Inverse spectral theory for Jacobi matrices and their almost periodicity (original) (raw)
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We solve the following physically motivated problem: to determine all finite Jacobi matrices J and corresponding indices i, j such that the Green's function e j , (zI − J) −1 e i is proportional to an arbitrary prescribed function f (z). Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map (J, i, j) −→ [ e j , (zI − J) −1 e i ] (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
Inverse spectral analysis for finite matrix-valued Jacobi operators
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Marchenko-Ostrovski mappings for periodic Jacobi matrices
Russian Journal of Mathematical Physics, 2007
We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain. Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.
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?
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.