C. Daskaloyannis

Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

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 into 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, that is for all two-dimensional superintegrable systems with quadratic integrals of motion.

Quantum groups and their applications in nuclear physics

Abstract 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 suq(2) rotator model and its extensions, the construction of deformed exactly soluble models (u(3)so(3) model, Interacting Boson Model, Moszkowski model), the 3-dimensional q-deformed harmonic oscillator imd its relation to the nuclear shell 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 the structure of molecules and of atomic clusters, as well as an outlook are also given.

Deformed oscillator algebras for two-dimensional quantum superintegrable systems

Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.

Generalized deformed oscillator and nonlinear algebras

The harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied. The eigenstates and eigenvalues of the deformed oscillator are calculated. This generalized deformed oscillator algebra is used to create nonlinear deformation of the classical SU(2) algebra. The deformed boson realizations of an arbitrary nonlinear SU(2) algebra are calculated. The algebra of the q-deformed oscillator is a special case of the generalized deformed oscillator algebra.

Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold

In this paper we prove that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes.

General deformation schemes and N = 2 supersymmetric quantum mechanics

Abstract The generalized deformed schemes. proposed initially as unified frameworks of various deformed oscillators, are proven to be equivalent. The unified representation of these generalized deformed schemes leads to the correspondence between the N = 2 supersymmetric quantum mechanics (SUSY-QM) scheme and the deformed oscillator.

Quantum superintegrable systems with quadratic integrals on a two dimensional manifold

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems, as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schrodinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but can be solved. Therefore, there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quan...There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schrödinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but the can be solved. Therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogues of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by −~2 plus quantum corrections of order ~4 and ~6. ∗e:mail address: daskalo@math.auth.gr

GENERALIZED DEFORMED VIRASORO ALGEBRAS

The deformed Virasoro algebra is a structure, generated by a generalized deformed oscillator algebra. The usual or the q-deformed centerless Virasoro algebras are special cases of this structure, and their properties can be reduced from the properties of the generalized deformed algebra. The construction of other deformed Virasoro algebras is given, using known deformation schemes other than the q-deformation.

Generalized deformed oscillator for the pairing correlations in a single-j shell

Abstract Using deformed bosons we construct a generalized deformed harmonic oscillator which satisfies the same commutation relations as a system of fermion pairs of zero angular momentum in a single- j shell pairing energy is also exactly reproduced by this mapping. The spectrum of the generalized deformed oscillator corresponds, up to first order perturbation theory, to a harmonic oscillator with an x 4 perturbation. A generalized deformed oscillator giving the same spectrum as a general anharmonic oscillator is also constructed.

WKB equivalent potentials for the q-deformed harmonic oscillator

WKB equivalent potentials (WKB-EP) giving the same spectrum as the q-deformed harmonic oscillator with the symmetry SUq(2) are determined. While in the case of q being real the WKB-EP goes to infinity as one moves away from the origin, in the case of q being a phase the WKB-EP goes to a finite limiting value, thus resembling, for example, the modified Poschl-Teller potential.