Large-ω Asymptotic Expansions and Padé Approximants for the Harmonic Oscillator

Abstract

An approximate solution to the Schrödinger equation for a particle in various perturbed harmonic‐oscillator potentials is obtained by utilizing large‐ω (ω is the natural frequency of unperturbed oscillator) asymptotic expansion theory. Both the basic asymptotic and perturbation theory solutions are obtained and are shown to be related. Explicit high‐order perturbation expansions for the energy and mean‐square displacement of the symmetric anharmonic oscillator are obtained and are seen to be divergent but asymptotic. It is shown that by rearranging these expansions as Padé approximants, accurate numerical values are obtained for these observables, even for nonperturbative values of the coupling constant.