Periodic orbits for an infinite family of classical superintegrable systems

Abstract

We show that all bounded trajectories in the two dimensional classical system with the potential V (r, ϕ) = ω 2 r 2 + and does not depend on k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.

Superintegrability and higher order integrals for quantum systems

“…This proves superintegrability and supports recent studies by Tremblay,Turbiner and Winternitz [1,8] of the potentials with k rational where it has been demonstrated that all the orbits are closed. We also studied a new class of systems (7) and showed that again the systems are superintegrable and demonstrated how to find a maximal set of constants polynomial in the momenta.…”

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

“…As an instructive example, we apply this technique to construct the two-dimensional version of the quantum harmonic oscillator (19), focusing the analysis on its superintegrability properties. To this aim, let us apply the coproduct map to the Hamiltonian Ĥ :…”

“…As recently showed, the coalgebraic analysis can also be used to give new insights about the already known classifications of superintegrable systems: in [17] it has been shown that a canonical transformation can generate different coalgebraic systems which, once embedded in higher dimensional spaces, generate genuinely new superintegrable systems as deformations, or generalizations, of TTW systems [18,19] to non-Euclidean spaces. The same philosophy has been reproposed in [20], involving a gauge transformation applied to a two-dimensional scalar Hamiltonian.…”

Comments on Dihedral and Supersymmetric Extensions of a Family of Hamiltonians on a Plane

“…Such an operator will coincide with the operator 4ωΓ, coming from the supersymmetric extension and given in (8), provided the coefficients of ab †…”