  In previous theoretical studies of phonon-mediated superconductors, the
electron-phonon coupling is treated by solving the Migdal-Eliashberg equations
under the bare vertex approximation, whereas the effect of Coulomb repulsion is
incorporated by introducing one single pseudopotential parameter. These two
approximations become unreliable in low carrier-density superconductors in
which the vertex corrections are not small and the Coulomb interaction is
poorly screened. Here, we shall go beyond these two approximations and employ
the Dyson-Schwinger equation approach to handle the interplay of
electron-phonon interaction and Coulomb interaction in a self-consistent way.
We first derive the exact Dyson-Schwinger integral equation of the full
electron propagator. Such an equation contains several unknown single-particle
propagators and fermion-boson vertex functions, and thus seems to be
intractable. To solve this difficulty, we further derive a number of identities
satisfied by all the relevant propagators and vertex functions and then use
these identities to show that the exact Dyson-Schwinger equation of electron
propagator is actually self-closed. This self-closed equation takes into
account not only all the vertex corrections, but also the mutual influence
between electron-phonon interaction and Coulomb interaction. Solving it by
using proper numerical methods leads to the superconducting temperature $T_{c}$
and other quantities. As an application of the approach, we compute the $T_{c}$
of the interfacial superconductivity realized in the one-unit-cell
FeSe/SrTiO$_{3}$ system. We find that $T_{c}$ can be strongly influenced by the
vertex corrections and the competition between phonon-mediated attraction and
Coulomb repulsion.
