Dynamical symmetry algebra of two superintegrable two-dimensional systems
A complete classification of 2D superintegrable systems on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems $E_{8}$ and $E_{10}$ from such a classification by a new approach based on extra sets of ladder operators. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the dynamical symmetry algebra explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states. Our approach provides a natural framework for such states.