Discrete Schrodinger Operators and Related Models

这个项目涉及薛定谔算子的谱理论，这是量子力学的一个中心话题。薛定谔算符描述电子在受无序系统影响的介质中的运动。薛定谔算符理论的发展有望增强对许多类型物理现象的理解，包括电导、量子霍尔效应、准晶体和石墨烯。PI打算开发新的工具，为这些现象提供严格的数学解释。将开发的工具也将在数学的其他分支中找到应用，包括谐波分析，概率论和数论。这个项目由几个部分组成。一是研究磁场中的量子图。PI旨在了解光谱的拓扑结构和光谱分解。第二部分涉及研究离散周期图上的拉普拉斯算子，目的是找到谱隙的存在与底层晶格的几何结构之间的联系。第三个目标涉及离散拟周期薛定谔算子，重点关注几个众所周知的问题，包括谱的测量、谱的连续性、本征函数在局域域中的结构和拟周期算子在奇异连续域中的量子动力学。另一个目标是理解由斜移产生的势的薛定谔算子。这些操作符，虽然是完全确定的，但期望类似于随机特征。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns the spectral theory of Schrodinger operators, a central topic in quantum mechanics. Schrodinger operators describe the movement of an electron in a medium subject to a disordered system. The development of the theory of Schrodinger operators is expected to enhance the understanding of many types of physical phenomena, including conductance, quantum Hall effect, quasicrystals, and graphene. The PI intends to develop new tools to provide rigorous mathematical explanations for these phenomena. The tools that will be developed will also find applications in other branches of mathematics, including harmonic analysis, probability and number theory.This project consists of several parts. One is to study quantum graphs in magnetic fields. The PI intends to understand the topological structure of the spectrum and spectral decompositions. The second part involves studying Laplacians on discrete periodic graphs with a goal of finding the connection between the presence of spectral gaps and the geometric structure of the underlying lattice. The third goal concerns discrete quasi-periodic Schrodinger operators, focusing on several well-known problems including measure of the spectrum, continuity of the spectrum, structure of eigenfunctions in the localization regime and quantum dynamics in the singular continuous regime for quasi-periodic operators. Another goal is to understand Schrodinger operators with potentials generated by skew-shift. These operators, although being completely deterministic, are expected to resemble random features.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.