Demkov-Fradkin tensor for curved harmonic oscillators

TL;DR

This paper constructs the Demkov-Fradkin tensor for quantum and classical curved harmonic oscillators in constant curvature spaces, revealing algebraic structures and explicit solutions for two-dimensional cases.

Contribution

It introduces a method to derive the Demkov-Fradkin tensor for curved oscillators, connecting basic operators with Lie algebra structures and providing explicit solutions.

Findings

Operators form an $so_4$ Lie algebra in 2D case

Spectrum and eigenfunctions are explicitly obtained

Classical trajectories are computed

Abstract

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. 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: 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.