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abstract: |
    The theory of the electron-phonon interaction ($EPI$) with strong forward scattering peak ($FSP$) in an extreme delta-peak limit [@Kulic-Zeyher1-2]-[@Dan-Dol-Kul-Oud] is recently applied in [@Lee-Interfacial-supplement]-[@Johnston1-2] for the explanation of high $T_{c}(\sim100$ $K)$ in a monolayer $FeSe$ grown on $SrTiO_{3}$ [@Lee-Interfacial-supplement] and $TiO_{2}$ [@Rebec-FeSe-TiO2] substrates. The $EPI$ is due to a long-range dipolar electric field created by the high-energy oxygen vibrations ($\Omega\sim90$ $meV$) at the interface [@Lee-Interfacial-supplement]-[@Johnston1-2]. We show that in leading order (with respect to $T_{c0}/\Omega$) the mean-field critical temperature $T_{c0}=\left\langle V_{epi}(q)\right\rangle _{q}/4)$ $\sim$ $(aq_{c})^{2}V_{epi}(0)$ and the gap $\Delta_{0}=2T_{c0\text{ }}$ are due to an interplay between the maximal $EPI$ pairing potential $V_{epi}(0)$ and the $FSP$-width $q_{c}$. For $T_{c0}\sim100$ $K$ one has $\Delta_{0}\sim16$ $meV$ in a satisfactory agreement with $ARPES$ experiments. We find that in leading order $T_{c0}$ is * mass-independent* and a very small oxygen isotope effect is expected in next to leading order. In clean systems $T_{c0}$ for $s$-wave and $d$-wave pairing is degenerate but both are affected by non-magnetic impurities, which are *pair-weakening* in the $s$-channel and *pair-breaking* in the $d$-channel.

    The self-energy and replica bands at $T=0$ and at the Fermi surface are calculated and compared with the corresponding results at $T>0$ [@Johnston1-2]. The $EPI$ 