4. Conclusions

We have considered the corrections due to the inclusion of the bound-electron pair production cross section (completely neglected in the Born approximation) in the optical theorem for the imaginary part of the forward Delbrück scattering amplitude. The real part is obtained via a dispersion relation. The usual single-electron partitioning of the elastic scattering amplitude implies that the cross section for bound-electron pair production into all bound states (both occupied and unoccupied) should be included. Numerical results for the correction to the real part of the forward Delbrück amplitude due to bound-electron pair production indicate a correction as large as $\approx$ 12% of the Born-approximation result in the region just above the pair production threshold. (The real part of the forward Delbrück amplitude dominates in this regime.) Thus these effects are comparable in this regime with the corrections due to Coulomb and screening effects in the ordinary (electron in continuum) pair production cross section (they become unimportant at higher energies), and both need to be considered for energies near and below the pair production threshold. The net correction to Born approximation is small well below threshold, significant well above threshold.