Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

Chen, Zhe; Marquette, Ian; Zhang, Yao-Zhong

doi:10.1016/j.aop.2019.167970

Cited by 12 publications

(14 citation statements)

References 45 publications

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“…We observe that the polynomials P 1 and Q 1 do not commute with H 1 at the level of the enveloping algebra, but only as the result of considering the realization (11). Thus the (infinite-dimensional) quadratic algebra determined by ( 16) is only valid for the given realization, and is not an algebraic consequence of the underlying hidden symmetry algebra.…”

Section: Polynomial Algebras Related To the Smorodinsky-winternitz Sy...mentioning

confidence: 92%

“…Other classes of quantum models and superintegrable systems have been studied over the years using different techniques, such as systems characterized by symmetry algebras related to non Abelian polynomial algebras (see e.g. [6][7][8][9][10][11]). The underlying polynomial algebras are constructed via integrals based on explicit differential operator realizations [11], a fact that again poses a more or less severe restriction for their classification and detailed study, as the relations among elements must be understood via a differential operator algebra.…”

Section: Introductionmentioning

confidence: 99%

“…[6][7][8][9][10][11]). The underlying polynomial algebras are constructed via integrals based on explicit differential operator realizations [11], a fact that again poses a more or less severe restriction for their classification and detailed study, as the relations among elements must be understood via a differential operator algebra.…”

Section: Introductionmentioning

confidence: 99%

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Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

Campoamor-Stursberg,

Marquette

2020

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The notion of hidden symmetry algebra used in the context of exactly solvable systems (typically a non semisimple Lie algebra) is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a new approach to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian. These polynomial algebras play an important role in context of superintegrability, but are still poorly understood from an algebraic point of view. Among the main results, we present polynomial quadratic algebras of dimensions 4, 5, 6 and 8, as well as cubic algebras of dimensions 3 and 5, and various Abelian algebras, all of dimension 3. Basing on the observation how superintegrability is associated with exact solvability, we propose a procedure that connects the underlying Lie algebra with algebraic integrals of motion. As the integrals constructed in such way are now independent on the realization, alternative choices of realizations can provide new explicit models with the same symmetry algebra. In this paper, we consider examples of such equivalent Hamiltonians in terms of differential operators for the three cases and connected to the underlying Lie algebra gl(2, R) ⋉ R 2 ⊕ T1 as well as to the maximal parabolic subalgebra of gl(3, R). We also point out differences between the enveloping algebra of Lie algebras and the enveloping algebra of the related differential operators realization.

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Section: Polynomial Algebras Related To the Smorodinsky-winternitz Sy...mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

Campoamor-Stursberg,

Marquette

2020

Preprint

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“…Recently it has been shown that quadratic algebras associated with n-dimensional systems are in general of higher rank [15,16,17,18]. These algebraic structures allow one to obtain useful information on quantum systems and their degenerate spectrum [19].…”

Section: Introductionmentioning

confidence: 99%

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Latini¹,

Marquette²,

Zhang³

2022

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We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

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“…Separation of variables is a powerful tool in mathematical physics. In [128] and [140], we have applied separation of variables to obtain quadratic algebras. In this chapter, we apply the separation of variables to construct high order integral of motion.…”

Section: Introductionmentioning

confidence: 99%

New families of exactly solvable many-body models and superintegrable systems

Chen¹

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In this thesis, we study various extensions of Calogero models and superintegrable systems. We construct a kN-body one-dimensional model which reduces to the familiar Calogero model when k = 1. We present a class of many-body systems that are equivalent to harmonic oscillators. We study interesting extensions of the D-dimensional Coulomb-Kepler system and show that when the extension satisfies certain conditions, then some components of the Laplace-Runge-Lenz vector can be extended to conserved quantity of the new models. By introducing block separation of variables, we construct the Kepler-singular oscillator type models which are a new family of superintegrable systems. We also use separation of variables to obtain the energy spectrum, eigenfunctions and corresponding quadratic algebraic structures. Separation of variables is not only used for solving eigenvalue problems but also provides us with new tools for generalizing superintegrable systems. We generalize the double harmonic-singular oscillators and Kepler-singular oscillator systems. The integrals of new models now can involve angles from block spherical coordinates. A derivation of more general quadratic algebraic structures is presented as well. We also give examples to show they can be solved in terms of so-called X 1 Jacobi polynomials. ABSTRACT iii Declaration by authorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

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Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

Cited by 12 publications

References 45 publications

Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

New families of exactly solvable many-body models and superintegrable systems

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