Abstract
We consider the dynamics of a 1D Bloch electron
subjected to a constant electric field
$E$, described in the temporal gauge
by the Hamiltonian
$${\tilde H}^{SW}(t)=(p-eEt)^2+V_{{\rm per }}$$
The periodic potential is supposed to be less singular
than the $\delta$-like potential (Dirac comb). The main result is a rigor
ous proof
of Ao's claim that for a large class of initial conditions
(high momentum regime) there is no dynamical localization. The proof is ba
sed
on the mathematical substantiation of the two simplifying assumptions
made in physical literature: the transitions between far away bands
can be neglected and the transitions at the quasi-crossing can be describe
d
by Landau-Zener like formulae. By Avron and Nemirovski connection
our results implies also the increase of energy for
weakly singular driven quantum rings.