Scattering Theory for Jacobi Operators with Quasi-Periodic Background
  We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely. 
Iryna Egorova, Johanna Michor, and Gerald Teschl