  In this article we study for $p\in (1,\infty)$ the $L^p$-realization of the
vector-valued Schr\"odinger operator $\mathcal{L}u := \mathrm{div} (Q\nabla u)
+ V u$. Using a noncommutative version of the Dore-Venni theorem due to
Monniaux and Pr\"uss, we prove that the $L^p$-realization of $\mathcal{L}$,
defined on the intersection of the natural domains of the differential and
multiplication operators which form $\mathcal{L}$, generates a strongly
continuous contraction semigroup on $L^p(\mathbb{R}^d; \mathbb{R}^m)$. We also
study additional properties of the semigroup such as extension to $L^1$,
positivity, ultracontractivity and prove that the generator has compact
resolvent.
