Study on possible molecular states composed of ${\mathrm{\Lambda }}_c\bar{D}$ (${\mathrm{\Lambda }}_{b}B$) and ${\mathrm{\Sigma }}_c\bar{D}$ (${\mathrm{\Sigma }}_{b}B$) within the Bethe-Salpeter framework

$P_c(4312)$ observed by the LHCb collaboration is confirmed as a pentaquark and its structure, production, and decay behaviors attract great attention from theorists and experimentalists. Since its mass is very close to sum of ${\mathrm{\Sigma }}_c$ and $\bar{D}$ masses, it is naturally tempted to be considered as a molecular state composed of ${\mathrm{\Sigma }}_c$ and $\bar{D}$. Moreover, $P_c(4312)$ is observed in the channel with $J\mathrm{/}\psi p$ final state, requiring that isospin conservation $P_c(4312)$ is an isospin-1/2 eigenstate. In literature, several groups used various models to estimate its spectrum. We systematically study the pentaquarks within the framework of the Bethe-Salpeter equation; thus $P_c(4312)$ is an excellent target because of the available data. We calculate the spectrum of $P_c(4312)$ in terms of the Bethe-Salpter equations and further study its decay modes. Some predictions on other possible pentaquark states that can be tested in future experiments are made.