Weighted sub-Laplacians on Métivier groups: Essential self-adjointness and spectrum

Abstract:

Let $G$ be a Métivier 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.