Representation Theory of Generalized Deformed Oscillator Algebras

Abstract

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators $1,a,a^{†},N$. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for $[a,a^{†}{]}_q=G(N)$, where $[a,b{]}_q=ab-qba$ 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 $a^{†}a$. 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.

8 pages, LaTeX, no figures, submitted to Proc. 5th Int. Coll. ``Quantum Groups and Integrable Systems'', Prague, 20-22 June 1996 (to be published in Czech. J. Phys.)

References in corpus (3)

• Generalized q-Oscillators and their Hopf Structures
• Recursively minimally-deformed oscillators
• A Classically Singular Representation of $s{u}_q(n)$

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• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• New algebraic structures in the $C_{\lambda }$-extended Hamiltonian system