Electron-phonon coupling and superconductivity in MgB$_2$ under hydrostatic pressure.

We have studied the dynamics and coupling of the $E_{2g}$ phonon mode with the $\sigma$-band in MgB$_2$ under pressure using the Frozen Phonon Approximation. The results were obtained by means of first-principles total-energy calculations using the full potential Linearized Augmented Plane Wave (LAPW) method and the Generalized Gradient Approximation (GGA) for the exchange-correlation potential. We present results for the evolution of the anharmonicity and phonon frequency of the $E_ {2g}$ mode, the electron-phonon coupling constant, and $T_c$ as a function of hydrostatic pressure in the range 0-40 GPa. We find that the phonon frequency increases monotonically with pressure, but the the anharmonicity, the electron-phonon coupling and $T_c$ decreases with pressure. We have obtained a very good agreement between the calculated $T_c(P)$ and the experimental data available in the literature, in particular with the experimental data corresponding to monocystalline samples. This work was supported by Consejo Nacional de Ciencia y Tecnolog{\'\i}a (CONACYT, M{\'e}xico) under Grant No. 43830-F.