Robustness of perturbation series at high energies

Completely integrable Hamiltonians submitted to a smooth perturbation are c onsidered in the high-energy limit. It is shown that certain perturbation s eries, when suitably truncated, keep their validity even if the perturbatio n amplitude is much larger then the distance between levels. This is relate d to the localization of the corresponding eigenfunctions in the space of q uantum numbers; such eigenstates are robust against level crossing. They ar e the natural analogues of classical KAM tori. Moreover, the method can als o be used in classically resonant regions.