Asymptotic Behavior of Bohmian Trajectories in Scattering Situations

We study the asymptotic behavior of Bohmian trajectories in a scattering situation with short range potential V and for wave functions Ψ = Ψ + Ψ ∈ L(R) with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long time behavior is governed by the scattering part Ψ of the wave function (scattering trajectories) and trajectories whose long time behavior is governed by the bound part Ψ of the wave function (bound trajectories). Furthermore, the scattering trajectories behave like trajectories in classical mechanics in the limit t → ∞. As an intermediate step we show that the asymptotic velocity v∞ := lim t→∞ Q/t exists almost surely and is randomly distributed with the density |Ψ̂ out |, where Ψ is the outgoing asymptote of the scattering part of the wave function.