Possible $K \bar{K}^*$ and $D \bar{D}^*$ resonances by solving Schr\"odinger equation

The one-pion exchange interaction between the kaon and the vector antikaon is investigated by solving the Schr\"odinger equation in the S-wave approximation. In addition to the particle $f_1(1285)$, another bound state of $K \bar{K}^*$ is obtained, 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 with different coupling constants fixed with the binding energies of $f_1(1285)$ 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 simultaneously. This method is extended to study the $D \bar{D}^*$ system analogously. When the particle $\chi_{c1}(3872)$ is treated as a bound state of $D \bar{D}^*$, the particles $T_{c\bar{c}1}(3900)$, $T_{c \bar{c}}(4020)$ and $X(3940)$ can be obtained as solutions of the Schr\"odinger equation under the outgoing wave condition, which implies the spin and parity of these particles are all $J^P=1^+$.