CAREER: Quantum Systems with Deterministic Disorder

无序量子系统是现代数学物理的基本对象，因为由于热的存在，任何现实生活中的系统都有一定程度的噪声。具有噪声/无序的系统通常用概率方法研究，并假设噪声是随机的。该项目的主要范围是研究噪声具有附加结构和确定性（非随机）性质的系统。在物理学中，强随机无序通常意味着系统成为绝缘体，并阻止其中的电子自由移动。对于一些非随机无序的系统，这也是真的吗？如果是的话，哪些属性使非随机系统表现得像随机的，它可以被测量吗？如果紊乱不严重，是否会有其他影响？该项目的主要目标是在许多层面上解决这些问题，其中包括与本科生和研究生合作，开发动力学和光谱理论的研究生课程，并为高中生开发一门课程，说明基本线性代数和物理学之间的联系，为他们提供技能和动机，以便他们在STEM领域继续深造。该项目包括分析一类模型的教学和研究活动，数学量子物理学，包括发展算子理论的抽象技术和建立更具体系统的严格结果。所有提出的模型都涉及无序，然而，与通常对无序系统的概率观点不同，主要目标将是在完全确定性的环境中研究无序，或者使用非常少量的随机/遍历参数。这种系统的一个例子是具有动力学定义势的薛定谔算子，其中底层动力学系统具有小维度和低混合度（例如，无理旋转）。通常，具有大的随机无序的量子系统倾向于阻止电子自由移动（安德森局域化）。为了回答确定性无序系统中电子输运的基本问题，人们必须用数论、遍历理论、半代数几何和其他深层次的方法来取代通常的概率方法。该项目的主要方向包括相互作用准周期粒子系统的局部化/离域分析和相互作用的影响，粗糙势单粒子算子的微扰方法，周期算子谱带的微扰性质，以及应用于几乎可换算子和矩阵的算子理论的抽象方法，该奖项反映了NSF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。

Disordered quantum systems are fundamental objects in modern mathematical physics, since, due to presence of heat, any real-life system has some level of noise in it. Systems with noise/disorder are often studied by probabilistic methods and assume that the noise is random. The main scope of the project is to study systems where the noise has additional structure and has deterministic (non-random) nature. In physics, strong random disorder often implies that the system becomes an insulator and prevents electrons in it from moving freely. Would it also be true for some systems with non-random disorder? If yes, which properties make non-random systems behave like random and can it be measured? Are there any additional effects if the disorder is not strong? The main goal of the project is to address these questions on many levels, which will include work with undergraduate and graduate students, development of graduate courses on dynamics and spectral theory, and developing a course for high school students that would illustrate connections between basic linear algebra and physics, providing them skills and motivation for possible further education in STEM.This project incorporates teaching and research activities on the analysis of a class of models of mathematical quantum physics, including developing abstract techniques of operator theory and establishing rigorous results on more concrete systems. All proposed models involve disorder, however, unlike usual probabilistic view on disordered systems, the main goal will be studying the disorder in a completely deterministic setting, or with a very small number of random/ergodic parameters. An example of such system would be a Schrodinger operator with dynamically-defined potential, where the underlying dynamical system has small dimension and low degree of mixing (for example, irrational rotation). Typically, quantum systems with large random disorder tend to prevent electrons from moving freely (Anderson localization). To answer even basic questions about electron transport in deterministic disordered systems, one must replace usual probabilistic methods by methods of number theory, ergodic theory, semi-algebraic geometry, and other deep areas. The main directions of the project involve analysis of localization/delocalization for systems of interacting quasiperiodic particles and the effect of interaction, perturbative methods for single-particle operators with rough potentials, perturbation properties for spectral bands of periodic operators, and abstract methods of operator theory applied to almost commuting operators and matrices, with applications to quantum systems.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.