Superintegrability with third-order integrals 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 toh 2 , so their classical limit is free motion.

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

“…24 It was shown that in the case of one first-order and one third-order integral, the only classical superintegrable systems were known ones ͑like V = ␣r 2 or V = ␣r −1 ͒ for which the third-order integral was the product, on the Poisson commutator, of a first-and second-order one. However, in the quantum case, a new superintegrable potential of this type exists, namely V 1 ͑x , y͒ = ͑ប ͒ 2 k 2 sn 2 ͑ x , k͒, where sn͑ x , k͒ is a Jacobi elliptic function.…”

“…The classical limit ͑ប → 0͒ is free motion! 24 All classical and all quantum potentials allowing a second-order integral of the form ͑1.2͒ and a third-order integral ͑of any form ͒ were found in Ref. 25.…”

“…II to all classical superintegrable potentials that separate in Cartesian coordinates and allow a third-order integral of motion. 24,25 They have the form ͑1.1͒-͑1.3͒, so that we have a͑q 1 q 2 ͒ = c͑q 1 q 2 ͒ =1, b͑q 1 q 2 ͒ =0 in ͑2.1͒. Eight such systems exist; four of them are well known, which were first presented in Ref.…”

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials

“…A systematic study of superintegrable classical and quantum systems with a third order integral is more recent. 8,9 All classical and quantum potentials with a second and a third order integral of motion that separate in Cartesian coordinates in the twodimensional Euclidean space were found in Ref. 9.…”

A System of n = 3 Coupled Oscillators with Magnetic Terms: Symmetries and Integrals of Motion

“…Linear integrals of motion be obtained from exact Noether symmetries and quadratic integrals from Hamilton-Jacobi (Schrödinger) separability but third-order integrals must be obtaining by other alternative procedures; because of this, the number of known integrable systems admitting cubic in the momenta constants is very limited [9,10,13,21,23]. For n = 2 we can point out the Fokas-Lagerstrom and the Holt potentials for the Euclidean plane [8,12], and three of the Drach potentials for the pseudoeuclidean plane [16,19].…”