Vector-valued Schr\"odinger operators on $L^p$-spaces

Abstract

In this paper we consider vector-valued Schr\"odinger operators of the form $\mathrm{div}(Q\nabla u)-Vu$, where $V=(v_{ij})$ is a nonnegative locally bounded matrix-valued function and $Q$ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $V$, we assume an that it is pointwise accretive and that its entries are in $L^\infty_{\mathrm{loc}}(\mathbb{R}^d)$. Under these assumptions, we prove that a realization of the vector-valued Schr\"odinger operator generates a $C_0$-semigroup of contractions in $L^p(\mathbb{R}^d; \mathbb{C}^m)$. Further properties are also investigated.