### Abstract

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n - 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.

Original language | English (US) |
---|---|

Pages (from-to) | 576-583 |

Number of pages | 8 |

Journal | Physics of Atomic Nuclei |

Volume | 70 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2007 |

## Fingerprint Dive into the research topics of 'Second-order superintegrable quantum systems'. Together they form a unique fingerprint.

## Cite this

*Physics of Atomic Nuclei*,

*70*(3), 576-583. https://doi.org/10.1134/S1063778807030192