  The vibrational density of states $D(\omega)$ of solids controls their
thermal and transport properties. In crystals, the low-frequency modes are
extended phonons distributed in frequency according to Debye's law, $D(\omega)
\propto \omega^2$. In amorphous solids, phonons are damped, and at low
frequency $D(\omega)$ comprises extended modes in excess over Debye's
prediction, leading to the so-called boson peak in $D(\omega)/\omega^2$ at
$\omega_{\rm bp}$, and quasi-localized (QLMs) ones. Here we show that boson
peak and phonon attenuation in the Rayleigh scattering regime are related, as
suggested by correlated fluctuating elasticity theory (corr-FET), and that
amorphous materials can be described as homogeneous isotropic elastic media
punctuated by QLMs acting as elastic heterogeneities. Our numerical results
resolve the conflict between theoretical approaches attributing amorphous
solids' vibrational anomalies to elastic disorder and localized defects.
