Vol. 5, No. 3, 2012

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

David Borthwick

Vol. 5 (2012), No. 3, 513–552

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.

resonances, hyperbolic surfaces, scattering theory
Mathematical Subject Classification 2000
Primary: 35P25, 58J50
Secondary: 47A40
Received: 31 July 2010
Accepted: 26 February 2011
Published: 15 October 2012
David Borthwick
Department of Mathematics and Computer Science
Emory University
400 Dowman Drive
Atlanta, GA 30322
United States