Mariano Santander

A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator

A nonpolynomial one-dimensional quantum potential representing an oscillator, which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends on a parameter a, is considered and then a particular case is studied with great detail. It is proven that it is Schr?dinger solvable and then the wavefunctions ?n and the energies En of the bound states are explicitly obtained. Finally, it is proven that the solutions determine a family of orthogonal polynomials related to the Hermite polynomials and such that: (i) every is a linear combination of three Hermite polynomials and (ii) they are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators

A non-linear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S2 and hyperbolic plane H2, is discussed.

Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2

The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2. The approach used is based on enforcing the conditions for the existence of two independent integrals (further than the energy). This is done in a way which allows us to discuss at once the cases of the sphere S2 and the hyperbolical plane H2, by considering the curvature κ as a parameter. Different superintegrable potentials are obtained as the solutions of certain systems of two κ-dependent second order partial differential equations. The Euclidean results are directly recovered for κ=0, and the superintegrable potentials on either the standard unit sphere (radius R=1) or the unit Lobachewski plane (“radius” R=1) appear as the particular values of the κ-dependent superintegrable potentials for the values κ=1 and κ=−1. Some new superintegrable potentials are f...

Lagrangian formalism for nonlinear second-order Riccati systems: One-dimensional integrability and two-dimensional superintegrability

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are non-natural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures

Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere S2, or on the hyperbolic plane H2, when particularized for κ>0, or κ<0, respectively; in addition, the Euclidean case arises as the particular case κ=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods, first by direct integration and second by obtaining the κ-dependent version of the Binet’s equation. The final part of the paper, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.

One-dimensional model of a quantum nonlinear harmonic oscillator*

Abstract In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the λ → 0 limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.

The quantum harmonic oscillator on the sphere and the hyperbolic plane

Abstract A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμλ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrodinger equation is studied. Two λ-dependent Sturm–Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψm,n and the energies Em,n of the bound states are exactly obtained in both the sphere S2 and the hyperbolic plane H2.

Maximal superintegrability on N-dimensional curved spaces

A unified algebraic construction of the classical Smorodinsky–Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO(N, 1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space N+1, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N + 1 oscillators. Furthermore, each Lie algebra generator provides an integral of motion and a set of 2N − 1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky–Winternitz system is shown for any value of the curvature.

Conformal symmetries of spacetimes

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley–Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton–Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Möbius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley–Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional ‘conformal ambient space’, which in turn leads to an explicit description of the ‘conformal completion’ or compactification of the nine spaces.In this paper, we give a unified and global new approach to the study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, and both Newton–Hooke and Galilean). We obtain general expressions within a Cayley–Klein framework, holding simultaneously for all these nine spaces, whose cycles (including geodesics and circles) are explicitly characterized in a new way. The corresponding cycle-preserving symmetries, which give rise to (Mobius-like) conformal Lie algebras, together with their differential realizations are then deduced without having to resort to solving the conformal Killing equations. We show that each set of three spaces with the same signature type and any curvature have isomorphic conformal algebras; these are related through an apparently new conformal duality. Laplace and wave-type differential equations with conformal algebra symmetry are finally constructed.

Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries

A simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two-dimensional Cayley-Klein geometries. Their central extensions are also considered under this unified pattern. In both cases some fundamental properties characterizing the classical CK geometries (as the existence of a set of commuting involutions, contractions and dualities relationships), remain in the quantum version.