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


Author: Semyon Dyatlov
Journal: J. Amer. Math. Soc. 28 (2015), 311-381
MSC (2010): Primary 35B34; Secondary 35S30, 37D05, 83C57
DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5
Published electronically: December 16, 2014
MathSciNet review: 3300697
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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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Additional Information

Semyon Dyatlov
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: dyatlov@math.mit.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5
Received by editor(s): February 21, 2013
Published electronically: December 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society