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 ${\bar{B}}_{s}D$, $\bar{B}{D}_s$, ${\bar{B}}_s^{\ast }D$, ${\bar{B}}^{\ast }{D}_s$, ${\bar{B}}_s{D}^{\ast }$, $\bar{B}{D}_s^{\ast }$, ${\bar{B}}_s^{\ast }{D}^{\ast }$, and ${\bar{B}}^{\ast }{D}_s^{\ast }$ 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 ${\bar{B}}_s^{\ast }D$-${\bar{B}}^{\ast }{D}_s$ threshold, one pole of $J^P=1^{+}$ below the ${\bar{B}}_s{D}^{\ast }$-$\bar{B}{D}_s^{\ast }$ threshold, and three poles of $J^P=0^{+}\mathrm{/}{1}^{+}\mathrm{/}{2}^{+}$ below the ${\bar{B}}_s^{\ast }{D}^{\ast }$-${\bar{B}}^{\ast }{D}_s^{\ast }$ threshold. Their binding energies are calculated to be about 10-20 MeV with the cut-off momentum $q_{\text{max}}=600\text{ MeV}$. Similarly, we study the hadronic molecular states with $bs\bar{c}\bar{q}$ by investigating the interactions of the $\bar{B}{\bar{D}}_s$, ${\bar{B}}_c\bar{K}$, ${\bar{B}}^{\ast }{\bar{D}}_s$, ${\bar{B}}_c^{\ast }\bar{K}$, $\bar{B}{\bar{D}}_s^{\ast }$, ${\bar{B}}_c{\bar{K}}^{\ast }$, ${\bar{B}}^{\ast }{\bar{D}}_s^{\ast }$, ${\bar{B}}_c^{\ast }{\bar{K}}^{\ast }$ systems, and the states with $bq\bar{c}\bar{s}$ by investigating the interactions of the ${\bar{B}}_s\bar{D}$, ${\bar{B}}_{c}K$, ${\bar{B}}_s^{\ast }\bar{D}$, ${\bar{B}}_c^{\ast }K$, ${\bar{B}}_s{\bar{D}}^{\ast }$, ${\bar{B}}_c{K}^{\ast }$, ${\bar{B}}_s^{\ast }{\bar{D}}^{\ast }$, ${\bar{B}}_c^{\ast }{K}^{\ast }$ systems. However, no deeply-bound poles are found in these systems.