  Based on BCS model with the external pair potential formulated in a work
Grigorishin (2017) [1], analogous model with electron-phonon coupling and
Coulomb coupling is proposed. The generalized Eliashberg equations in the
regime of renormalization of the order parameter are obtained. High temperature
asymptotics and effect of Coulomb pseudopotential on them are investigated: as
in the BCS model the order parameter asymptotically tends to zero as
temperature rises, but the accounting of the Coulomb pseudopotential leads to
existence of critical temperature. The effective Ginzburg-Landau theory is
formulated for such model.
