Weighted sub-laplacians on métivier groups: Essential self-adjointness and spectrum

Let G be a Métivier group and let N be any homogeneous norm on G. For α > 0 denote by wα the function e−N α and consider∫ the weighted sub-Laplacian Lwα associated with the Dirichlet form φ→G ‖∇Hφ(y)‖2 wα (y) dy, where ∇H is the horizontal gradient on G. Consider Lwα with domain Cc∞. We prove that Lwα is essentially self-adjoint when α ≥ 1. For a particular N, which is the norm appearing in L’s fundamental solution when G is an H-type group, we prove that Lwα has purely discrete spectrum if and only if α > 2, thus proving a conjecture of J. Inglis.