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Black hole dynamical instabilities have been mostly studied in specificmodels. We here study the general properties of the complex-frequency modesresponsible for such instabilities, guided by the example of a charged scalarfield in an electrostatic potential. We show that these modes are squareintegrable, have a vanishing conserved norm, and appear in mode doublets orquartets. We also study how they appear in the spectrum and how their complexfrequencies subsequently evolve when varying some external parameter. Whenworking on an infinite domain, they appear from the reservoir of quasi-normalmodes obeying outgoing boundary conditions. This is illustrated bygeneralizing, in a non-positive definite Krein space, a solvable model(Friedrichs model) which originally describes the appearance of a resonancewhen coupling an isolated system to a mode continuum. In a finite spatialdomain instead, they arise from the fusion of two real frequency modes withopposite norms, through a process that closely resembles avoided crossing.

Dynamical instabilities and quasi-normal modes, a spectral analysis with applications to black-hole physics