Resonance projectors and asymptotics for 𝑟-normally hyperbolic trapped sets

Abstract

We prove a Weyl law for scattering resonances in a strip near the real axis when the trapped set is r r -normally hyperbolic with r r large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and are motivated by wave dynamics for black holes. The key step is a construction of a Fourier integral operator which microlocally projects onto the resonant states. In addition to the Weyl law, this operator provides new information about microlocal properties of resonant states.