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、谱理论和偏微分方程的一系列核心问题，重点是物理学中一些基本现象（例如粗糙或随机介质中的电磁波和声波传播）的数学模型的严格基础。目标是开发具有一系列应用的分析工具，包括概率领域和随机过程理论，该项目将采用谐波和复分析工具，包括波包分解、奇异积分算子的加权估计和时间分析。频率指导学生和初级研究人员将是该项目的一个组成部分，该项目将研究几个经典正交系统及其生成的算子，其中包括研究不同类型测量的正交多项式的大小。树上雅可比矩阵的谱理论与多重正交性概念之间的联系，以及将一维结果推广到多重正交性的情况，该项目的一个重要部分将集中于开发 Szegö。还将研究 de Branges 正则系统和 Kerin 弦的理论以及这些结果在高维椭圆方程的散射理论中的可能应用，并考虑线性演化方程中能量级联的相关问题。该奖项反映了通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。

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.