\({L^p}\)-theory for Schr\"odinger systems

Abstract

In this article we study for \(p\in (1,\infty)\) the \(L^p\)-realization of the vector-valued Schr\"odinger operator \(\mathcal{L}u := \mathrm{div} (Q\nabla u) + V u\). Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Pr\"uss, we prove that the \(L^p\)-realization of \(\mathcal{L}\), defined on the intersection of the natural domains of the differential and multiplication operators which form \(\mathcal{L}\), generates a strongly continuous contraction semigroup on \(L^p(\mathbb{R}^d; \mathbb{R}^m)\). We also study additional properties of the semigroup such as extension to \(L^1\), positivity, ultracontractivity and prove that the generator has compact resolvent.