Bethe-Peierls boundary condition (${\lim }_{r\to 0}{\partial }_r(r\psi )/(r\psi )=-1/a$)

This zero-range boundary condition replaces a short-range interaction by its S-wave scattering length. If $\psi =A/r+B+O(r)$, it requires $B=-A/a$. An outgoing spherical wave also has a constant $ik$ term; retaining that term gives the unitary scattering-length amplitude $f_0=-a/(1+ika)$. The model neglects effective-range corrections and is applied outside the microscopic range, not literally inside a finite potential.

Two separated short-range scatterers are coupled because the outgoing wave from either centre supplies part of the regular incoming field at the other. Applying the Bethe-Peierls boundary condition at both centres gives the displayed two-by-two linear system, where $s_j$ is the incident plane-wave phase. matrix inversion sums repeated scattering between the centres. Near a small determinant, effective-range corrections may matter even if the nominal microscopic range is short.

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.