Inverse scattering for impedance Schrödinger operators, I. Step-like impedance lattice

We study the scattering problem for Schrödinger operators in impedance form, with step-like impedance functions. In this paper, the case where the jumps are located at the vertices of periodic lattice is considered. The set of reflection coefficients is completely described and the scattering map is shown to be homeomorphic. Also, an algorithm reconstructing the impedance function from the reflection coefficient is developed.