the condition $\operatorname{Im}{(\omega_n)}<0$.

Multiplying eq. (\[eq:master2\]) by the complex conjugate of $\psi_n$ and integrating over the tortoise coordinate, we obtain after an integration by parts $$\begin{aligned}
\label{eq:master3}
-\bar \psi_n \frac{d\psi_n}{dr^*}\Bigl |_{-\infty}^{+\infty}+\int_{\mathbb{R}}{\rm d}r^* \Bigl[\Bigl |\frac{d\psi_n}{dr^*}\Bigr |^2+V |\psi_n |^2\Bigr]=\omega_n^2 A^2\,,\end{aligned}$$ where $\bar \psi_n$ is the complex conjugate of $\psi_n$ and $A^2=\int_{\mathbb{R}}{\rm d}r^* \frac{ |\psi_n |^2}{c_r^2}>0$. In the first situation, where the black hole is asymptotically flat or dS, the choice of the boundary conditions are ingoing near the horizon $(r^*=-\infty)$ and outgoing at infinity or at the cosmological horizon $(r^*=+\infty)$. Therefore, $$\begin{aligned}
\psi_n(r^*)&\propto e^{-{{i\mkern1mu}}\omega_n r^*}\,,~~\text{when  }r^*=-\infty~~\text{and}\nonumber\\
\psi_n(r^*)&\propto e^{+{{i\mkern1mu}}\omega_n r^*}\,,~~\text{when  }r^*=+\infty\,.\end{aligned}$$ This implies that there is no signal which comes from the black hole or from any source at infinity. The conditions for asymptotically AdS are the same at the horizon but the Dirichlet one at infinity, given by $$\begin{aligned}
\psi_n(r^*)&\propto e^{-{{i\mkern1mu}}\omega_n r^*}\,,~~\text{when  }r^*=-\infty~~\text{and}\nonumber\\
\psi_n(r^*)&=0\,,~~\text{when  }r^*=0\end{aligned}$$ where $r^*=0$ corresponds to $r=\infty$.

We therefore have $$\begin{aligned}
-\bar \psi_n \frac{d\psi_n}{dr^*}\Bigl |_{-\infty}^{+\infty}=-{{i\mkern1mu}}\omega_n B^2\,,\end{aligned}$$ where $B^2=|\psi_n(+\infty)|^2+|\psi_n(-\infty)|^2$, for an asymptotically flat or dS black hole and $B^2=|\psi_n(-\infty)|^2$ for an asymptotically AdS spacetime. Hence, $B^2$ is always a real positive number. Therefore, the eq. (\[eq:master3\]) becomes $$\begin{aligned}
\label{eq:AB}
\int_{\mathbb{R}}{\rm d}r^* \Bigl[\Bigl |\frac{d\psi_n}{dr^*}\Bigr |^2+V |\psi_n |^2\Bigr]=\omega_n^2 A^2+{{i\mkern1mu}}\omega_n B^2\,.\end{aligned}$$ The imaginary 