  An analytical derivation of the vibrational density of states (DOS) of
liquids, and in particular of its characteristic linear in frequency low-energy
regime, has always been elusive because of the presence of an infinite set of
purely imaginary modes -- the instantaneous normal modes (INMs). By combining
an analytic continuation of the Plemelj identity to the complex plane with the
purely overdamped dynamics of the INMs, we derive a closed-form analytic
expression for the low-frequency DOS of liquids. The obtained result explains
from first principles the widely observed linear in frequency term of the DOS
in liquids, whose slope appears to increase with the average lifetime of the
INMs. The analytic results are robustly confirmed by fitting simulations data
for Lennard-Jones liquids, and they also recover the Arrhenius law for the
average relaxation time of the INMs, as expected.
