self-adjoint operators, and estimates for operator and commutator moduli of continuity. The project hopes to reveal new connections with Schur multipliers, Hankel operators, and Hankel tensors; comprehend the role of Hankel and Toeplitz operators in noncommutative analysis; develop techniques of Schur multipliers in operator theory and harmonic analysis; find new applications in mathematical physics, control theory, prediction theory, and approximation theory. The problems of perturbation theory that are the focus of this project are important in applications in quantum mechanics, mathematical physics, electrical engineering, and control theory, and in other fields of science. The importance of perturbation theory in applications stems from the fact that if one is faced with an equation that describes a given situation (say, a physical one), the equation is often very complicated. One can replace it with a simpler (perturbed) equation that is reasonably close to the original equation and solve the new equation. Then it is time to invoke the latest results of perturbation theory to find out whether the solutions of the perturbed equation are reasonably close to the solutions of the initial equation that is needed for applications. This project considers important problems in perturbation theory that can help in such commonly occurring situations. Other problems described in the proposal are also important in applications such as prediction theory, control theory, and approximation theory.

该项目致力于研究非交换分析中的各种问题。特别是，它集中于微扰理论中的问题。该提案最重要的问题之一是估计自伴算子的扰动非交换对的函数的问题。这个问题在许多应用中非常重要，特别是在量子力学中。微扰理论中的其他重要问题涉及微扰算子函数的迹公式、几乎交换自伴算子的泛函计算以及算子和交换器连续性模的估计。该项目希望揭示与 Schur 乘子、Hankel 算子和 Hankel 张量的新联系；理解 Hankel 和 Toeplitz 算子在非交换分析中的作用；发展算子理论和调和分析中的 Schur 乘法器技术；在数学物理、控制理论、预测理论和逼近理论中找到新的应用。作为该项目重点的微扰理论问题在量子力学、数学物理、电气工程和控制理论以及其他科学领域的应用中具有重要意义。微扰理论在应用中的重要性源于这样一个事实：如果一个人面临一个描述给定情况（例如，物理情况）的方程，那么该方程通常非常复杂。人们可以用一个相当接近原始方程的更简单（扰动）方程来替换它，并求解新方程。 然后是时候调用摄动理论的最新结果来找出摄动方程的解是否相当接近应用所需的初始方程的解。该项目考虑了微扰理论中的重要问题，可以帮助解决这种常见情况。提案中描述的其他问题在预测理论、控制理论和逼近理论等应用中也很重要。