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

SIGMA 7 (2011), 035, 21 pages      arXiv:1104.0294      https://doi.org/10.3842/SIGMA.2011.035
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

Christiane Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received January 17, 2011, in final form March 25, 2011; Published online April 02, 2011

Abstract
The D-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing D auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and w(2D)⊕ssp(4D,R) dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky-Winternitz system. The action of generators on wavefunctions is given in explicit form for D=2.

Key words: Schrödinger equation; superintegrability; potential algebras; dynamical potential algebras.

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