Spectral Theory and Quantum Dynamics

谱理论和量子动力学

• 批准号: 2054752
• 负责人: David Damanik
• 金额: $ 29.4万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054752&HistoricalAwards=false
• 关键词: Spectral; Theory; Quantum; Dynamics

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.

该项目旨在提高对环境中存在的无序程度如何促进或抑制系统中的传输的理解，该问题是在原子水平的量子力学背景下进行研究的，包括开发量子系统的新见解。该项目通过研究生培训、本科生研究监督以及常微分方程入门教科书的编写和出版来支持教育和多样性，旨在向本科生介绍该领域的严格研究。 .该项目致力于光谱分析薛定谔算子和雅可比算子的酉类似物在许多领域都具有相关性，主要是在量子力学和逼近理论中，其目标是开发这些算子的谱分析的新方法，所采用的方法包括从泛函分析到调和。该项目还研究了传输和色散现象方面的薛定谔时间演化。研究人员寻求对薛定谔算子进行完整的谱分析。双曲变换产生的势、单位圆正交多项式中几个猜想的证明、证明多频薛定谔算子零测度谱的方法，以及从输运指数和量子动力学角度研究量子动力学分散的估计。这反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Simon’s OPUC Hausdorff dimension conjecture

Simon 的 OPUC Hausdorff 维数猜想

• DOI: 10.1007/s00208-021-02283-7
• 发表时间: 2022
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Damanik, David;Guo, Shuzheng;Ong, Darren C.
• 通讯作者: Ong, Darren C.

The Spectrum of Schrödinger Operators with Randomly Perturbed Ergodic Potentials

具有随机扰动遍历势的薛定谔算子谱

• DOI: 10.1007/s00039-023-00632-z
• 发表时间: 2023
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: Avila, Artur;Damanik, David;Gorodetski, Anton
• 通讯作者: Gorodetski, Anton

The Almost Sure Essential Spectrum of the Doubling Map Model is Connected

几乎可以肯定，倍增图模型的基本谱是相连的

• DOI: 10.1007/s00220-022-04607-3
• 发表时间: 2023
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Damanik, David;Fillman, Jake
• 通讯作者: Fillman, Jake