  The anharmonic self energy of two zone boundary phonons were computed to
lowest order for AlH$_3$ in the $Pm\bar 3n$ structure at 110 GPa. The
wavevector and branch index corresponding to these modes are situated in a
region of phase space providing most of the electron-phonon coupling. The self
energies are found to be very large and the anharmonic contribution to the
linewidth of one of the modes studied could be distinguished from the
electron-phonon linewidth. It is found that anharmonicity suppresses the
electron-phonon coupling parameter $\lambda$, providing a possible explanation
for the disagreement between experiment and previous theoretical studies of
superconductivity in this system.
