Discrete-state continuum scattering amplitude ($f_k=-A/(E-\widetilde{ϵ}+iAk)$)

For a real point coupling $g$ between a three-dimensional continuum and one discrete state, $A=m{g}^2/(2\pi {\hslash }^2)$. After the short-distance level shift is absorbed, the isotropic amplitude has the displayed form. The imaginary part is fixed by the outgoing boundary condition, and $\mathrm{Im}(1/f_k)=-k$ supplies elastic unitarity.