Exchange amplitudes and forward dispersion relations for electron-atomic hydrogen scattering

The analytic properties of the forward elastic exchange amplitude g(E) are studied as a function of the incident electron energy E for electron scattering by atomic hydrogen. The authors first prove that the bound-state of g B2 (E), the second Born contribution to g(E), has an infinite number of singularities due to the analytic structure of the off-shell first-order exchange matrix element. These singularities are (i) poles of order (n-1) at E=-1/2n 2 (n>or=2) associated with excited intermediate states of the target and (ii) cuts at E=-1/2n 2 (n>or=1) coming from all intermediate states. In addition, the bound-state part of g B2 (E) also exhibits poles and cuts at E=-2(1+1/2n) 2 , n>or=1, arising from the integration over the intermediate momentum of the projectile. They then show that the continuum contributions to g B2 (E) lead to singularities having the same location as those coming from the bound states, so that the interplay between bound and continuum singularities is very important.