  A problem to account for the direct electron-positron annihilation in
positron-hydrogen scattering above the positronium formation threshold has been
resolved within the time independent formalism. The generalization of the
optical theorem is derived for the case when an absorption potential is present
in the Hamiltonian. With this theorem the annihilation cross section is fully
determined by scattering amplitudes. This allows us to separate out
analytically the contribution of the positronium formation from the overall
annihilation cross section. The rest is determined as the direct annihilation
cross section. It is done uniformly below as well as above the positronium
formation threshold. The multichannel three-body theory for scattering states
in the presence of an imaginary absorption potential is developed in order to
compute the direct $e^+ e^-$ annihilation amplitude. Special attention has been
paid to an accurate definition of the coordinate part of the absorption
potential as the properly constructed zero-range potential, which corresponds
to the delta-function originated from the first order perturbation theory. The
calculated direct annihilation cross section below the positronium formation
threshold is in good agreement with results of other authors. The direct
annihilation cross section computed with the formalism of the paper shows
nonsingular behavior at the positronium formation threshold and is in good
agreement with existing data. A number of $e^+ e^-$ direct annihilation cross
sections and positronium formation cross sections in the energy gap between
Ps$(1s)$ and H$(n=2)$ thresholds are reported. A sharp increase in the
calculated direct annihilation cross section across the resonant energy is
found for all first S and P-wave Feshbach resonances.
