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We first deduce the analytical continuation in the complex planes of theinitial and final three-momenta of the Lippmann-Schwinger equation in coupledor uncoupled partial-wave amplitudes. This result allows us to deduce a masterequation whose solution is the exact discontinuity of the on-shell partial-waveamplitudes along the left-hand cut. This equation is always a linearnon-singular integral equation whose solution is fixed exclusively by theknowledge of the potential, applicable to either regular or singularpotentials. The capability of calculating exactly this discontinuity allows oneto settle the exact $N/D $ method in two-body non-relativistic scattering forcoupled and uncoupled waves. We exemplify this new advance in scattering theoryby explicitly checking the agreement between the Lippmann-Schwinger equationwith the corresponding solutions of the exact $N/D$ method for some examplesthat involve regular and singular potentials, either attractive or repulsive.

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