On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on R

Abstract

We discuss a q-analogue of the linear harmonic oscillator in quantum mechanics based on a q-extension of the classical Hermite polynomials H,(x) recently introduced by us in R. Alvarez-Nodarse et al.: Boletin de la Sociedad Matematica Mexicana (3) 8 (2002) 127. The wave functions in this q-model of the quantum harmonic oscillator possess the continuous orthogonality property on the whole real line R with respect to a positive weight function. A detailed description of the corresponding q-system is carried out.