The anisotropic oscillator on curved spaces: A new exactly solvable model

TL;DR

This paper introduces a new exactly solvable classical and quantum anisotropic oscillator model on curved spaces, generalizing known flat-space systems to spherical and hyperbolic geometries with novel superintegrability properties.

Contribution

It develops a curved-space anisotropic oscillator model that is integrable and superintegrable for specific frequency ratios, extending known flat-space systems to curved geometries with explicit quantum solutions.

Findings

The model is exactly solvable in both classical and quantum cases.

Superintegrability is established for commensurate frequencies.

The quantum spectrum is explicitly derived, showing maximal degeneracy.

Abstract

We present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies $\omega_x$ and $\omega_y$. The new curved Hamiltonian ${H}_\kappa$ depends on the curvature $\kappa$ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of ${H}_\kappa$ relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies $\omega_x: \omega_y$, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the $\kappa\to 0$ limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known $1:1$ and $2:1$ anisotropic curved oscillators are recovered as particular cases of ${H}_\kappa$, meanwhile all the remaining $\omega_x: \omega_y$ curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian $\hat {H}_\kappa$ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the Hamiltonian $\hat {H}_\kappa$ turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as an analytical deformation of the Euclidean eigenvalues in terms of both the curvature $\kappa$ and the Planck constant $\hbar$. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) P\"osch-Teller set of eigenvalues.