Decomposition of Schwarzschild Green's Function

We present a formulation of the spherically decomposed Green's function for a Schwarzschild black hole, based on a decomposition into two components, $G^+$ and $G^-$, based on their large-frequency behaviour. While similar decompositions have been considered previously, here we systematically apply it to Schwarzschild spacetime and analyze its implications for the analytic structure of the Green's function in the complex-frequency plane. We show that both $G^+$ and $G^-$ possess branch cuts along the imaginary axis, which give rise to the direct part and the late-time tail, while the poles of $G^+$ correspond to the quasinormal mode spectrum. This allows us to identify a $\textit{branch-cut direct part}$, a quasinormal-mode contribution, and a late-time tail through contours adapted to different causal spacetime regions. This is in sharp contrast to Leaver's original formulation, where the prompt response is tied to a technically difficult large-arc contribution. We validate our decomposition with independent time-domain Regge-Wheeler simulations finding excellent agreement. Our results provide a practical and physically transparent framework for disentangling the distinct pieces of the Schwarzschild response, and offer a natural starting point for extensions to Kerr perturbations and non-linear ringdown physics.