  We study the properties of the hadron-hadron potentials and quark-antiquark
potentials from the viewpoint of the channel coupling. We demonstrate that, for
finite quark masses, the coupling to the two-hadron continuum induces the
imaginary part of the quark-antiquark potential, in contrast to the
string-breaking phenomena in the static limit. It is also shown that the
elimination of the different degrees of freedom induces the nonlocality and
energy dependence of the effective potentials. For the obtained nonlocal
potentials, we apply two methods of the local approximation proposed
previously, the formal derivative expansion and the derivative expansion in the
HAL QCD method, by carefully examining the energy dependence of the potential.
As an application, we construct a coupled-channel model of $c\bar{c}$ and
$D^{0}\bar{D}^{*0}$ to describe $X(3872)$, and discuss the property of the
effective $D^{0}\bar{D}^{*0}$ potentials. We confirm that the local
approximation by the HAL QCD method works better than the formal derivative
expansion also for the energy-dependent potential. At the same time, we show
that, in the HAL QCD method, the resulting phase shift is sensitive to the
choice of the wavefunction to construct the local potential when the system has
a shallow bound state such as $X(3872)$.
