Vector-valued Schrödinger operators in <i>L<sup>p</sup></i>-spaces

Abstract

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