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

Ian Marquette; Christiane Quesne

doi:10.1063/1.4901006

paper

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

arXiv:1402.6380 · doi:10.1063/1.4901006

Abstract

Type III multi-step rationally-extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of $k$ integers $m_{1}$ , $m_{2}$ , \ldots, $m_{k}$ , such that $m_{1} < m_{2} < \dots < m_{k}$ with $m_{i}$ 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 $m_{k} + 1$ 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.

46 pages

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

Abstract

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