Scattering in one dimension: The coupled Schrödinger equation, threshold behaviour and Levinson’s theorem

We formulate scattering in one dimension due to the coupled Schrödinger equation in terms of the S matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson’s theorem is seen to have the form η(0)=π(nb+1/2n−1/2N), where η(0) is the phase of the S matrix at zero energy, nb the number of bound states with nonzero binding energy, n the number of half-bound states, and N the number of coupled equations. In view of the effects due to the half-bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is also illustrated by means of particular potential models.