Third order superintegrable systems separating in polar coordinates

Abstract

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlevé transcendent or in terms of the Weierstrass elliptic function.

References in corpus (11)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Superintegrability and higher order constants for classical and quantum systems
• Periodic orbits for an infinite family of classical superintegrable systems
• Superintegrable Systems with a Third Order Integrals of Motion
• Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$
• Superintegrability and higher order constants for quantum systems
• Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Families of classical subgroup separable superintegrable systems
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

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