Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David Borthwick, Colin Guillarmou

Type: Article

Publication Date: 2016-03-26

Citations: 3

DOI: https://doi.org/10.4171/jems/607

Abstract

On geometrically finite hyperbolic manifolds \Gamma\backslash\mathbb H^{d} , including those with non-maximal rank cusps, we give upper bounds on the number N(R) of resonances of the Laplacian in disks of size R as R \to \infty . In particular, if the parabolic subgroups of \Gamma satisfy a certain Diophantine condition, the bound is N(R)=\mathcal O(R^d (\mathrm {log} R)^{d+1}) .

Locations

arXiv (Cornell University) - View - PDF
Journal of the European Mathematical Society - View - PDF

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Works Cited by This (30)

Action	Title	Year	Authors
+	The analysis of linear partial differential operators	1990	Lars Hörmander
+	Scattering theory for geometrically finite groups	1999	Ulrich Bunke Martin Olbrich
+	Geometric bounds on the density of resonances for semiclassical problems	1990	Johannes Sjöstrand
+	Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature	1987	Rafe Mazzeo Richard Melrose
+	Scattering Asymptotics for Riemann Surfaces	1997	Laurent Guillopé Maciej Zworski
+	The Wave Trace for Riemann Surfaces	1999	L. Guillop� Maciej Zworski
+	Sharp Bounds on the Number of Resonances for Conformally Compact Manifolds with Constant Negative Curvature Near Infinity	2003	Claudio Cuevas Georgi Vodev
+	Sharp polynomial bounds on the number of scattering poles	1989	Maciej Zworski
+	Geometrical Finiteness for Hyperbolic Groups	1993	Brian H. Bowditch
+ PDF Chat	Group Cohomology and the Singularities of the Selberg Zeta Function Associated to a Kleinian Group	1999	Ulrich Bunke Martin Olbrich