Bound-state poles of a partial-wave S-matrix

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

For the exterior wavethe partial-wave scattering matrix isThus $\tan {\delta }_0=-k/\lambda$, the scattering length is $1/\lambda$, and the pole at $k=i\lambda$ represents a bound state of energy $-{\hslash }^2{\lambda }^2/(2m)$ whose exterior wavefunction is proportional to $\mathrm{sech}(\lambda r)/r$.