  We study a recently introduced and exactly solvable mean-field model for the
density of vibrational states $\mathcal{D}(\omega)$ of a structurally
disordered system. The model is formulated as a collection of disordered
anharmonic oscillators, with random stiffness $\kappa$ drawn from a
distribution $p(\kappa)$, subjected to a constant field $h$ and interacting
bilinearly with a coupling of strength $J$. We investigate the vibrational
properties of its ground state at zero temperature. When $p(\kappa)$ is gapped,
the emergent $\mathcal{D}(\omega)$ is also gapped, for small $J$. Upon
increasing $J$, the gap vanishes on a critical line in the $(h,J)$ phase
diagram, whereupon replica symmetry is broken. At small $h$, the form of this
pseudogap is quadratic, $\mathcal{D}(\omega)\sim\omega^2$, and its modes are
delocalized, as expected from previously investigated mean-field spin glass
models. However, we determine that for large enough $h$, a quartic pseudogap
$\mathcal{D}(\omega)\sim\omega^4$, populated by localized modes, emerges, the
two regimes being separated by a special point on the critical line. We thus
uncover that mean-field disordered systems can generically display both a
quadratic-delocalized and a quartic-localized spectrum at the glass transition.
