Self-adjoint extensions of differential operators on Riemannian manifolds

We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.