Coupling constant metamorphosis, the Stäckel transform and superintegrability

Post, Sarah

doi:10.1063/1.3537855

Cited by 13 publications

(14 citation statements)

References 23 publications

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“…A discussion of similarities (and more importantly, differences) between the CCM and the Stäckel transform can be found in Ref. [22]. Proof.…”

Section: Proof By Hypothesis the Hamiltonian Is Of The Formmentioning

confidence: 99%

Invariant classification of second-order conformally flat superintegrable systems

Capel¹,

Kress²

2014

J. Phys. A: Math. Theor.

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In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Stäckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a 7-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three-dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Gröbner basis calculations using the computer algebra software packages Maple and Singular. *

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“…A discussion of similarities (and more importantly, differences) between the CCM and the Stäckel transform can be found in Ref. [22]. Proof.…”

Section: Proof By Hypothesis the Hamiltonian Is Of The Formmentioning

confidence: 99%

Invariant classification of second-order conformally flat superintegrable systems

Capel¹,

Kress²

2014

J. Phys. A: Math. Theor.

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“…In this sense, the coupling constant α has been exchanged with the energy, which can be represented by −α, and the Stäckel transform coincides with coupling constant metamorphosis [70] for second-order superintegrable systems. See [147] for more information on the two transformations. We explain the process in greater detail in the case of a 2D classical system; for notational purposes, we denote H 0 = 1 λ (p 2 x +p 2 y ) and L 0 = a ij p i p j .…”

Section: Stäckel Transform: Proof Of Constant Curvaturementioning

confidence: 99%

Classical and quantum superintegrability with applications

Miller¹,

Post²,

Winternitz³

2013

J. Phys. A: Math. Theor.

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291

497

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A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of secondorder superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space E 2 are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.

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“…It was further shown that any second-order superintegrable system in 2D can be related, via the Stäckel transform, to a superintegrable system on a space of constant curvature and a complete list was given of all second-order superintegrable systems on 2D Euclidean space, E 2,C , and on the two sphere, S 2,C . Since we will not use the Stäckel transform explicitly in this paper, we refer the reader to [7,8] and references therein for a complete exposition. We only note here that the Stäckel transform is a mapping between Hamiltonian systems, possibly on different manifolds, which preserves superintegrability and the algebra structure of the integrals up to a permutation of the parameters and the energy.…”

Section: Introductionmentioning

confidence: 99%

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Post¹

2011

SIGMA

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Abstract. In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of Stäckel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system.

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Coupling constant metamorphosis, the Stäckel transform and superintegrability

Cited by 13 publications

References 23 publications

Invariant classification of second-order conformally flat superintegrable systems

Invariant classification of second-order conformally flat superintegrable systems

Classical and quantum superintegrability with applications

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

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