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.

该研究项目属于谱渐近领域，研究流形(曲面和有曲率的空间)上高频拉普拉斯特征函数的行为。本征函数的物理模拟是驻波，本征值可以被认为是它们对应的频率。高频本征函数与它们所在的流形几何之间的相互依赖关系是从量子物理到数论的广泛领域的核心。实际上，本征函数是薛定谔方程的稳态解，它们的本征值就是相应的能量。为了说明与数论的联系，准确计算平坦环面上给定频率的特征函数的数量相当于计算整数可以表示为(比方说)两个平方和的方法的数量。这个项目旨在开发新的工具来促进对光谱渐近性的理解，光谱渐近性与数学和科学中看似不同的领域的相互联系使其研究特别有价值。作为研究项目的一部分，PI打算开发和使用微局域分析和傅立叶积分算子理论的工具来提炼各种描述高频特征函数行为的公式，特别是描述基础几何对这些公式的影响。PI打算改进关于流形乘积的Weyl定律的余项的猜想，发展用于谱渐近和几何测度论的傅立叶积分算子的一般多线性理论，并进一步探索谱量与流形中相应几何构型的存在之间的联系。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。