Experimental Mathematics

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
http://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. Experimental Mathematics 5 (1996), no. 1, 57--80. http://projecteuclid.org/euclid.em/1047591148.


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