Asymptotics of Linear Waves and Resonances with Applications to Black Holes

We describe asymptotic behavior of linear waves on Kerr(-de Sitter) black holes and more general Lorentzian manifolds, providing a quantitative analysis of the ringdown phenomenon. In particular we prove that if the initial data is localized at frequencies , then the energy norm of the solution is bounded by is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an remainder. This splitting generalizes expansions into quasi-normal modes available in completely integrable settings. In the case of generalized Kerr(-de Sitter) black holes satisfying certain natural conditions, quasi-normal modes are localized in bands and satisfy a precise counting law.