The absolute definition of the phase-shift in potential scattering

The variable phase approach to potential scattering with regular sphericall y symmetric potentials satisfying Eq. (1), and studied by Calogero in his b ook [Variable Phase Approach to Potential Scattering (Acadamic, New York, 1 967)] is revisited, and we show directly that it gives the absolute definit ion of the phase-shifts, i.e., the one which defines delta (l)(k) as a cont inuous function of k for all k greater than or equal to0, up to infinity, w here delta (l)(infinity)=0 is automatically satisfied. This removes the usu al ambiguity +/-n pi, n integer, attached to the definition of the phase-sh ifts through the partial wave scattering amplitudes obtained from the Lippm ann-Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definiti on of the phase-shifts is very general, and applies as well to all potentia ls which have a strong repulsive singularity at the origin, for instance th ose which behave like gr(-m), g >0, m greater than or equal to2, etc. We al so give an example of application to the low-energy behavior of the S-wave scattering amplitude in two dimensions, which leads to an interesting resul t. (C) 2001 American Institute of Physics.