Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 35E (Applications of Quantum Mechanics)

For real $k$, conservation of the radial probability current, equivalently constancy of the radial Wronskian, equates the incoming and outgoing fluxes and givesFor a real potential, the regular radial solution depends on $k$ through $k^2$. Comparing its asymptotic form at $k$ and $-k$ therefore givesThe first identity places $S_l(k)$ on the unit circle, sofor a real phase shift. Choosing the phase continuously, the second identity givesThese are the unitarity and reflection identities for a partial-wave S-matrix.

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.