# A Sturm Liouville theorem for quadratic operator pencils

## TL;DR

This paper extends Sturm-Liouville theory to quadratic operator pencils, providing tools to count unstable roots and analyze wave stability, with applications in reducing eigenvalue problems to scalar forms.

## Contribution

It introduces a Sturm-Liouville theorem for quadratic operator pencils, enabling the counting of unstable roots and aiding stability analysis of waves.

## Key findings

- Established a Sturm-Liouville theorem for quadratic operator pencils.
- Applied the theorem to stability analysis of wave solutions.
- Provided a method to reduce eigenvalue systems to scalar problems.

## Abstract

We establish a Sturm{Liouville theorem for quadratic operator pencils counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.