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混乱量如何促进或抑制系统中的运输的理解。在原子水平的量子力学的背景下研究了这个问题。有关量子系统的新见解的应用包括开发量子计算设备和量子算法。该项目通过研究生培训，本科研究的监督以及旨在将本科生对该领域的严格处理的普通微分方程进行介绍和发表的介绍和出版的教育和发表的介绍和出版。该项目介绍了SchrödingerOperators和Jacobi Operators的单位类似物的光谱分析。这些操作员在许多领域都相关，主要是量子力学和近似理论。目的是为这些运算符的光谱分析开发新方法，而采用的方法范围从通过谐波分析的功能分析到动态系统和奇异理论。该项目还根据运输和分散现象研究了Schrödinger时间的演变。研究人员寻求对Schrödinger运算符的完整光谱分析，其潜在的潜力是由双曲转换产生的，这证明了单位圆上正交多项式的几个猜测的证明，一种证明零量频谱的方法，用于零句号的频谱，用于多夸克斯·施罗辛格式奖励者的估算，以及估算的数量研究，以及分配的动力学。法定使命，并使用基金会的知识分子优点和更广泛的影响标准通过评估被认为是宝贵的支持。

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