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Emerging Applications of Number Theory pp 617–641Cite as

Eigenvalues of the Laplacian for Bianchi Groups

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 109))

Abstract

We present results concerning the computation of Laplace eigenvalue spectra for certain arithmetic, discrete subgroups of PSL(2, ℂ) acting on the hyperbolic upper half space. These subgroups are PSL(2, O), where O is the ring of integers of an imaginary quadratic number field. Special attention is devoted to the cases D = 1, 2, 3, 7, 11, 19 having fundamental domains with one cusp. It is proved that the spectra are not simple. We sketch a method showing how such eigenvalues and associated eigenfunctions can be computed. Some of the eigenvalues are recognized as being lifts from the modular group PSL(2, ℤ). As an application to quantum chaos, we demonstrate that the spectra exhibit random fluctuations close to Poissonian. Samples of eigenvalues are listed.

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Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Hamburg, Bundesstraße 55, D-20146, Hamburg, Germany

    Gunther Steil

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  1. Gunther Steil
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Editor information

Editors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Dennis A. Hejhal

  2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada

    Joel Friedman

  3. IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Martin C. Gutzwiller

  4. AT&T Labs—Research, Florham Park, NJ, 07932-0971, USA

    Andrew M. Odlyzko

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© 1999 Springer Science+Business Media New York

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Steil, G. (1999). Eigenvalues of the Laplacian for Bianchi Groups. In: Hejhal, D.A., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds) Emerging Applications of Number Theory. The IMA Volumes in Mathematics and its Applications, vol 109. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1544-8_27

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  • DOI: https://doi.org/10.1007/978-1-4612-1544-8_27

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7186-4

  • Online ISBN: 978-1-4612-1544-8

  • eBook Packages: Springer Book Archive

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