Upper bounds on the number of resonances for non-compact Riemann surfaces
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.
scientific article; zbMATH DE number 764031