Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Post, Sarah

doi:10.3842/sigma.2011.036

Cited by 26 publications

(27 citation statements)

References 28 publications

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“…In particular, the wave functions of superintegrable systems have been expressed in terms of OPs [15] and all superintegrable systems are conjectured to have this exactly-solvable nature [26]. More recently, this connection has been extended by studying the representation theory of the symmetry algebras associated with superintegrable systems and interbasis expansion coefficients [5,6,7,8,9,16,18,25]. In [19], the Askey scheme of classical hypergeometric OPs in one variable as well as their limits were related to superintegrable systems in 2D and the contractions of their symmetry algebras.…”

Section: Resultsmentioning

confidence: 99%

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Capel¹,

Kress²,

Post³

2015

SIGMA

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Abstract. The invariant classification of superintegrable systems is reviewed and utilized to construct singular limits between the systems. It is shown, by construction, that all superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be obtained from singular limits of a generic system on the sphere. By using the invariant classification, the limits are geometrically motivated in terms of transformations of roots of the classifying polynomials.

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Section: Resultsmentioning

confidence: 99%

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Capel¹,

Kress²,

Post³

2015

SIGMA

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“…The best known of them are the quadratically superintegrable systems, i.e., those allowing two (at most) second-order integrals of motion. Their study began in the mid 1960s [56] and by now they have been completely classified in conformally flat spaces [57,58,59,60,61,62,63,64,65,66]. In the higher-order case, the direct approach for determining integrals of motion becomes more and more difficult as their order increases, as it has recently been shown for third [67,68,69,70] and quartic order [71].…”

Section: Introductionmentioning

confidence: 99%

Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability

Marquette

Quesne

2014

Journal of Mathematical Physics

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Articles you may be interested inNew ladder operators for a rational extension of the harmonic oscillator and superintegrability of some twodimensional systems J. Math. Phys. 54, 102102 (2013) Type III multi-step rationally extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of k integers m 1 , m 2 , · · · , m k , such that m 1 < m 2 < · · · < m k with m i even (resp. odd) for i odd (resp. even), are considered. The state-adding and state-deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators. The eigenstates of the Hamiltonians are shown to separate into m k + 1 infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebras. These ladder operators are then used to build a higher-order integral of motion for seven new infinite families of superintegrable two-dimensional systems separable in cartesian coordinates. The finite-dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one-dimensional Hamiltonian eigenstates and provide an algebraic derivation of the superintegrable systems whole spectrum including the level total degeneracies. C 2014 AIP Publishing LLC.[http://dx

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“…However, there are ways that these abstract contractions can have practical significance. In the paper [32] Post shows that the structure equations for all of the quantum 2D quadratic algebras can be represented by either differential or difference operators depending on one complex variable.…”

Section: Contraction Description Of the Top Half Of The Askey Schemementioning

confidence: 99%

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Ruiz

Kalnins

Miller

et al. 2017

SIGMA

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Abstract. Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra so(4, C) to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of e(2, C) and so(3, C) and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.

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Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Cited by 26 publications

References 28 publications

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

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