Star product for deformed oscillator algebra Aq(2, ν)

An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula exp[iεαβ ∂α∂β]. Using Pochhammer formula [1], integration over these parameters is carried over a Riemann surface associated with the expression of the type zx(1 − z)y where x and y are arbitrary real numbers.

Minimal bosonization of double-graded quantum mechanics

The superalgebra of [Formula: see text]-graded supersymmetric quantum mechanics is shown to be realizable in terms of a single bosonic degree of freedom. Such an approach is directly inspired by a description of the corresponding [Formula: see text]-graded superalgebra in the framework of a Calogero–Vasiliev algebra or, more generally, of a generalized deformed oscillator algebra. In the case of the [Formula: see text]-graded superalgebra, the central element [Formula: see text] has the property of distinguishing between degenerate eigenstates of the Hamiltonian.

Vibrational molecular and nuclear spectra in terms of quantum algebras

A generalized deformed oscillator giving the same spectrum as the Morse potential is con­ structed through the use of quantum algebraic techniques. The model of n coupled anharmonic oscillators of Iachello and Oss, suitable for the description of vibrational spectra of polyatomic molecules, is subsequently written in terms of such generalized deformed oscilla­tors. In addition to clarifying the relation of the model of Iachello and Oss to other models using coupled oscillators for the description of vibrational molecular spectra, the present formalism allows for the construction of a large class of exactly soluble models with no extra computational effort. As an example, the way of including a coupling of the Darling-Dennison type is shown. Nuclear models giving good descriptions of vibrational spectra are reviewed and the need of modifying the SUq(2) model in order to extend its region of applicability towards the vibrational limit is discussed.

Quasi-Exactly Soluble Potentials and Deformed Oscillators

It is proved that quasi-exactly soluble potentials corresponding to an oscillator with harmonic, quartic and sextic terms, for which the n+1 lowest levels of a given parity can be determined exactly, may be approximated by WKB equivalent poten- tials corresponding to deformed anharmonic oscillators of SUq(1,1) symmetry, which have been used for the description of vibrational spectra of diatomic molecules. This connection allows for the immediate approximate determination of the levels of the same parity lying above the fewest n+1 known levels, as well as of all levels of the opposite parity. Such connections are not possible in the cases of the q-deformed oscillator, the Q-deformed oscillator, and the modified Pöschl-Teller potential with SU(1,1) symmetry.

Quantum algebraic symmetries in nuclear physics

The pairing correlations in a single-j nuclear shell are considered. It is proven that a simple boson mapping in terms of q-defonned bosons exists, which reproduces correctly both the commutation relations and the energy up to first order corrections, the parameter q being connected to the size of the shell. An exact solution in terms of a generalized deformed oscillator is also found

Photon added coherent states of the parity deformed oscillator

Using the parity deformed Heisenberg algebra (RDHA), we first establish associated coherent states (RDCSs) for a pseudo-harmonic oscillator (PHO) system that are defined as eigenstates of a deformed annihilation operator. Such states can be expressed as superposition of an even and odd Wigner cat states.[Formula: see text] The RDCSs minimize a corresponding uncertainty relation, and resolve an identity condition through a positive definite measure which is explicitly derived. We introduce a class of single-mode excited coherent states (PARDCS) of the PHO through “m” times application of deformed creation operators to RDCS. For the states thus constructed, we analyze their statistical properties such as squeezing and sub-Poissonian statistics as well as their uncertainty relations.