Scattering theory for the perturbations of periodic Schrödinger operators

Abstract

In this article, we study the short- and long-range perturbations of periodic Schrödinger operators. The asymptotic completeness is proved in the short-range case by referring to known results on the stationary approach and more explicitly with the time-dependent approach. In the long-range case, one is able to construct modified wave operators. In both cases, the asymptotic observables can be defined as elements of a commutative $C^{*}$-algebra of which the spectrum equals or is contained in the Bloch variety. Especially, the expression of the mean velocity as the gradient of the Bloch eigenvalues is completely justified in this framework, even when the Bloch variety presents singularities.