A Tauberian Approach to an Analog of Weyl's law for the Kohn Laplacian on Compact Heisenberg Manifolds

TL;DR

This paper establishes an analog of Weyl's law for the Kohn Laplacian on compact Heisenberg manifolds, providing precise asymptotics for eigenvalue counts using a Tauberian approach.

Contribution

It introduces a Tauberian method to derive eigenvalue asymptotics for the Kohn Laplacian on compact Heisenberg quotients, extending Weyl's law to this setting.

Findings

Eigenvalue counting function asymptotics for differential operators on Heisenberg manifolds.

Weyl's law analog for the Kohn Laplacian on compact quotients.

Explicit constants depending on dimension and parameters.

Abstract

Let $M= \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We show that the eigenvalue counting function $N(\lambda)$ for any fixed element of a family of second order differential operators $\left\{\mathcal{L}_\alpha\right\}$ on $M$ has asymptotic behavior $N\left(\lambda\right) \sim C_{d,\alpha} \operatorname{vol}\left(M\right) \lambda^{d + 1}$, where $C_{d,\alpha}$ is a constant that only depends on the dimension $d$ and the parameter $\alpha$. As a consequence, we obtain an analog of Weyl's law (both on functions and forms) for the Kohn Laplacian on $M$. Our main tools are Folland's description of the spectrum of $\mathcal{L}_{\alpha}$ and Karamata's Tauberian theorem.