Inequalities and separation for covariant Schr\"odinger operators

We consider a differential expression $L^{\nabla}_{V}=\nabla^{\dagger}\nabla+V$, where $\nabla$ is a metric covariant derivative on a Hermitian bundle $E$ over a geodesically complete Riemannian manifold $(M,g)$ with metric $g$, and $V$ is a linear self-adjoint bundle map on $E$. In the language of Everitt and Giertz, the differential expression $L^{\nabla}_{V}$ is said to be separated in $L^p(E)$ if for all $u\in L^p(E)$ such that $L^{\nabla}_{V}u\in L^p(E)$, we have $Vu\in L^p(E)$. We give sufficient conditions for $L^{\nabla}_{V}$ to be separated in $L^2(E)$. We then study the problem of separation of $L^{\nabla}_{V}$ in the more general $L^p$-spaces, and give sufficient conditions for $L^{\nabla}_{V}$ to be separated in $L^p(E)$, when $1<p<\infty$.