# Trace estimation of a family of periodic Sturm-Liouville operators with application to Robe's restricted three-body problem

## TL;DR

This paper develops trace estimation methods for a family of periodic Sturm-Liouville operators and applies these results to analyze the linear stability of equilibrium points in Robe's restricted three-body problem.

## Contribution

It introduces trace formula techniques for periodic Sturm-Liouville operators and applies them to stability analysis in celestial mechanics.

## Key findings

- Elliptic regions are estimated using trace formulas.
- Bifurcation analysis with respect to parameters is performed.
- Linear stability of equilibrium points along the z-axis is established.

## Abstract

In this paper, we consider a family of Sturm-Liouville operators on the $\omega$-periodic domain. The bifurcation with respect to the parameter region is studied, and the elliptic regions are estimated by trace formula. At last, these results are used to study the linear stability of the elliptic equilibrium point along $z$-axis in Robe's restricted three-body problem.