Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: applications to ladder operators and superintegrability

Type III multi-step rationally extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of k integers m, m, ..., m, such that m < m < ... < m with m even (resp. odd) for i odd (resp. even), are considered. The state-adding and state-deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators. The eigenstates of the Hamiltonians are shown to separate into m + 1 infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebras. These ladder operators are then used to build a higher-order integral of motion for seven new infinite families of superintegrable two-dimensional systems separable in cartesian coordinates. The finite-dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one-dimensional Hamiltonian eigenstates and provide an algebraic derivation of the superintegrable systems whole spectrum including the level total degeneracies