Discrete Schrodinger Operators and Related Models

离散薛定谔算子及相关模型

基本信息

批准号：
1800689
负责人：
Rui Han
金额：
$ 12.43万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2020-10-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800689&HistoricalAwards=false
关键词：
Discrete Schrodinger Operators Related Models

项目摘要

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.

该项目涉及Schrodinger Operators的光谱理论，Schrodinger操作员是量子力学中的核心主题。 Schrodinger操作员描述了电子在受无序系统的培养基中的运动。预计Schrodinger操作员理论的发展将增强对许多类型的物理现象的理解，包括电导，量子霍尔效应，准晶体和石墨烯。 PI打算开发新工具，为这些现象提供严格的数学解释。将要开发的工具还将在数学的其他分支中找到应用程序，包括谐波分析，概率和数理论。该项目由几个部分组成。一种是研究磁场中的量子图。 PI打算了解光谱和光谱分解的拓扑结构。第二部分涉及在离散的周期图上研究拉普拉斯人，其目的是找到光谱间隙的存在与基础晶格的几何结构之间的联系。第三个目标涉及离散的准周期施罗宾格运营商，重点关注几个众所周知的问题，包括测量频谱的测量，频谱的连续性，本地化方案中本本特征的结构以及准周期运算符的奇异连续性方案中的量子动态。另一个目标是了解具有偏移偏移产生的潜力的施罗宾格运营商。这些操作员虽然完全确定性，但有望类似于随机功能。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。