Analysis & PDE
- Anal. PDE
- Volume 3, Number 2 (2010), 207-225.
Lower bounds for resonances of infinite-area Riemann surfaces
For infinite-area, geometrically finite surfaces , we prove new omega lower bounds on the local density of resonances when lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension of the limit set of . The first bound is valid when and shows logarithmic growth of the number of resonances at high energy, that is, when . The second bound holds for and if 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.
Anal. PDE, Volume 3, Number 2 (2010), 207-225.
Received: 24 September 2009
Accepted: 10 February 2010
First available in Project Euclid: 20 December 2017
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Jakobson, Dmitry; Naud, Frédéric. Lower bounds for resonances of infinite-area Riemann surfaces. Anal. PDE 3 (2010), no. 2, 207--225. doi:10.2140/apde.2010.3.207. https://projecteuclid.org/euclid.apde/1513731057