Motivated by a physical model of a system of quantum oscillators with nonlinear interactions, we study spectral properties of certain matrices which arise as perturbations of some tridiagonal k-Toeplitz matrices. Concretely, we are interested in the spectral properties of general tridiagonal k-Toeplitz matrices (which are k-periodic Jacobi matrices) and certain perturbations of them. We will show in this talk how the theory of orthogonal polynomials (and in particular the polynomial mappings) can be used for solving the unperturbed case. For the perturbed case we will focus our attention on the localization of the eigenvalues of such matrices, as well as on the distance between two consecutive eigenvalues.
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