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

When solving the inverse scattering problem for a discrete Sturm-Liouville operator with a rapidly decreasing potential, one gets reflection coefficients s_± and invertible operators I + ℋ_s±, where ℋ_s± is the Hankel operator related to the symbol s_±. The Marchenko-Faddeev theorem (in the continuous case, for the discrete case see [4, 6]), guarantees the uniqueness of the solution of the inverse scattering problem. In this article we ask the following natural question - can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators I + ℋ_s± are invertible? Can one claim that uniqueness implies invertibility or vise versa? Moreover, we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials. So we merge here two mostly developed cases of the inverse problem for Sturm-Liouville operators: the inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.