Coherent states of the one-dimensional Dunkl oscillator for real and complex variables and the Segal–Bargmann transformation of Dunkl-type

Abstract

Abstract In this paper, we construct coherent states of a parity deformation of the Heisenberg algebra and we examine some of its properties. We show that these states minimize the uncertainty principle and obey the classical equations of motion for the harmonic oscillator. Also they constitute a non-orthogonal over-complete system which yields a resolution of the identity operator. As a concrete realization of this algebra and its coherent states, we treat the quantum systems governed by the one-dimensional Dunkl oscillator for real and complex variables. We show that these quantum systems are unitary equivalent and the unitary equivalence is a Segal–Bargmann transformation of Dunkl-type.