The dynamics of one-dimensional Bloch electrons in constant electric fields

We study the dynamics of a one-dimensional Bloch electron subjected to a constant electric field. The periodic potential is supposed to be less singular than the δ-like potential (Dirac comb). We give a rigorous proof of Ao's result that for a large class of initial conditions (high momentum regime) there is no localization in momentum space. The proof is based 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 quasicrossing can be described by Landau-Zener-type formulas. Using the connection between the above model and the driven quantum ring (DQR) shown by Avron and Nemirovski, our results imply the increase of energy for weakly singular such DQR and appropiate initial conditions