  The retarded Green function for linear field perturbations in Schwarzschild
black hole space-time possesses a branch cut in the complex-frequency plane.
This branch cut has remained largely unexplored: only asymptotic analyses
either for small-frequency (yielding the known tail decay at late times of an
initial perturbation of the black hole) or for large-frequency (quasinormal
modes close to the branch cut in this regime have been linked to quantum
properties of black holes) have been carried out in the literature. The regime
along the cut inaccessible to these asymptotic analyses has so far remained
essentially unreachable. We present a new method for the analytic calculation
of the branch cut directly on the cut for general-spin fields in Schwarzschild
space-time. This method is valid for any values of the frequency on the cut and
so it provides analytic access to the whole branch cut for the first time. We
calculate the modes along the cut and investigate their properties and
connection with quasinormal modes. We also investigate the contribution from
these branch cut modes to the self-force acting on a point particle on a
Schwarzschild background space-time.
