Superintegrability with third order invariants in quantum and classical mechanics

Abstract

We consider here the coexistence of first- and third-order integrals of motion in two dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to \hbar^2, so their classical limit is free motion.

15 pages

References in corpus (5)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Quantum superintegrability and exact solvability in N dimensions
• A unified treatment of quartic invariants at fixed and arbitrary energy
• Superintegrable models of Winternitz type

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