An asymptotic approximation applied to a simple model

An asymptotic approximation scheme based on a sequence of solutions is applied to a simple model problem of a harmonic oscillator coupled to a scalar field. We evaluate explicitly the evolution of the field off the initial hypersurface and discuss an appropriate choice of the initial data for the field in order for a sequence of solutions to have a Newtonian limit. The study may help to understand the more complicated situation in general relativity. We also discuss an alternative sequence of solutions that may be useful in studying instabilities that occur for large values of the coupling constant.