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 $C\left(3\right)\oplus L_1\oplus L_2$ 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. (paper)