Lp-theory for Schrödinger systems

In this article, we study for p ∈ (1, ∞) the Lp-realization of the vector-valued Schrödinger operator L u := div(Q∇u) + V u. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Pr ̈uss, we prove that the Lp-realization of L, defined on the intersection of the natural domains of the differential and multiplication operators which form L, generates a strongly continuous contraction semigroup on Lp(R^d; C^m). We also study additional properties of the semigroup such as extension to L1, positivity, ultracontractivity and prove that the generator has compact resolvent.