Lower bounds for resonances of infinite-area Riemann surfaces

Jakobson, Dmitry; Naud, Frédéric

doi:10.2140/apde.2010.3.207

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Lower bounds for resonances of infinite-area Riemann surfaces

Dmitry Jakobson and Frédéric Naud

Vol. 3 (2010), No. 2, 207–225

DOI: 10.2140/apde.2010.3.207

Abstract

For infinite-area, geometrically finite surfaces $X = Γ ∖ ℍ^{2}$ , we prove new omega lower bounds on the local density of resonances $D (z)$ when $z$ lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension $δ$ of the limit set of $Γ$ . The first bound is valid when $δ > \frac{1}{2}$ and shows logarithmic growth of the number $D (z)$ of resonances at high energy, that is, when $| Re (z) | \to + \infty$ . The second bound holds for $δ > \frac{3}{4}$ and if $Γ$ is an infinite-index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillopé and Zworski on the existence of a fractal Weyl law for resonances.

Keywords

Laplacian, resonances, arithmetic fuchsian groups

Mathematical Subject Classification 2000

Primary: 11F72, 58J50

Milestones

Received: 24 September 2009

Accepted: 10 February 2010

Published: 8 June 2010

Authors

Dmitry Jakobson
Department of Mathematics and Statistics McGill Uuniversity Montreal, QC H3A 2K6 Canada

Frédéric Naud
Laboratoire d’Analyse Non-linéaire et Géométrie (EA 2151) Université d’Avignon et des pays de Vaucluse 84018 Avignon France

Vol. 3, No. 2, 2010