Upper and Lower Bounds on Resonances for Manifolds Hyperbolic Near Infinity

• Type of Material: Article
• Authors: David Borthwick, Emory University
• Language: English
• Date: 2008-08
• Publisher: Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Publication Version: Preprint (Prior to Peer Review)
• Copyright Statement: © Taylor & Francis Group, LLC
• Final Published Version (URL): http://dx.doi.org/10.1080/03605300802031598
• Title of Journal or Parent Work: Communications in Partial Differential Equations
• ISSN: 0360-5302
• Volume: 33
• Issue: 8
• Start Page: 1507
• End Page: 1539
• Research Categories: Mathematics

Abstract

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(rn+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an rn+1 lower bound on the counting function for scattering poles.