Upper bounds on the number of resonances for non-compact Riemann surfaces
type: Item
instance of: scholarly article
title: Upper bounds on the number of resonances for non-compact Riemann surfaces
zbMATH Open document ID: 0841.58063
author: Laurent Guillopé
author: Maciej Zworski
published in: Journal of Functional Analysis
publication date: 1995-07-31T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated Fuchsian group. The authors consider the kernel of the resolvent of the Laplace-Beltrami operator for \(\Gamma\). First of all they give a fairly simple proof that this kernel can be analytically continued to the complex plane (in the usual \(s\)-variable), even when there are parabolic elements in the group. The method uses the fact that this is known for an elementary group, and a perturbation argument is used to glue together the ends and the compact core of the group. This argument then also supplies the basis of the estimate for the number of singularities (counted with multiplicities) of the resolvent kernel taken in a large circle \(|s |< R\); this estimate is \(O (R^2)\). This estimate is sharp but it is in general unknown how the singularities are distributed in regions other than circles.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 764031
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: number of singularities
zbMATH Keywords: resolvent kernel
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: jfan.1995.1055
OpenAlex ID: W2056996244
DOI: 10.1006/JFAN.1995.1055
description: scientific article; zbMATH DE number 764031