Dynamical system approach to the spectral instability of black holes under localized potential perturbations

The aim of this work is to improve understanding of the resonant spectra of black holes under perturbations arising from e.g. compact objects or accretion disks in their vicinity. It is known that adding a weak perturbation to the radial potential can strongly disrupt the spectrum of quasinormal modes and Regge poles of a black hole spacetime. Here we examine the effect of (weak or strong) localized delta-function perturbations on the resonant spectra of spherically symmetric systems, to address fundamental questions around linear and nonlinear spectral stability. We examine two cases: the Nariai spacetime with a Poschl-Teller potential and the Schwarzschild spacetime. We show that, in either case, the spectrum deforms in a smooth and continuous manner as the position and strength of the perturbation is varied. As the strength of the perturbation is increased, resonances migrate along trajectories in the complex plane which ultimately tend toward attracting points determined by a hard-wall scenario. However, for weak perturbations the trajectory near the unperturbed resonance is typically strongly influenced by a set of repelling points which, for perturbations far from the system, lie very close to the unperturbed resonances; hence there arises a nonlinear instability (i.e. the failure of a linearized approximation). Taking a dynamical systems perspective, the sets of attracting and repelling spectral points follow their own trajectories as the position of the perturbation is varied, and these are tracked and understood.