On harmonic oscillators on the two-dimensional sphere S 2 and the hyperbolic plane H 2 . II.Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a twodimensional 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 nonlinear 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 nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere S 2 and hyperbolic plane H 2 , is discussed.MSC Classification: 37J35, 34A34, 34C15, 70H06 a)

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 Schrödinger solvable and then the wavefunctions Ψn 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.

The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx 2 ) −1 and with a λ-dependent nonpolynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. Firstly, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.

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A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as 'curvature/signature (in)dependent trigonometry' and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an 'absolute trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton-Hooke and galilean spacetimes follow as particular instances of the general approach.Distinctive traits of the method are 'universality' and '(self-)duality': every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical (riemannian) spaces also has a version for the remaining six 'spacetimes'; in most cases these equations are new. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail.The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry; this can be successfully applied for other rank-one spaces as well (e.g. the complex type, as the quantum space of states), and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.1

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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 nonnatural 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 behaviour 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