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参数h，称为普朗克常数; h的观测值被认为是我们宇宙的基本常数之一。 量子力学系统和它们的经典对应物之间的关系可以通过研究当参数h的值在某种意义上趋于零时量子力学方程的行为来探索。 然而，严格地追求这一想法涉及微妙之处。 因为h不是无量纲量，所以参数值的“小”必须用其他相关量来表示。 本研究课题以普朗克常数h趋于零的极限为中心，通过严密的数学分析，加深对量子力学与经典力学之间的关系的理解。 所研究的问题对数学家和物理学家都有很大的兴趣。这个研究项目主要涉及量子系统的定态。 整体调和分析是使用黎曼流形的测地流的长时间动力学来理解本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。 利用最近的结果有关几何控制和限制的特征函数超曲面，调查员的目的是研究节点问题在非遍历的情况下，可能的关系，分形不确定性原则。 该项目还包括探索其他几个主题，包括高维流形上本征函数的节点域的数目;本征函数的Lp范数和微局部Kakeya-Nikodym范数及其与Grauert管解析延拓的关系;以及各种哈密顿量在自然界面（如能量表面）上的光谱投影态密度行为。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，被认为值得支持。