Harmonic-Counting Measures and Spectral Theory of Lens Spaces

TL;DR

This paper introduces harmonic-counting measures linked to lattices and applies them to analyze the asymptotic behavior of Laplace-Beltrami eigenfunctions on lens spaces, connecting spectral properties with lattice geometry.

Contribution

It defines harmonic-counting measures for lattices and uses them to study eigenfunction asymptotics on lens spaces, advancing spectral geometry understanding.

Findings

Harmonic-counting measures are associated with lattices.

Asymptotic behavior of eigenfunctions is characterized.

Results connect lattice structure with spectral properties.

Abstract

In this article, associated with each lattice $T\subseteq \mathbb{Z}^n$ the concept of a harmonic-counting measure $\nu_T$ on a sphere $S^{n-1}$ is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of the cardinality of the set of independent eigenfunctions associated with the elements of $T$ which lie in a cone is determined when $T$ is the lattice of a lens space.