Let G be a M\'etivier group and let N be any homogeneous norm on G. For
α>0 denote by wα the function e−Nα and consider the
weighted sub-Laplacian Lwα associated with the Dirichlet
form ϕ↦∫G∣∇Hϕ(y)∣2wα(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.Comment: 15 pages; to appear on Proc. Amer. Math. So