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The Schrodinger equation with a Yukawa type of potential is solvedanalytically. When different boundary conditions are taken into account, aseries of solutions are indicated as Bessel function, the first kind of Hankelfunction and the second kind of Hankel function, respectively. Subsequently,the scattering processes of $K \bar{K}^*$ and $D \bar{D}^*$ are investigated.In the $K \bar{K}^*$ sector, the $f_1(1285)$ particle is treated as a $K\bar{K}^*$ bound state, therefore, the coupling constant in the $K \bar{K}^*$Yukawa potential can be fixed according to the binding energy of the$f_1(1285)$ particle. Consequently, a $K \bar{K}^*$ resonance state isgenerated by solving the Schrodinger equation with the outgoing wave condition,which lie at $1417-i18$MeV on the complex energy plane. It is reasonable toassume that the $K \bar{K}^*$ resonance state at $1417-i18$MeV might correspondto the $f_1(1420)$ particle in the review of Particle Data Group(PDG). In the$D \bar{D}^*$ sector, since the $X(3872)$ particle is almost located at the $D\bar{D}^*$ threshold, the binding energy of it equals to zero approximately.Therefore, the coupling constant in the $D \bar{D}^*$ Yukawa potential isdetermined, which is related to the first zero point of the zero order Besselfunction. Similarly to the $K \bar{K}^*$ case, four resonance states areproduced as solutions of the Schrodinger equation with the outgoing wavecondition. It is assumed that the resonance states at $3885-i1$MeV,$4328-i191$MeV and $4772-i267$MeV might be associated with the $Zc(3900)$, the$\chi_{c1}(4274)$ and $\chi_{c1}(4685)$ particles, respectively. As to thestate at $4029-i108$ MeV, no counterpart has been found in the PDG data. It isnoted that all solutions are independent on the isospin.

The possible $K \bar{K}^*$ and $D \bar{D}^*$ bound and resonance states by solving Schrodinger equation

The possible $K \bar{K}^$ and $D \bar{D}^$ bound and resonance states by solving Schrodinger equation