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Title: Factorization approach to superintegrable systems: Formalism and applications

Abstract

The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super) integrable anisotropic oscillators on the sphere are constructed. The Tremblay–Turbiner–Winternitz system on the Euclidean plane is also studied from this viewpoint.

Authors:
;  [1];  [2];  [3]
  1. Universidad de Burgos, Departamento de Física (Spain)
  2. Ankara University, Department of Physics, Faculty of Science (Turkey)
  3. Universidad de Valladolid, Departamento de Física Teórica, Atómica y Óptica (Spain)
Publication Date:
OSTI Identifier:
22613974
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Atomic Nuclei; Journal Volume: 80; Journal Issue: 2; Other Information: Copyright (c) 2017 Pleiades Publishing, Ltd.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; ANISOTROPY; EUCLIDEAN SPACE; FACTORIZATION; HAMILTONIANS; INTEGRAL CALCULUS; OSCILLATORS

Citation Formats

Ballesteros, Á., E-mail: angelb@ubu.es, Herranz, F. J., E-mail: fjherranz@ubu.es, Kuru, Ş., E-mail: kuru@science.ankara.edu.tr, and Negro, J., E-mail: jnegro@fta.uva.es. Factorization approach to superintegrable systems: Formalism and applications. United States: N. p., 2017. Web. doi:10.1134/S1063778817020053.
Ballesteros, Á., E-mail: angelb@ubu.es, Herranz, F. J., E-mail: fjherranz@ubu.es, Kuru, Ş., E-mail: kuru@science.ankara.edu.tr, & Negro, J., E-mail: jnegro@fta.uva.es. Factorization approach to superintegrable systems: Formalism and applications. United States. doi:10.1134/S1063778817020053.
Ballesteros, Á., E-mail: angelb@ubu.es, Herranz, F. J., E-mail: fjherranz@ubu.es, Kuru, Ş., E-mail: kuru@science.ankara.edu.tr, and Negro, J., E-mail: jnegro@fta.uva.es. Wed . "Factorization approach to superintegrable systems: Formalism and applications". United States. doi:10.1134/S1063778817020053.
@article{osti_22613974,
title = {Factorization approach to superintegrable systems: Formalism and applications},
author = {Ballesteros, Á., E-mail: angelb@ubu.es and Herranz, F. J., E-mail: fjherranz@ubu.es and Kuru, Ş., E-mail: kuru@science.ankara.edu.tr and Negro, J., E-mail: jnegro@fta.uva.es},
abstractNote = {The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super) integrable anisotropic oscillators on the sphere are constructed. The Tremblay–Turbiner–Winternitz system on the Euclidean plane is also studied from this viewpoint.},
doi = {10.1134/S1063778817020053},
journal = {Physics of Atomic Nuclei},
number = 2,
volume = 80,
place = {United States},
year = {Wed Mar 15 00:00:00 EDT 2017},
month = {Wed Mar 15 00:00:00 EDT 2017}
}
  • We classify the superintegrable potentials in the Euclidean plane by means of an orbit analysis of the space of valence two Killing tensors under the action of the group of rigid motions. Our approach generalizes the classical approach of Winternitz and collaborators by considering pairs of Killing tensors that are not both in canonical form.
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  • We present a study of non-relativistic superintegrable systems whose invariants are quadratic in the momenta. In two dimensions, there exist only two inequivalent classes of such systems. The symmetries responsible for the accidental degeneracies of those problems are investigated and shown to be best described in terms of polynomial algebras. We also determine the quasi-exactly solvable (QES) systems that can be obtained by dimensional reduction from the two- and three-dimensional superintegrable models, establishing in each case the equivalence between the QES Schr{umlt o}dinger equation and the spectral problem associated to a quadratic element in the questions of a Lie algebra.more » {copyright} 1995 Academic Press, Inc.« less