Generation of semigroup for symmetric matrix Schrödinger operators in $L^p$-spaces

Abstract: 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.