  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
$\alpha ^{2}F_{\sigma \sigma}(\omega)$\textbf{+}$\alpha ^{2}F_{\sigma \pi
}(\omega)$, together with a minor $\alpha ^{2}F_{\pi \pi}(\omega
)$\textbf{+}$\alpha ^{2}F_{\pi \sigma} (\omega)$ 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 $\alpha^2F(\omega)$
for tunneling along the c axis. Our results are compared with recent
experimental tunneling and point-contact data.
