Time-Dependent Quantum Perturbations Uniform in the Semiclassical Regime

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