Can one extract the electron-phonon-interaction from tunneling data in case of the multigap superconductor MgB${}_2$?

In the present work we calculate the tunneling density of states (DOS) of MgB% ${}_2$ for different tunneling directions by directly solving the two-band Eliashberg equations (EE) in the real-axis formulation. This procedure reveals the fine structures of the DOS due to the optical phonons. Then we show that the numeric inversion of the standard \emph{single-band} EE (the only available method), when applied to the \emph{two-band} DOS of MgB${}_2$, may lead to wrong estimates of the strength of certain phonon branches (e.g. the $E_{2g}$) in the extracted electron-phonon spectral function ${\alpha }^{2}F(\omega )$. The fine structures produced by the two-band interaction at energies between 20 and 100 meV turn out to be clearly observable only for tunneling along the $ab$ planes, when the extracted ${\alpha }^{2}F(\omega )$ contains the combination ${\alpha }^2{F}_{\sigma \sigma }(\omega )$$\mathbf{\text{+}}$$\alpha ^{2}F_{\sigma \pi }(\omega)$,togetherwithaminor$\alpha ^{2}F_{\pi \pi}(\omega )$\mathbf{\text{+}}$\alpha ^{2}F_{\pi \sigma} (\omega)$ component. Only in this case it is possible to extract information on the $\sigma$-band contribution to the spectral functions. For any other tunneling direction, the $\pi$-band contribution (which does not determine the superconducting properties of MgB${}_2$) is dominant and almost coincides with the whole ${\alpha }^{2}F(\omega )$ for tunneling along the c axis. Our results are compared with recent experimental tunneling and point-contact data.