Deformed Harmonic Oscillator Algebras defined by their Bargmann representations

M. Irac-Astaud; G. Rideau

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arXiv · q-alg/9712043

Deformed Harmonic Oscillator Algebras defined by their Bargmann representations

M. Irac-Astaud·G. Rideau

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Abstract

Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint $a$ and $a^\dagger$, and two self-adjoint $N$ and the unity $1$ such as: $[a,N] = a, [a^\dagger, N]= -a^\dagger, a^\dagger a = ψ(N)$ and $aa^\dagger =ψ(N+1)$. The Bargmann Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of a Deformed Harmonic Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the annihilation operator) is the multiplication by $z$. We discuss the conditions of existence of Deformed Harmonic Oscillator Algebras assumed to admit a given Bargmann representation.

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BibTeX RIS

M. Irac-Astaud, G. Rideau. 1997-12-17. Deformed Harmonic Oscillator Algebras defined by their Bargmann representations. https://arxiv.org/abs/q-alg/9712043

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