Asch, J., Duclos, P. and Exner, P.
Stability of driven systems with growing gaps,
Quantum rings and Wannier ladders
(42K, latex2e)
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.