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Formal solutions to the mode equations for both spherically symmetric blackholes and Bose-Einstein condensate acoustic black holes are obtained by writingthe spatial part of the mode equation as a linear Volterra integral equation ofthe second kind. The solutions work for a massless minimally coupled scalarfield in the s-wave or zero angular momentum sector for a spherically symmetricblack hole and in the longitudinal sector of a 1D Bose-Einstein condensateacoustic black hole. These solutions are used to obtain in a rigorous wayanalytic expressions for the scattering coefficients and gray-body factors inthe zero frequency limit. They are also used to study the infrared behaviors ofthe symmetric two-point function and two functions derived from it: thepoint-split stress-energy tensor for the massless minimally coupled scalarfield in Schwarzschild-de Sitter spacetime and the density-density correlationfunction for a Bose-Einstein condensate acoustic black hole.

Low frequency gray-body factors and infrared divergences: rigorous results