Tools for Verifying Classical and Quantum Superintegrability - Mathematical PhysicsReport as inadecuate

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Abstract: Recently many new classes of integrable systems in n dimensions occurring inclassical and quantum mechanics have been shown to admit a functionallyindependent set of 2n-1 symmetries polynomial in the canonical momenta, so thatthey are in fact superintegrable. These newly discovered systems are allseparable in some coordinate system and, typically, they depend on one or moreparameters in such a way that the system is superintegrable exactly when someof the parameters are rational numbers. Most of the constructions to date arefor n=2 but cases where n>2 are multiplying rapidly. In this article weorganize a large class of such systems, many new, and emphasize the underlyingmechanisms which enable this phenomena to occur and to provesuperintegrability. In addition to proofs of classical superintegrability weshow that the 2D caged anisotropic oscillator and a Stackel transformed versionon the 2-sheet hyperboloid are quantum superintegrable for all rationalrelative frequencies, and that a deformed 2D Kepler-Coulomb system is quantumsuperintegrable for all rational values of a parameter k in the potential.

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


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