  In the context of a geodesically complete Riemannian manifold $M$, we study
the self-adjointness of $\nabla^{\dagger}\nabla+V$ where $\nabla$ is a metric
covariant derivative (with formal adjoint $\nabla^{\dagger}$) on a Hermitian
vector bundle $\mathcal{V}$ over $M$, and $V$ is a locally square integrable
section of $\textrm{End }\mathcal{V}$ such that the (fiberwise) norm of the
"negative" part $V^{-}$ 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 $\varepsilon \in [0,1]$ and a
positive function $q$ on $M$ satisfying certain growth conditions, such that
$\varepsilon \nabla^{\dagger}\nabla+V\geq -q$, the inequality being understood
in the quadratic form sense over $C_{c}^{\infty}(\mathcal{V})$. In the first
result, which pertains to the case $\epsilon \in [0,1)$, we use the elliptic
equation method. In the second result, which pertains to the case
$\varepsilon=1$, we use the hyperbolic equation method.
