Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 35E / b / Solution

At small $k$,Since $S_0=e^{2i{\delta }_0}$,The scattering length and low-energy total cross-section are therefore

The poles at $k=2i\lambda$ and $k=3i\lambda$ lie on the positive imaginary axis. They represent two bound states, with energiesThere are no resonance poles away from the imaginary axis in the lower half-plane. Positivity of $\lambda$ puts the poles on the physical bound-state axis, where $e^{ikr}$ decays; changing its sign would move them to the unphysical half-plane. This illustrates the bound-state poles of a partial-wave S-matrix.