Does $f_1(1420)$ have a double-peak structure?

The one-pion exchange interaction between the kaon and the vector antikaon is investigated by solving the Schrodinger equation in the S-wave approximation. In addition to the $f_1(1285)$ particle, another bound state of $K \bar{K}^*$ is found, which is approximately 9 MeV below the threshold of $K \bar{K}^*$ and labeled $f_1(1378)$ for convenience in this manuscript. Under the outgoing wave condition, two resonance states of $K \bar{K}^*$ are produced by solving the Schrodinger equation with different coupling constants in the one-pion-exchange potential fixed with the binding energies of $K \bar{K}^*$ bound states $f_1(1285)$ and $f_1(1378)$ respectively. Both of the resonance states are located in the vicinity of 1400 MeV, and thus it is reasonable to assume that these two resonance states correspond to the $f_1(1420)$ particle in the review of the Particle Data Group although they arise from different couplings of the kaon and vector antikaon, respectively.