Pseudospectrum and Black Hole Quasinormal Mode Instability

We revisit and numerically demonstrate the stability of the black hole slowest decaying quasi-normal mode (QNM), also providing evidence of the instability of black hole highly damped QNMs under small scale perturbations, as identified by Nollert. A reconciliation between the unstable aspects of the QNM spectrum and the robustness of ringing down frequencies is given in terms of the underlying analytical structure of the problem. Methodologically, a compactified hyperboloidal approach to QNMs is adopted. This casts QNMs in terms of the spectral problem of a non-selfadjoint operator where specific tools, namely the pseudospectrum, can be applied. After illustrating the approach with the P\"oschl-Teller potential, we study the physically relevant Schwarzschild case and consider some of the astrophysical and fundamental physics implications of QNM (in)stabilities.