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

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 $ω_x$ and $ω_y$. The new curved Hamiltonian ${H}_κ$ depends on the curvature $κ$ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of ${H}_κ$ 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 $ω_x: ω_y$, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the $κ\to 0$ 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 ${H}_κ$, meanwhile all the remaining $ω_x: ω_y$ curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian $\hat {H}_κ$ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the Hamiltonian $\hat {H}_κ$ 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 $κ$ and the Planck constant $\hbar$. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.