2001/04/14 by Alexander Volberg, A. Volberg, Peter Yuditskii +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP
paper · pdf · doi:10.1007/s002200200623
38 pages, AMS-TeX
arxiv created 2001/04/14 · openalex publication_date 2002/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Solving inverse scattering problem for a discrete Sturm-Liouville operator with the fast decreasing potential one gets reflection coefficients s_± and invertible operators I+Hs_±, where Hs_± is the Hankel operator related to the symbol s_±. 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+Hs_± 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.