Essential self?adjointness of perturbed quadharmonic operators on Riemannian manifolds with an application to the separation problem

We consider perturbed quadharmonic operators, ? 4 + V , acting on sections of a Hermitian vector bundle over complete Riemannian manifold, with the potential satisfying bound from below by non-positive function depending distance point. Under bounded geometry assumption and underlying we give sufficient condition for essential self-adjointness such operators. then apply this to prove separation property in L2 when operator acts functions.