Spectral Theory and Integrable Systems

该项目研究谱理论中的问题，谱理论是描述物理概念的数学理论，如量子系统的能级和机械系统的振动频率。所考虑的数学模型是在相互作用中的无序和一些远程秩序（如空间慢衰减相互作用和准周期相互作用）存在竞争影响的制度下。一个重点是应用谱理论来解释某些非线性系统中的守恒定律，并通过可积性的数学概念来描述它们的行为中隐藏的可预测性。该项目专注于量子力学的核心——薛定谔算子，所开发的数学方法有可能照亮其他数学模型和物理应用，如无序材料中的电子电导率和使用孤子的信号传输。本项目的一个主要焦点是几乎周期薛定谔算子的谱理论和具有几乎周期初始数据的Korteweg-de Vries方程的可积性。虽然Lax对表示将方程正式改写为等谱流，但可积性的严格表征，例如逆散射变换的构造，高度依赖于所考虑的相空间，并且要求并激发了直接和逆谱理论中深刻的新结果。在这个项目中考虑的其他主题包括根据光谱数据估计解的大小和连续性，具有慢衰减势的薛定谔算符，高阶Szego定理，以及薛定谔算符和量子自旋系统的输运性质。

This project studies problems in spectral theory, the mathematical theory that describes physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. The mathematical models considered are in regimes where there are competing influences from a disorder in the interaction and some long-range order, such as spatially slowly decaying interactions and quasi-periodic interactions. One focus is on applications of spectral theory to explain conservation laws in certain nonlinear systems and find otherwise hidden predictability in their behavior, described through the mathematical notion of integrability. The project focuses on Schrodinger operators, central to quantum mechanics, and the mathematical methods developed have the potential to illuminate other mathematical models and physical applications, such as electron conductivity in disordered materials and signal transmission using solitons. One main focus of this project is the spectral theory of almost periodic Schrodinger operators and integrability of the Korteweg-de Vries equation with almost periodic initial data. While a Lax pair representation formally rewrites the equation as an isospectral flow, rigorous characterizations of integrability, such as construction of an inverse scattering transform, are highly dependent on the phase space under consideration and require and motivate deep new results in direct and inverse spectral theory. Other topics considered in this project include estimates for the size and continuity of the solution in terms of spectral data, Schrodinger operators with slowly decaying potentials, higher-order Szego theorems, and transport properties of Schrodinger operators and quantum spin systems.