Quadratic algebras and dynamics in curved spaces. I. Oscillator

The dynamical symmetry of a three-dimensional oscillator in a space of constant curvature is described by three operators formed from the components of the Fradkin–Higgs tensor and the generators of the quadratic Racah algebra $QR(3)$. This atgebra makes it possible to find all dynamical characteristics of the problem: the spectrum, degeneracy of the energy levels, and the overlap coefficients of the wave functions in different coordinate systems. The algebra that generates the spectrum is constructed and found to be the quadratic Jacobi algebra $QJ(3)$.