Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

Borthwick, David

doi:10.2140/apde.2012.5.513

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

David Borthwick

Vol. 5 (2012), No. 3, 513–552

DOI: 10.2140/apde.2012.5.513

Abstract

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.

Keywords

resonances, hyperbolic surfaces, scattering theory

Mathematical Subject Classification 2000

Primary: 35P25, 58J50

Secondary: 47A40

Milestones

Received: 31 July 2010

Accepted: 26 February 2011

Published: 15 October 2012

Authors

David Borthwick
Department of Mathematics and Computer Science Emory University 400 Dowman Drive Atlanta, GA 30322 United States

Vol. 5, No. 3, 2012