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We develop direct and inverse scattering theory for Jacobi operators (doublyinfinite second order difference operators) with steplike coefficients whichare asymptotically close to different finite-gap quasi-periodic coefficients ondifferent sides. We give necessary and sufficient conditions for the scatteringdata in the case of perturbations with finite second (or higher) moment.

Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data