Inequalities and separation for covariant Schr\"odinger operators

We consider a differential expression $L_V^{\mathrm{\nabla }}={\mathrm{\nabla }}^{†}\mathrm{\nabla }+V$, where $\mathrm{\nabla }$ 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 $L_V^{\mathrm{\nabla }}$ is said to be separated in $L^p(E)$ if for all $u\in L^p(E)$ such that $L_V^{\mathrm{\nabla }}u\in L^p(E)$, we have $Vu\in L^p(E)$. We give sufficient conditions for $L_V^{\mathrm{\nabla }}$ to be separated in $L^2(E)$. We then study the problem of separation of $L_V^{\mathrm{\nabla }}$ in the more general $L^p$-spaces, and give sufficient conditions for $L_V^{\mathrm{\nabla }}$ to be separated in $L^p(E)$, when $1<p<\mathrm{\infty }$.