Meixner Oscillators

TL;DR

This paper introduces Meixner oscillators, a two-parameter family of difference operator-based models with equally spaced energy spectra, connecting classical and relativistic harmonic oscillators through special functions and symmetry groups.

Contribution

It explicitly constructs Meixner oscillators, their coherent states, and reproducing kernels, linking them to classical, relativistic, and special function models.

Findings

Meixner oscillators include Charlier, Kravchuk, and Hermite oscillators as special cases.

Explicit coherent states and reproducing kernels are derived.

Connections to relativistic models via Meixner-Pollaczek polynomials are established.

Abstract

Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,$\Re$). The reproducing kernel for the wavefunctions of these models is also found.