Deformed Harmonic Oscillator Algebras defined by their Bargmann representations
Abstract
Description
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.
27 pages, Latex
27 pages, Latex