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

TL;DR

This paper re-examines the D-dimensional Smorodinsky-Winternitz system, revealing new algebraic structures such as potential and dynamical potential algebras, and explicitly describing generator actions for D=2.

Contribution

It introduces and constructs potential and dynamical potential algebras for the system, extending the understanding of its symmetry properties from an algebraic perspective.

Findings

Identifies potential and dynamical potential algebras for the system.

Transforms known symmetry and dynamical algebras into new algebraic structures.

Provides explicit generator actions on wavefunctions for D=2.

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 ${\rm w}(2D) \oplus_s {\rm sp}(4D,{\mathbb R})$ 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.