Scattering in one dimension: The coupled Schroedinger 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 $η(0) = π(n_b + 1/2 n - 1/2 N)$, where $η(0)$ 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.

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