Quasi-Normal Modes from Bound States: The Numerical Approach

TL;DR

This paper introduces a numerical method to compute black hole quasi-normal modes using bound state spectra of related potentials, overcoming limitations of analytical approaches and improving accuracy especially for overtones.

Contribution

The authors propose a new numerical approach that leverages bound state spectra of similar potentials to accurately determine quasi-normal modes, expanding computational options beyond analytically solvable cases.

Findings

The new method yields more accurate quasi-normal mode estimates than traditional potentials.

It is numerically stable and applicable to potentials without known analytical solutions.

The approach improves approximation especially for the first overtone.

Abstract

It is known that the spectrum of quasi-normal modes of potential barriers is related to the spectrum of bound states of the corresponding potential wells. This property has been widely used to compute black hole quasi-normal modes, but it is limited to a few "approximate" potentials with certain transformation properties for which the spectrum of bound states must be known analytically. In this work we circumvent this limitation by proposing an approach that allows one to make use of potentials with similar transformation properties, but where the spectrum of bound states can also be computed numerically. Because the numerical calculation of bound states is usually more stable than the direct computation of the corresponding quasi-normal modes, the new approach is also interesting from a technical point of view. We apply the method to different potentials, including the P\"oschl-Teller potential for which all steps can be understood analytically, as well as potentials for which we are not aware of analytic results but provide independent numerical results for comparison. As a canonical test, all potentials are chosen to match the Regge-Wheeler potential of axial perturbations of the Schwarzschild black hole. We find that the new approximate potentials are more suitable to approximate the exact quasi-normal modes than the P\"oschl-Teller potential, particularly for the first overtone. We hope this work opens new perspectives to the computation of quasi-normal modes and finds further improvements and generalizations in the future.