Quantum superintegrability and exact solvability in n dimensions

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space. Two different sets of n commuting second-order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.

[1]  E. Kalnins.  Separation of variables for Riemannian spaces of constant curvature , 1986 .

[2]  Completeness of superintegrability in two-dimensional constant-curvature spaces , 2001, math-ph/0102006.

[3]  Peter J. Olver,et al.  Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators , 1993 .

[4]  W. PauliJr..  Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik , 1926 .

[5]  A. Turbiner.  Quasi-exactly-solvable problems andsl(2) algebra , 1988 .

[6]  W. Miller,et al.  The Group $O(4)$, Separation of Variables and the Hydrogen Atom , 1976 .

[7]  Jamil Daboul,et al.  The Hydrogen algebra as centerless twisted Kac-Moody algebra , 1993 .

[8]  P. Winternitz,et al.  A systematic search for nonrelativistic systems with dynamical symmetries , 1967 .

[9]  W. Miller,et al.  Completeness of multiseparable superintegrability in E2,C , 2000 .

[10]  Evans,et al.  Superintegrability in classical mechanics. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[11]  M. Shifman.  NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM) , 1989 .

[12]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[13]  P. Winternitz,et al.  ON HIGHER SYMMETRIES IN QUANTUM MECHANICS , 1965 .

[14]  Superintegrability in a two-dimensional space of nonconstant curvature , 2001, math-ph/0108015.

[15]  Jamil Daboul,et al.  From hydrogen atom to generalized Dynkin diagrams , 1998 .

[16]  Alexander Turbiner.  Lie-algebras and linear operators with invariant subspaces , 1993 .

[17]  E. L. Hill,et al.  On the Problem of Degeneracy in Quantum Mechanics , 1940 .

[18]  P. Winternitz,et al.  A new basis for the representations of the rotation group. Lamé and Heun polynomials , 1973 .

[19]  V. Fock,et al.  Zur Theorie des Wasserstoffatoms , 1935 .

[20]  H. McIntosh.  Symmetry and Degeneracy , 1971 .

[21]  P. Winternitz,et al.  Separation of variables and subgroup bases on n-dimensional hyperboloids , 2002 .

[22]  N. Evans.  Group theory of the Smorodinsky-Winternitz system , 1991 .

[23]  W. Jr. Pauli,et al.  Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik , 1926 .

[24]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[25]  P. Winternitz,et al.  Contractions of Lie algebras and separation of variables. The n -dimensional sphere , 1999 .

[26]  W. Miller,et al.  Subgroups of Lie groups and separation of variables , 1981 .

[27]  Ernest M. Loebl,et al.  Group theory and its applications , 1968 .

[28]  Exact solvability of superintegrable systems , 2000, hep-th/0011209.

[29]  W. Miller,et al.  Completeness of multiseparable superintegrability on the complex 2-sphere , 2000 .

[30]  P. Tempesta,et al.  Superintegrable systems in quantum mechanics and classical Lie theory , 2001 .

[31]  N. Evans.  Super-integrability of the Winternitz system , 1990 .

[32]  M. J. Englefield.  Group Theory and the Coulomb Problem , 1974 .

[33]  Y. Smorodinskii,et al.  SYMMETRY GROUPS IN CLASSICAL AND QUANTUM MECHANICS , 1966 .

[34]  M. Moshinsky,et al.  The harmonic oscillator in modern physics , 1996 .

[35]  V. Bargmann,et al.  Zur Theorie des Wasserstoffatoms , 1936 .

[36]  B. Dorizzi,et al.  Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems , 1984 .

[37]  W. Miller,et al.  Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions , 1996 .