Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Abstract

The maximal superintegrability of the intrinsic harmonic oscillator potential on N-dimensional spaces with constant curvature is revisited from the point of view of sl(2)-Poisson coalgebra symmetry. It is shown how this algebraic approach leads to a straightforward definition of a new large family of quasi-maximally superintegrable perturbations of the intrinsic oscillator on such spaces. Moreover, the generalization of this construction to those N-dimensional spaces with non-constant curvature that are endowed with sl(2)-coalgebra symmetry is presented. As the first examples of the latter class of systems, both the oscillator potential on an N-dimensional Darboux space as well as several families of its quasi-maximally superintegrable anharmonic perturbations are explicitly constructed.

MICZ-Kepler systems in noncommutative space and duality of force laws

“…Well known result for the central potentials that give rise to bounded closed orbits have been generalised to the general class of Hamiltonians with either a Kepler-type or a harmonic oscillator-type potential [18,19]. Investigation to analyse the notion of integrability beyond classical systems are pursued in recent times [20][21][22][23][24][25][26][27]. Generalisation of the concept of integrability to the relativistic domain was initiated by the study of Bertrand space-time which serves as a platform for the generalisation of Bertrand's theorem [28].…”

From ordinary to discrete quantum mechanics: The Charlier oscillator and its coalgebra symmetry

“…Moreover, since we showed that the coalgebra symmetry approach can also be used in the realm of discrete quantum mechanics, we will investigate the possibility of constructing (quasi)-Maximally and, hopefully, Maximally Superintegrable generalization of one-dimensional discrete system with an underlying coalgebra symmetry. Along this direction, another interesting point is related to the fact that our discrete representation can also be used to introduce superintegrable discrete versions of oscillators that are defined on curved spaces [33]. Work on all these lines is in progress.…”

Classical and Quantum Superintegrability of Stäckel Systems

“…(iii) For m = n + 1 there is no constant solution of L Y G = 0. This equation has a simple linear in q solution (3.11) provided that b 0 = b 1 = 0 but Y is not a symmetry for any nontrivial potential V (k) 1 . In consequence, the function…”

“…This paper is devoted to n-dimensional maximally superintegrable classical and quantum Stäckel systems with all constants of motion quadratic in momenta. Although superintegrable systems of second order, both classical and quantum, have been intensively studied (see for example [1,2,11,14,16,17] and the review paper [19]), nevertheless all the results about superintegrable Stäckel systems (including the important classification results) were mainly restricted to two or three dimensions or focused on the situation when the Hamiltonian is a sum of one degree of freedom terms and therefore itself separates in the original coordinate arXiv:1608.04546v2 [nlin.SI] 30 Jan 2017 system (see for example [3,12] or [15]). Here we present some general results concerning ndimensional classical separable superintegrable systems in flat spaces, constant curvature spaces and conformally flat spaces.…”