Time dependent quantum perturbations uniform in the semiclassical regime

We present a time dependent quantum perturbation result, uniform in the Planck constant for potential whose gradient is bounded a.e..We show also that the classical limit of the perturbed quantum dynamics remains in a tubular neighborhood of the classical unperturbed one, the size of this neighborhood being of the order of the square root of the size of the perturbation. We treat both Schr\"odinger and von Neumann-Heisenberg equations.