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THE SPECTRA OF THE SPHERICAL AND EUCLIDEAN TRIANGLE GROUPS

Part of: Algebraic groups General theory of linear operators Representation theory of groups

Published online by Cambridge University Press:  01 April 2007

MARK HARMER
MARK HARMER*
Affiliation:
Department of Mathematics, Australian National University, Canberra, Australia (email: [email protected])
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Abstract

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We derive the spectrum of the Laplace–Beltrami operator on the quotient orbifold of the nonhyperbolic triangle groups.

Keywords

triangle groupsspectral multiplicity

MSC classification

Secondary: 14L35: Classical groups (geometric aspects) 47A75: Eigenvalue problems 20C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 2 , April 2008 , pp. 217 - 227
Copyright
Copyright © 2007 Australian Mathematical Society

References

[1]Cooper, S. and Lam, H. Y., ‘Sums of two, four, six and eight squares and triangular numbers: an elementary approach’, Indian J. Math. 44 (2002), 21–40.Google Scholar
[2]Eisenstein, G., ‘Aufgaben und Lehrsätze’, Crelle’s J. 27 (1844), 281–283.Google Scholar
[3]Grosswald, E., Representations of Integers as Sums of Squares (Springer, New York, 1985).CrossRefGoogle Scholar
[4]Hardy, G. H., Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work (Cambridge University Press, Cambridge, 1940).Google Scholar
[5]Hejhal, D. A., ‘Eigenvalues of the Laplacian for Hecke triangle groups’, Mem. Amer. Math. Soc. 97(469) (1992), vi+165 .Google Scholar
[6]Hirschhorn, M. D., ‘Three classical results on representations of a number’, Sém. Lothar. Combin. 42 (1999), 8 (electronic).Google Scholar
[7]Lamé, M. G., Leçons sur la Théorie Mathématique de l’Elasticité des Corps Solides (Gauthier-Villars, Paris, 1866).Google Scholar
[8]Maass, H., ‘Über eine neue art von nichtanalytischen automorphen funktionen und die bestimmung Dirichletscher reihen durch funktionalgleichungen’, Math. Ann. 121 (1949), 141–183.CrossRefGoogle Scholar
[9]McCartin, B. J., ‘Eigenstructure of the equilateral triangle, part 1: the Dirichlet problem’, SIAM Rev. 45(2) (2003), 267–287.CrossRefGoogle Scholar
[10]Pinsky, M. A., ‘The eigenvalues of an equilateral triangle’, SIAM J. Math. Anal. 11(5) (1980), 819–827.CrossRefGoogle Scholar
[11]Pinsky, M. A., ‘Completeness of the eigenfunctions of the equilateral triangle’, SIAM J. Math. Anal. 16(4) (1985), 848–851.CrossRefGoogle Scholar
[12]Práger, M., ‘Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle’, Appl. Math. 43(4) (1998), 311–320.CrossRefGoogle Scholar
[13]Selberg, A., ‘Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series’, J. Indian Math. Soc. 20 (1956), 47–87.Google Scholar
[14]Vilenkin, N. J., Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar