On self-adjointness of a Schrödinger operator on differential forms

Let $M$ be a complete Riemannian manifold and let $\Omega ^{\bullet }(M)$ denote the space of differential forms on $M$. Let $d:\Omega ^{\bullet}(M)\to \Omega ^{\bullet+1}(M)$ be the exterior differential operator and let $\Delta =dd^{*}+d^{*}d$ be the Laplacian. We establish a sufficient condition for the Schrödinger operator $H=\Delta +V(x)$ (where the potential $V(x):\Omega ^{\bullet}(M)\to \Omega ^{\bullet}(M)$ is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by I. Oleinik about self-adjointness of a Schrödinger operator which acts on the space of scalar valued functions.