On self-adjointness of a Schroedinger operator

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