  The low-frequency vibrations of glasses are markedly different from those of
crystals. These vibrations have recently been categorized into two types:
spatially extended vibrations, whose vibrational density of states (vDOS)
follows a non-Debye quadratic law, and quasilocalized vibrations (QLVs), whose
vDOS follows a quartic law. The former are explained by elasticity theory with
quenched disorder and microscopic replica theory as being a consequence of
elastic instability, but the origin of the latter is still debated. Here, we
show that the latter can also be directly derived from elasticity theory with
quenched disorder. We find another elastic instability that the theory
encompasses but that has been overlooked so far, namely, the instability of the
system against a local dipolar force. This instability gives rise to an
additional contribution to the vDOS, and the spatial structure and energetics
of the mode originating from this instability are consistent with those of the
QLVs. Finally, we construct a model in which the additional contribution to the
vDOS follows a quartic law.
