Scattering theory with finite-gap backgrounds: Transformation operators   and characteristic properties of scattering data
  We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment. 
Iryna Egorova, Johanna Michor, and Gerald Teschl
math.SP math-ph math.MP