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

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.Comment: 38 pages, AMS-Te

Asymptotics of orthogonal polynomials via the Koosis theorem

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szeg\"o measure on the unit circumference is perturbed by an arbitrary measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk