On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space

We give a graphical prescription for obtaining and characterizing all separable coordinates for which the Schr?dinger equation admits separable solutions for one of the superintegrable potentials Here xn+1 is a distinguished Cartesian variable. The algebra of second-order symmetries of the resulting Schr?dinger equation is given and, for the first potential, the closure relations of the corresponding quadratic algebra. These potentials are particularly interesting because they occur in all dimensions n ? 1, the separation of variables problem is highly nontrivial for them, and many other potentials are limiting cases.

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

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

[3]  W. Miller,et al.  Superintegrability in three-dimensional Euclidean space , 1999 .

[4]  Y. Hakobyan,et al.  On a Generalized D-Dimensional Oscillator: Interbasis Expansions , 1998 .

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

[6]  A. Macfarlane.  The quantum Neumann model with the potential of Rosochatius , 1992 .

[7]  W. Miller,et al.  Separable coordinates, integrability and the Niven equations , 1992 .

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

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

[10]  W. Miller,et al.  Sta¨kel equivalent integrable Hamiltonian systems , 1986 .

[11]  W. Miller,et al.  Separation of variables on n‐dimensional Riemannian manifolds. I. The n‐sphere Sn and Euclidean n‐space Rn , 1986 .

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

[13]  W. Miller,et al.  Killing Tensors and Variable Separation for Hamilton-Jacobi and Helmholtz Equations , 1980 .

[14]  Willard Miller,et al.  Symmetry and Separation of Variables , 1977 .

[15]  W. Miller,et al.  Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials , 1975 .

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

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