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We provide a rigorous definition of quasi-normal modes for a rotating blackhole. They are given by the poles of a certain meromorphic family of operatorsand agree with the heuristic definition in the physics literature. If the blackhole rotates slowly enough, we show that these poles form a discrete subset ofthe complex plane. As an application we prove that the local energy of linearwaves in that background decays exponentially once orthogonality to the zeroresonance is imposed.

Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole