## Experimental Mathematics

- Experiment. Math.
- Volume 5, Issue 1 (1996), 57-80.

### A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp

Fritz Grunewald and Wolfgang Huntebrinker

#### Abstract

Let $\hz_{3}$ be three-dimensional hyperbolic space and $\Gamma$ a group of isometries of $\hz_3$ that acts discontinuously on $\hz_{3}$ and that has a fundamental domain of finite hyperbolic volume. The Laplace operator $\MinusDelta$ of $\hz_{3}$ gives rise to a positive, essentially selfadjoint operator on $L^2(\Gamma \backslash \hz_{3})$. The nature of its discrete spectrum $\dspec (\Gamma)$ is still not well understood if $\Gamma$ is not cocompact.

This paper contains a report on a numerical study of $\dspec (\Gamma)$ for various noncocompact groups $\Gamma$. Particularly interesting are the results for some nonarithmetic groups $\Gamma$.

#### Article information

**Source**

Experiment. Math., Volume 5, Issue 1 (1996), 57-80.

**Dates**

First available in Project Euclid: 13 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1047591148

**Mathematical Reviews number (MathSciNet)**

MR1420720

**Zentralblatt MATH identifier**

0870.65092

**Subjects**

Primary: 11F72: Spectral theory; Selberg trace formula

Secondary: 11Y35: Analytic computations 35P99: None of the above, but in this section 58G25 65N25: Eigenvalue problems

#### Citation

Grunewald, Fritz; Huntebrinker, Wolfgang. A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp. Experiment. Math. 5 (1996), no. 1, 57--80. https://projecteuclid.org/euclid.em/1047591148