A first step towards effectively nonperturbative scattering amplitudes in the perturbative regime

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 λ 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.