creation operators, respectively. The Fourier transformed potential $g_{epi}(\mathbf{q})(=(g_{0}/\sqrt{N})e^{-q/q_{c}})$ is given by $$g_{epi}(\mathbf{q})=\frac{2\pi n_{d}eq_{eff}}{\varepsilon_{\perp}^{eff}}\sqrt{\frac{\hbar}{M\Omega N}}e^{-q/q_{c}}, \label{g-epi}$$ $g_{0}=(2\pi n_{d}eq_{eff}/\varepsilon_{\perp}^{eff})(\hbar/M\Omega)^{1/2}$, $e$ is the electronic charge. Here, the screening momentum $q_{c}=(\varepsilon_{\perp}^{eff}/\varepsilon_{\parallel}^{eff})^{1/2}h_{0}^{-1}$ characterizes the range of the $EPI$ potential, i.e. for $q_{c}\ll k_{F}$ ($k_{F}$ is the Fermi momentum) the $EPI$ is sharply peaked at $\mathbf{q}=0$ - the forward scattering peak ($FSP$), and the potential in real space is long-ranged, while for $q_{c}\sim k_{F}$ it is short-ranged, like in the standard $EPI$ theory. Since we are interested in the $T_{c}$ dependence on the effective parameters $\varepsilon_{\parallel}^{eff},\varepsilon_{\perp
}^{eff},q_{eff},h_{0}$, then an explicit dependence of the potential is important. We shall see below, that in order that this approach is applicable to $1ML$ $FeSe/SrTiO_{3}$ (and $1ML$ $FeSe/TiO_{2}$) $\varepsilon_{\parallel
}^{eff},\varepsilon_{\perp}^{eff}$ must be very different from the bulk values of $\varepsilon$ in the bulk $SrTiO_{3}$ - where $\varepsilon\sim500-10^{4}$, or in the rutile $TiO_{2}$ structure where $\varepsilon<260$ [@Rebec-FeSe-TiO2].

Self-energy effects and $ARPES$ replica bands
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The general self-energy $\Sigma_{epi}(\mathbf{k}_{F},\omega)$ at $T=0$ in the extreme $FSP$ $\delta$-peak limit with the width $q_{c}\ll k_{F}$) is given by (see Appendix)$$\Sigma_{epi}(\mathbf{k}_{F},\omega)\approx-\lambda_{m}\frac{\omega}{1-(\omega/\Omega)^{2}}, \label{Sigma-epi}$$ where $\lambda_{m}=\left\langle V_{epi}(\mathbf{q})\right\rangle _{q}/2\Omega$ is the *mass-dependent coupling constant*. Here, the average $EPI$ potential is given by $\left\langle V_{epi}(\mathbf{q})\right\rangle
_{q}=Ns_{c}(2\pi)^{-2}\int d^{2}qV_{epi}(\mathbf{q},0)\approx(1/4\pi
)(aq_{c})^{2}V_{epi}^{0}$, where $s_{c}=2a^{2}$ 