Green's Function for the Quartic Oscillator

Abstract

In this paper, a quantum mechanical Green's function $G_{qo}(y_b,t_b;$ $y_a,t_a)$ for the quartic oscillator is presented. This result is built upon two previous papers: first [1], detailing the linearization of the quartic oscillator $(qo)$ to the harmonic oscillator $(ho)$, second [2], the integration of the classical action function for the quartic oscillator. Here an equivalent form for the quartic oscillator action function $S_{qo}(y_b,t_b;$ $y_a,t_a)$ in terms of harmonic oscillator variables is derived in order to facilitate the derivation of the quartic oscillator Green's Function amplitude. Thus, the papers [1] and [2] and this paper, taken together, result in the incorporation of the quartic oscillator into the non-relativistic quantum mechanical physics literature consisting of those single particle systems whose properties are described in terms of trig functions, their quadratures or both.