Communications in Mathematical Physics

, Volume 359, Issue 2, pp 699–731 | Cite as

Resonances for Obstacles in Hyperbolic Space

  • Peter Hintz
  • Maciej Zworski


We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound \({ {\rm Im} \lambda \leq - \frac12 }\), which is optimal in dimension 2. In odd dimensions we also show that \({ {\rm Im} \lambda \leq - \frac{\mu}{\rho} }\) for a universal constant \({\mu}\), where \({ \rho }\) is the radius of a ball containing the obstacle; this gives an improvement for small obstacles. In dimensions 3 and higher the proofs follow the classical vector field approach of Morawetz, while in dimension 2 we obtain our bound by working with spaces coming from general relativity. We also show that in odd dimensions resonances of small obstacles are close, in a suitable sense, to Euclidean resonances.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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