  We study the linear excitations around typical energy minima of a mean-field
disordered model with continuous degrees of freedom undergoing a Random First
Order Transition (RFOT). Contrary to naive expectations, the spectra of linear
excitations are ungapped and we find the presence of a pseudogap corresponding
to localized excitations with arbitrary low excitation energy. Moving to deeper
minima in the landscape, the excitations appear increasingly localized while
their abundance decreases. Beside typical minima, there also exist rare
ultra-stable minima, with an energy gap and no localised excitations.
