Hadronic molecular states with the quark contents $bc\bar{s}\bar{q}$, $b\bar{c}s\bar{q}$, and $b\bar{c}\bar{s}q$

We study the hadronic molecular states with the quark content $bc\bar{s}\bar{q}$ by investigating the interactions of the $\bar{B}_s D$, $\bar{B} D_s$, $\bar{B}_s^* D$, $\bar{B}^* D_s$, $\bar{B}_s D^*$, $\bar{B} D_s^*$, $\bar{B}_s^* D^*$, and $\bar{B}^* D_s^*$ systems. By solving the Bethe-Salpeter equation within the extended local hidden gauge formalism, we find altogether six poles qualifying as possible hadronic molecular states: one pole of $J^P=0^+$ below the $\bar{B}_s D$-$\bar{B}D_s$ threshold, one pole of $J^P=1^+$ below the $\bar{B}_s^* D$-$\bar{B}^* D_s$ threshold, one pole of $J^P=1^+$ below the $\bar{B}_s D^*$-$\bar{B}D_s^*$ threshold, and three poles of $J^P=0^+/1^+/2^+$ below the $\bar{B}_s^* D^*$-$\bar{B}^* D_s^*$ threshold. Their binding energies are calculated to be about 10-20 MeV with the cut-off momentum $q_\textrm{max}=600\textrm{ MeV}$. Similarly, we study the hadronic molecular states with $bs\bar{c}\bar{q}$ by investigating the interactions of the $\bar{B}\bar{D}_s$, $\bar{B}_c\bar{K}$, $\bar{B}^*\bar{D}_s$, $\bar{B}_c^*\bar{K}$, $\bar{B}\bar{D}_s^*$, $\bar{B}_c\bar{K}^*$, $\bar{B}^*\bar{D}_s^*$, $\bar{B}_c^*\bar{K}^*$ systems, and the states with $bq\bar{c}\bar{s}$ by investigating the interactions of the $\bar{B}_s\bar{D}$, $\bar{B}_cK$, $\bar{B}_s^*\bar{D}$, $\bar{B}_c^*K$, $\bar{B}_s\bar{D}^*$, $\bar{B}_cK^*$, $\bar{B}_s^*\bar{D}^*$, $\bar{B}_c^*K^*$ systems. However, no deeply-bound poles are found in these systems.