On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications

In a recent paper (J.R. Morris, Quant. Stud. Math. Found. 2 (2015) 359), an inhomogeneous compactification of the extra dimension of a five-dimensional Kaluza-Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. In this paper (arXiv:1605.06829) we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Finally, it is shown that the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.

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

We present (arXiv:1605.02384) 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. The new curved Hamiltonian depends on the curvature κ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of the system 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, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the κ0 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, meanwhile all the remaining ωx:ωy curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the quantum Hamiltonian 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 . In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.