Strong gravitational lensing and black hole quasinormal modes: Towards a semiclassical unified description

We examine in a semiclassical framework the deflection function of strong gravitational lensing, for static and spherically symmetric black holes, endowed with a photon sphere. From a first-order WKB analysis near the maximum of the Regge-Wheeler potential, we extract the real phase shifts from the S-matrix elements and then we derive the associated semiclassical deflection function, characterized by a logarithmic divergent behavior. More precisely, using the complex angular momentum techniques, we show that the Regge poles and the associated greybody factor residues, for a massless scalar field theory, from which one can recover the black hole quasinormal complex frequencies as well as the fluctuations of the high energy absorption cross section, play naturally the role of critical parameters in the divergent behavior of the semiclassical deflection function. For very high frequencies, we finally recover the logarithmic part of the classical strong deflection limit, which clarifies analytically the fundamental link between quasinormal modes and strong gravitational lensing, suggested in recent works.