  Black hole dynamical instabilities have been mostly studied in specific
models. We here study the general properties of the complex-frequency modes
responsible for such instabilities, guided by the example of a charged scalar
field in an electrostatic potential. We show that these modes are square
integrable, have a vanishing conserved norm, and appear in mode doublets or
quartets. We also study how they appear in the spectrum and how their complex
frequencies subsequently evolve when varying some external parameter. When
working on an infinite domain, they appear from the reservoir of quasi-normal
modes obeying outgoing boundary conditions. This is illustrated by
generalizing, in a non-positive definite Krein space, a solvable model
(Friedrichs model) which originally describes the appearance of a resonance
when coupling an isolated system to a mode continuum. In a finite spatial
domain instead, they arise from the fusion of two real frequency modes with
opposite norms, through a process that closely resembles avoided crossing.
