Preservation of absolutely continuous spectrum of periodic Jacobi operators under perturbations of square--summable variation

TL;DR

This paper proves that the absolutely continuous spectrum of certain periodic Jacobi operators remains stable under square-summable perturbations, extending previous results to a broader class of operators.

Contribution

It demonstrates the preservation of the essential support of the absolutely continuous spectrum for Jacobi operators with square-summable perturbations converging to periodic sequences.

Findings

The absolutely continuous spectrum remains unchanged under specified perturbations.

The essential support of the spectrum matches that of the asymptotic periodic operator.

Extends previous results to more general classes of Jacobi operators.

Abstract

We study self-adjoint bounded Jacobi operators of the form: (J \psi)(n) = a_n \psi(n + 1) + b_n \psi(n) +a_{n-1} \psi(n - 1) on $\ell^2(\N)$. We assume that for some fixed q, the q-variation of $\{a_n\}$ and $\{b_n\}$ is square-summable and $\{a_n\}$ and $\{b_n\}$ converge to q-periodic sequences. Our main result is that under these assumptions the essential support of the absolutely continuous part of the spectrum of J is equal to that of the asymptotic periodic Jacobi operator. This work is an extension of a recent result of S.A.Denisov.