Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

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.

Computation of Scattering Matrices and their Derivatives for Waveguides

“…Figure 2: Inequivalent paths that lift to paths with endpoints in sheets (1,2,3,4), (1,3,4), and (1), of Z respectively.…”

“…We will also show that the scattering matrix, as well as its derivatives, can be obtained from this data directly. Levitin and Strohmaier have already used this technique to obtain the scattering matrix on finite volume, non-compact hyperbolic surfaces [1]. Due to the more complicated nature of the Riemann surface and the fact that the rank of the scattering matrix jumps each time the spectrum is crossed, the problem determining the scattering matrix for waveguides is more complex.…”

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This paper describes the calculation of the stationary scattering matrix and its derivatives for Euclidean waveguides. This is an adaptation and extension to a procedure developed by Levitin and Strohmaier which was used to compute the stationary scattering matrix [1]. On Euclidean waveguides, the scattering matrix can be meromorphically continued from the complex plane to a Riemann surface with a countably infinite number of sheets.

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