Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

Debergh, N.; Bossche, Bruno Van Den

doi:10.1016/s0003-4916(03)00176-3

Cited by 7 publications

(11 citation statements)

References 17 publications

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“…and for which choice of m, n, f (x) do these operators posses a relation with the enveloping algebra of some Lie (or "deformed" Lie) algebra ?. This question generalizes the cases of monomials adressed in [3] and more recently in [4] At the moment the question is, to our knowledge, not solved in its generality but we present a few non trivial solutions in the next section.…”

Section: Introductionmentioning

confidence: 74%

“…The general cases of vector spaces constructed over monomials was first adressed in [3] and the particular subcase f (x) = x a was reconsidered recently [4]. The corresponding vector space was denoted V (1) in [4]; here we will reconsider this case and extend the discussion of the operators which leave it invariant. For later convenience, it is usefull to introduce more precise notations, setting P n ≡ P(n, x) and…”

Section: Case F (X) =mentioning

confidence: 99%

“…in passing, note that the notations of [4] are n = s and m = N − s − 2. The vector space above is clearly constructed as the direct sum of two subspaces.…”

Section: Case F (X) =mentioning

confidence: 99%

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Qes Systems, Invariant Spaces and Polynomials Recursions

2007

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Let us denote V, the finite dimensional vector spaces of functions of the form ψ(x) = p n (x) + f (x)p m (x) where p n (x) and p m (x) are arbitrary polynomials of degree at most n and m in the variable x while f (x) represents a fixed function of x. Conditions on m, n and f (x) are found such that families of linear differential operators exist which preserve V. A special emphasis is accorded to the cases where the set of differential operators represents the envelopping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurence relations. When the operator is quasi exactly solvable, it possesses a finite dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

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Section: Introductionmentioning

confidence: 74%

Section: Case F (X) =mentioning

confidence: 99%

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Qes Systems, Invariant Spaces and Polynomials Recursions

2007

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“…Moreover, if we realize this Casimir in terms of Poincaré or Beltrami coordinates, we obtain that C sl (3) (2,R) = K 2 , which is just the square of the coupling constant of the KC potential. It is worth to stress that the Higgs algebra has been deeply studied and applied to different quantum physical models (beyond integrable systems) with an underlying nonlinear angular momentum symmetry (see [24,25,26,27,28,29,30] and references therein).…”

Section: Nonlinear Angular Momentum Symmetrymentioning

confidence: 99%

Maximal superintegrability of the generalized Kepler–Coulomb system onN-dimensional curved spaces

Ballesteros¹,

Herranz²

2009

J. Phys. A: Math. Theor.

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The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the N D spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N centrifugal terms, is shown to be endowed with 2N − 1 functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler-Coulomb potential, with its associated quadratic Laplace-Runge-Lenz N -vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic), and coalgebra symmetries in all these systems is highlighted.

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“…The search for physical inspired Hamiltonians, which may display definite features about polynomial algebras is still open. Concrete applications of the formalism have been explored more recently in the context of schematic models [6], to complement previous mathematical efforts, like the studies of [7,8,9]. Among the already studied applications of the concept of nonlinear algebras, the results of [9] can be taken as definite motivations for our present effort.…”

Section: Introductionmentioning

confidence: 99%

Generalized rotational Hamiltonians from nonlinear angular momentum algebras

et al. 2007

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Higgs algebras are used to construct rotational Hamiltonians. The correspondence between the spectrum of a triaxial rotor and the spectrum of a cubic Higgs algebra is demonstrated. It is shown that a suitable choice of the parameters of the polynomial algebra allows for a precise identification of rotational properties. The harmonic limit is obtained by a contraction of the algebra, leading to a linear symmetry.

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Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

Cited by 7 publications

References 17 publications

Qes Systems, Invariant Spaces and Polynomials Recursions

Qes Systems, Invariant Spaces and Polynomials Recursions

Maximal superintegrability of the generalized Kepler–Coulomb system onN-dimensional curved spaces

Generalized rotational Hamiltonians from nonlinear angular momentum algebras

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