Spectral instability of black holes: relating the frequency domain to the time domain

Recent work has shown that the quasinormal mode spectrum of black holes is unstable under small perturbations (of order $\epsilon$) of the radial potential, while the early time-domain ringdown waveform is only marginally affected. In this paper we provide further insight into the apparent tension between the frequency-domain and the time-domain descriptions by analyzing the scattering properties of the problem. In the frequency domain, we study analytically the solutions corresponding to the perturbed potential. We show that there are two qualitatively different classes of instabilities, and that both Schwarzschild and Kerr black holes are affected by what we call a "Type II" instability, i.e., an exponential migration of the mode frequencies away from their unperturbed value as the perturbing "bump" moves away from the peak of the unperturbed potential. In the time domain, we elucidate the effect of the spectral instability in terms of the causal structure of the Green's function. By using an equivalent scattering problem we confirm analytically (and show numerically) that the deviation from the unperturbed waveform in the early ringdown stage is proportional to $\epsilon$ when $\epsilon\lesssim10^{-2}$.