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

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 $\varphi \mapsto {\int }_G\mathrm{\mid }{\mathrm{\nabla }}_{\mathcal{H}}\varphi (y){\mathrm{\mid }}^2{w}_{\alpha }(y)dy$, where ${\mathrm{\nabla }}_{\mathcal{H}}$ is the horizontal gradient on $G$. Consider ${\mathcal{L}}^{w_{\alpha }}$ with domain $C_c^{\mathrm{\infty }}$. We prove that ${\mathcal{L}}^{w_{\alpha }}$ is essentially self-adjoint when $\alpha \ge 1$. 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.