Weighted sub-Laplacians on M\'etivier Groups: Essential Self-Adjointness and Spectrum

TL;DR

This paper investigates the mathematical properties of weighted sub-Laplacians on Métivier groups, establishing conditions for their self-adjointness and spectrum discreteness, and confirming a conjecture related to spectral behavior.

Contribution

It proves essential self-adjointness of weighted sub-Laplacians for certain weights and characterizes when their spectrum is purely discrete, confirming a conjecture by Inglis.

Findings

Essential self-adjointness for α ≥ 1

Purely discrete spectrum for α > 2 in specific cases

Confirmation of Inglis's conjecture on spectrum discreteness

Abstract

Let $G$ be a M\'etivier group and let $N$ be any homogeneous norm on $G$. For $\alpha>0$ denote by $w_\alpha$ the function $e^{-N^\alpha}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_\alpha}$ associated with the Dirichlet form $\phi \mapsto \int_{G} |\nabla_\mathcal{H}\phi(y)|^2 w_\alpha(y)\, dy$, where $\nabla_\mathcal{H}$ is the horizontal gradient on $G$. Consider $\mathcal{L}^{w_\alpha}$ with domain $C_c^\infty$. We prove that $\mathcal{L}^{w_\alpha}$ is essentially self-adjoint when $\alpha \geq 1$. For a particular $N$, which is the norm appearing in $\mathcal{L}$'s fundamental solution when $G$ is an H-type group, we prove that $\mathcal{L}^{w_\alpha}$ has purely discrete spectrum if and only if $\alpha>2$, thus proving a conjecture of J. Inglis.