Phase Shifts for Scattering by a One-Dimensional Delta-Function Potential

I. Richard Lapidus

Clinton DeW. Van Siclen

A one−dimensional calculation is carried out to determine the phase shifts for scattering from an attractive delta−function potential. (AIP)

V.I. Karloukovski

Rocco Schiavilla

Scattering in one dimension is discussed in terms of wave packets; reflection, transmission, and tunneling probabilities are defined; the WKB method to calculate these probabilities is introduced; the S-matrix for scattering in one dimension is defined; the phase-shift method for one-dimensional parity-invariant potentials is introduced; applications to various combinations of finite and infinite barriers with delta-function potentials are examined.

K. Chadan, Kobayashi, R., Kobayashi, T.

The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (1), and studied by Calogero in his book5, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines δℓ(k) as a continuous function of k for all k ≥ 0, up to infinity, where δℓ(∞) = 0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g> 0, m ≥ 2, etc. We also give an example of application to the low-energy behaviour of the S-wave scattering amplitude in two dimensions, which leads to an interesting result.

Chandan Kumar