Stability of driven systems with growing gaps, Quantum rings and Wannier ladders

We consider a quantum particle in a periodic structure submitted to a constant external electromotive force. The periodic background is given by a smooth potential plus singular point interactions and has the property that the gaps between its bands are growing with the band index. We prove that the spectrum is pure point--i.e. trajectories of wave packets lie in compact sets in Hilbert space-- if the Bloch frequency is non-resonant with the frequency of the system and satisfies a Diophantine type estimate, or if it is resonant. Furthermore it is shown that the KAM method employed in the non-resonant case produces uniform bounds on the growth of energy for driven systems.