# Quasinormal modes of black holes and Borel summation

## TL;DR

This paper introduces a novel, efficient method for computing black hole quasinormal modes using Borel summation and perturbation theory, accurately reproducing known results and revealing the series' divergence properties.

## Contribution

It presents a new approach linking quasinormal modes to bound states via analytic continuation, enabling high-order WKB series computations and demonstrating their Borel summability.

## Key findings

- Borel sums match numerical quasinormal frequencies
- WKB series are divergent but Borel summable
- Method applies to Schwarzschild and Reissner-Nordström black holes

## Abstract

We propose a simple and efficient way to compute quasinormal frequencies of spherically symmetric black holes. We revisit an old idea that relates them to bound state energies of anharmonic oscillators by an analytic continuation. This connection enables us to achieve remarkable high-order computations of WKB series by Rayleigh-Schr\"odinger perturbation theory. The known WKB results are easily reproduced. Our analysis shows that the perturbative WKB series of the quasinormal frequencies turn out to be Borel summable divergent series both for the Schwarzschild and for the Reissner-Nordstr\"om black holes. Their Borel sums reproduce the correct numerical values.