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

Abstract

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 gB2(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=-1/2n2 (n>or=2) associated with excited intermediate states of the target and (ii) cuts at E=-1/2n2 (n>or=1) coming from all intermediate states. In addition, the bound-state part of gB2(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 gB2(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.