Topics in Analysis, Spectral Theory, and Partial Differential Equations

分析、谱理论和偏微分方程主题

• 批准号: 2054465
• 负责人: Sergey Denisov
• 金额: $ 25.31万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054465&HistoricalAwards=false
• 关键词: Topics; Analysis; Spectral; Theory; Partial

This research project covers a range of questions central to classical analysis, spectral theory, and partial differential equations. The project focuses on rigorous underpinning of mathematical models for some essential phenomena in physics, such as electromagnetic and acoustic wave propagation in rough or random media. The goal is to develop analytical tools with a range of applications, including the fields of probability and the theory of stochastic processes. The project will employ tools from harmonic and complex analysis including wave packet decomposition, weighted estimates for singular integral operators, and time-frequency analysis. Mentoring students and junior researchers will be an integral part of the project. This project will investigate the properties of several classical orthogonal systems and the operators generated by them. This includes studying the size of orthogonal polynomials for different types of measures, understanding the connection between spectral theory of Jacobi matrices on trees and the concept of multiple orthogonality, and generalization of one-dimensional results to the case of multiple orthogonality. A significant part of the project will focus on developing the Szegö theory for de Branges canonical systems and Krein strings and possible application of these results to nonlinear evolution equations. Scattering theory for elliptic equations in higher dimensions will also be investigated, and related questions of energy cascade in linear evolution equations will be considered.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目涵盖了经典分析，光谱理论和部分微分方程的一系列问题。该项目着重于对物理学某些基本现象的数学模型的严格基础，例如在粗或随机培养基中的电磁波和声波传播。目的是开发具有一系列应用的分析工具，包括概率和随机过程理论。该项目将采用谐波和复杂分析的工具，包括波数据包分解，单数积分运算符的加权估计以及时频分析。指导学生和初级研究人员将是该项目不可或缺的一部分。该项目将调查几个经典正交系统的属性以及它们生成的操作员。这包括研究不同类型措施的正交多项式的大小，了解树木上的雅各比矩阵的光谱理论与多重正交性的概念以及对多重正交性的一维结果的概括。该项目的重要组成部分将着重于开发用于规范系统和凯林字符串的Szegö理论，并将这些结果应用于非线性演化方程。还将研究较高维度中椭圆方程的散射理论，并将考虑线性进化方程中能量级联的相关问题。该奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响审查标准来评估NSF的法定任务。

项目成果

Szegő condition, scattering, and vibration of Krein strings

Kerin 弦的 SzegÅ 条件、散射和振动

• DOI: 10.1007/s00222-023-01201-9
• 发表时间: 2023
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Bessonov, R.;Denisov, S.
• 通讯作者: Denisov, S.