  To study the internal structure of the exotic hadron $X(3872)$, considering
the coupled-channel potential between quarks and hadrons is necessary because
$X(3872)$ is regarded as a mixture state of $c\bar{c}$ and $D^0\bar{D}^{*0}$.
In this work, we construct the hadron-hadron potentials coupled to the quark
channel and study the properties of the $D^0\bar{D}^{*0}$ potentials for
$X(3872)$. In particular, we compare various local approximations by focusing
on the scattering phase shifts.
