# On the absolutely continuous spectrum of generalized indefinite strings

## TL;DR

This paper studies the absolutely continuous spectrum of generalized indefinite strings, demonstrating stability under perturbations and applying results to the Camassa-Holm flow to identify its spectral support.

## Contribution

It establishes the stability of the absolutely continuous spectrum for generalized indefinite strings and applies this to the spectral analysis of the Camassa-Holm flow.

## Key findings

- Stability of the absolutely continuous spectrum under perturbations.
- Spectrum of the Camassa-Holm flow is supported on [1/4, ∞).
- Extension of Deift and Killip's approach to generalized indefinite strings.

## Abstract

We 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 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 conservative Camassa-Holm flow in the dispersive regime is essentially supported on the interval $[1/4,\infty)$.