Deformed harmonic oscillators : coherent states and Bargmann representations

Details Deformed harmonic oscillators : coherent states and Bargmann representations Deformed harmonic oscillators : coherent states and Bargmann representations

1997-07-17

arxiv.org/abs/q-alg/9707023v1

Mathematics

Quantum Algebra

8 pages, Latex, Presented at the 6th Colloquium "Quantum Groups and Integrables Systems", Prague, 19-21 June 1997

Scientific paper

10.1023/A:1021670419793

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators $a, a^\dagger, N$ and the unity 1 such as $[a,N] = a, [a^\dagger,N] = -a^\dagger$, $a^\dagger a = \psi(N)$ and $aa^\dagger =\psi(N+1)$. We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of $a$ (or $a^\dagger$). We give various examples, in particular we consider functions $\psi$ that are linear combinations of $q^N$, $q^{-N}$ and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.

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