We revisit the peculiar electromagnetic quasinormal mode spectrum of an
asymptotically anti-de Sitter Schwarzschild black hole. Recent numerical
calculations have shown that some quasinormal mode frequencies become purely
overdamped at some critical black hole sizes, where the spectrum also
bifurcates. In this paper, we shed light on unnoticed and unexplained
properties of this spectrum by exploiting some novel analytic results for the
large black hole limit. We demonstrate, both numerically and analytically, that
the quasinormal mode spectra of large black holes become approximately
isospectral, and refer to this new symmetry property as spectral similarity. We
take advantage of this spectral similarity to derive a precise analytic
expression for the locations of the bifurcations, in which a surprising
Feigenbaum-like constant appears. We derive an exact solution for its spectrum
and eigenfunctions, and find that large black holes cannot be made to vibrate
with electromagnetic perturbations, independently of the boundary conditions
imposed at spatial infinity. Finally, we characterize the insensitivity of the
spectrum to different boundary conditions by analyzing the expansion of the
quasinormal mode spectrum around the large black hole limit.

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