Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler-Coulomb model with non-central terms and the double singular oscillators of type (n, N - n). We show how the integrals of motion generate higher rank cubic algebra C(3) circle plus L-1 circle plus L-2 with structure constants involving Casimir operators of the Lie algebras L-1 and L-2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.