Superposition of super-integrable pseudo-Euclidean potentials in N = 2 with a fundamental constant of motion of arbitrary order in the momenta

Abstract

It is shown that for any \documentclass[12pt]{minimal}\begin{document}$\alpha ,\beta \in \mathbb {R}$\end{document}α,β∈R and \documentclass[12pt]{minimal}\begin{document}$k\in \mathbb {Z}$\end{document}k∈Z, the Hamiltonian \documentclass[12pt]{minimal}\begin{document}$H_{k}=p_{1}p_{2}\break-\alpha q_{2}^{(2k+1)}q_{1}^{(-2k-3)}-\frac{\beta }{2} q_{2}^{k}q_{1}^{(-k-2)}$\end{document}Hk=p1p2−αq2(2k+1)q1(−2k−3)−β2q2kq1(−k−2) is super-integrable, possessing fundamental constants of motion of degrees 2 and 2k + 2 in the momenta.

Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

“…Fris et al studied [1] the two-dimensional Euclidean systems, which admit separability in more than one coordinate systems and obtained four families of potentials V r , r = a, b, c, d, possessing three functionally independent integrals of motion (they were mainly interested in the quantum two-dimensional Schrödinger equation but the results obtained are also valid at the classical level). Then other authors studied similar problems on higher-dimensional Euclidean spaces [2]- [4], on two-dimensional spaces with a pseuo-Euclidean metric (Drach potentials) [5]- [8], or on curved spaces [9]- [15] (see [16] for a recent review on superintegrability that includes a long list of references).…”

On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

“…Hence, by solving As immediate corollaries of the fact that the Hamiltonian (1.2) is an extension of (3.8), we get additional information about (1.2) (i) the function L is a quadratic first-integral of H, in accordance with the results of [2];…”

“…The procedure of Laplace-Beltrami quantization of extended Hamiltonians and their associated first integrals of high-degree developed in [10] on flat manifolds is applied to the present case, so the quantum superintegrability of (1.2) and its generalisations on flat manifolds is proved. The problem of the quantum superintegrability of (1.2) was left open in [2]. The quantization of extended Hamiltonians on curved manifolds, such as (4.10) and (4.11), is still an open problem.…”

“…A partial generalization of this Hamiltonian has been later proved to be again superintegrable in [2]. This is the Hamiltonian…”

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

“…Second, the three κ-dependent Runge-Lenz functions (24) obtained in the previous section, and characterizing to the potential k/ Tκ(r), are no longer integrals of motion. Now we prove that this system admits three quartic constants of motion.…”