The relation between polynomial deformations of<i>sl</i>(2,<i>R</i>) and quasi-exact solvability

Debergh, N.

doi:10.1088/0305-4470/33/40/308

Cited by 20 publications

(36 citation statements)

References 22 publications

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“…The non-negative integer d, introduced above for convenience, has to take specific values according to J and l. These values have been given in [7]. Table 2.…”

Section: Discussionmentioning

confidence: 99%

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On the exact solutions of the Lipkin–Meshkov–Glick model

Debergh¹,

Stancu²

2001

J. Phys. A: Math. Gen.

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We present the many-particle Hamiltonian model of Lipkin, Meshkov and Glick in the context of deformed polynomial algebras and show that its exact solutions can be easily and naturally obtained within this formalism. The Hamiltonian matrix of each j multiplet can be split into two submatrices associated to two distinct irreps of the deformed algebra. Their invariant subspaces correspond to even and odd numbers of particle-hole excitations.

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“…The non-negative integer d, introduced above for convenience, has to take specific values according to J and l. These values have been given in [7]. Table 2.…”

Section: Discussionmentioning

confidence: 99%

“…3. One of the aims of [7] was to construct finite dimensional representations of the polynomial deformed algebra …”

Section: Appendixmentioning

confidence: 99%

On the exact solutions of the Lipkin–Meshkov–Glick model

Debergh¹,

Stancu²

2001

J. Phys. A: Math. Gen.

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“…[12][13][14][15][16][17][18][19][20] An extension of q-deformed algebras with the right-hand side of their commutators as nonlinear expressions in terms of the generators, was soon introduced. [21][22][23][24][25][26][27][28][29][30][31][32] The azimuthal and magnetic quantum numbers l and m of spherical harmonics have performed a remarkable role in many branches of physics and the other grounds such as chemistry. It is the aim of this paper to consider the role of them in realization of nonlinear deformations of su(2) algebra via transitions between spherical harmonics Y m l (θ, φ) whose 2l ∓ m or 3l ∓ m or etc are given values.…”

Section: Motivationmentioning

confidence: 99%

APPROACH OF THE AZIMUTHAL AND MAGNETIC QUANTUM NUMBERS l AND m TO REPRESENTATION OF A CUBIC DEFORMATION OF su(2) ALGEBRA

Fakhri

Hashemzadeh

2009

Int. J. Mod. Phys. A

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It is shown that the space of spherical harmonics Y m l (θ, φ) whose 2l − m = p − 1 is given, represent irreducibly a cubic deformation of su(2) algebra, the so-called su Φp (2), with deformation function as Φp(x) = 27 2 x 2 + 3(7 − 3p 2 )x. The irreducible representation spaces are classified in three different bunches, depending on one of values 3k − 2, 3k − 1 and 3k, with k as a positive integer, to be chosen for p. So, three different methods for generating the spectrum of spherical harmonics are presented by using the cubic deformation of su(2). Moreover, it is shown that p plays the role of deformation parameter.The group structure of spectrum-generating algebras are used to determine quantum structure, including energy levels and allowed transitions in the bound states. Quantum algebras (deformations of Lie algebras) and their representations have grown into a major area of research in quantum dynamical systems, statistical mechanics, conformal quantum field theory, quantum optics, atomic and nuclear spectroscopies, etc. They generally depend on one or more phenomenological parameters, the so-called deformation parameters. The known Lie algebras can be obtained as a limit case of the quantum algebras, when deformation parameter tends to a limiting value. For example, su q (2) with q near to 1, as a q-analogue of angular momentum algebra describes the fine structure effects. q-oscillator algebras and their representations were first proposed by Arik and Coon 1 and subsequently considered by other authors (for example, see ). Also, Mcfarlane 4727 Int. J. Mod. Phys. A 2009.24:4727-4736. Downloaded from www.worldscientific.com by MCGILL UNIVERSITY on 06/26/16. For personal use only.

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“…In this respect, notice that, in fact, such an identity operator should also be introduced in the commutation relations of sl (3) (2, R) (1) multiplying the parameter δ, but due to the above reason this is not usually written explicitly in the literature [1,2,3].…”

Section: B Harmonic Limit Of the Triaxial Rotormentioning

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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The relation between polynomial deformations ofsl(2,R) and quasi-exact solvability

Cited by 20 publications

References 22 publications

On the exact solutions of the Lipkin–Meshkov–Glick model

On the exact solutions of the Lipkin–Meshkov–Glick model

APPROACH OF THE AZIMUTHAL AND MAGNETIC QUANTUM NUMBERS l AND m TO REPRESENTATION OF A CUBIC DEFORMATION OF su(2) ALGEBRA

Generalized rotational Hamiltonians from nonlinear angular momentum algebras

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