Self-adjointness of non-semibounded covariant Schrodinger operators on Riemannian manifolds

In the context of a geodesically complete Riemannian manifold M, we study the self-adjointness of backward difference & DAG; backward difference +V$\nabla <^>{\dagger }\nabla +V$, where backward difference is a metric covariant derivative (with formal adjoint backward difference & DAG;$\nabla <^>{\dagger }$) on a Hermitian vector bundle V$\mathcal {V}$ over M, and V is a locally square integrable section of EndV$\operatorname{End}\mathcal {V}$ such that the (fiberwise) norm of the "negative" part V-$V<^>{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number & epsilon;& ISIN;[0,1]$\varepsilon \in [0,1]$ and a positive function q on M satisfying certain growth conditions, such that & epsilon; backward difference & DAG; backward difference +V & GE;-q$\varepsilon \nabla <^>{\dagger }\nabla +V\ge -q$, the inequality being understood in the quadratic form sense over Cc & INFIN;(V)$C_{c}<^>{\infty }(\mathcal {V})$. In the first result, which pertains to the case & epsilon;& ISIN;[0,1)$\varepsilon \in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case & epsilon;=1$\varepsilon =1$, we use the hyperbolic equation method.