The direct inversion of the Eliashberg equations ͑EE͒ in case of a multiband superconductor is a mathematically ill-defined problem, because it is not possible to obtain several band splitted electron-phonon spectral functions ␣ 2 F i j () from a single function of the tunnel current. In the present work we follow another direction and calculate the tunneling density of states ͑DOS͒ of MgB 2 for different tunneling directions by directly solving the two-band 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 single-band EE, when applied to the two-band DOS of MgB 2 , underestimates the strength of certain phonon branches ͑e.g., the E 2g ) in the extracted ␣ 2 F(). 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 and at very low temperature. Only in this case it is possible to extract some information on the -band contribution to the spectral functions. For any other tunneling direction, the -band contribution is dominant and almost coincides with the whole ␣ 2 F() for tunneling along the c axis. Our results are compared with recent experimental tunneling and point-contact data. There is a growing consensus that the superconductivity in MgB 2 with a critical temperature T c Ӎ40 K ͑Ref. 1͒ is driven by the electron-phonon interaction ͑EPI͒ ͑for a recent review see Ref.2͒. An important subject to address for a proper understanding of the surprising physical properties of this material is the character of the order parameter ͑or superconducting gap͒: is it constant over the whole Fermi surface, or strongly momentum dependent? The idea of multiband superconductivity in MgB 2 ͑Refs. 3-9͒ is supported by many recent experimental results from tunneling, 10-12 point contact,13-15 and specific-heat capacity measurements. 6,16 These data directly support the picture that the superconducting gap has two different values on two qualitatively different parts of the Fermi surface, one ⌬ for the two quasi-twodimensional bands and another one ⌬ for the pair of three-dimensional ͑3D͒ bands.
3,5While, within first-principles calculations of the electronic structure and the EPI in this compound, there is an agreement 17 on this qualitative picture, still disagreement is present about the precise values of characteristic frequencies and coupling constants. According to most calculations, 3,18,19,8 the EPI or, equivalently, the Eliashberg spectral function ␣ 2 F() ͑EF͒ is dominated by the optical boron bond-stretching E 2g phonon branch of around 60-70 meV.In principle, photoemission lineshapes or the deviation of the far-IR absorption from the Drude law are controlled by the same Eliashberg functions as tunneling or their transport counterparts, so that optical measurements can also deliver information on the EPI. 20,21 Unfortunately, the extraction of the spectral functions from these experiments, fo...