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Scattering Phase Asymptotics with Fractal Remainders

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Abstract

For a Riemannian manifold (M, g) which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate. For Axiom A geodesic flows, this gives a polynomial improvement over the known remainders. We also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis, and provide similar asymptotics for hyperbolic quotients using the Selberg zeta function.

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Authors and Affiliations

  1. Department of Mathematics, Evans Hall, University of California, Berkeley, CA, 94720, USA

    Semyon Dyatlov

  2. DMA, U.M.R. 8553 CNRS, École Normale Superieure, 45 Rue D’Ulm, 75230, Paris Cedex 05, France

    Colin Guillarmou

Authors
  1. Semyon Dyatlov
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  2. Colin Guillarmou
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Correspondence to Semyon Dyatlov.

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Communicated by S. Zelditch

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Dyatlov, S., Guillarmou, C. Scattering Phase Asymptotics with Fractal Remainders. Commun. Math. Phys. 324, 425–444 (2013). https://doi.org/10.1007/s00220-013-1809-8

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  • DOI: https://doi.org/10.1007/s00220-013-1809-8

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