# Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

Judge, Edmund, Naboko, Serguei, Wood, Ian (2018) Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique. Studia Mathematica, . ISSN 0039-3223. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:65787)

## Abstract

For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solutions. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l^1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.

Item Type: Article Jacobi operators, Levinson techniques, periodic operators, Wigner-von Neumann potentials, subordinate solutions Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Ian Wood 25 Jan 2018 12:44 UTC 16 Feb 2021 13:52 UTC https://kar.kent.ac.uk/id/eprint/65787 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7181-7075