  It has been recently shown [E. Lerner, G. D\"uring, and E. Bouchbinder, Phys.
Rev. Lett. 117, 035501 (2016)] that the non-phononic vibrational modes of
structural glasses at low-frequencies $\omega$ are quasi-localized and follow a
universal density of states $D(\omega)\!\sim\!\omega^4$. Here we show that the
gapless nature of the observed density of states depends on the existence of
internal stresses which generically emerge in glasses due to frustration, thus
elucidating a basic element underlying this universal behavior. Similarly to
jammed particulate packings, low-frequency modes in structural glasses emerge
from a balance between a local elasticity term and an internal stress term in
the dynamical matrix, where the difference between them is orders of magnitude
smaller than their typical magnitude. By artificially reducing the magnitude of
internal stresses in a computer glass former in three dimensions, we show that
a gap is formed in the density of states below which no vibrational modes
exist, thus demonstrating the crucial importance of internal stresses. Finally,
we show that while better annealing the glass upon cooling from the liquid
state significantly reduces its internal stresses, the self-organizational
processes during cooling render the gapless $D(\omega)\!\sim\!\omega^4$ density
of state unaffected.
