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2006). R )−1 on a state of a particle localized in position very far Note that the action of ( + away and moving towards the scattering center is close to the action of the scattering operator S R . In this sense the theorem gives the Joos and Zeh formula modified for the presence of the internal motion of the heavy particle. Let us briefly discuss how the decoherence effect on the heavy particle can be derived from the above result. 20) It should be stressed that the asymptotic dynamics of the heavy particle described by ρa (t) is a free evolution and the only effect of the interaction is to induce a sudden change of the initial condition from the product state ϕ(R)ϕ(R ) to the entangled state ϕ(R)ϕ(R )I(R, R ).
S. 37) is a strictly positive constant. Then, proceeding as in the previous ε,n is negligible for ε → 0. 38) where Rk (ε) = O(ε2k−2 ) for any k ∈ N. ε,s . It turns out that, for u ˆ ∈ C0 , the phase has a manifold of Let us analyze G12 critical points in the integration region, parametrized by a vector in R2 . 46) It is relevant that both the phase and the Hessian of the phase at the critical point are strictly positive and do not depend on the parameters (η1 , η2 ). This fact is crucial to apply the stationary phase theorem (Fedoryuk 1971; Hörmander 1983; Bleinstein and Handelsman 1975) to the oscillatory integral I ε (η1 , η2 ) and to derive its asymptotic expansion for ε → 0.
4) describes the transition from a product state to an entangled state for the two-particle system. Such a final state is computed in a zero-th order adiabatic approximation, with the light particle instantaneously scattered far away by the heavy one considered as a fixed scattering center. 4) the evolution in time of the system is completely neglected, in the sense that time zero for the heavy particle corresponds to infinite time for the light one. 4), in Joos and Zeh (1985) the authors compute the reduced density matrix of the heavy particle and then add the contributions of a large number of scattering events.