Representation Theory of Generalized Deformed Oscillator Algebras

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators ${1,a,a^{\dag},N}$. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for $[a, a^{\dag}]_q = G(N)$, where $[a,b]_q = a b - q b a$ and $G(N)$ is a Hermitian, analytic function. The unitary irreductible representations are obtained by means of a Casimir operator $C$ and the semi-positive operator $a^{\dag} a$. They may belong to one out of four classes: bounded from below (BFB), bounded from above (BFA), finite-dimentional (FD), unbounded (UB). Some examples of these different types of unirreps are given.