Antiadiabatic phonons, Coulomb pseudopotential and superconductivity in Eliashberg - McMillan theory

The influence of antiadiabatic phonons on the temperature of superconducting transition is considered within Eliashberg - McMillan approach in the model of discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature $T_c$ is proposed, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We study the contribution of such phonons into the Coulomb pseudopotential $\mu^{\star}$. It is shown, that antiadiabatic phonons do not contribute to Tolmachev's logarithm and its value is determined by partial contributions from adiabatic phonons only. The results obtained are discussed in the context of the problem of unusually high superconducting transition temperature of FeSe monolayer on STO.