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被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。当参数h的值在某种意义上趋于零时，可以通过研究量子力学方程的行为来探索量子力学系统和它们的经典对应系统之间的关系。然而，严格追求这一理念涉及到微妙之处。因为h不是无量纲的量，参数值的“小”必须用其他相关的量来表示。这一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，加深对量子力学和经典力学之间关系的理解。正在研究的问题是数学家和物理学家都非常感兴趣的问题。这个研究项目主要是关于量子系统的稳态态。全局调和分析是利用黎曼流形上测地线流的长时间动力学来理解特征函数的高频渐近性。该项目的两个最重要的目标是(1)分析特征函数的Lp范数和(2)分析节点集和临界点集。利用最近关于几何控制和特征函数对超曲面的限制的结果，研究者旨在研究非遍历情况下的节点问题以及可能与分形测不准原理的关系。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的数目；本征函数和微局部Kakeya-Nikodym范数的Lp范数及其与Grauert管的分析连续性的关系；以及各种哈密顿人跨自然界面(如能量面)的光谱投影的状态密度行为。该奖项反映了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。