  We present a time dependent quantum perturbation result, uniform in the
Planck constant, for perturbations of potentials whose gradients are Lipschitz
continuous by potentials whose gradients are only bounded a.e.. Though this low
regularity of the full potential is not enough to provide the existence of the
classical underlying dynamics, at variance with the quantum one, our result
shows that the classical limit of the perturbed quantum dynamics remains in a
tubular neighbourhood of the classical unperturbed one of size of order of the
square root of the size of the perturbation. We treat both Schr\"odinger and
von Neumann-Heisenberg equations.
