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.