A Numerical Study of Eigenvalues of the Hyperbolic Laplacian for Polyhedra with One Cusp

Wolfgang Huntebrinker

Abstract

Let ℍ_{3} be three-dimensional hyperbolic space and Γ a group of isometries of ℍ_{3} that acts discontinuously on ℍ_{3} and that has a fundamental domain of finite hyperbolic volume. The Laplace operator −Δ of ℍ_{3} gives rise to a positive, essentially selfadjoint operator on L^{2}(Γ\ℍ_{3}). The nature of its discrete spectrum is still not well understood if Γ is not cocompact.

This paper contains a report on a numerical study of the discrete spectrum of Γ for various noncocompact groups Γ. Particularly interesting are the results for some nonarithmetic groups Γ.