# $\ell^2$ bounded variation and absolutely continuous spectrum of Jacobi matrices

## TL;DR

This paper disproves a conjecture about the absolutely continuous spectrum of Jacobi matrices with $ ext{l}^2$ bounded variation, showing the spectrum exists on a smaller set than previously conjectured and establishing the optimality of this result.

## Contribution

The authors disprove a conjecture regarding the spectrum of Jacobi matrices with $ ext{l}^2$ bounded variation, providing a more precise characterization of the spectrum's support.

## Key findings

- Disproved the Breuer-Last-Simon conjecture.
- Established the existence of absolutely continuous spectrum on a smaller set.
- Proved the optimality of the new spectral set.

## Abstract

We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal.