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.