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

## TL;DR

This paper uses discrete Levinson techniques to analyze how Wigner-von Neumann type perturbations create spectral singularities in the absolutely continuous spectrum of periodic Jacobi matrices, revealing stability and quantization conditions.

## Contribution

It introduces a novel application of discrete Levinson methods to construct spectral singularities for perturbed periodic Jacobi matrices, extending spectral analysis techniques.

## Key findings

- Spectral singularities are constructed on the absolutely continuous spectrum.
- Spectral singularities are stable under $l^1$-perturbations.
- Quantization conditions relate potential oscillation frequency to quasi-momentum.

## 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 solution. 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.