Spectral Asymptotics of the Laplacian on Surfaces of Constant Curvature

TL;DR

This paper investigates the asymptotic behavior of Laplacian eigenvalues on 2D surfaces of constant curvature, providing computational evidence supporting a conjecture about their spectral distribution.

Contribution

It offers the first extensive computational analysis of Laplacian spectra on various constant curvature surfaces, supporting a specific spectral asymptotics conjecture.

Findings

Eigenvalue counting function aligns with the 3-term asymptotic prediction

Spectral differences exhibit expected regular behavior

Computational methods are validated across multiple geometries

Abstract

The purpose of this paper is to explore the asymptotics of the eigenvalue spectrum of the Laplacian on 2 dimensional spaces of constant curvature, giving strong experimental evidence for a conjecture of the second author \cite{strichartz2016}. We computed and analyzed the eigenvalue spectra of several different regions in Euclidean, Hyperbolic, and Spherical space under Dirichlet, Neumann, and mixed boundary conditions and in particular we found that the average of the difference between the eigenvalue counting function and a 3-term prediction has the expected nice behavior. All computational code and data is available on our companion website \cite{murray2015results}