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

Asymptotic Expansion for the Wave Function in a one-dimensional Model of Inelastic Interaction