English

Deformed Harmonic Oscillator Algebras defined by their Bargmann representations

q-alg 2007-05-23 v1 Quantum Algebra

Abstract

Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint aa and aa^\dagger, and two self-adjoint NN and the unity 11 such as: [a,N]=a,[a,N]=a,aa=ψ(N)[a,N] = a, [a^\dagger, N]= -a^\dagger, a^\dagger a = \psi(N) and aa=ψ(N+1)aa^\dagger =\psi(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 zz. We discuss the conditions of existence of Deformed Harmonic Oscillator Algebras assumed to admit a given Bargmann representation.

Keywords

Cite

@article{arxiv.q-alg/9712043,
  title  = {Deformed Harmonic Oscillator Algebras defined by their Bargmann representations},
  author = {M. Irac-Astaud and G. Rideau},
  journal= {arXiv preprint arXiv:q-alg/9712043},
  year   = {2007}
}

Comments

27 pages, Latex

R2 v1 2026-07-22T19:22:24.714Z