Superintegrability and higher order polynomial algebras

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painlevé transcendent