# Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow

## TL;DR

This paper develops Floquet theory and trace formulas for the periodic conservative Camassa-Holm flow's spectral problem, demonstrating how spectra depend continuously on the coefficients in a weak* topology.

## Contribution

It introduces a Floquet theory framework and trace formulas for the isospectral problem of the Camassa-Holm flow, analyzing spectral dependence on coefficients.

## Key findings

- Established Floquet theory for the spectral problem.
- Derived trace formulas for the spectra.
- Proved continuous dependence of spectra on coefficients.

## Abstract

This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = z\omega f + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where $\omega$ is a periodic real distribution in $H^{-1}_{\mathrm{loc}}(\mathbb{R})$ and $\upsilon$ is a periodic non-negative Borel measure on $\mathbb{R}$. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak$^\ast$ topology.