coupling constant $\lambda_{m}=\left\langle
    V_{epi}(q)\right\rangle _{q}/2\Omega$, which enters the self-energy $\Sigma(k,\omega)$, is *mass-dependent* ($M^{1/2}$) which at $\omega(\ll\Omega)$ makes the slope of $\ \Sigma(k,\omega)(\approx-\lambda
    _{m}\omega)$ and the replica intensities $A_{i}(\sim\lambda_{m})$ *mass-dependent*. This result, overlooked in the literature, is contrary to the prediction of its mass-independence in the standard Migdal-Eliashberg theory for $EPI$. The small oxygen isotope effect in $T_{c0}$ and pronounced isotope effect in $\Sigma(k,\omega)$ and $ARPES$ spectra $A_{i}$ of replica bands in $FeSe$ films on $SrTiO_{3}$ and $TiO_{2}$ is a smoking-gun experiment for testing an applicability of the $EPI-FSP$ theory to these systems. The $EPI-FSP$ theory predicts a large number of low-laying pairing states (above the ground state) thus causing internal pair fluctuations. The latter reduce $T_{c0}$ additionally, by creating a pseudogap state for $T_{c}<T<T_{c0}$.

    Possibilities to increase $T_{c0}$, by designing novel structures are discussed in the framework of the $EPI-FSP$ theory.
address: |
    $^{1}$Institute for Theoretical Physics, Goethe-University D-60438 Frankfurt am Main, Germany\
    $^{2}$Institute of Physics, Pregrevica 118, 11080 Belgrade (Zemun), Serbia\
    $^{3}$Max-Planck-Institut für Festkörperphysik,70569 Stuttgart, Germany
author:
- '$^{1,2}$M. L. Kulić, $^{3}$O. V. Dolgov'
title: 'The electron-phonon interaction with forward scattering peak is a relevant approach to high $T_{c}$ superconductivity in $FeSe$ films on $SrTiO_{3}$ and $TiO_{2}$ '
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