Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System in $D$ Dimensions

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 $\operatorname{su}(2D)$ symmetry and $\operatorname w(2D)\oplus_s\operatorname{sp}(4D,\mathbb R)$ 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 $D=2$.