Quantum superintegrability and exact solvability in n dimensions

Journal of Mathematical Physics

Volumn 43, Issue 3, 2002, Pages 1309-1322

  Rodríguez, Miguel A ^a   Winternitz, Pavel ^b  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035981863     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1435077     Document Type: Article

References (41)

1. J. Jauch and E. Hill, "On the problem of degeneracy in quantum mechanics," Phys. Rev. 51, 641-645 (1940).
2. M. Moshinsky and Yu.F. Smirnov, The Harmonic Oscillator in Modern Physics (Harwood, Amsterdam, 1996).
3. W. Pauli, "Über das Wasserstoffspektrum von Standpunk der neuen Quanten-mechanik," Z. Phys. 36, 336-363 (1926).
4. V.A. Fock, "Zur theorie des Wasserstoffatoms," Z Phys. 98, 145-154 (1935).
5. V. Bargmann, "Zur theorie des Wasserstoffatoms," Z. Phys. 99, 576-582 (1936).
6. M.J. Englefield, Group Theory and the Coulomb Problem (Wiley, New York, 1972).
7. H.V. MacIntosh, "Symmetry and Degeneracy," in Group Theory and its Applications, edited by E.M. Loebl (Academic, New York, 1971), Vol. II, pp. 75-144.
8. E.G. Kalnins, W. Miller, Jr., and P, Winternitz, "The group O(4), separation of variables and the hydrogen atom," SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 30, 630-664 (1976).
9. J. Bertrand, "Théorème relatif au mouvement d'un point attiré vers un centre fixe" C.R. Ac. Sci. 77, 849-853 (1873).
10. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1990).
11. I. Friš, V. Mandrosov, J, Smorodinsky, M. Uhlíř, and P. Winternitz, "On higher symmetries in quantum mechanics," Phys. Lett. 16, 354-356 (1965).
12. P. Winternitz, J, Smorodinsky, and M. Uhlíř, "Symmetry groups in classical and quantum mechanics," Yad. Fiz. 4, 625 (1966); Sov. J. Nucl. Phys. 4, 1326 (1967).
13. P. Winternitz, J, Smorodinsky, and M. Uhlíř, "Symmetry groups in classical and quantum mechanics," Yad. Fiz. 4, 625 (1966); Sov. J. Nucl. Phys. 4, 1326 (1967).
14. A. Makarov, J. Smorodinsky, Kh. Valiev, and P. Winternitz, "A systematic search for nonrelativistic systems with dynamical symmetries," Nuovo Cimento A 52, 1061-1084 (1967).
15. N.W. Evans, "Superintegrability in classical mechanics," Phys. Rev. A 41, 5666-5676 (1990).
16. N.W. Evans, "Group theory of the Smorodinsky-Winternitz system," J. Math. Phys. 32, 3369-3375 (1991).
17. N.W. Evans, "Superintegrability of the Winternitz system," Phys. Lett. A 147, 483-486 (1990).
18. 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).
19. 2,C," J. Phys. A 33, 4105-4120 (2000).
20. E.G. Kalnins, W. Miller, Jr., and G.S. Pogosyan, "Completeness of multiseparability on the complex 2-sphere," J. Phys. A 33, 6791-6806 (2000).
21. E.G. Kalnins, W. Miller, Jr., and G.S. Pogosyan, "Completeness of multiseparability in two-dimensional constant curvature spaces," J. Phys. A 34, 4705-4720 (2001).
22. E.G. Kalnins, J.M. Kress, and P. Winternitz, "Superintegrability in a two-dimensional space of non-constant curvature," (LANL preprint archives math-ph/0108015). J. Math. Phys. (to be published).
23. P. Tempesta, A.V. Turbiner, and P. Winternitz, "Exact solvability of superintegrable systems," J. Math. Phys. 42, 4248-4257 (2001).
24. A.V. Turbiner, "Lie algebras and linear operators with invariant subspaces," in Lie Algebras, Cohomologies and New Findings in Quantum Mechanics, edited by N. Kamran and P.J. Olver (American Mathematical Society, Providence, 1994), Vol. 160, pp. 263-310.
25. A.V. Turbiner, "Quasi-exactly solvable problems and sl(2) algebra," Commun. Math. Phys. 118, 467-474 (1988).
26. M.B. Sheftel, P. Tempesta, and P. Winternitz, "Superintegrable systems in quantum mechanics and classical Lie theory," J. Math. Phys. 42, 659-673 (2001).
27. J. Daboul, P. Slodowy, and C. Daboul, "The hydrogen algebra as a centerless twisted Kac-Moody algebra," Phys. Lett. B 317, 321-328 (1993).
28. C. Daboul and J. Daboul, "From hydrogen atom to generalized Dynkin diagrams," Phys. Lett. B 425, 135-144 (1998).
29. P. Winternitz and I. Friš, "Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group," Sov. J. Nucl. Phys. 1, 636-643 (1965).
30. P. Winternitz, I. Lukac, and Ya. Smorodinskii, "Quantum numbers in the little groups of the Poincaré group," Sov. J. Nucl. Phys. 7, 139-145 (1968).
31. J. Patera and P. Winternitz, "A new basis for representations of the rotation group. Lamé and Heun polynomials," J. Math. Phys. 14, 1130-1139 (1973).
32. W. Miller, Jr., J. Patera, and P. Winternitz, "Subgroups of Lie groups and separation of variables," J. Math. Phys. 22, 251-260 (1991).
33. W. Miller, Jr., Symmetry and the Separation of Variables (Addison-Wesley, Reading, MA, 1997).
34. E.G. Kalnins, Separation of Variables for Riernannian Spaces of Constant Curvature (Longman, Burnt Mill, 1986).
35. 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).
36. A. González-López, N. Kamran, and P.J. Olver, "Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators," Commun. Math. Phys. 153, 117-146 (1993).
37. M. Shifman, "New findings in quantum mechanics (partial algebraization of the spectral problem)," Int. J. Mod. Phys. A 4, 2852-2897 (1989).
38. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, 1968).
39. N.Ya. Vilenkin, G.I. Kuznetsov, and Ya.A. Smorodinskii, "Eigenfunctions of the Laplace operator realizing representations of the groups U(2), SU(2), SO(3), U(3), and SU(3) and the symbolic method," Sov. J. Nucl. Phys. 2, 645-655 (1965).
40. A.A. Izmest'ev, G.S. Pogosyan, A.N. Sissakian, and P. Winternitz, "Contractions of Lie algebras and the separation of variables. The n-dimensional sphere," J. Math. Phys. 40, 1549-1573 (1999).
41. G.S. Pogosyan and P. Winternitz, "Separation of variables and subgroup bases on n-dimensional hyperboloids" J. Math. Phys. (unpublished).