Possible hadronic molecular states composed of $S$-wave heavy-light mesons

We perform a systematical study of possible molecular states composed of the $S$ wave heavy light mesons, where the $S-D$ mixing and {$\eta-\eta^{\prime}$ mixing} are explicitly included. Our calculation indicates that the observed $X(3872)$ could be a loosely shallow molecular state composed of $D\bar{D}^\ast +h.c$, while neither $Z_c(3900)/Z_c(4020)$ nor $Z_b(10610)/Z_b(10650)$ is supported to be a molecule. Some observed possible molecular states are predicted, which could be searched for by further experimental measurements.