Spectral Asymptotics of Laplace Eigenfunctions

该研究项目福尔斯谱渐近学领域，研究流形（曲面和曲率空间）上高频拉普拉斯本征函数的行为。本征函数的物理类似物是驻波，本征值可以被认为是它们相应的频率。高频本征函数和它们所处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，它们的本征值是相应的能量。为了说明与数论的联系，精确计算平坦环面上给定频率的本征函数的数量的任务等价于计算一个整数可以表示为两个平方之和的方式的数量。该项目旨在开发新的工具，以促进对谱渐近的理解，其与看似不同的数学和科学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部分析和傅立叶积分算子理论的工具来改进描述高频本征函数行为的各种公式，特别是描述基础几何对这些公式的影响。PI打算对流形乘积的Weyl定律的余项进行猜想，发展傅里叶积分算子的一般多线性理论，用于谱渐近和几何测度理论，并进一步探索光谱量和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，并被认为是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，

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.