even if a wide range of cut offs are used. In Fig. \[Fig:dgamr\], we show the results for five different, higher values of $q_{\rm max}$. The magnitude of the $a_0(980)$ production grows a bit with $q_{\rm
max}$, with the prescription taken above, but the strength of the $f_0(980)$ production also grows as a consequence of an increase in the width. One can also see that the peak of the $f_0(980)$ moves to lower energies, what puts constraints on $q_{\rm max}$, but we see that, even within this broad range of values of $q_{\rm max}$, the ratio of Eq. (\[ratioth\]) remains within the errors of this equation and is a solid prediction.

![(Color online) The $\pi^+ \pi^-$ (black curves) and $\pi^0
\eta$ (blue curves) invariant mass distributions with different cut off $q_{\rm max}$ for the $D^0 \to \bar{K}^0 \pi^+ \pi^-$ decay and $D^0 \to \bar{K}^0 \pi^0 \eta$ decay, respectively.[]{data-label="Fig:dgamr"}](dgamr.eps)

It should not go unnoticed that we also predict a sizeable fraction of the decay width into $D^0 \to \bar{K}^0 f_0(500)$, with a strength several times bigger than for the $f_0(980)$. The $\pi^+
\pi^-$ distributions is qualitatively similar to that obtained in Ref. [@weihong] for the $\bar{B}^0 \to J/\psi \pi^+\pi^-$ decay, although 