Second-order superintegrable quantum systems
- University of Minnesota, School of Mathematics (United States)
- University of Waikato, Department of Mathematics and Statistics (New Zealand)
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 Schroedinger 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.
- OSTI ID:
- 21075911
- Journal Information:
- Physics of Atomic Nuclei, Vol. 70, Issue 3; Other Information: DOI: 10.1134/S1063778807030192; Copyright (c) 2007 Nauka/Interperiodica; Article Copyright (c) 2007 Pleiades Publishing, Ltd; Country of input: International Atomic Energy Agency (IAEA); ISSN 1063-7788
- Country of Publication:
- United States
- Language:
- English
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