  Solving inverse scattering problem for a discrete Sturm-Liouville operator
with the fast decreasing potential one gets reflection coefficients $s_\pm$ and
invertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operator
related to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuous
case) and the Guseinov theorem (in the discrete case), guarantees the
uniqueness of solution of the inverse scattering problem. In this article we
asks the following natural question --- can one find a precise condition
guaranteeing that the inverse scattering problem is uniquely solvable and that
operators $I+H_{s_\pm}$ 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 merege
here two mostly developed cases of inverse problem for Sturm-Liouville
operators: the inverse problem with (almost) periodic potential and the inverse
problem with the fast decreasing potential.
