Complex Methods in Spectral and Scattering Problems

光谱和散射问题的复杂方法

• 批准号: 2244801
• 负责人: Alexei Poltoratski
• 金额: $ 29.93万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2244801&HistoricalAwards=false
• 关键词: Complex; Methods; Spectral; Scattering; Problems

Applications of complex function theory in spectral and scattering problems for differential equations present examples of the undisputed relevance of pure mathematics in the areas of theoretical physics and engineering. One of the focal points will be the non-linear Fourier transform (NLFT), closely related to scattering transform for the Dirac system of differential equations. The Dirac system is the quantum electrodynamical law governing spin ½ particles and should be thought of as a relativistic generalization of the Schrödinger equation. Better understanding of NLFT can lead to progress in several areas of mathematical analysis and non-linear differential equations. Questions of convergence and maximal estimates for NLFT will be studied. These are considered by the experts to be among the main problems of non-linear harmonic analysis. The second part will concern extensions and generalizations of classical results of spectral theory, which can be obtained via a new approach developed recently based on the use of so-called Toeplitz operators. Some related questions will be investigated jointly with graduate students. The materials of the project will be used in minicourses and a special topics course aiming at junior researchers and graduate students.A large part of this project concerns problems of convergence of the scattering data for a Dirac system of differential equations. Scattering is commonly viewed as a non-linear version of the classical Fourier transform, which connects this project to the maximal estimates for NLFT. These connections lead to natural questions of establishing versions of the classical results of Fourier analysis in the non-linear settings of scattering. They have been appearing in various forms for most of the last century and remain an object of active research today. As an example, one can look at the non-linear version of Parseval's identity, which can be traced as far back as the work of Verblunski in the 1930s, and a non-linear analog of Hausdorff-Young inequality, which appears in more recent work of Christ and Kiselev. Several of such questions will be studied, as well as ones in the areas of inverse spectral theory and completeness in various spaces of analytic functions.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.

复变函数理论在微分方程的光谱和散射问题中的应用，提供了理论物理和工程领域中纯数学无可争议的相关性的例子。重点之一将是非线性傅立叶变换（NLFT），密切相关的散射变换的狄拉克系统的微分方程。狄拉克系统是控制自旋1/2粒子的量子电动力学定律，应该被认为是薛定谔方程的相对论推广。更好地理解NLFT可以在数学分析和非线性微分方程的几个领域取得进展。将研究NLFT的收敛性和最大估计问题。专家们认为这些是非线性谐波分析的主要问题。第二部分将涉及扩展和推广的经典结果的谱理论，这可以通过一个新的方法，最近开发的基础上使用所谓的Toeplitz运营商。一些相关问题将与研究生共同研究。本计画的资料将用于针对初级研究人员与研究生的小型课程与专题课程。本计画的大部分内容是关于Dirac微分方程组的散射资料的收敛问题。散射通常被视为经典傅立叶变换的非线性版本，它将该项目与NLFT的最大估计相联系。这些连接导致自然的问题，建立版本的经典结果的傅立叶分析中的非线性设置的散射。它们在上个世纪的大部分时间里以各种形式出现，今天仍然是积极研究的对象。作为一个例子，我们可以看看Parseval恒等式的非线性版本，它可以追溯到20世纪30年代Verblunski的工作，以及Hausdorff Young不等式的非线性模拟，它出现在Christ和Kiselev最近的工作中。一些这样的问题将被研究，以及在反频谱理论和完整性领域的各种空间的解析functions.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。