From Bargmann representations to Deformed Harmonic Oscillator Algebras

Deformed Harmonic Oscillator Algebras (DHOA) are generated by four operators: two mutually adjoint a and a†, self-adjoint N, and the unity 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 DHOA act on a Bargmann-Hilbert space, and the creation (or the annihilation operator) is the multiplication by z. We discuss conditions for the existence of DHOA assumed to admit a given Bargmann representation.