  We calculate the tunneling density of states (DOS) of MgB2 for different
tunneling directions, by directly solving the real-axis, two-band Eliashberg
equations (EE). Then we show that the numeric inversion of the standard
single-band EE, if applied to the DOS of the two-band superconductor MgB2, 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 turn out to be clearly
observable only for tunneling along the ab planes in high-quality single
crystals. The results are compared to recent experimental data.
