# Eliashberg theory with the external pair potential

## TL;DR

This paper extends the Eliashberg theory by incorporating an external pair potential into a model with electron-phonon and Coulomb interactions, deriving generalized equations and analyzing high-temperature behavior.

## Contribution

It introduces a novel Eliashberg framework with an external pair potential and derives generalized equations considering Coulomb effects.

## Key findings

- Order parameter tends to zero at high temperature
- Coulomb pseudopotential influences critical temperature
- Effective Ginzburg-Landau theory formulated

## Abstract

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.