theory is recovered. A pedestrian derivation of (\[eq:Trueman\]) is given in Appendix A. The presence of $a$ instead of $\lambda$ in (\[eq:Trueman\]) indicates that the strong potential $\lambda v(r)$ acts many times, so that the shift is by no mean a perturbative effect.

The Deser–Trueman formula has sometimes been blamed for being inaccurate. In fact, if the scattering length is calculated with Coulomb interference effects, it is usually extremely good., see, e.g., [@Carbonell:1992wd] for a discussion and [@Ericson:2003ju] for higher-order corrections. However, this approximation obviously breaks down if the scattering length becomes very large, i.e., if the potential $\lambda v(r)$ approaches the situation of supporting a bound state.

Now the pattern in Fig. \[CoulSW\] can be read as follows. For small positive $\lambda$, the additional potential is deeply attractive but produces a small scattering length and hence a small energy shift. As the critical strength $\lambda=\lambda_1$ for binding in $\lambda v(r)$ is approached, the scattering length increases rapidly, and there is a sudden change of the energies. The ground-state of the system, which is an atomic 1S level for small $\lambda$ and a deeply bound nuclear state for $\lambda \gs \lambda_1$ evolves continuously (from first principles it should be a 