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We study scalar, electromagnetic and gravitational perturbations of aReissner-Nordstr\"om-anti-de Sitter (RN-AdS) spacetime, and compute itsquasinormal modes (QNM's). We confirm and extend results previously found forSchwarzschild-anti-de Sitter (S-AdS) black holes. For ``large'' black holes,whose horizon is much larger than the AdS radius, different classes ofperturbations are almost exactly {\it isospectral}; this isospectrality isbroken when the black hole's horizon radius is comparable to the AdS radius. Weprovide very accurate fitting formulas for the QNM's, which are valid for blackholes of any size and charge $Q<Q_{ext}/3$.  Electromagnetic and axial perturbations of large black holes arecharacterized by the existence of pure-imaginary (purely damped) modes. Thedamping of these modes tends to infinity as the black hole charge approachesthe extremal value; if the corresponding mode amplitude does not tend to zeroin the same limit, this implies that {\it extremally charged RN-AdS black holesare marginally unstable}. This result is relevant in view of the AdS/CFTconjecture, since, according to it, the AdS QNM's give the timescales forapproach to equilibrium in the corresponding conformal field theory.Comment: 13 papges, 5 figures, Phys. Rev. D in press, minor changes and new  reference

Quasinormal modes of Reissner–Nordström–anti-de Sitter black holes: Scalar, electromagnetic, and gravitational perturbations