Inverse scattering for impedance Schrödinger operators. I: Step-like impedance lattice (Q1674371)

The main aim of this paper is to study the inverse scattering problem for the Schrödinger equation on the line in impedance form \[ ({1\over p^{2}(x)} {d\over dx} p^{2}(x) {d\over dx}+\omega^{2})u(x,\omega)=0, \] where the impedance function \(p\) is uniformly positive, uniformly bounded on the whole line and has jumps located at the vertices of a periodic lattice. A direct method -- that can be considered as discrete analogous of the classical Marchenko approach for the full line potential scattering -- for reconstructing piece-wise constant impedance functions \(p\) is developed.