  We consider the problem of quasinormal modes (QNM) for strongly hyperbolic
systems on stationary, asymptotically anti-de Sitter black holes, with very
general boundary conditions at infinity. We argue that for a time slicing
regular at the horizon the QNM should be identified with certain H^k
eigenvalues of the infinitesimal generator of the solution semigroup. Using
this definition we are able to prove directly that the quasinormal frequencies
form a discrete, countable subset of the complex plane, which in the globally
stationary case accumulates only at infinity. We avoid any need for meromorphic
extension, and the quasinormal modes are honest eigenfunctions of an operator
on a Hilbert space. Our results apply to any of the linear fields usually
considered (Klein-Gordon, Maxwell, Dirac etc.) on a stationary black hole
background, and do not rely on any separability or analyticity properties of
the metric. Our methods and results largely extend to the locally stationary
case. We provide a counter-example to the conjecture that quasinormal modes are
complete. We relate our approach directly to the approach via meromorphic
continuation.
