Rigorous calculation of scalar scattering in the Schwarzschild background: The convergence of the partial-wave series and the Poisson spot

Black hole (BH) perturbation theory and the scattering models provide a powerful framework for studying gravitational lensing at the wave-optics level. However, conventional calculations encountered two issues: the divergence of the partial-wave series and the divergence of the Poisson spot near the optical axis. These issues hinder the accurate calculation of lensed waveforms and the study of polarization and wave characteristics in the lensing process, especially near the optical axis. This work demonstrates that both divergences stem from the asymptotic expansion of the radial wave function. By computing the scattered wave function at finite radii and avoiding the asymptotic expansion, we naturally obtain convergent results. We compute scalar waves scattered by (1) a weak-gravity body with Newtonian potential and (2) a Schwarzschild BH with Regge-Wheeler potential. In both cases, we analyze the convergence of the partial-wave series and present finite-luminosity diffraction patterns, with a bright Poisson spot. The above calculations are compared with the Kirchhoff diffraction integral in the near-axis regions and give consistent results. Our investigations provide a foundation for studying gravitational wave scattering by BHs and understanding lensing at the wave-optics level.