Global Harmonic Analysis

全局谐波分析

• 批准号: 1810747
• 负责人: Jared Wunsch
• 金额: $ 22.6万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810747&HistoricalAwards=false
• 关键词: Global; Harmonic; Analysis

The fundamental equations of quantum mechanics contain a parameter h known as Planck's constant; the observed value of h is understood to be one of the fundamental constants of our universe. Relationships between quantum-mechanical systems and their classical counterparts can be explored by investigating how the quantum-mechanical equations behave as the value of the parameter h tends to zero in some sense. Pursuing this idea rigorously, however, involves subtleties. Because h is not a dimensionless quantity, the "smallness" of values of the parameter must be expressed in terms of other relevant quantities. This research project aims to deepen understanding of relationships between quantum mechanics and classical mechanics through rigorous mathematical analysis centered on the limit when the Planck constant h tends to zero. The questions under study are of great interest both to mathematicians and to physicists.This research project primarily concerns stationary states of quantum systems. Global harmonic analysis is the use of the long-time dynamics of the geodesic flow of a Riemannian manifold to understand the high-frequency asymptotics of eigenfunctions. Two of the most important goals of the project are (1) to analyze Lp norms of eigenfunctions and (2) to analyze nodal sets and critical point sets. Making use of recent results relating geometric control and restrictions of eigenfunctions to hypersurfaces, the investigator aims to study nodal problems in non-ergodic situations and possible relations to the fractal uncertainty principle. The project also includes exploration of several other topics, including the numbers of nodal domains of eigenfunctions on manifolds of higher dimension; Lp norms of eigenfunctions and microlocal Kakeya-Nikodym norms and their relations to analytic continuation to Grauert tubes; and the behavior of density of states of spectral projections for various Hamiltonians across natural interfaces such as energy surfaces.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.

量子力学的基本方程包含一个称为普朗克常数的参数 h； h 的观测值被理解为我们宇宙的基本常数之一。 通过研究当参数 h 的值在某种意义上趋于零时量子力学方程如何表现，可以探索量子力学系统与其经典对应系统之间的关系。 然而，严格追求这一想法涉及微妙之处。 由于 h 不是无量纲量，因此参数值的“小”必须用其他相关量来表示。 该研究项目旨在通过以普朗克常数h趋于零时的极限为中心的严格数学分析，加深对量子力学与经典力学之间关系的理解。 数学家和物理学家都对所研究的问题非常感兴趣。该研究项目主要涉及量子系统的稳态。 全局调和分析是利用黎曼流形测地流的长期动力学来理解特征函数的高频渐近。 该项目的两个最重要的目标是 (1) 分析本征函数的 Lp 范数，以及 (2) 分析节点集和临界点集。 利用与几何控制和超曲面特征函数限制相关的最新结果，研究人员旨在研究非遍历情况下的节点问题以及与分形不确定性原理的可能关系。 该项目还包括对其他几个主题的探索，包括高维流形上本征函数的节点域数量；本征函数的 Lp 范数和微局域 Kakeya-Nikodym 范数及其与 Grauert 管解析延拓的关系；该奖项反映了 NSF 的法定使命，并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。