Demkov–Fradkin tensor for curved harmonic oscillators

In this work, we obtain the Demkov–Fradkin tensor of symmetries for the quantum curved harmonic oscillator in a space with constant curvature given by a parameter \(\kappa\). In order to construct this tensor, we have firstly found a set of basic operators which satisfy the following conditions: (i) Their products give symmetries of the problem; in fact, the Hamiltonian is a combination of such products; (ii) they generate the space of eigenfunctions as well as the eigenvalues in an algebraic way; (iii) in the limit of zero curvature, they come into the well-known creation/annihilation operators of the flat oscillator. The appropriate products of such basic operators will produce the curved Demkov–Fradkin tensor. However, these basic operators do not satisfy Heisenberg commutators but close another Lie algebra which depends on \(\kappa\). As a by-product, the classical Demkov–Fradkin tensor for the classical curved harmonic oscillator has been obtained by the same method. The case of two dimensions has been worked out in detail: Here, the operators close a \(so_\kappa (4)\) Lie algebra; the spectrum and eigenfunctions are explicitly solved in an algebraic way and in the classical case the trajectories have been computed.