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2010 Lower bounds for resonances of infinite-area Riemann surfaces
Dmitry Jakobson, Frédéric Naud
Anal. PDE 3(2): 207-225 (2010). DOI: 10.2140/apde.2010.3.207

Abstract

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>12 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>34 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.

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Dmitry Jakobson. Frédéric Naud. "Lower bounds for resonances of infinite-area Riemann surfaces." Anal. PDE 3 (2) 207 - 225, 2010. https://doi.org/10.2140/apde.2010.3.207

Information

Received: 24 September 2009; Accepted: 10 February 2010; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1243.11064
MathSciNet: MR2657455
Digital Object Identifier: 10.2140/apde.2010.3.207

Subjects:
Primary: 11F72, 58J50

Rights: Copyright © 2010 Mathematical Sciences Publishers

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