Recursively minimally-deformed oscillators

Abstract

A recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator [a, a † ] = f k (· · · ;n) into [a, a † ] q k+1 = f k (· · · ;n), where · · · stands for the set of deformation parameters that f k depends on, followed by a transformation into the commutator [a, a † ] = f k+1 (· · · , q k+1 ;n) to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation [a, a † ] = 1 we obtain the Arik-Coon and the Macfarlane-Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The "generic" form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.

Non-linear coherent states associated with conditionally exactly solvable problems

“…This, however, will in general not be the case for non-Fock space representations of the algebra (14) [10,11,12].…”

“…More explicitly, the above algebra (14) is a polynomial deformed su(1, 1) algebra and has first been discussed in some detail by Rocek [10]. For a discussion within a more general approach see also Karassiov [11] and Katriel and Quesne [12]. The quadratic Casimir operator for the non-linear (cubic) algebra (14) reads…”

Generalized coherent states for polynomial Weyl-Heisenberg algebras

“…Polynomial Weyl-Heisenberg algebra Following many works on possible extensions of the usual Weyl-Heisenberg algebras [17][18][19][20][21][22][23][24][25][26][27], let us consider the algebra spanned by an annihilation operator (a − ), a creation operator (a + ) and a number operator (N = a + a − ) satisfying the commutation relations…”

Even and odd nonlinear charge coherent states and their nonclassical properties

“…Choose θ to be π 2 or 0, so that ± cos 2θ = −1. Conditions (78) and (79) may or may not be realized, depending on the assumed functional (19), (71) and (72), we have…”