Vector-valued Schrödinger operators on $L^p$-spaces

Abstract

In this paper we consider vector-valued Schrödinger 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 $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^d)$. Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a $C_0$-semigroup of contractions in $L^p({\mathbb{R}}^d;{\mathbb{C}}^m)$. Further properties are also investigated.

11 pages, no figures