Schrödinger operators withL loc p -potentials

Abstract

We discuss the question of when the closure of the Schrodinger operator, −Δ+V, acting inLp(Rl,dlx), generates a strongly continuous contraction semigroup. We prove a series of theorems proving the stability for −Δ:Lp→Lp of the property of having am-accretive closure under perturbations by functions inLlocq (1<p≦q). The connection with form sums and the Trotter product formula are considered. These results generalize earlier results of Kato, Kalf-Walter, Semenov and Beliy-Semenov in that we allow more general local singularities, including arbitrary singularities at one point, and arbitrary growth at infinity. We exploit bilinear form methods, Kato's inequality and certain properties of infinitesimal generators of contractions.