Analyticity of the Black Hole S-Matrix

We establish the analytic structure of the S-matrix in the complex-frequency plane for classical wave scattering on a Schwarzschild background in four space-time dimensions. Our argument relies on the analytic continuation of the gravitational potential, with the singularity behind the horizon playing a crucial role. We find that in the lower half-plane the partial-wave amplitudes are analytic except for the quasinormal-mode poles and the branch cut associated with late-time tails. As a direct consequence of causality, the retarded Green's function and absorption amplitude are analytic in the upper-half plane. We show, however, that Stokes phenomena can obstruct this analyticity domain from carrying over to the elastic amplitude, which instead develops a branch-cut in the upper-half plane. We also determine the effect of infrared (IR) regulators on the analytic structure, showing that polynomial boundedness requires a sharp lower bound on the IR cutoff in terms of the Schwarzschild radius.