Fine Structure for Second Order Superintegrable Systems

Kalnins, E. G.; Kress, Jonathan M.; MillerJr., Willard

doi:10.1007/978-0-387-73831-4_5

Cited by 2 publications

(6 citation statements)

References 37 publications

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“…Table 3 shows the pattern of vanishing ideals for each of the representative systems and hence for every system in that class. The results obtained allow us to confidently state the systems listed elsewhere in the literature [5,9,13] form a complete list of Stäckel equivalent 3-dimensional, maximum-parameter, superintegrable potentials over conformally-flat spaces. By considering specialisations of the potentials above that give Stäckel multipliers for a flat metric, we reprove the result of Ref.…”

Section: Subcase [111111]mentioning

confidence: 76%

“…The classification of second-order superintegrable systems, both in classical and quantum mechanics, is a topic which is often revisited [1][2][3][4][5][6][7][8], usually with the goal of increasing our understanding of superintegrable systems in general. Superintegrablility has deep links to quasi-exact solvability, and it has been conjectured that all quasi-exact solvable systems arise from superintegrable systems [9].…”

Section: Introductionmentioning

confidence: 99%

“…The classification of two-dimensional second-order systems is complete [3,11,12] and substantial steps have been taken towards the classification of three-dimensional second-order systems [8,13]. Most of the results obtained have relied on the use of separation of variables, however some recent work in this area has focused on classifying systems based on the structure of their symmetry algebra [5,14,15] or by investigating their integrability conditions [8,13,16]. Both of these approaches may provide techniques that can be applied to non-separable systems.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Subcase [111111]mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Invariant classification of second-order conformally flat superintegrable systems

Capel¹,

Kress²

2014

J. Phys. A: Math. Theor.

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“…The structure and classification theory for these systems has just begun, with initial results for three-parameter FLI systems. 21,29 Sometimes, a quadratic algebra structure exists and sometimes it does not. Extension of these methods to complete the fine structure analysis for 3D systems appears relatively straightforward.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…In Ref. 21, we have begun the study of fine structure for second order 3D superintegrable systems, i.e., the structure and classification theory of systems with various types of degenerate potentials.…”

Section: Introductionmentioning

confidence: 99%

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

Kalnins

Kress

Miller

2007

Journal of Mathematical Physics

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A classical ͑or quantum͒ second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate ͑i.e., four-parameter͒ potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials.

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Fine Structure for Second Order Superintegrable Systems

Cited by 2 publications

References 37 publications

Invariant classification of second-order conformally flat superintegrable systems

Invariant classification of second-order conformally flat superintegrable systems

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

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