  The low-temperature properties of amorphous solids are widely believed to be
controlled by low-frequency quasi-localized modes. What governs their spatial
structure and density is however debated. We study these questions numerically
in very large systems as the jamming transition is approached and the pressure
p vanishes. We find that these modes consist of an unstable core in which
particles undergo the buckling motions and decrease the energy, and a stable
far-field component which increases the energy and prevents the buckling of the
core. The size of the core diverges as $p^{-1/4}$ and its characteristic volume
as $p^{-1/2}$ These features are precisely those of the anomalous modes known
to cause the Boson peak in the vibrational spectrum of weakly-coordinated
materials. From this correspondence we deduce that the density of
quasi-localized modes must go as $g_{\mathrm{loc}}(\omega) \sim \omega^4/p^2$ ,
in agreement with previous observations. Our analysis thus unravels the nature
of quasi-localized modes in a class of amorphous materials.
