Asymptotic observables on scattering states

In quantum mechanical potential scattering theory we use selected observables to describe the asymptotic properties of scattering states for long times. E.g., we show for the position and momentum operators that for Ωε?^cont(H), $$left( {mfrac{x}{t} - p} right)e^{ - iHt} Psi to 0,$$ and that the set of outgoing states is absorbing. This is obtained easily without any detailed analysis of the interacting time evolution. The class of forces includes highly singular and very long range potentials. The results may serve as an intermediate step in a proof of asymptotic completeness; as a particular application we present a simple proof of completeness for Coulomb systems.