Divergence of the Green's Function Series for Rearrangement Collisions

The convergence of the Born series for rearrangement collisions is investigated in a potential model. For a certain class of potentials it is shown that the iterated series for the full two particle Green's function, $\left({{\mathrm{k}}_1}^{\prime }{{\mathrm{k}}_2}^{\prime }\mathrm{}\right|G\left(E\right)\left|{\mathrm{k}}_1{\mathrm{k}}_2\right)$, in terms of either the free-particle Green's function, the initial state Green's function, or the final state Green's function, diverges for some continuous range of the variables ${\mathbf{k}}_1$, ${\mathbf{k}}_2$, ${\mathbf{k}}_1$′, and ${\mathbf{k}}_2$′ independent of the energy, $E$, of the incident particle. It is suggested that the usual Born series, which is an integral over this Green's function series, therefore, also diverges for rearrangement collisions independent of the incident energy.