Built on

Eisenhart L.P., Separable systems of Stäckel,
1934
Earlier work this paper cites.
Tremblay F., Winternitz P., Third order superintegrable systems separating in polar coordinates,
Original
1989
Earlier work this paper cites.
Andrews G.E., Askey R., Roy R., Special functions,
1999
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W. Jr., Completeness of superintegrability in two-dimensional constant curvature spaces,
2001
Earlier work this paper cites.
Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G. (Editors), Superintegrability in classical and quantum systems (Montréal, 2002),
2004
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform,
2005
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems,
2006
Earlier work this paper cites.
Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold,
2006
Earlier work this paper cites.

Similar

Kalnins E.G., Kress J.M., Miller W. Jr., Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties,
Original
2007
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Kalnins E.G., Miller W. Jr., Post S., Wilson polynomials and the generic superintegrable system on the 2-sphere,
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Eastwood M., Miller W. Jr. (Editors), Symmetries and overdetermined systems of partial differential equations, Summer Program (Minneapolis, 2006),
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Kalnins E.G., Miller W. Jr., Post S., Quantum and classical models for quadratic algebras associated with second order superintegrable systems,
Original
2008
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Then

Kalnins E.G., Kress J.M., Miller W. Jr., Post S., Structure theory for second order 2D superintegrable systems with 1-parameter potentials,
Original
2009
Later among the works it cites.
Tremblay F., Turbiner A., Winternitz P., Periodic orbits for an infinite family of classical superintegrable systems,
Original
2010
Closest in time.
Quesne C., Superintegrability of the Tremblay–Turbiner–Winternitz quantum Hamiltonians on a plane for odd
Original
2010
Closest in time.
Post S., Winternitz P., An infinite family of superintegrable deformations of the Coulomb potential,
Original
2010
Closest in time.
Kalnins E.G., Miller W. Jr., Post S., Coupling constant metamorphosis and
Original

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2010

Tools for Verifying Classical and Quantum Superintegrability

Kalnins, Ernest G., Kress, Jonathan M., Miller Jr, Willard

Understand

Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n-1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable.

These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers.
Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly.
In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability.

Built on

Eisenhart L.P., Separable systems of Stäckel,
1934
Earlier work this paper cites.
Tremblay F., Winternitz P., Third order superintegrable systems separating in polar coordinates,
Original
1989
Earlier work this paper cites.
Andrews G.E., Askey R., Roy R., Special functions,
1999
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W. Jr., Completeness of superintegrability in two-dimensional constant curvature spaces,
2001
Earlier work this paper cites.
Tempesta P., Winternitz P., Harnad J., Miller W. Jr., Pogosyan G. (Editors), Superintegrability in classical and quantum systems (Montréal, 2002),
2004
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Miller W. Jr., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform,
2005
Earlier work this paper cites.
Kalnins E.G., Kress J.M., Miller W. Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems,
2006
Earlier work this paper cites.
Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold,
2006
Earlier work this paper cites.