Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Journal of Mathematical Physics

Volumn 46, Issue 5, 2005, Pages

  Kalnins, E G ^a   Kress, J M ^b   Miller Jr , W ^c  

^a UNIVERSITY OF WAIKATO   (New Zealand)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18844407093     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1897183     Document Type: Article

References (48)

1. S. Wojciechowski, " Superintegrability of the Calogero-Moser System., " Phys. Lett. 95A, 279 -281 (1983).
2. N. W. Evans, " Superintegrability in classical mechanics., " Phys. Rev. A 41, 5666 -5676 (1990)
3. N. W. Evans, " Group theory of the Smorodinsky-Winternitz system., " J. Math. Phys. 32, 3369 (1991).
4. N. W. Evans, " Super-integrability of the Winternitz system., " Phys. Lett. A 147, 483 -486 (1990).
5. J. Friš, V. Mandrosov, Ya. A. Smorodinsky, M. Uhlír, and P. Winternitz, " On higher symmetries in quantum mechanics., " Phys. Lett. 16, 354 -356 (1965).
6. J. Friš, Ya. A. Smorodinskii, M. Uhlír, and P. Winternitz, " Symmetry groups in classical and quantum mechanics., " Sov. J. Nucl. Phys. 4, 444 -450 (1967).
7. A. A. Makarov, Ya. A. Smorodinsky, Kh. Valiev, and P. Winternitz, " A systematic search for nonrelativistic systems with dynamical symmetries., " Nuovo Cimento A 52, 1061 -1084 (1967).
8. F. Calogero, " Solution of a three-body problem in one dimension., " J. Math. Phys. 10, 2191 -2196 (1969).
9. A. Cisneros and H. V. McIntosh, " Symmetry of the two-dimensional hydrogen atom., " J. Math. Phys. 10, 277 -286 (1969).
10. E. K. Sklyanin, " Separation of variables in the Gaudin model., " J. Sov. Math. 47, 2473 -2488 (1989).
11. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987).
12. A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics (IoP, Bristol, 1993).
13. J. Harnad, " Loop groups, R-matrices and separation of variables., " in Integrable Systems: From Classical to Quantum, edited by, J. Harnad, G. Sabidussi, and, P. Winternitz, [CRM Proceedings and Lecture Notes 26, 21-54, (2000)].
14. M. Karlovini and K. Rosquist, " A unified treatment of cubic invariants at fixed and arbitrary energy., " J. Math. Phys. 41, 370-384 (2000).
15. L. P. Eisenhart, " Enumeration of potentials for which one-particle Schrödinger equations are separable., " Phys. Rev. 74, 87 -89 (1948).
16. L. P. Eisenhart, Riemannian Geometry, 2nd printing (Princeton University Press, Princeton, 1949).
17. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Providence, RI, 1977).
18. E. G. Kalnins and W. Miller, Jr., " Killing tensors and variable separation for Hamilton-Jacobi and Helmholtz equations., " SIAM J. Math. Anal. 11, 1011-1026 (1980).
19. W. Miller, " The technique of variable separation for partial differential equations., " Proceedings of School and Workshop on Nonlinear Phenomena, Oaxtepec, Mexico, 29 November-17 December, 1982, Lecture Notes in Physics Vol. 189 (Springer, New York, 1983) pp. 184 -208.
20. E. G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics, Vol. 28 (Longman, Essex, 1986).
21. W. Miller, Jr., " Mechanisms for variable separation in partial differential equations and their relationship to group theory., " in Symmetries and Non-linear Phenomena (World Scientific, Singapore, 1988), pp. 188 -221.
22. A. Erd́lyi, W. Magnus, F. Oberhëttinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II.
23. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, 1958).
24. F. M. Arscott, Periodic Differential Equations (Macmillan, New York, 1964).
25. Ya. A. Granovsky, A. S. Zhedanov, and I. M. Lutzenko, " Quadratic algebra as a 'hidden' symmetry of the Hartmann potential., " J. Phys. A 24, 3887 -3894 (1991).
26. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, " Elliptic basis for a circular oscillator., " Nuovo Cimento Soc. Ital. Fis., B 0369-4100 88, 43 -56 (1985)
27. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, " Two-dimensional hydrogen atom. I. Elliptic bases., " Theor. Math. Phys. 61, 1021 -1039 (1984)
28. L. G. Mardoyan, G. S. Pogosyan, A. N. Sissakian, and V. M. Ter-Antonyan, " Hidden symmetry, separation of variables and interbasis expansions in the two-dimensional hydrogen atom., " J. Phys. A 18, 455 -466 (1985).
29. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, " Superintegrability and associated polynomial solutions. Euclidean space and the sphere in two dimensions., " J. Math. Phys. 37, 6439 -6467 (1996).
30. D. Bonatos, C. Daskaloyannis, and K. Kokkotas, " Deformed oscillator algebras for two-dimensional quantum superintegrable systems., " Phys. Rev. A 50, 3700 -3709 (1994).
31. C. Daskaloyannis, " Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associate algebras of quantum superintegrable systems., " J. Math. Phys. 42, 1100-1119 (2001).
32. S. P. Smith, " A class of algebras similar to the enveloping algebra of sl (2)., " Trans. Am. Math. Soc. 322, 285-314 (1990).
33. E. G. Kalnins, W. Miller, and M. V. Tratnik, " Families of orthogonal and biorthogonal polynomials on the n -sphere., " SIAM J. Math. Anal. 22, 272-294 (1991).
34. P. Letourneau and L. Vinet, " Superintegrable systems: Polynomial algebras and quasi-exactly solvable Hamiltonians., " Ann. Phys. 243, 144-168 (1995).
35. P. Tempesta, A. V. Turbiner, and P. Winternitz, " Exact solvability of superintegrable systems., " J. Math. Phys. 42, 4248-4257 (2001).
36. C. Grosche, G. S. Pogosyan, and A. N. Sissakian, " Path integral discussion for Smorodinsky-Winternitz potentials. I. Two- and three dimensional Euclidean space., " Fortschr. Phys. 43, 453 -521 (1995).
37. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and G. S. Pogosyan, " Completeness of superintegrability in two-dimensional constant curvature spaces., " J. Phys. A 34, 4705 -4720 (2001).
38. E. G. Kalnins, J. M. Kress, and P. Winternitz, " Superintegrability in a two-dimensional space of non-constant curvature., " J. Math. Phys. 43, 970-983 (2002).
39. E. G. Kalnins, J. M. Kress, W. Miller, Jr., and P. Winternitz, " Superintegrable systems in Darboux spaces., " J. Math. Phys. 44, 5811-5848 (2003).
40. M. F. Rañada, " Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach., " J. Math. Phys. 38, 4165 -4178 (1997).
41. E. G. Kalnins, W. Miller, Jr., G. C. Williams, and G. S. Pogosyan, " On superintegrable symmetry-breaking potentials in n -dimensional Euclidean space., " J. Phys. A 35, 4655 -4720 (2002).
42. C. P. Boyer, E. G. Kalnins, and W. Miller, " Stäckel-equivalent integrable Hamiltonian systems., " SIAM J. Math. Anal. 17, 778 -797 (1986).
43. J. Hietarinta, B. Grammaticos, B. Dorizzi, and A. Ramani, " Coupling-constant metamorphosis and duality between integrable Hamiltonian systems., " Phys. Rev. Lett. 53, 1707 -1710 (1984).
44. G. Koenigs, " Sur les ǵod́siques a int́grales quadratiques.. " A note appearing in Lecons sur la th́orie ǵńrale des surfaces, edited by, G. Darboux, (Chelsea, New York, 1972), Vol. 4, pp. 368-404.
45. G. H. Katzin and J. Levine, " Quadratic first integrals of the geodesics in spaces of constant curvature., " Tensor 16, 97-104 (1965).
46. I. Hauser and R. J. Malhiot, " Structural equations for Killing tensors of order two, II., " J. Math. Phys. 16, 1625-1629 (1975).
47. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, " Completeness of multiseparable superintegrability in E2,C., " J. Phys. A 33, 4105 -4120 (2000).
48. E. G. Kalnins, W. Miller, Jr., and G. S. Pogosyan, " Completeness of multiseparable superintegrability on the complex 2-sphere., " J. Phys. A 33, 6791-6806 (2000).