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.

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 κ→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, 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 ℏ. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

In the recent paper arXiv:1601.03357, a method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rössler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson-Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.

In this paper (arXiv:1512.06610) the factorization technique to superintegrable systems is revisited. We recall that if an integrable classical Hamiltonian H can be separated in a certain coordinate system, it is well known that each coordinate leads to an integral of the motion. Then, for each coordinate two sets of “ladder” and “shift” functions can be found. It is shown that, if certain conditions are fulfilled, additional constants of motion can be explicitly constructed in a straightforward manner by combining these functions, and such integrals are, in the general case, of higher-order on the momenta. We apply this technique to both known and new classical integrable systems, and we stress that the very same procedure can also be applied to quantum Hamiltonians leading to ladder and shift operators. In particular, we study the factorization of the classical anisotropic oscillators on the Euclidean plane and by making use of this technique we construct new classical (super)integrable anisotropic oscillators on the sphere. Finally, we also illustrate this approach through the well-known Tremblay-Turbiner-Winternitz (TTW) system on the Euclidean plane.

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. In this paper (arXiv:1511.08908) we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

In this work (arXiv:1503.09187) the constant curvature analogue on the two-dimensional sphere and the hyperbolic space of an integrable Hénon-Heiles Hamiltonian of KdV type, is revisited. The resulting integrable curved Hamiltonian depends on a parameter κ which is just the curvature of the underlying space and allows one to recover the initial Hamiltonian under the smooth flat/Euclidean limit κ→0. This system can be regarded as an integrable cubic perturbation of a specific curved 1:2 anisotropic oscillator, which was already known in the literature. The Ramani series of potentials associated to the curved Hamiltonian is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of H that are separable in parabolic coordinates. Integrable perturbations are also presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hénon-Heiles Hamiltonian. It is explicitly shown that the latter perturbations can be understood as the “negative index” counterpart of the curved Ramani series of potentials. Furthermore, it is shown that the integrability of the curved Hénon-Heiles Hamiltonian is preserved under the simultaneous addition of curved analogues of “positive” and “negative” families of Ramani potentials.

In this work (arXiv:1502.07518) we present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern–Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel’d double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding
principle towards the description of (3+1) gravity. To this aim, a canonical classical r-matrix arising from a Drinfel’d double structure for the three (3+1) Lorentzian algebras is obtained. This r-matrix turns out to be a twisted version of the one corresponding to the (3+1) kappa-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the kappa-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

A Lie system is a system of differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. In this paper (arXiv:11412.0300) we define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on R and R^2. Our results shall be illustrated through examples of physical and mathematical interest.

In this paper (arXiv:1411.7569) we review two maximally superintegrable Hamiltonian systems that are defined, respectively, on an N-dimensional spherically symmetric generalization of the Darboux surface of type III and on an N-dimensional Taub-NUT space. Afterwards, we show that their quantization leads, respectively, to exactly solvable deformations of the two basic quantum mechanical systems: the harmonic oscillator and the Coulomb problem. In both cases the quantization is performed in such a way that the maximal superintegrability of the classical  Hamiltonian is fully preserved.  In particular, we prove that this strong condition is fulfilled by  applying the so-called conformal Laplace-Beltrami quantization prescription. In this way, the eigenvalue problems for the quantum counterparts of these two Hamiltonians can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out that the maximal degeneracy of both systems is preserved under the deformation induced by the curvature. Finally, new multiparametric generalizations of both systems that preserve their superintegrability are envisaged.

In this paper (arXiv:1411.2033) we construct a constant curvature analogue  on the two-dimensional sphere and  the hyperbolic space of the integrable Hénon-Heiles Hamiltonian of KdV type. The curved integrable Hamiltonian so obtained depends on a real parameter which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system is smoothly obtained in the zero-curvature limit. On the other hand, the Hamiltonian that we propose can be regarded as an integrable perturbation of a known curved integrable 1:2 anisotropic oscillator. We stress that in order to obtain the curved  Hénon-Heiles Hamiltonian,  the preservation of the full integrability structure of the flat Hamiltonian under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani series of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved Ramani potentials will be also explicitly presented.