Overdamped QNM for Schwarzschild black holes

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 Jézéquel 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.

We have modified and simplified Section 2.2