Dynamical invariance algebra of the Hartmann potential

The separability and Runge-Lenz-type dynamical symmetry of the internal dynamics of certain two-electron Quantum Dots, found by Simonović et al. [1], is traced back to that of the perturbed Kepler problem. A large class of axially symmetric perturbing potentials which allow for separation in parabolic coordinates can easily be found. Apart of the 2:1 anisotropic harmonic trapping potential considered in [1], they include a constant electric field parallel to the magnetic field (Stark effect), the ring-shaped Hartmann potential, etc. The harmonic case is studied in detail.

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“…[17,18]. The spherical Stäckel quantities are (2.10) except for the last contribution to w which, fixed by the potential,…”

Section: Further Separable Perturbationsmentioning

confidence: 99%

Separability and dynamical symmetry of Quantum Dots

et al. 2014

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“…Among the simplest superintegrable ones are the Coulomb system and the harmonic oscillator in spaces of any dimension. The celebrated many-body Calogero model is superintegrable as well as the Hartmann potential of quantum chemistry [10,9].…”

Section: Introductionmentioning

confidence: 99%

Exact solvability of superintegrable systems

Tempesta

Turbiner

Winternitz

2001

Journal of Mathematical Physics

136

204

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It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the sl(3)-algebra, realized by first order differential operators.

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“…Let us consider a canonical extension, the polar-polar transformation (q 1 , q 2 , q 3 , q 4 ) → (ρ 1 , ρ 2 , α 1 , α 2 ), considered previously by Kibler and Winternitz (1987), and given by…”

Section: Still Another Canonical Extension Superintegrabilitymentioning

confidence: 99%

“…When we take Q = 0 in potential V 1 we have the Hartmann (1972) model. Continuing the study done by Kibler and Negali (1984a), the solution was detailed by Kibler and Winternitz (1987) in parabolic coordinates. With respect to potential V 2 , the case P = 0 has been studied by Quesne (1988).…”

Section: Introductionmentioning

confidence: 99%

Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Periodic orbits

Tresaco

Ferrer

2010

Celest Mech Dyn Astr

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To cite this version:Eva Tresaco, Sebastián Ferrer. Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Abstract A generalized integrable biparametric family of 4-D isotropic oscillators is proposed. It allows to treat in a unified way, Pöschl-Teller, Hartmann and other ring-shaped systems. This approach, based in the use of two canonical extensions, helps to simplify the studies of classical aspects of those systems. As an illustration, an analysis of the periodic solutions of those system is presented.

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Dynamical invariance algebra of the Hartmann potential

Cited by 108 publications

References 19 publications

Separability and dynamical symmetry of Quantum Dots

Separability and dynamical symmetry of Quantum Dots

Exact solvability of superintegrable systems

Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Periodic orbits

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