Self-energy Effect of Superconducting Energy Gap in Point-Contact Spectra of MgB${}_2$

In strong-coupling superconductors with a short electron mean free path the self-energy effects in the superconducting order parameter play a major role in the phonon manifestation of the point-contact spectra at the above-gap energies. We derive asymptotic expressions of the phonon structure in MgB$$ in the case $eV\gg \mathrm{\Delta }$ for tunnel, ballistic, and diffusive point-contacts and show that these expressions not only qualitatively, but also semi-quantitatively correspond to the measurements of the phonon structure in the point-contact spectra for the $\pi$-band of MgB${}_2$ c-axis oriented thin films.