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The wave equation, O(2, 2), and separation of variables on hyperboloids

Published online by Cambridge University Press:  14 November 2011

E. G. Kalnins  and
W. Miller Jr
E. G. Kalnins
Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand
W. Miller Jr
Affiliation:
School of Mathematics, University of Minnesota
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Synopsis

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 3-4 , 1978 , pp. 227 - 256
Copyright
Copyright © Royal Society of Edinburgh 1978

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