Papers by Costas Daskaloyannis
We study harmonic maps between Riemann surfaces, when the curvature in the target is a negative c... more We study harmonic maps between Riemann surfaces, when the curvature in the target is a negative constant. In [7], harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.
Geometric Methods in Physics, 2009
We demonstrate that quasi-exactly solvable models of quantum mechanics can be used in nonlinear o... more We demonstrate that quasi-exactly solvable models of quantum mechanics can be used in nonlinear optical processes for a down conversion or second-harmonic generation processes. After the reduction we use the Turbiner and Bender -Dunne polynomial approach. The eigenvalues of Hamiltonian for low number of photons are calculated and the approximation formula is found out.
Generalized Lie Theory in Mathematics, Physics and Beyond, 2009
Definitions of the parastatistics algebras and known results on their Lie (super)algebraic struct... more Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category H M (H: a quasitriangular Hopf algebra) into an ordinary Hopf algebra is presented and it is applied in the case of the parabosonic algebra. A bosonisation-like construction is also introduced for the same algebra and the differences are discussed.
AIP Conference Proceedings, 2010
The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafer... more The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will be investigated and a braided group structure will be described for it. A new family of realizations of an arbitrary Lie superalgebra will be presented and it will be shown that these realizations possess the valuable representation-theoretic property of transferring invariably the super-Hopf structure. Finally two classes of virtual applications will be outlined: The first is of interest for both mathematics and mathematical physics and deals with the representation theory of infinite dimensional Lie superalgebras, while the second is of interest in theoretical physics and has to do with attempts to determine specific classes of solutions of the Skyrme model.
Modern Physics Letters B, 1996
The algebra of observables of a system of two identical vortices in a superfluid thin film is des... more The algebra of observables of a system of two identical vortices in a superfluid thin film is described as a generalized deformed oscillator with a structure function containing a linear (harmonic oscillator) term and a quadratic term. In contrast to the deformed oscillators occurring in other physical systems (correlated fermion pairs in a single-j nuclear shell, Morse oscillator), this oscillator is not amenable to perturbative treatment and cannot be approximated by quons. From the mathematical viewpoint, this oscillator provides a novel boson realization of the algebra su (1,1).
Chemical Physics Letters, 1991
ABSTRACT
In the three-dimensional flat space, a classical Hamiltonian, which has five functionally indepen... more In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages) have proved that, in the case of nondegenerate potentials, i.e. potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler-Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8 pages) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the sa...
Journal of Physics A: Mathematical and General, 1999
AbstracL Pairing in a single-j shell is described in terms of two Q-oscillators, one describing t... more AbstracL Pairing in a single-j shell is described in terms of two Q-oscillators, one describing the J= 0 fermion pairs and the other conesponding to the J# 0 pairs, the deformation parameter T= In Q being related to the inverse of the size of the shell. Using these two oscillators an SUp (2) algebra is constructed, while a pairing Hamiltonian aving the comt energy eigenvalues up to terms of first order in the small parameter can be written in terms of the Casimir operators of the algebras appearing in the up@) 3 Up (1) chain, thus ...
In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf di... more In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefore is a harmonic function. The real part of the logarithm of the Beltrami function satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is an elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way. The harmonic maps are therefore classified by the classification of the solutions of the sinh-Gordon equation.
Arxiv preprint hep-th/9411218, Nov 29, 1994
Abstract: The symmetry algebra of the N-dimensional anisotropic quantum harmonic oscillator with ... more Abstract: 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.
Journal of Mathematical Physics, 2001
The integrals of motion of the classical two dimensional superintegrable systems with quadratic i... more The integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The quadratic Poisson algebra is deformed to a quantum associative algebra, the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is shown that, the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for all two dimensional superintegrable systems with quadratic integrals of motion.
arXiv: Mathematical Physics, 2009
The three dimensional superintegrable systems with quadratic integrals of motion have five functi... more The three dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non degenerate potentials there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. We show that in all the non degenerate cases (with one exception) there are at least two subalgebras of three integrals having a Poisson quadratic algebra structure, which is similar to the two dimensional case.
A rigorous algebraic description of the notion of realization, specialized in the case of Lie sup... more A rigorous algebraic description of the notion of realization, specialized in the case of Lie superalgebras is given. The idea of the Relative Parabose set PBF is recalled together with some recent developments and its braided group structure is established together with an extended discussion of its (Z2 ×Z2)-grading. The final result of the paper employs PBF in order to realize an arbitrary Lie superalgebra. It is furthermore shown that the constructed realization is a Z2-graded Hopf algebra homomorphism.
arXiv: Mathematical Physics, 2009
In the three dimensional flat space any classical Hamiltonian, which has five functionally indepe... more In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller [1] have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. The Kepler Coulomb potential that was introduced by Verrier and Evans [2] is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the non degenerate case of systems with quadratic integrals.
Definitions of the parastatistics algebras and known results on their Lie (super)algebraic struct... more Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category ${}_{H}\mathcal{M}$ ($H$: a quasitriangular Hopf algebra) into an ordinary Hopf algebra is presented and it is applied in the case of the parabosonic algebra. A bosonisation-like construction is also introduced for the same algebra and the differences are discussed.
The general properties of the ordinary and generalized parafermionic algebras are discussed. The ... more The general properties of the ordinary and generalized parafermionic algebras are discussed. The generalized parafermionic algebras are proved to be polynomial algebras. The ordinary parafermionic algebras are shown to be connected to the Arik–Coon oscillator algebras. The study of systems of many spins is of interest in many branches of physics. This study is in many cases facilitated through boson mapping procedures (see [1] for a comprehensive review). Some well-known examples are the Holstein–Primakoff mapping of the spinor algebra onto the harmonic oscillator algebra [2] and the Schwinger mapping of Lie algebras (or of qdeformed algebras) onto the usual (or onto the q-deformed) oscillator algebras [3]. In parallel, in addition to bosons and fermions, parafermions of order p have been introduced [4, 5] (with p being a positive integer), having the characteristic property that at most p identical particles of this kind can be found in the same state. Ordinary fermions clearly cor...
Parabosonic $P_{B}^{(n)}$ and parafermionic $P_{F}^{(n)}$ algebras are described as quotients of ... more Parabosonic $P_{B}^{(n)}$ and parafermionic $P_{F}^{(n)}$ algebras are described as quotients of the tensor algebras of suitably choosen vector spaces. Their (super-) Lie algebraic structure and consequently their (super-) Hopf structure is shortly discussed. A bosonisation-like construction is presented, which produces an ordinary Hopf algebra $P_{B(K^{\pm})}^{(n)}$ starting from the super Hopf algebra $P_{B}^{(n)}$.
The finite dimensional representations of associative quadratic algebras with three generators ar... more The finite dimensional representations of associative quadratic algebras with three generators are investigated by using a technique based on the deformed parafermionic oscillator algebra. One application on the calculation of the eigenvalues of the two-dimensional superintegrable systems is discussed.
The symmetry algebra of the N-dimensional anisotropic quantum harmonic oscillator with rational r... more 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.
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Papers by Costas Daskaloyannis