Overdamped QNM for Schwarzschild black holes

TL;DR

This paper establishes a lower bound on the number of quasinormal modes for Schwarzschild black holes, demonstrating the sharpness of recent bounds and providing a refined spectral analysis method for their evaluation.

Contribution

It introduces a spectral asymptotics approach to bound and describe QNM for black holes, explaining their distribution and enabling accurate evaluation in complex regimes.

Findings

Lower bound of c r^3 for QNM count

Sharpness of Jézéquel's upper bound confirmed

Provides a method for accurate QNM evaluation deep in the complex plane

Abstract

We prove that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild-de Sitter black holes in a disc of radius $ r $ is bounded from below by $ c r^3 $. This shows that the recent upper bound by J\'ez\'equel is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM and explains the emergence of a distorted lattice on which they lie. Our presentation gives a general result about exponentially accurate Bohr-Sommerfeld quantization rules for one dimensional problems. The description of QNM allows their accurate evaluation ``deep in the complex" where numerical methods break down due to pseudospectral effects.