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Black hole spectroscopy is a powerful tool to probe the Kerr nature ofastrophysical compact objects and their environment. The observation ofmultiple ringdown modes in gravitational waveforms could soon lead tohigh-precision gravitational spectroscopy, so it is critical to understand ifthe quasinormal mode spectrum is stable against perturbations. It was recentlyshown that the pseudospectrum can shed light on the spectral stability of blackhole quasinormal modes. We study the pseudospectrum of Reissner-Nordstr\"omspacetimes and we find a spectral instability of scalar and gravitoelectricquasinormal modes in subextremal and extremal black holes, extending similarfindings for the Schwarzschild spacetime. The asymptotic structure ofpseudospectral contour levels is the same for scalar and gravitoelectricperturbations. By making different gauge choices in the hyperboloidal slicingof the spacetime, we find that the broad features of the pseudospectra areremarkably gauge-independent. The gravitational-led and electromagnetic-ledquasinormal modes of extremal Reissner-Nordstr\"om black holes exhibit "strong"isospectrality: not only their spectrum coincides, but the whole pseudospectrumis the same for both classes of perturbations. We observe that a conformalduality between the extremal horizon and spacetime boundaries at infinity isresponsible for such "strong" isospectrality property.

Pseudospectrum of Reissner-Nordstr\"om black holes: quasinormal mode instability and universality