Analysis & PDE

  • Anal. PDE
  • Volume 5, Number 3 (2012), 513-552.

Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

David Borthwick

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We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary conditions.

Article information

Anal. PDE, Volume 5, Number 3 (2012), 513-552.

Received: 31 July 2010
Accepted: 26 February 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P25: Scattering theory [See also 47A40] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]

resonances hyperbolic surfaces scattering theory


Borthwick, David. Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends. Anal. PDE 5 (2012), no. 3, 513--552. doi:10.2140/apde.2012.5.513.

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