  In this paper we establish generation of analytic strongly continuous
semigroup in $L^p$--spaces for the symmetric matrix Schr\"odinger operator
$div(Q\nabla u)-Vu$, where, for every $x\in\mathbb{R}^d$, $V(x)=(v_{ij}(x))$ is
a semi-definite positive and symmetric matrix. The diffusion matrix $Q(\cdot)$
is supposed to be strongly elliptic and bounded and the potential $V$ satisfies
the weak condition $v_{ij}\in L^1_{loc}(\mathbb{R}^d)$, for all
$i,j\in\{1,\dots,m\}$. We also characterize positivity of the semigroup and we
investigate on its compactness.
