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

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 $R_0 >a$.