Nonlinear Coherent States Associated with a Measure on the Positive Real Half Line

Abstract

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated Bargmann-type transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this method with the measure $$r^\beta e^{-r}dr$$, where $$\beta $$ is a non-negative constant, which leads to a new generalization of the true-polyanalytic Bargmann transform.