Spectral Theory and Quantum Dynamics

该项目旨在提高对环境中存在的障碍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背景下研究的。关于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。该项目支持教育和多样性，通过研究生培训，本科生研究的监督，以及编写和出版一本介绍性教科书，旨在介绍本科生对这一领域的严格治疗常微分方程。该项目涉及薛定谔算子和雅可比算子的酉类似物的谱分析。这些算子在许多领域都有相关性，主要是量子力学和近似理论。我们的目标是开发新的方法，这些运营商的频谱分析，并采用的方法范围从功能分析，通过谐波分析，动力系统和遍历理论。该项目还研究了薛定谔时间演化的传输和色散现象。研究人员寻求一个完整的谱分析薛定谔算子与双曲变换产生的潜力，证明几个命题的上下文中的正交多项式的单位圆，一种方法来证明零测度谱多频薛定谔算子，以及从输运指数和色散估计的角度研究量子动力学。该奖项反映了NSF的法定使命，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，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.