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 nonresonant with the frequency of the system and satisfies a Diophantine-type estimate, or if it is resonant. Furthermore, we show that the KAM method employed in the nonresonant case produces uniform bounds on the growth of energy for driven systems.