Radial Mirror Scattering and the QNM Convergence Region

Abstract: We revisit the convergence region of the quasinormal modes expansion of Schwarzschild retarded Green functions from a radial scattering viewpoint. The tortoise coordinate admits a natural reflection about a distinguished point, which maps the original Regge-Wheeler problem to a mirror radial problem with the same quasinormal mode spectrum. Although this reflection is not a spacetime symmetry and does not leave the potential invariant, it gives a simple image interpretation of the second lightcone distance that controls convergence. Equivalently, after folding the radial line at the reflection point, the direct and mirror contributions arise as diagonal and off-diagonal propagation channels of a two-component half-line problem. We also relate this structure to the AdS$_2$ Green function, where the same direct-plus-image lightcone structure arises from a genuine boundary-bouncing null geodesic. This provides a spectral interpretation of the convergence condition and clarifies the role of the reflection point in the Schwarzschild radial Green function.