Prediction of possible DK${}_1$ bound states

Stimulated by recent experimental observation of X${}_1$(2900) just below the ${\overline{D}}_1$K threshold, we extend our previous study of ${\overline{D}}_1$D S-wave bound state by vector meson exchange to ${\overline{D}}_1$K system as well as similar $\overline{D}$K${}_1$, D${}_1$K and DK${}_1$ systems to look for possible bound states. We find that the potential of DK${}_1$ is attractive and strong enough to form bound states with mass around 3110 MeV for DK${}_1$(1270) and 3240 MeV for DK${}_1$(1400). D${}_1$K is also attractive but weaker, hardly enough to form bound states. While $\overline{D}$K${}_1$ becomes further less attractive, the potential between ${\overline{D}}_1$K is the weakest, definitely too weak to form any bound state, which excludes the recently observed X${}_1$(2900) to be a ${\overline{D}}_1$K bound state. We also give the decay properties of the predicted DK${}_1$ bound states.