condition that the solution behaves as an incoming wave in the near horizon limit $ z \to z_h$. The absence of outgoing solutions represents the absorption by the black hole horizon. In order to implement this condition one can use the Regge-Wheeler tortoise coordinate that makes it explicit the decomposition of the solutions of the equations of motion in incoming plus outgoing solutions. One introduces the coordinate $ r_\ast $ such that $ \partial_{r_\ast} = - f(z) \partial_z $ that implies $$r_\ast  = \frac{1}{2} z_h \Big[ -\tan^{-1} \left( \frac{z}{z_h} \right)  +
  \frac{1}{2}  \ln \left(  \frac{z_h - z}{z_h + z} \right)  \Big]\,,$$ in the interval $ z \le z_h $ where $z$ is defined.

Performing a Bogoliubov transformation $ v (\omega , z) = e^{B/2} \psi  (\omega , z) $ with $ e^B = z e^{k^ 2 z^ 2}  $ one finds that the equation of motion (\[EOMfiniteT\]) takes the form

$$\partial^2_{r_\ast}  \psi + \omega^2 \psi = U \psi \,,$$

where the potential $$U (z) \,=\, \left( 1 - \frac{z^ 4}{z_h^4} \right) \left[ \left(  k^4 z^2 + \frac{3}{4z^ 2} \right)   \left( 1 - \frac{z^ 4}{z_h^4} \right) +   2 z^2 \frac{(1 + 2 k^2 z^2 )}{z_h^4} \right]$$ vanishes at the horizon. Thus, 