WEIGHTED SUB-LAPLACIANS ON METIVIER GROUPS: ESSENTIAL SELF-ADJOINTNESS AND SPECTRUM

Let G be a Metivier 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 subLaplacian L-w alpha associated with the Dirichlet form phi bar right arrow integral(G) parallel to del(H)phi(y) parallel to(2) w alpha(y) dy, where del(H) is the horizontal gradient on G. Consider L-w alpha with domain C-c(infinity). We prove that L-w alpha is essentially self-adjoint when alpha >= 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 L-w alpha has purely discrete spectrum if and only if alpha > 2, thus proving a conjecture of J. Inglis.