In the present work, we investigate the molecular properties of the hidden charm pentaquark states $P_c$ and $P_{cs}$ with a coupled-channel framework that combines heavy quark spin symmetry and the local hidden gauge formalism. By solving the Bethe-Salpeter equation with the cutoff method, we obtain the pole trajectories, wave functions, and root-mean-square radii. For the hidden charm system, the full coupled-channel interactions respecting the heavy quark spin symmetry are essential to generate the $P_c$ states, which significantly affect the poles’ widths. The dominant bound channels are $\overline{D}{\mathrm{\Sigma }}_c$ and ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$, which couple strongly to the lower decay channels. In contrast, for the hidden charm strange system, the full heavy quark spin symmetry treatment is not necessary, where the splitting pseudoscalar meson-baryon (PB) and vector meson-baryon (VB) sectors yield similar results. The main bound channels $\overline{D}{\mathrm{\Xi }}_c$ and ${\overline{D}}^{*}{\mathrm{\Xi }}_c$ couple strongly to ${\overline{D}}_s{\mathrm{\Lambda }}_c$ and ${\overline{D}}_s^{*}{\mathrm{\Lambda }}_c$, respectively, but weakly to the lower decay channels, different from the hidden charm case. The trajectories of the pole widths for the loosely bound channels $\overline{D}{\mathrm{\Xi }}_c^{\prime }$, ${\overline{D}}^{*}{\mathrm{\Xi }}_c^{\prime }$, and ${\overline{D}}^{*}{\mathrm{\Xi }}_c^{*}$ exhibit distinct behaviors. Notably, all the primary bound channels have similar binding energies in the single-channel interactions due to equally attractive potentials. Furthermore, we also calculate the wave functions and root-mean-square radii of the corresponding poles. The wave functions are localized within $0\sim 6\mathrm{fm}$ and vanish fast beyond 4 fm. The root-mean-square radii, evaluated by two consistent methods, typically lie between 0.5 and 2 fm, comparable to the characteristic scale of molecular states. The root-mean-square radii depend on the pole trajectories and differ among the full coupled-channel case, the split PB and VB sectors, and the single-channel interactions.

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• Multiquark bound states
• Particle interactions
• Phenomenology