  In this paper, we study vector--valued elliptic operators of the form
$\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$
acting on vector-valued functions $f:\mathbb{R}^d\to\mathbb{R}^m$ and involving
coupling at zero and first order terms. We prove that $\mathcal{L}$ admits
realizations in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $1<p<\infty$, that
generate analytic strongly continuous semigroups provided that
$V=(v_{ij})_{1\le i,j\le m}$ is a matrix potential with locally integrable
entries satisfying a sectoriality condition, the diffusion matrix $Q$ is
symmetric and uniformly elliptic and the drift coefficients $F=(F_{ij})_{1\le
i,j\le m}$ and $C=(C_{ij})_{1\le i,j\le m}$ are such that
$F_{ij},C_{ij}:\mathbb{R}^d\to\mathbb{R}^d$ are bounded.
  We also establish a result of local elliptic regularity for the operator
$\mathcal{L}$, we investigate on the $L^p$-maximal domain of $\mathcal{L}$ and
we characterize the positivity of the associated semigroup.
