  Scattering amplitudes involving three-particle scattering processes are
investigated within the isobar approximation which respects constraints from
two- and three-body unitarity. The particular system considered is the
$D^0D^{*+}-D^+D^{*0}$, where the $D^{*+}$ $(D^{*0})$ enters as a $p$-wave
$D^+\pi^0$ or $D^0\pi^+$ ($D^0\pi^0$ or $D^+\pi^-$) resonance. The interaction
potentials in the coupled-channel $D^0D^{*+}-D^+D^{*0}$ system contain the
$\sigma$, $\rho$, $\omega$ and $\pi$-exchange. The analytic continuation of the
amplitudes across the three-body unitary cuts is investigated to search for
poles on the unphysical Riemann sheets. Associated with an unstable particle
$D^{*+}$ $(D^{*0})$ is a complex two-body unitarity cut, through which one can
further analytically continue into another unphysical Riemann sheet. Dynamical
singularities emerged from the $\pi$-exchange potential are stressed. The pole
generated from the $D^0D^{*+}-D^+D^{*0}$ interaction and its line shape in
$D^0D^0\pi^+$ break-up production are in agreement with double-charmed
tetraquark $T_{cc}^+$ observed by the LHCb Collaboration.
