A unified approach to quantum and classical TTW systems based on factorizations

Celeghini, E.; Kuru, Ş.; Negro, J.; Olmo, M. A. del

doi:10.1016/j.aop.2013.01.008

Cited by 29 publications

(66 citation statements)

References 17 publications

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“…Examples of periodic and non-periodic orbits for this pasted potential are given in figure 2. For 'global' superintegrable systems all the bounded motions are periodic and the trajectories look like deformed Lissajous curves [10,11]. We should remark that such trajectories are smooth and they do not present 'angles' when crossing the boundary of a domain.…”

Section: (A) Superintegrable Hamiltonians By Adding Two Heisenberg Hamentioning

confidence: 99%

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

et al. 2017

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Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in Cartesian coordinates. A few particular cases of these types of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of n R , and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the 0 ħ → limit.

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Section: (A) Superintegrable Hamiltonians By Adding Two Heisenberg Hamentioning

confidence: 99%

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

et al. 2017

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“…In this section we review the Hamiltonian (1.1) from the classical factorization viewpoint introduced in [15][16][17][18]. Although the results here presented are quite elementary, they are useful in order to present the approach that we will follow in the curved case.…”

Section: The Classical Factorization Methodsmentioning

confidence: 99%

“…In order to get a unified approach to find the symmetries for both, the classical and quantum systems, we have applied a factorization approach [15,18,36,44]. This method turns out to be helpful in order to highlight the correspondence between the classical and quantum algebraic symmetries.…”

Section: Discussionmentioning

confidence: 99%

“…For κ > 0 we have the system defined on the sphere, for κ < 0 on the hyperboloid, while in the limit κ → 0 we will recover all the well-known Euclidean results. In Sections 2 and 3, we start by reviewing the integrability properties of classical and quantum anisotropic oscillators on the Euclidean plane E 2 through the factorization approach introduced in [10][11][12][13][14][15] (see also [16][17][18] and references therein). In Section 4, we will revisit the well-known 1 : 1 and 2 : 1 curved oscillators [6,19,20], and we will show that both Hamiltonians can be written in a simple and unified way if we express them in terms of the so called geodesic parallel coordinates on S 2 and H 2 (see [6,[21][22][23]).…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2016

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a b s t r a c tWe present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the twodimensional sphere and to the hyperbolic space of the twodimensional 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 → 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 HamiltonianĤ κ is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the HamiltonianĤ 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 * Corresponding author. Á. Ballesteros et al. / Annals of Physics 373 (2016) an analytical deformation of the Euclidean eigenvalues in terms of both the curvature κ and the Planck constanth. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

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“…Our aim is to find additional constants of motion by means of the factorization method in order to show the maximal superintegrability of the system. We recall that the application of the factorisation method to the classical realm was first proposed in [12] and then developed in [13], [14], [15], [16], [17], [18]. A first pair of constants X ± will be obtained from the effective Hamiltonians H ξ and H θ , and a second pair Y ± from the next two effective Hamiltonians, H θ and H ϕ .…”

Section: Constants Of Motionmentioning

confidence: 99%

Integrals of motion and trajectories for the Perlick system I: an algebraic approach

Ragnisco

Kuru

Negro

2018

J. Phys.: Conf. Ser.

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Abstract. In this paper we tersely recall the main algebraic and geometric properties of the maximally superintegrable system known as "Perlick System Tipe I", considering all possible values of the relevant parameters. We will follow a classical variant of the so called factorization method, emphasizing the role played the Poisson Algebra of the constants of motion in sheding light on the geometric features of the trajectories.

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A unified approach to quantum and classical TTW systems based on factorizations

Cited by 29 publications

References 17 publications

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

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

Integrals of motion and trajectories for the Perlick system I: an algebraic approach

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