The exact discontinuity of a partial wave along the left-hand cut and the exact $N/D$ method in non-relativistic scattering

TL;DR

This paper derives an exact method to compute the discontinuity of partial-wave amplitudes along the left-hand cut in non-relativistic scattering, enabling an exact $N/D$ approach for coupled and uncoupled waves.

Contribution

It introduces a master integral equation for the exact discontinuity of partial-wave amplitudes, applicable to regular and singular potentials, advancing the $N/D$ method in scattering theory.

Findings

Exact discontinuity computed for various potentials.

Confirmed agreement between Lippmann-Schwinger and $N/D$ solutions.

Applicable to both regular and singular potentials.

Abstract

We first deduce the analytical continuation in the complex planes of the initial and final three-momenta of the Lippmann-Schwinger equation in coupled or uncoupled partial-wave amplitudes. This result allows us to deduce a master equation whose solution is the exact discontinuity of the on-shell partial-wave amplitudes along the left-hand cut. This equation is always a linear non-singular integral equation whose solution is fixed exclusively by the knowledge of the potential, applicable to either regular or singular potentials. The capability of calculating exactly this discontinuity allows one to settle the exact $N/D $ method in two-body non-relativistic scattering for coupled and uncoupled waves. We exemplify this new advance in scattering theory by explicitly checking the agreement between the Lippmann-Schwinger equation with the corresponding solutions of the exact $N/D$ method for some examples that involve regular and singular potentials, either attractive or repulsive.