GROUP-THEORY OF THE SMORODINSKY-WINTERNITZ SYSTEM

The three degrees of freedom Smorodinsky-Winternitz system is a degenerate or super-integrable Hamiltonian that possesses five functionally independent globally defined and single-valued integrals of the motion in both classical and quantum mechanics. This is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the Hamiltonian under a group of symmetry transformations isomorphic to the three-dimensional unitary unimodular group, SU(3). In turn, this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other, the so-called dynamical group. All the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group. So, in common with the hydrogen atom and the harmonic oscillator, the quantum mechanics of the Smorodinsky-Winternitz system may be completely solved within the framework of group theory alone.