# Analytical proof of the isospectrality of quasinormal modes for Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter spacetimes

## TL;DR

This paper provides an analytical proof explaining why axial and polar perturbations of Schwarzschild black holes share the same quasinormal mode spectrum, extending previous numerical results to include spacetimes with a cosmological constant.

## Contribution

It offers a clearer, extended analytical proof of isospectrality for Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter spacetimes, beyond prior numerical studies.

## Key findings

- Analytical proof of isospectrality for these spacetimes
- Extension of results to include cosmological constants
- Clarification of the underlying reasons for spectral equivalence

## Abstract

The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective.