 Delta-shell Born amplitude

For a real shell potential $V(r)=V_0\delta (r-R)$, define $g=2m{V}_{0}R/{\hslash }^2$ and $\mathrm{sinc}u=\sin u/u$, with value one at zero. Angular integration of its three-dimensional Fourier transform gives $4\pi V_0{R}^2\mathrm{sinc}(qR)$, where the momentum transfer has wave-number magnitude $q=2k\sin (\theta /2)$. Substitution in the first Born approximation proves the displayed scattering amplitude. It is real. Its squared modulus is a nonzero order-$V_0^2$ differential scattering cross-section, whose optical theorem counterpart requires the order-$V_0^2$ imaginary part of the forward amplitude from the next Born term. Thus squaring the first-order amplitude does not supply a consistently unitary second-order amplitude.