SIGMA 8 (2012), 105, 11 pages      arXiv:1212.6123      https://doi.org/10.3842/SIGMA.2012.105
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Separation of Variables and Contractions on Two-Dimensional Hyperboloid

Ernie Kalnins ^a, George S. Pogosyan ^{b, c} and Alexander Yakhno ^c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
^b) International Center for Advanced Studies, Yerevan State University, A. Manougian 1, Yerevan, 0025, Armenia
^c) Departamento de Matemáticas, CUCEI, Universidad de Guadalajara, Jalisco, Mexico

Received August 09, 2012, in final form December 19, 2012; Published online December 26, 2012

Abstract
In this paper analytic contractions have been established in the R→∞ contraction limit for exactly solvable basis functions of the Helmholtz equation on the two-dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.

Key words: analytic contraction; separation of variables; Lie group; Helmholtz equation.

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References

1. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. I, McGraw-Hill Book Company, New York, 1953.
2. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. II, McGraw-Hill Book Company, New York, 1953.
3. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. III, McGraw-Hill Book Company, New York, 1955.
4. Grosche C., Path integrals, hyperbolic spaces, and Selberg trace formulae, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.
5. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables, J. Phys. A: Math. Gen. 29 (1996), 5949-5962.
6. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. The n-dimensional sphere, J. Math. Phys. 40 (1999), 1549-1573.
7. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid, Internat. J. Modern Phys. A 12 (1997), 53-61.
8. Kalnins E.G., Miller Jr. W., Lie theory and separation of variables. IV. The groups SO(2,1) and SO(3), J. Math. Phys. 15 (1974), 1263-1274.
9. Kalnins E.G., Miller Jr. W., Pogosyan G.S., Contractions of Lie algebras: applications to special functions and separation of variables, J. Phys. A: Math. Gen. 32 (1999), 4709-4732.
10. Olevski M.N., Triorthogonal systems in spaces of constant curvature in which the equation Δ₂u+λu=0 allows a complete separation of variables, Mat. Sb. 27 (1950), 379-426.
11. Olver F.W.J., Asymptotics and special functions, Academic Press, New York - London, 1974.
12. Pogosyan G.S., Sissakian A.N., Winternitz P., Separation of variables and Lie algebra contractions. Applications to special functions, Phys. Part. Nuclei 33 (2002), 235-276, math-ph/0310011.
13. Pogosyan G.S., Winternitz P., Separation of variables and subgroup bases on n-dimensional hyperboloids, J. Math. Phys. 43 (2002), 3387-3410.
14. Pogosyan G.S., Yakhno A., Lie algebra contractions and separation of variables. Three-dimensional sphere, Phys. Atomic Nuclei 72 (2009), 836-844.
15. Pogosyan G.S., Yakhno A., Lie algebra contractions on two-dimensional hyperboloid, Phys. Atomic Nuclei 73 (2010), 499-508.
16. Pogosyan G.S., Yakhno A., Two-dimensional imaginary Lobachevsky space. Separation of variables and contractions, Phys. Atomic Nuclei 74 (2011), 1062-1072.
17. Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.
18. Winternitz P., Lukac I., Smorodinskii Y.A., Quantum numbers in the little group of the Poincaré group, Sov. J. Nuclear Phys. 7 (1968), 139-145.