Complex Methods in Spectral and Scattering Problems

复变函数理论在微分方程式的光谱和散射问题中的应用提供了纯数学在理论物理和工程领域中无可争辩的相关性的例子。其中一个焦点将是与狄拉克微分方程组的散射变换密切相关的非线性傅立叶变换(NLFT)。狄拉克系统是控制自旋1/2粒子的量子电动力学定律，应该被视为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式的几个领域取得进展。将研究NLFT的收敛和最大估计问题。这些都被专家认为是非线性调和分析的主要问题之一。第二部分将讨论谱理论的经典结果的推广和推广，这些结果可以通过最近发展起来的基于所谓Toeplitz算子的新方法来获得。一些相关问题将与研究生一起进行调查。该项目的材料将用于微型课程和一门针对初级研究人员和研究生的专题课程。该项目的很大一部分涉及狄拉克微分方程组的散射数据的收敛问题。散射通常被视为经典傅里叶变换的非线性版本，它将该项目与NLFT的最大估计联系在一起。这些联系导致了在散射的非线性设置中建立经典傅立叶分析结果的版本的自然问题。在上个世纪的大部分时间里，它们一直以各种形式出现，至今仍是活跃研究的对象。作为一个例子，我们可以看看Parseval身份的非线性版本，它可以追溯到Verblunski在20世纪30年代的工作，以及Hausdorff-Young不等的非线性模拟，它出现在基督和基塞列夫最近的工作中。其中几个问题将被研究，以及在逆谱理论和各种解析函数空间中的完备性领域的问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.