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Solving inverse scattering problem for a discrete Sturm-Liouville operatorwith the fast decreasing potential one gets reflection coefficients $s_\pm$ andinvertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operatorrelated to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuouscase) and the Guseinov theorem (in the discrete case), guarantees theuniqueness of solution of the inverse scattering problem. In this article weasks the following natural question --- can one find a precise conditionguaranteeing that the inverse scattering problem is uniquely solvable and thatoperators $I+H_{s_\pm}$ are invertible? Can one claim that uniqueness impliesinvertibility or vise versa?  Moreover we are interested here not only in the case of decreasing potentialbut also in the case of asymptotically almost periodic potentials. So we meregehere two mostly developed cases of inverse problem for Sturm-Liouvilleoperators: the inverse problem with (almost) periodic potential and the inverseproblem with the fast decreasing potential.Comment: 38 pages, AMS-Te

communications in mathematical physics/communications in mathematical physics

On the Inverse Scattering Problem for Jacobi Matrices¶with the Spectrum on an Interval, a Finite System¶of Intervals or a Cantor Set of Positive Length