Bloch function in an external electric field and Berry-Buslaev phase

The study of the behavior of Bloch electrons in an uniform external electric field is as old as the quantum theory of solids. Analysis of the motion of electrons in such external fields turned out, perhaps rather surprisingly, to be quit complicated and even at present the subject is very much alive. The source of the difficulties of this problem is that no matter how small the electron field strength is, for sufficiently large distances, the perturbed potential becomes arbitraly strong. In fact the perturbation created by electric field is singular from the spectral theoretic point of view. In consequence, strightforward application of the naive perturbation method is dangerous and rigorous results are hard to come by. For a better understanding of the difficulties, let us remined here the case of the Stark effect in atomic physics : although the Stark effects were the first example of quantum mechanical perturbation theory, it needed half a centry to develop a satisfactory mathematical description. Actually the existence of the Stark-Wannier resonance states, a quantum mechanical concept proposed by Wannier in solid state physics about 40 years ago, was strongly debated until recently.