  A partial width formula is proposed using the analytical extension of the
wave function in momentum space. The distinction of the Riemann sheets is
explained from the perspective of the Schrodinger equation. The analytical form
in coordinate space and the partial width are derived subsequently. Then a
coupled-channel analysis is performed to investigate the open-charm branching
ratios of the $P_c$ states, involving the contact interactions and
one-pion-exchange potential with the three-body effects. The low energy
constants are fitted using the experimental masses and widths as input. The
$P_c(4312)$ is found to decay mainly to $\Lambda_c\bar{D}^*$, while the
branching ratios of the $P_c(4440)$ and $P_c(4457)$ in different channels are
comparable. Under the reasonable assumption that the off-diagonal contact
interactions are small, the $J^P$ quantum numbers of the $P_c(4440)$ and the
$P_c(4457)$ prefer $\frac{1}{2}^-$ and $\frac{3}{2}^-$ respectively. Three
additional $P_c$ states at 4380 MeV, 4504 MeV and 4516 MeV, together with their
branching ratios, are predicted. A deduction of the revised one-pion-exchange
potential involving the on-shell three-body intermediate states is performed.
