Scattering in one dimension: The coupled Schrodinger equation, threshold behaviour and Levinson's theorem

We formulate scattering in one dimension due to the coupled Schrodinger 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 eta(0) = pi(n(b) + 1/2n - 1/2N), where eta(0) is the phase of the S matrix at zero energy, n(b) 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. (C) 1996 American Institute of Physics.