# On the absolutely continuous spectrum of generalized indefinite strings II

## TL;DR

This paper extends the analysis of the absolutely continuous spectrum of generalized indefinite strings, demonstrating stability under perturbations and applying results to the two-component Camassa-Holm system in a specific dispersive regime.

## Contribution

It introduces new stability results for the absolutely continuous spectrum of generalized indefinite strings and applies these to the two-component Camassa-Holm system.

## Key findings

- Stability of the absolutely continuous spectrum under wide perturbations.
- Support of the spectrum for the two-component Camassa-Holm system in a dispersive regime.
- Extension of previous spectral analysis methods.

## Abstract

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$.