Factorization approach to superintegrable systems: Formalism and applications

In this paper (arXiv:1512.06610) the factorization technique to superintegrable systems is revisited. We recall that if an integrable classical Hamiltonian H can be separated in a certain coordinate system, it is well known that each coordinate leads to an integral of the motion. Then, for each coordinate two sets of “ladder” and “shift” functions can be found. It is shown that, if certain conditions are fulfilled, additional constants of motion can be explicitly constructed in a straightforward manner by combining these functions, and such integrals are, in the general case, of higher-order on the momenta. We apply this technique to both known and new classical integrable systems, and we stress that the very same procedure can also be applied to quantum Hamiltonians leading to ladder and shift operators. In particular, we study the factorization of the classical anisotropic oscillators on the Euclidean plane and by making use of this technique we construct new classical (super)integrable anisotropic oscillators on the sphere. Finally, we also illustrate this approach through the well-known Tremblay-Turbiner-Winternitz (TTW) system on the Euclidean plane.