  We consider a two-body quantum system in dimension one composed by a test
particle interacting with an harmonic oscillator placed at the position $a>0$.
At time zero the test particle is concentrated around the position $R_0$ with
average velocity $\pm v_0$ while the oscillator is in its ground state. In a
suitable scaling limit, corresponding for the test particle to a semi-classical
regime with small energy exchange with the oscillator, we give a complete
asymptotic expansion of the wave function of the system in both cases $R_0 <a$
and $R_0 >a$.
