Dynamical symmetry algebras of two superintegrable two-dimensional systems

A complete classification of 2D quadratically superintegrable systems with scalar potential 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. They correspond in fact to two of those equivalence classes. 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 underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore 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 which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states.