A systematic search for nonrelativistic systems with dynamical symmetries

Makarov, Alexander; Smorodinsky, J.; Valiev, Kh.; Winternitz, P.

doi:10.1007/bf02755212

Cited by 361 publications

(389 citation statements)

References 6 publications

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“…3,4 A systematic search for superinte-grable systems in E 2 and E 3 was conducted in the 1960s. [5][6][7] Like much of the later work on superintegrable systems, it was restricted to the case of second-order integrals of motion. [8][9][10][11][12][13] This case turned out to have an intimate connection with the separation of variables in the HamiltonJacobi and Schrödinger equations.…”

Section: ͑14͒mentioning

confidence: 99%

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Marquette

Winternitz

2007

Journal of Mathematical Physics

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Section: ͑14͒mentioning

confidence: 99%

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Marquette

Winternitz

2007

Journal of Mathematical Physics

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“…But we could have also started at the other end, i.e., with the pure anisotropic oscillator, which is separable, for 2 : 1 ratio of the frequencies, in both Cartesian and (semi)parabolic coordinates [10,11]. Then we could have observed that separability in (semi)parabolic coordinates is consistent with a Kepler potential of arbitrary strength, viewed as a perturbation of our initial oscillator.…”

Section: Resultsmentioning

confidence: 95%

“…(iii) The third quantity is, once again, the half of the squared z component of the angular momentum. The familiar Keplerian quantities [7] and those of the 2 : 1 anisotropic oscillator [10,11] are recovered when V osc = 0 or when the Kepler potential is switched off, a = 0, respectively. Some classical trajectories will be presented in Sect.…”

Section: )mentioning

confidence: 97%

Separability and dynamical symmetry of Quantum Dots

et al. 2014

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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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“…A classical superintegrable system H = ͚ ij g ij p i p j + V͑x͒ on an n-dimensional local Riemannian manifold is one that admits 2n − 1 functionally independent symmetries ͑i.e., constants of the motion͒ S k , k =1, ... ,2n − 1 with S 1 = H. That is, ͕H , S k ͖ = 0 where ͕f,g͖ = ͚ j=1 n ‫ץ͑͑‬ x j f‫ץ‬ p j g − ‫ץ‬ p j f‫ץ‬ x j g͒͒ is the Poisson bracket for functions f͑x , p͒ , g͑x , p͒ on phase space. [1][2][3][4][5][6][7][8] Note that 2n − 1 is the maximum possible number of functionally independent symmetries and, locally, such symmetries always exist. The main interest is in symmetries that are polynomials in the p k and are globally defined, except for lower dimensional singularities such as poles and branch points.…”

Section: Introduction and Examplesmentioning

confidence: 99%

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

Kalnins

Kress

Miller

2005

Journal of Mathematical Physics

112

193

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This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general twodimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2 ϫ 2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

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A systematic search for nonrelativistic systems with dynamical symmetries

Cited by 361 publications

References 6 publications

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

Separability and dynamical symmetry of Quantum Dots

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

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