Gaussian bounds for propagators perturbed by potentials

We develop the perturbation theory for propagators, with the objective to prove Gaussian bounds. Let U be a strongly continuous propagator, i.e., a family of operators describing the solutions of a non-autonomous evolution equation, on an Lp-space, and assume that U is positive and satisfies Gaussian upper and lower bounds. Let V be a (time-dependent) potential satisfying certain Miyadera conditions with respect to U. We show that then the perturbed propagator enjoys Gaussian upper and lower bounds as well. To prepare the necessary tools, we extend the perturbation theory of strongly continuous propagators and the theory of absorption propagators. © 2006 Elsevier Inc. All rights reserved.