Unstable circular null geodesics of static spherically symmetric black holes, Regge poles, and quasinormal frequencies

We consider a wide class of static spherically symmetric black holes of arbitrary dimension with a photon sphere (a hypersurface on which a massless particle can orbit the black hole on unstable circular null geodesics). This class includes various spacetimes of physical interest such as Schwarzschild, Schwarzschild-Tangherlini, and Reissner-Nordstr\"om black holes, the canonical acoustic black hole, or the Schwarzschild--de Sitter black hole. For this class of black holes, we provide general analytical expressions for the Regge poles of the S matrix associated with a massless scalar field theory. This is achieved by using third-order WKB approximations to solve the associated radial wave equation. These results permit us to obtain analytically the nonlinear dispersion relation and the damping of the ``surface waves'' lying close to the photon sphere as well as, from Bohr-Sommerfeld--type resonance conditions, formulas beyond the leading-order terms for the complex frequencies corresponding to the weakly damped quasinormal modes.