Spectral Theory and Quantum Dynamics

该项目旨在提高对环境中存在的无序量如何促进或抑制系统中的运输的理解。这个问题是在原子水平的量子力学的背景下研究的。有关量子系统的新见解的应用包括量子计算设备和量子算法的开发。该项目通过研究生培训、对本科生研究的监督以及编写和出版一本关于常微分方程式的入门教科书来支持教育和多样性，旨在向本科生介绍该领域的严格处理。本项目涉及薛定谔算子和雅可比算子的么正类似物的频谱分析。这些运算符在许多领域都是相关的，主要是在量子力学和近似理论中。我们的目标是为这些算子的谱分析开发新的方法，所用的方法范围从泛函分析到调和分析到动力系统和遍历理论。该项目还从输运和色散现象的角度研究薛定谔时间演化。研究人员寻求对具有由双曲变换产生的势的薛定谔算符的完整谱分析，在单位圆上的正交多项式的背景下证明几个猜想，证明多频率薛定谔算符的零测量谱的方法，并从传输指数和色散估计的角度研究量子动力学。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to improve understanding of how the amount of disorder present in an environment can promote or suppress transport in a system. This issue is studied in the context of quantum mechanics at the atomic level. Applications of new insights about quantum systems include the development of quantum computing devices and quantum algorithms. The project supports education and diversity though graduate student training, the supervision of undergraduate research, and the writing and publication of an introductory textbook on ordinary differential equations aimed at introducing undergraduate students to a rigorous treatment of this field.This project addresses the spectral analysis of Schrödinger operators and unitary analogues of Jacobi operators. These operators are relevant in many areas, primarily in quantum mechanics and approximation theory. The objective is to develop new approaches for the spectral analysis of these operators, and the methods employed range from functional analysis via harmonic analysis to dynamical systems and ergodic theory. The project also investigates the Schrödinger time evolution in terms of transport and dispersion phenomena. The investigator seeks a complete spectral analysis of Schrödinger operators with potentials generated by hyperbolic transformations, a proof of several conjectures in the context of orthogonal polynomials on the unit circle, an approach to proving zero-measure spectrum for multi-frequency Schrödinger operators, and a study of quantum dynamics from the perspective of transport exponents and dispersive estimates.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.