$L^p$-theory for Schrödinger systems

Abstract

In this article we study for $p\in (1,\infty )$ the $L^p$-realization of the vector-valued Schrödinger operator $\mathcal{L}u:=\mathrm{div}(Q\nabla u)+Vu$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, 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.

15 pages, no figures