Title: Eliashberg theory with the external pair potential,
Abstract: Based on BCS model with the external pair potential formulated in a work
\emph{K.V. Grigorishin} arXiv:1605.07080, 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 influence 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, where the temperature dependencies near
$T_{c}$ of the basic characteristics of a superconductor (coherence length,
magnetic penetration depth, GL parameter, the thermodynamical critical field,
the first and the second critical fields) recovers to the temperature
dependencies as in the ordinary GL theory after the BCS model with the external
pair potential.