This intuitive picture is not quite right, because   is an eigenfunction of the Hamiltonian and so at different times only differs by a phase. Thus, in particular, the physical state does not evolve and so it cannot become noninteracting.  This problem is easily circumvented by assembling  and  into wavepackets with some distribution  of energies  over a characteristic scale .  The uncertainty principle now allows the interactions of the asymptotic states to occur over a timescale  and in particular it is no longer inconceivable that the interactions may turn off outside of this interval.  The following argument suggests that this is indeed the case.
Plugging the Lippmann–Schwinger equations into the definitions
and
of the wavepackets we see that, at a given time, the difference between the  and  wavepackets is given by an integral over the energy .