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Type III multi-step rationally-extended harmonic oscillator and radialharmonic oscillator potentials, characterized by a set of $k$ integers $m_1$,$m_2$, \ldots, $m_k$, such that $m_1 < m_2 < \cdots < m_k$ with $m_i$ even(resp.\ odd) for $i$ odd (resp.\ even), are considered. The state-adding andstate-deleting approaches to these potentials in a supersymmetric quantummechanical framework are combined to construct new ladder operators. Theeigenstates of the Hamiltonians are shown to separate into $m_k+1$infinite-dimensional unitary irreducible representations of the correspondingpolynomial Heisenberg algebras. These ladder operators are then used to build ahigher-order integral of motion for seven new infinite families ofsuperintegrable two-dimensional systems separable in cartesian coordinates. Thefinite-dimensional unitary irreducible representations of the polynomialalgebras of such systems are directly determined from the ladder operatoraction on the constituent one-dimensional Hamiltonian eigenstates and providean algebraic derivation of the superintegrable systems whole spectrum includingthe level total degeneracies.

journal of mathematical physics

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