Vol. 3, No. 2, 2010

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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

For infinite-area, geometrically finite surfaces X = Γ2, 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 > 1 2 and shows logarithmic growth of the number D(z) of resonances at high energy, that is, when |Re(z)| +. The second bound holds for δ > 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.

Laplacian, resonances, arithmetic fuchsian groups
Mathematical Subject Classification 2000
Primary: 11F72, 58J50
Received: 24 September 2009
Accepted: 10 February 2010
Published: 8 June 2010
Dmitry Jakobson
Department of Mathematics and Statistics
McGill Uuniversity
Montreal, QC H3A 2K6
Frédéric Naud
Laboratoire d’Analyse Non-linéaire et Géométrie (EA 2151)
Université d’Avignon et des pays de Vaucluse
84018 Avignon