2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165172We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.23 pagesSpectral TheoryMathematical Physics47B36, 81U40; 34L25, 39A11text