Resonances for Normally Hyperbolic Trapped Sets

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Semyon Dyatlov, UC Berkely

(PLEASE NOTE SPECIAL TIME.)  Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition.  Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions and revisit exponential decay of highly oscillating waves on Kerr-de Sitter and Kerr black holes.