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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.

Information

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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References

1Kalnins, E. G. and Miller, W. Jr.Lie theory and the wave equation in space-time, 1. The Lorentz group. J. Mathematical Phys. 18 (1977), 1–16.CrossRefGoogle Scholar
2Kalnins, E. G. and Miller, W. Jr. Lie theory and the wave equation in space-time, 2. The group SO (4, c). SIAM. J. Math. Anal., to appear.Google Scholar
3Kalnins, E. G. and Miller, W. Jr.Lie theory and the wave equation in space-time, 3. Semisubgroup coordinates. J. Mathematical Phys. 18 (1977), 271–280m.Google Scholar
4Kalnins, E. G. and Miller, W. Jr.Lie theory and separation of variables, II. The EPD equation. J. Mathematical Phys. 17 (1976), 369–377.Google Scholar
5Erdelyi, A. et al. Higher Transcendental Functions, 1 and 2 (New York: McGraw-Hill, 1951).Google Scholar
6Limic, N., Niederle, J., and Raczka, R.Eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones. J. Mathematical Phys. 8 (1967), 1079–1093.Google Scholar
7Strichartz, R. S.Harmonic analysis on hyperboloids. J. Functional Analysis 12 (1973), 341–383.Google Scholar
8Gel'fand, I. M. and Shilov, G. E.Generalized Functions, 1 (New York, Academic Press, 1964).Google Scholar
9Gel'fand, I. M., Graev, M. I. and Vilenkin, N. Y.Generalized functions, 5, ch. 7 (New York: Academic Press, 1966).Google Scholar
10Ruhl, W.The Lorentz Group and Harmonic Analysis, ch. 5 and 6 (New York: Benjamin, 1970).Google Scholar
11Miller, W. Jr., Symmetry, separation of variables and special functions. In Theory and Applications of Special Functions. Ed. Askey, R. (New York: Academic Press, 1975).Google Scholar
12Miller, W. Jr,. Symmetry and Separation of Variables (Reading, Mass.: Addison-Wesley, 1977).Google Scholar
13Bargmann, V.Irreducible unitary representations of the Lorentz group. Ann. of Math. 48 (1947), 568–640.Google Scholar
14Kalnins, E. G.Mixed basis matrix elements for the subgroup reductions of SO(2, 1). J. Mathematical Phys. 14 (1973), 654–657.Google Scholar
15Vilenkin, N. Y.Special Functions and the Theory of Group Representations (Providence, R. I.: Amer. Math. Soc. Transl., 1968).Google Scholar
16Kalnins, E. G. and Miller, W. Jr.Lie theory and separation of variables 3. The equation fu−Fss = r2f. J. Mathematical Phys. 15 (1974), 1025–1032. Erratum, J. Mathematical Phys. 16 (1975), 1531.Google Scholar
17Kalnins, E. G.On the separation of variables for the Laplace equation in two- and threedimensional Minkowski space. SIAM. J. Math. Anal. 6 (1975), 340–374.Google Scholar
18Kalnins, E. G. and Miller, W. Jr.Lie theory and separation of variables. 8. Semisubgroup coordinates for γu − Δ2 γ = 0. J. Mathematical Phys. 16 (1975), 2507–2516.Google Scholar
19Kalnins, E. G. and Miller, W. Jr.Lie theory and separation of variables, 4. The groups SO(2, 1) and SO(3). J. Mathematical Phys. 15 (1974), 1263–1274.CrossRefGoogle Scholar
20Kalnins, E. G., Miller, W. Jr., and Winternitz, P.The group O(4), separation of variables and the hydrogen atom. SIAM. J. Appl. Math. 30 (1976), 630–664.Google Scholar
21Winternitz, P., Lukač, I., and Smorodinsky, Y.Quantum numbers in the little groups of the Poincaré groups. Soviet J. Nuclear Phys. 7 (1968), 139–145.Google Scholar
22Olevski, P.The separation of variables in the equation Δ2u + λu − 0 for spaces of constant curvature in two and three dimensions. Mat. Sb. 27 (1950), 379–426.Google Scholar