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.
Categories: math.AP