Resonance projectors and asymptotics for r-normally hyperbolic trapped sets

TL;DR

This paper establishes an asymptotic formula for scattering resonances near the real axis in systems with r-normally hyperbolic trapped sets, relevant to black hole physics, using advanced microlocal analysis techniques.

Contribution

It introduces a new asymptotic formula for resonances in systems with r-normally hyperbolic trapped sets and develops a Fourier integral operator to analyze resonant states.

Findings

Derived an asymptotic resonance count near the real axis.

Provided new insights into microlocal concentration of resonant states.

Demonstrated stability of assumptions under smooth perturbations.

Abstract

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