Inspired by the invariant mass distributions for the decays $D^{ 0} \rightarrow K_S^{ 0} \pi^{ 0} \eta$ and $D^{ 0} \rightarrow K_S^{ 0} \pi^{ 0} \pi^{ 0}$ reported by the BESIII Collaboration, we investigate these processes with an unified final state interaction formalism by incorporating both the $S$-wave pseudoscalar meson-pseudoscalar meson interactions within a chiral unitary approach and the $P$-wave contribution from the intermediate resonance $\bar{ K}^*(892)$. By performing a combined fit to the invariant mass spectra and taking into account the coherence between the $S$- and $P$-waves, our results are in agreement with the experimental data. For the decay $D^{ 0} \rightarrow K_S^{ 0} \pi^{ 0} \eta$, the structure near 1.0 GeV in the $\pi^0 \eta$ invariant mass distribution corresponds to the signal of the $a_{ 0}(980)$, which is dynamically generated from the $S$-wave interactions. In the case of $D^{ 0} \rightarrow K_S^{ 0} \pi^{ 0} \pi^{ 0}$, the near-threshold enhancement in the $\pi^0\pi^0$ mass distribution arises from the combined contributions of the $f_0(500)$ and the intermediate $\bar{ K}^*(892)$, while the cusp-like structure around 1 GeV$^2$ is associated with the $f_0(980)$.
Categories: hep-ph