Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

Author:

Levitin Michael1,Strohmaier Alexander2

Affiliation:

1. Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, UK

2. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

Abstract

Abstract In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM. All the videos accompanying this paper are available with its online version, or externally either at http://michaellevitin.net/hyperbolic.html or as a dedicated YouTube playlist.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference37 articles.

1. Deformation of ${\varGamma }_0(5)$-cusp forms;Avelin;Math. Comp,2007

2. Computations of Eisenstein series on Fuchsian groups;Avelin;Math. Comp.,2008

3. Numerical computations of Green’s function and its Fourier coefficients on $\textrm {PSL}\left (2,\mathbbm {Z}\right )$;Avelin;Exp. Math.,2010

4. Effective computation of Maass cusp forms;Booker;Int. Math. Res. Not.,2006

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