Positron-atom dispersion relations

The singularity structure of the forward positron-atom scattering amplitude is studied within the framework of perturbation theory with the use of the Landau rules. The unphysical and anomalous regions in the positron-atom forward dispersion relations are discussed. It is shown that the unphysical region starts at $E_0=-\left|E\right(\mathrm{Ps}\left)\right|+\left|\mathrm{}{E}_1\right|$) and the anomalous region at $E_a=-[\sqrt{2\left|E\right(\mathrm{Ps}\left)\right|}-\sqrt{\left|E_1\right|}{]}^2$, where $E\left(\mathrm{Ps}\right)$ and $E_1$ are correspondingly the positronium and the atomic-electron binding energies. The anomalous region is on the physical sheet if $\frac{\left|E\right(\mathrm{Ps}\left)\right|}{2}>\left|E_1\right|$.