= 0\,,
\label{fi0}$$ where $r^*$ is given in Eq. (\[tortoise\]) and $$V_0[r^*(r)] = 
 f\frac{l(l+p-1)}{r^2}
-\frac{f}{h}\left[\frac{p(2-p)}{4r^2} - \frac{f^{\prime 2}}{2f^2}
+ \frac{p}{4r}\left( \frac{f'}{f} + \frac{h'}{h}\right) + \frac{f''}{2f}
- \frac{f'h'}{4fh}\right] \,.$$ For spacetimes where $f(r)$ and $h(r)^{-1}$ have simple zeroes at $r=r_H$, we find $V_0 [r^*(r_H)] \neq 0$ in general. If $V_0[r^*(r_H)] > 0$, then the non-physical modes coming out from the horizon have frequencies satisfying $\omega^2 > V_0[r^*(r_H)]$. In particular, for Reissner-Nordström spacetime in (p+2)-dimensions [@MP], where $$f(r) = h(r)^{-1} =\left[ 1- \left(\frac{r_+}{r}\right)^{p-1}\right]
\left[ 1- \left(\frac{r_-}{r}\right)^{p-1}\right]$$ with $r_\pm^{p-1} = M \pm \sqrt{M^2-Q^2\,}$ (so that $r_H = r_+$), we find $${V_0 [r^*(r_H)] } = \frac{(p-1)^2}{(2r_+)^2}
\left[ 1- \left(\frac{r_-}{r_+}\right)^{p-1}\right]^2
\,.$$ As expected, we have $\langle \varphi^{(0\rightarrow)}_{\omega l},
\varphi^{(0\leftarrow)}_{\omega' l}\rangle = 0$, i.e. the solutions of Eq. (\[fi0\]) incoming from $H^{-}$ and those incoming from ${\cal  J}^-$ are orthogonal to one another with respect to the inner product defined by Eq. (\[2dinner\]). By normalizing these solutions so that $$\begin{aligned}
\varphi^{(0\rightarrow)}_{\omega l} & \approx &
\sqrt{{\omega}/{\tilde\omega}\,}
\left( e^{i\tilde\omega r^*} + 
{\cal R}^{(0\rightarrow)}_{\omega l} e^{-i\tilde\omega r^*} \right)
\,\ \ \ (r^* \to -\infty)\,, \\
\varphi^{(0\leftarrow)}_{\omega l} & \approx &
e^{-i\omega r^*} + {\cal R}^{(0\leftarrow)}_{\omega l} e^{i\omega r^*}
\,\ \ \ (r^* \to +\infty)\,, \end{aligned}$$ up to a 