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

TL;DR

This paper studies quantum particles in periodic structures with growing energy gaps under external forces, proving spectral properties and energy bounds using KAM methods for both resonant and non-resonant cases.

Contribution

It demonstrates spectral localization and energy bounds in driven quantum systems with growing gaps, extending understanding of stability in such structures.

Findings

Spectrum is pure point for non-resonant frequencies.

Wave packet trajectories remain in compact sets in Hilbert space.

Uniform energy bounds are established using KAM techniques.

Abstract

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.