Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials

Abstract

We consider a two dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) nor V_2(y) satisfy a linear ODE) allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.

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Cited by in corpus (5)

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• General $N^{th}$-order superintegrable systems separating in polar coordinates
• On superintegrable systems separable in Cartesian coordinates
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra