  We numerically study the evolution of the vibrational density of states
$D(\omega)$ of zero-temperature glasses when their kinetic stability is varied
over an extremely broad range, ranging from poorly annealed glasses obtained by
instantaneous quenches from above the onset temperature, to ultrastable glasses
obtained by quenching systems thermalised below the experimental glass
temperature. The low-frequency part of the density of states splits between
extended and quasi-localized modes. Extended modes exhibit a boson peak
crossing over to Debye behaviour ($D(\omega) \sim \omega^2$) at low-frequency,
with a strong correlation between the two regimes. Quasi-localized modes
instead obey $D(\omega) \sim \omega^4$, irrespective of the glass stability.
However, the prefactor of this quartic law becomes smaller in more stable
glasses, and the corresponding modes become more localized and sparser. Our
work is the first numerical observation of quasi-localized modes in a regime
relevant to experiments, and it establishes a direct connection between glass
stability and soft vibrational motion in amorphous solids.
