Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 015, 17 pages      nlin.SI/0511035      https://doi.org/10.3842/SIGMA.2005.015

Second Order Superintegrable Systems in Three Dimensions

Willard Miller
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received October 28, 2005; Published online November 13, 2005

Abstract
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n-1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, the system is second order superintegrable. Such systems have remarkable properties. Typical properties are that 1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems, 2) they are multiseparable, 3) the second order symmetries generate a closed quadratic algebra and in the quantum case the representation theory of the quadratic algebra yields important facts about the spectral resolution of the Schrödinger operator and the other symmetry operators, and 4) there are deep connections with expansion formulas relating classes of special functions and with the theory of Exact and Quasi-exactly Solvable systems. For n = 2 the author, E.G. Kalnins and J. Kress, have worked out the structure of these systems and classified all of the possible spaces and potentials. Here I discuss our recent work and announce new results for the much more difficult case n = 3. We consider classical superintegrable systems with nondegenerate potentials in three dimensions and on a conformally flat real or complex space. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We describe the Stäckel transformation, an invertible conformal mapping between superintegrable structures on distinct spaces, and give evidence indicating that all our superintegrable systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We also indicate how to extend the classical 2D and 3D superintegrability theory to include the operator (quantum) case.

Key words: superintegrability; quadratic algebra; conformally flat spaces.

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References

1. Wojciechowski S., Superintegrability of the Calogero-Moser system, Phys. Lett. A, 1983, V.95, 279-281.
2. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A, 1990, V.41, 5666-5676; Group theory of the Smorodinsky-Winternitz system, J. Math. Phys., 1991, V.32, 3369-3375.
3. Evans N.W., Super-integrability of the Winternitz system, Phys. Lett. A, 1990, V.147, 483-486.
4. Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett., 1965, V.16, 354-356.
5. Fris J., Smorodinskii Ya.A., Uhlír M., Winternitz P., Symmetry groups in classical and quantum mechanics, Sov. J. Nucl. Phys., 1967, V.4, 444-450.
6. Makarov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries, Nuovo Cimento, 1967, V.52, 1061-1084.
7. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys., 1969, V.10, 2191-2196.
8. Cisneros A., McIntosh H.V., Symmetry of the two-dimensional hydrogen atom, J. Math. Phys., 1969, V.10, 277-286.
9. Sklyanin E.K., Separation of variables in the Gaudin model, J. Sov. Math., 1989, V.47, 2473-2488.
10. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Berlin, Springer, 1987.
11. Harnad J., Loop groups, R-matrices and separation of variables, in "Integrable Systems: From Classical to Quantum", Editors J. Harnad, G. Sabidussi and P. Winternitz, CRM Proceedings and Lecture Notes, 2000, V.26, 21-54.
12. Eisenhart L.P., Riemannian geometry, Princeton University Press, 2nd printing, 1949.
13. Miller W.Jr., Symmetry and separation of variables, Providence, Rhode Island, Addison-Wesley Publishing Company, 1977.
14. Kalnins E.G., Miller W.Jr., Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations, SIAM J. Math. Anal., 1980, V.11, 1011-1026.
15. Miller W., The technique of variable separation for partial differential equations, in Proceedings of School and Workshop on Nonlinear Phenomena (November 29 - December 17, 1982, Oaxtepec, Mexico), Lecture Notes in Physics, Vol. 189, New York, Springer-Verlag, 1983, 184-208.
16. Kalnins E.G., Separation of variables for Riemannian spaces of constant curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics, Vol. 28, Essex, England, Longman, 1986, 184-208,
17. Miller W.Jr., Mechanisms for variable separation in partial differential equations and their relationship to group theory, in "Symmetries and Non-linear Phenomena", World Scientific, 1988, 188-221
18. Kalnins E.G., Kress J.M., Miller W.Jr., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys., 2005, V.46, 053509, 28 pages.
19. Kalnins E.G., Kress J.M., Miller W.Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys., 2005, V.46, 053510, 15 pages.
20. Kalnins E.G., Kress J.M., Miller W.Jr., Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory, J. Math. Phys., 2005, V.46, 103507, 28 pages.
21. Kalnins E.G., Kress J.M., Miller W.Jr., Second order superintegrable systems in conformally flat spaces. IV. The classical three-dimensional Stäckel transform, submitted.
22. Kalnins E.G., Miller W.Jr., Pogosyan G.S., Superintegrability in three dimensional Euclidean space, J. Math. Phys., 1999, V.40, 708-725.
23. Kalnins E.G., Miller W.Jr., Pogosyan G.S., Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions, J. Math. Phys., 1996, V.37, 6439-6467.
24. Bonatos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A, 1994, V.50, 3700-3709, hep-th/9309088.
25. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associate algebras of quantum superintegrable systems, J. Math. Phys., 2001, V.42, 1100-1119, math-ph/0003017.
26. Smith S.P., A class of algebras similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc., 1990, V.322, 285-314.
27. Kalnins E.G., Miller W., Tratnik M.V., Families of orthogonal and biorthogonal polynomials on the n-sphere, SIAM J. Math. Anal., 1991, V.22, 272-294.
28. Ushveridze A.G., Quasi-exactly solvable models in quantum mechanics, Bristol, Institute of Physics, 1993.
29. Letourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Phys., 1995, V.243, 144-168.
30. Kalnins E.G., Miller W.Jr., Pogosyan G.S., Exact and quasi-exact solvability of second order superintegrable systems. I. Euclidean space preliminaries, submitted.
31. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials: I. Two- and three-dimensional Euclidean space, Fortschritte der Physik, 1995, V.43, 453-521, hep-th/9402121.
32. Kalnins E.G., Kress J.M., Miller W.Jr., Pogosyan G.S., Completeness of superintegrability in two-dimensional constant curvature spaces, J. Phys. A: Math. Gen., 2001, V.34, 4705-4720, math-ph/0102006.
33. Kalnins E.G., Kress J.M., Winternitz P., Superintegrability in a two-dimensional space of non-constant curvature, J. Math. Phys., 2002, V.43, 970-983, math-ph/0108015.
34. Kalnins E.G., Kress J.M., Miller W.Jr., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys., 2003, V.44, 5811-5848, math-ph/0307039.
35. Rañada M.F., Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys., 1997, V.38, 4165-4178.
36. Kalnins E.G., Miller W.Jr., Williams G.C., Pogosyan G.S., On superintegrable symmetry-breaking potentials in n-dimensional Euclidean space, J. Phys. A: Math. Gen., 2002, V.35, 4655-4720.
37. Boyer C.P., Kalnins E.G., Miller W., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal., 1986, V.17, 778-797.
38. Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, Phys. Rev. Lett., 1984, V.53, 1707-1710.
39. Kalnins E.G., Miller W., Reid G.K., Separation of variables for Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for Sc and EnC, Proc. R. Soc. Lond. A, 1984, V.39, 183-206.