Group theory of the Smorodinsky–Winternitz system

Abstract

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.