e^+ + H direct annihilation above the positronium formation threshold

A long-standing problem with the solution of the Schr\"odinger equation has been its inability to account for the electron-positron annihilation in positron hydrogen scattering above the positronium formation threshold. This letter shows that this problem has been resolved by the use of the modified Faddeev equations. A number of $e^- e^+$ annihilation cross sections in the energy gap between $Ps(1s)$ and $H(n=2)$ thresholds are reported for both the positron plus hydrogen incoming channel as well as the proton plus positronium incoming channel. However the indirect annihilation cross sections after formation of the positronium themselves are well known, they will not be included in this report.