Resonance projectors and asymptotics for 𝑟-normally hyperbolic trapped sets

Dyatlov, Semyon

doi:10.1090/S0894-0347-2014-00822-5

Resonance projectors and asymptotics for $r$-normally hyperbolic trapped sets
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by Semyon Dyatlov

J. Amer. Math. Soc. 28 (2015), 311-381

DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5

Published electronically: December 16, 2014

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Abstract:

We prove a Weyl law for 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 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.

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