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The absolute definition of the phase-shift in potential scattering

The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying (\ref{1e}), and studied by Calogero in his book$^{5}$, is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e. the one which defines $δ_{\ell}(k)$ as a continuous function of $k$ for all $k \geq 0$, up to infinity, where $δ_{\ell}(\infty)=0$ is automatically satisfied. This removes the usual ambiguity $\pm n π$, $n$ 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 $gr^{-m}$, $g > 0$, $m \geq 2$, 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.