Analysis & PDE

  • Anal. PDE
  • Volume 3, Number 2 (2010), 207-225.

Lower bounds for resonances of infinite-area Riemann surfaces

Dmitry Jakobson and Frédéric Naud

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 3, Number 2 (2010), 207-225.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731057

Digital Object Identifier
doi:10.2140/apde.2010.3.207

Mathematical Reviews number (MathSciNet)
MR2657455

Zentralblatt MATH identifier
1243.11064

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Keywords
Laplacian resonances arithmetic fuchsian groups

Citation

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


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