Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability
Abstract
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.
- Authors:
-
- School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072 (Australia)
- Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels (Belgium)
- Publication Date:
- OSTI Identifier:
- 22403047
- Resource Type:
- Journal Article
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 11; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; CARTESIAN COORDINATES; EIGENSTATES; HAMILTONIANS; HARMONIC OSCILLATORS; IRREDUCIBLE REPRESENTATIONS; POLYNOMIALS; POTENTIALS; QUANTUM MECHANICS
Citation Formats
Marquette, Ian, and Quesne, Christiane. Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability. United States: N. p., 2014.
Web. doi:10.1063/1.4901006.
Marquette, Ian, & Quesne, Christiane. Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability. United States. https://doi.org/10.1063/1.4901006
Marquette, Ian, and Quesne, Christiane. 2014.
"Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability". United States. https://doi.org/10.1063/1.4901006.
@article{osti_22403047,
title = {Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability},
author = {Marquette, Ian and Quesne, Christiane},
abstractNote = {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.},
doi = {10.1063/1.4901006},
url = {https://www.osti.gov/biblio/22403047},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 11,
volume = 55,
place = {United States},
year = {Sat Nov 15 00:00:00 EST 2014},
month = {Sat Nov 15 00:00:00 EST 2014}
}