Physics – Mathematical Physics
Scientific paper
2004-10-30
J. Math. Phys. 46, 043505 (2005)
Physics
Mathematical Physics
51 pages, 1 figure
Scientific paper
10.1063/1.1870732
We study the dynamics of a 1D Bloch electron subjected to a constant electric field. The periodic potential is supposed to be less singular than the $\delta $-like potential (Dirac comb). We give a rigorous proof of Ao's result \cite{Ao} 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 quasi-crossing can be described by Landau-Zener like formulae. Using the connection between the above model and the driven quantum ring (DQR) shown by Avron and Nemirovski \cite{AvN}, our results imply the increase of energy for weakly singular such DQR and appropiate initial conditions.
Bentosela Francois
Duclos Pierre
Moldoveanu Valeriu
Nenciu Gh.
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