  The Schrodinger equation with a Yukawa type of potential is solved
analytically. When different boundary conditions are taken into account, a
series of solutions are indicated as Bessel function, the first kind of Hankel
function 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 is
generated by solving the Schrodinger equation with the outgoing wave condition,
which lie at $1417-i18$MeV on the complex energy plane. It is reasonable to
assume that the $K \bar{K}^*$ resonance state at $1417-i18$MeV might correspond
to 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 is
determined, which is related to the first zero point of the zero order Bessel
function. Similarly to the $K \bar{K}^*$ case, four resonance states are
produced as solutions of the Schrodinger equation with the outgoing wave
condition. It is assumed that the resonance states at $3885-i1$MeV, $4029-i108$
MeV, $4328-i191$MeV and $4772-i267$MeV might be associated with the $Zc(3900)$,
the $X(3940)$, the $\chi_{c1}(4274)$ and $\chi_{c1}(4685)$ particles,
respectively. It is noted that all solutions are isospin degenerate.
