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

Journal Article · Sat Nov 15 04:00:00 EST 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4901006· OSTI ID:22403047

Marquette, Ian ^[1]; Quesne, Christiane ^[2]

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1. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072 (Australia)
2. Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels (Belgium)

Type III multi-step rationally extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of k integers m{sub 1}, m{sub 2}, ⋯, m{sub k}, such that m{sub 1} < m{sub 2} < ⋯ < m{sub k} with m{sub 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{sub 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.

• OSTI ID: 22403047
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 11 Vol. 55; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

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