  We propose an effectively nonperturbative approach to calculating scattering
amplitudes in the perturbative regime. We do this in a discretized momentum
space by using the QSE method to calculate all the contributions (to all orders
in perturbation theory) to the scattering eigenstates that are above a
precision cutoff. We then calculate the scattering amplitude by directly taking
the inner product between these eigenstates. In the current work we have
analyzed this procedure for a $\lambda\phi^4$ theory in one spatial dimension
and compared our results with perturbation theory obtaining favorable results
suggestive that further research in this direction might be worthwhile. In
particular, we show that the efficiency of our method scales much better than
second- and higher-order perturbation theory as the momentum lattice spacing
decreases and as the eigenstate energy increases.
