  We consider perturbed quadharmonic operators, $\Delta^4 + V$, acting on
sections of a Hermitian vector bundle over a complete Riemannian manifold, with
the potential $V$ satisfying a bound from below by a non-positive function
depending on the distance from a point. Under a bounded geometry assumption on
the Hermitian vector bundle and the underlying Riemannian manifold, we give a
sufficient condition for the essential self-adjointness of such operators. We
then apply this to prove the separation property in $L^2$ when the perturbed
operator acts on functions.
