onance. In the first extreme, we assume a featureless density of states in the vicinity of the Fermi energy over an interval $[-2\w_0,2\w_0]$, i. e. $\rho_{\mu,\mu'\sigma}(\w)\to {\rm const.}$ in , and $\beta\w_0\gg 1$ so that the Bose function can be ignored. Then $\tau^{(2)}_{\sigma}(\w) $ will be dominated by the Fermi functions, introducing two threshold contributions in the overall differential conductance $dI/dV$ at $\pm \w_0$. These are the typical $dI/dV$ steps that are often encountered in inelastic tunnel spectroscopy as shown in Fig. 1 of Ref. [@REED2008].

In the second extreme, we consider an electronic DOS of the sample system S which possesses a sharp spectral peak located at $\w=0$ with a width $\Gamma\ll \w_0$. Then $\tau^{(2)}_{\sigma}(\w)$ exhibits again a sharp threshold behavior at $\pm \w_0$, but instead of a plateau the spectral function decreases with increasing $|\w|$ on a scale given by the peak broadening $\Gamma$. In this case, two “replicas” of the peak at $\w=0$ can be found at $\pm \w_0$ in the overall differential conductance $dI/dV$. However, the Fermi function $f_{\rm S}(\w_0\pm\w)$ cuts away the halves of the replicas on the low-$|\w|$ side and modifies them to a threshold behavior: a minimal energy transfer for $eV=\pm\w_0$ is required to generate an inelastic contribution replicating the standard picture [@MolecularVibrationTunnel1968]. These truncated replicas of the $\w=0$ peak are generated by the inelastic tunneling process due to the change of the distance between system S and STM tip. In case of a large electron-phonon coupling in S, the approximation is invalid and the proper Green’s function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'}  d^\dagger_{\mu' \sigma}}(t)$ must be calculated, along with $G_{X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}}$ and $G_{d_{\mu\sigma},  X_\nu d^\dagger_{\mu'\sigma}}$.

### Discussion

The presented tunneling theory combines different limits [@MolecularVibrationTunnel1968; @Mahan81; @PerssonBaratoff1987; @LorentePersson2000; @Pa