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P. Kolokotronis

Publications and source records attributed to P. Kolokotronis.

16 recordsLinked to original sources
arXiv1997

Nonlinear deformed su(2) algebras involving two deforming functions

The most common nonlinear deformations of the su(2) Lie algebra, introduced by Polychronakos and Ro\v cek, involve a single arbitrary function of J_0 and include the quantum algebra su_q(2) as a special case. In the present contribution, less common nonlinear deformations of su(2), introduced by Delbecq and Quesne and involving two deforming functions of J_0, are reviewed. Such algebras include Witten's quadratic deformation of su(2) as a special case. Contrary to the former deformations, for which the spectrum of J_0 is linear as for su(2), the latter give rise to exponential spectra, a property that has aroused much interest in connection with some physical problems. Another interesting algebra of this type, denoted by ${\cal A}^+_q(1)$, has two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed. The resulting algebraic structure, referred to as a two-colour quasitriangular Hopf algebra, is described.

q-alg↗
arXiv1997

A nonlinear deformed su(2) algebra with a two-colour quasitriangular Hopf structure

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as ${\cal A}^+_q(1)$. This algebra is shown to possess two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, su_q(2) and ${\cal A}^+_q(1)$, is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of su_q(2) is carried over to ${\cal A}^+_q(1)$, thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and R-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix is a solution of the coloured Yang-Baxter equation, where the `colour' parameters take two discrete values associated with the two series of finite-dimensional representations.

q-alg↗
arXiv1997

Symmetries of anisotropic harmonic oscillators with rational ratios of frequencies and their relations to U(N) and O(N+1)

The concept of bisection of a harmonic oscillator or hydrogen atom, used in the past in establishing the connection between U(3) and O(4), is generalized into multisection (trisection, tetrasection, etc). It is then shown that all symmetries of the N-dimensional anisotropic harmonic oscillator with rational ratios of frequencies (RHO), some of which are underlying the structure of superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry of the corresponding isotropic oscillator with an appropriate combination of multisections. Furthermore, it is seen that bisections of the N-dimensional hydrogen atom, which possesses an O(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of the hydrogen atom lead to the symmetries of the N-dim RHO. The opposite is in general not true, i.e. multisections of U(N) do not lead to O(N+1) symmetries, the only exception being the occurence of O(4) after the bisection of U(3).

nucl-th↗
arXiv1997

Symmetries in nuclei and molecules

Recent progress in two different fronts is reported. First, the concept of bisection of a harmonic oscillator (HO) or hydrogen atom (HA), used in the past in establishing the connection between U(3) and O(4), is generalized into multisection (trisection, tetrasection, etc). It is then shown that all symmetries of the N-dim anisotropic HO with rational ratios of frequencies (RHO), some of which are underlying the structure of superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry of the corresponding isotropic HO with an appropriate combination of multisections. Furthermore, it is seen that bisections of the N-dim HA, which possesses an O(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of the HA lead to the symmetries of the N-dim RHO. Second, it is shown that there is evidence that the recently observed in superdeformed nuclear bands $ΔI=4$ bifurcation is also occuring in rotational bands of diatomic molecules. In addition there is evidence that a $ΔI=8$ bifurcation, of the same order of magnitude as the $ΔI=4$ one, is observed in superdeformed nuclear bands and rotational bands of diatomic molecules.

nucl-th↗
arXiv1997

Coupled Q-oscillators as a model for vibrations of polyatomic molecules

The system of two $Q$-deformed oscillators coupled so that the total Hamiltonian has the su$_Q$(2) symmetry is proved to be equivalent, to lowest order approximation, to a system of two identical Morse oscillators coupled by the cross-anharmonicity usually used empirically in describing vibrational spectra of triatomic molecules. The symmetry also imposes a connection between the self-anharmonicity of the Morse oscillators and the cross-anharmonicity strength, which can be removed by replacing the $Q$-oscillators by deformed anharmonic oscillators. The generalization to $n$ oscillators is straightforward. The applicability of the formalism to highly symmetric polyatomic molecules is discussed.

quant-ph↗
arXiv1996

Representations of the deformed U(su(2)) and U(osp(1,2)) algebras

The polynomial deformations of the Witten extensions of the U(su(2)) and U(osp(1,2)) algebras are three generator algebras with normal ordering, admitting a two generator subalgebra. The modules and the representations of these algebras are based on the construction of Verma modules, which are quotient modules, generated by ideals of the original algebra. This construction unifies a large number of the known algebras under the same scheme. The finite dimensional representations show new features such as the multiplicity of representations of the same dimensionality, or the existence of finite dimensional representations only for some dimensions.

q-alg↗
arXiv1995

Generalized Deformed Oscillators and Algebras

The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2 supersymmetric quantum mechanics (SUSY-QM) scheme. In addition, several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to possess the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.

hep-th↗
arXiv1995

Quantum Algebras in Nuclear Structure

Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools ($q$-numbers, $q$-analysis, $q$-oscillators, $q$-algebras), the su$_q$(2) rotator model and its extensions, the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model), the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to molecular structure and an outlook are also given.

nucl-th↗
arXiv1995

Quantum Algebraic Symmetries in Nuclei and Molecules

Various applications of quantum algebraic techniques in nuclear and molecular physics are briefly reviewed. Emphasis is put in the study of the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, the Bloch--Brink $α$-cluster model and possibly the shell structure in deformed atomic clusters.

nucl-th↗
arXiv1995

Quantum Algebraic Symmetries in Nuclear Structure

Various applications of quantum algebraic techniques in nuclear structure physics, such as the su$_q$(2) rotator model and its extensions, the use of deformed bosons in the description of pairing correlations, and the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model) are briefly reviewed. Emphasis is put in the study of the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underlie the structure of superdeformed and hyperdeformed nuclei, the Bloch--Brink $α$-cluster model and possibly the shell structure in deformed atomic clusters.

nucl-th↗
arXiv1995

The Use of Quantum Groups in Nuclear Structure Problems

Various applications of quantum algebraic techniques in nuclear structure physics, such as the su$_q$(2) rotator model and its extensions, the use of deformed bosons in the description of pairing correlations, and the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model) are briefly reviewed. Emphasis is put in the study of the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, the Bloch--Brink $α$-cluster model and possibly the shell structure in deformed atomic clusters.

q-alg↗
arXiv1995

Generalized Deformed su(2) Algebras, Deformed Parafermionic Oscillators and Finite W Algebras

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to posses the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.

hep-th↗
arXiv1994

Nonlinear extension of the u(2) algebra as the symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies and ``pancake'' nuclei

The symmetry algebra of the two-dimensional anisotropic quantum harmonic oscillator with rational ratio of frequencies, which is characterizing ``pancake'' nuclei, is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are determined using algebraic methods of general applicability to quantum superintegrable systems. For labelling the degenerate states an ``angular momentum'' operator is introduced, the eigenvalues of which are roots of appropriate generalized Hermite polynomials. In the special case with frequency ratio 2:1 the resulting algebra is identified as the finite W algebra W$_3^{(2)}$.

nucl-th↗
arXiv1994

The symmetry algebra of the N-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies and the Nilsson model

The symmetry algebra of the N-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies is constructed by a method of general applicability to quantum superintegrable systems. The special case of the 3-dim oscillator is studied in more detail, because of its relevance in the description of superdeformed nuclei and nuclear and atomic clusters. In this case the symmetry algebra turns out to be a nonlinear extension of the u(3) algebra. A generalized angular momentum operator useful for labeling the degenerate states is constructed, clarifying the connection of the present formalism to the Nilsson model in nuclear physics.

hep-th↗