Overdamped QNM for Schwarzschild Black Holes

Abstract

We show 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 cr³, proving that the recent upper bound by Jézéquel [Anal. PDE 17, 2024,] is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM, explaining the emergence of a distorted lattice and generalizing the lattice structure in strips described by Sá Barreto-Zworski [Math. Res. Lett. 4, 1997] (see Fig. 1). As a by-product we obtain an exponentially accurate Bohr–Sommerfeld quantization rule for one dimensional problems. The resulting description of QNM allows their accurate evaluation “deep in the complex” where numerical methods break down due to pseudospectral effects (see Fig. 2).