A New Method for Quasinormal Modes From Bound States and Homotopy deformations

Abstract

Inspired by Mashhoon's bound state method, we propose a new bound state method for computing quasinormal modes (QNMs). By a two-step coordinate transformation where a real parameter $\alpha$ is introduced, a QNM problem is mapped to a bound state problem, whose eigenvalues $E_n(\alpha)$ are inversely mapped to the QNM frequencies $\omega_n$ via analytic continuation. With this method, we numerically calculate various QNM frequencies for a Schwarzschild black hole directly from the bound state spectrum of the inverted Regge-Wheeler potential for the first time. It is found that the method yields QNM frequencies of high accuracy for low-lying modes with overtone $n\leq\ell$ ($\ell$ is the multipole number), while the accuracy degrades or the calculation fails for higher overtones. To identify the origin of this limitation, we analyze the singularity structure of the eigenvalues $E_n(\alpha)$ using Pad\'e approximants in the complex $\alpha$-plane. For higher overtones, the singularities of $E_n(\alpha)$ lie within the analytic continuation circle, providing a direct explanation for the limitation of the method. To mitigate this limitation, we suggest a homotopy deformation to the potential, which improves the method and enable us to compute a few more high overtone modes reliably.