It is well know that when one calculates the spectral function from correlators at the conformal boundary $z \to 0$, the spectral function in the limit $\omega \to \infty$ grows up as $\rho \sim  \omega^2 $. This results comes from conformal invariance and dimensional analyisis (see for example ref. [@Myers:2007we]).

In the present case we do not calculate the correlators at the $z\to 0$ conformal limit. There is an extra dimensionfull quantity, the position $z_0$, that appears in the calculation of the spectral function. So, the argument of simple dimensional analysis does not hold in the same way here. The numerical results obtained show a behavior that is different from the conformal case. For large frequencies the spectral function grows linearly with the frequency: $ \rho \sim \omega $. We present in appendix [**B**]{} an analysis of this behavior. We show there that if in the present model one takes the limit of $z_0 \to 0 $ one finds spectral functions growing with $\omega^2$, as expected in the conformal case. But for the finite value of $ z_0$ explored here they grow with $\omega$ for large $\omega$. So, we analyzed the behavior of the relevant (normalized) quantity: $$\frac{ \rho(\omega ) 