LEAPS-MPS: Ergodic Jacobi Matrices

量子系统的研究是现代数学物理学的基础。该项目旨在通过在直接和反向方法之间建立新的桥梁来研究量子系统的长期行为。直接方法要求描述给定系统的属性，而反向方法要求描述哪些系统可能表现出特定的属性。该项目计划通过开设数学物理暑期学校、指导本科生研究、编写和出版旨在向研究生介绍该领域的遍历式Schrödinger运算符的研究生教科书等方式，支持教育和多样性。该项目解决了Schrödinger， Jacobi和Dirac算子的频谱分析，其系数是通过沿遍历拓扑动力系统的轨道连续采样获得的。这个框架包括许多感兴趣的模型，如晶体、准晶体和无序介质。该项目将研究与这些模型的量子演化相关的开放问题，以及伪随机模型的频谱结构、图上的周期算子和非周期平铺。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。

The study of quantum systems is fundamental in modern mathematical physics. This project aims to study the long-time behavior of quantum systems by building new bridges between direct and inverse approaches. The direct approach asks one to describe the attributes of a given system, while the inverse approach asks what systems may exhibit specified attributes. The project plans to support education and diversity though a summer school in mathematical physics, the supervision of undergraduate research, and the writing and publication of a graduate textbook on ergodic Schrödinger operators aimed at introducing graduate students to this field. This project addresses the spectral analysis of Schrödinger, Jacobi, and Dirac operators with coefficients obtained by continuously sampling along the orbits of an ergodic topological dynamical system. This framework includes many models of interest, such as crystals, quasicrystals, and disordered media. The project will study open questions related to the quantum evolution for such models as well as the structure of the spectrum for pseudo-random models, periodic operators on graphs, and aperiodic tilings.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.