Discrete Schrodinger Operators and Related Models

这个项目涉及薛定谔算子的光谱理论，这是量子力学的一个中心课题。薛定谔算符描述电子在无序系统中的运动。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，包括调和分析，概率和数论。一个是研究磁场中的量子图。PI旨在了解频谱的拓扑结构和频谱分解。第二部分涉及研究离散周期图上的拉普拉斯算子，目标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准周期薛定谔算子，集中在几个著名的问题，包括测量的频谱，频谱的连续性，结构的本征函数的本地化制度和量子动力学在奇异连续制度的准周期运营商。另一个目标是了解薛定谔算子与潜在的斜移产生的。这些运营商，虽然是完全确定性的，预计类似于随机features.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。

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.