Scattering theory for Jacobi operators with general steplike quasi-periodic background (original) (raw)
Scattering theory for Jacobi operators with a steplike quasi-periodic background
Inverse Problems, 2007
We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.
Scattering Theory for Jacobi Operators with Quasi-Periodic Background
Communications in Mathematical Physics, 2006
We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.
Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem
2011
We consider a periodic Jacobi operator H with finitely supported perturbations on Z. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of R_-(λ)+1, where R_- is the reflection coefficient.
Periodic Jacobi operators with finitely supported perturbations. arXiv
2016
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
Resonances for periodic Jacobi operators with finitely supported perturbations
Journal of Mathematical Analysis and Applications, 2012
We describe the spectral properties of the Jacobi operator (Hy) n = a n−1 y n−1 +a n y n+1 + b n y n , n ∈ Z, with a n = a 0 n + u n , b n = b 0 n + v n , where sequences a 0 n > 0, b 0 n ∈ R are periodic with period q, and sequences u n , v n have compact support. In the case u n ≡ 0 we obtain the asymptotics of the spectrum in the limit of small perturbations v n .
Periodic Jacobi operator with finitely supported perturbation on the half-lattice
Inverse Problems, 2011
We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.
Communications in Mathematical Physics, 2002
Solving 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) [6] and the Guseinov theorem (in the discrete case) [4], guarantees the uniqueness of 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 inverse problem for Sturm-Liouville operators: the inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.
Journal d'Analyse Mathématique, 2008
We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.
Communications in Mathematical Physics, 2002
Solving 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) [6] and the Guseinov theorem (in the discrete case) [4], guarantees the uniqueness of 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?