Spectral Asymptotics of Laplace Eigenfunctions

拉普拉斯本征函数的谱渐近

• 批准号: 2422900
• 负责人: Emmett Wyman
• 金额: $ 8.37万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-03-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2422900&HistoricalAwards=false
• 关键词: Spectral; Asymptotics; Laplace; Eigenfunctions

The research project falls within the field of spectral asymptotics, which studies the behavior of high-frequency Laplace eigenfunctions on manifolds (surfaces and spaces with curvature). The physical analogues of eigenfunctions are standing waves, and the eigenvalues may be thought of as their corresponding frequencies. The interdependence between high-frequency eigenfunctions and the geometry of the manifold on which they live is central to a broad range of fields from quantum physics to number theory. Indeed, eigenfunctions are steady-state solutions to the Schrödinger equation, and their eigenvalues are the corresponding energies. To illustrate the connection to number theory, the task of accurately counting the number of eigenfunctions of a given frequency on the flat torus is equivalent to counting the number of ways an integer can be expressed as the sum of, say, two squares. This project aims to develop new tools to advance understanding in spectral asymptotics, whose interconnectedness to seemingly disparate areas of mathematics and science make its study particularly valuable. As part of the research project, the PI intends to develop and use tools from microlocal analysis and the theory of Fourier integral operators to refine a variety of formulas describing the behavior of high-frequency eigenfunctions, and in particular describing what effect the underlying geometry has on these formulas. The PI intends to make advancements towards a conjecture on the remainder term of the Weyl law for products of manifolds, to develop a general multilinear theory of Fourier integral operators for use in both spectral asymptotics and geometric measure theory, and to further explore the connection between spectral quantities and the presence of corresponding geometric configurations in the manifold.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.

光谱渐进型领域的研究项目，该项目研究了司法ds的高频特征性（具有曲率的表面和空间）高频特征函数和它们所生活的流形的几何形状是从量子物理学到数字理论的广泛领域的核心。理论对扁平的圆环进行了相当于计算整数的数量，例如两个正方形的总和。 theTegral的运算符，以完善各种福尔马斯描述高频特征的行为n特殊描述了基础几何形状的tormulas对tormulas具有pi的概念。测量理论以及光谱量与歧管中的Corresspond GeoMetri Ons的存在。