Keywords: Inverse Scattering, Toda Hierarchy.

Abstract: In this thesis 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. Then we apply this knowledge to solve the associated initial value problem of the Toda hierarchy via the inverse scattering transform.

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