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

Abstract

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.

38 pages, AMS-TeX

Cited by in corpus (17)

• Scattering Theory for Jacobi Operators with Quasi-Periodic Background
• Inverse Scattering Transform for the Toda Hierarchy with Quasi-Periodic Background
• Algebro-Geometric Constraints on Solitons with Respect to Quasi-Periodic Backgrounds
• Entropic fluctuations in XY chains and reflectionless Jacobi matrices
• Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds
• Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
• Scattering Theory for Jacobi Operators with Steplike Quasi-Periodic Background
• The inverse scattering transform for weak Wigner-von Neumann type potentials
• Remarks on Nehari's problem, matrix $A_2$ condition, and weighted bounded mean oscillation
• On Scattering for CMV Matrices
• Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data
• Failure of necessity of the energy condition
• On a new asymptotic problem in the scattering setting
• Entropic fluctuations of XY quantum spin chains
• CMV matrices with asymptotically constant coefficients. Szegö over Blaschke class, Scattering Theory
• Inverse scattering problem for a special class of canonical systems and non-linear Fourier integral. Part I: asymptotics of eigenfunctions
• Necessary/sufficient conditions in weighted theory