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Invoked by the recent observation of $Y(4390)$ at BESIII, which is about 40MeV below the $D^*(2010)\bar{D}_1(2420)$ threshold, we investigate possiblebound and resonance states from the $D^*(2010)\bar{D}_1(2420)$ interaction withthe one-boson-exchange model in a quasipotential Bethe-Salpeter equationapproach. A bound state with quantum number $0^-(1^{--})$ is produced at 4384MeV from the $D^ (2010)\bar{D}_1(2420)$ interaction, which can be related toexperimentally observed $Y(4390)$. Another state with quantum number$1^+(1^{+})$ is also produced at $4461+i39$ MeV from this interaction.Different from the $0^-(1^{--})$ state, the $1^+(1^{+})$ state is a resonancestate above the $D^*(2010)\bar{D}_1(2420)$ threshold. This resonance state canbe related to the first observed charged charmonium-like state $Z(4430)$, whichhas a mass about 4475 MeV measured above the threshold as observed at Belle andLHCb. Our result suggests that $Y(4390)$ is an isoscalar partner of the$Z(4430)$ as a hadronic-molecular state from the $D^*(2010)\bar{D}_1(2420)$interaction.

Interpretation of $Y(4390)$ as an isoscalar partner of $Z(4430)$ from $D^*(2010)\bar{D}_1(2420)$ interaction