  We show that the low-frequency regime of the density of states of structural
glass formers is crucially sensitive to the stress-ensemble from which the
configurations are sampled. Specifically, in two dimensions, an exactly
isotropic ensemble with zero shear stress fluctuations, displays a
$D(\omega_{\min}) \sim \omega^{5}_{\min}$ regime, as opposed to the
$\omega^{4}_{\min}$ regime observed under unstrained conditions. We also study
an ensemble of strained amorphous solids near a plastic event. We show that the
minimum eigenvalue distribution at a strain $\gamma$ near the plastic event
occurring at $\gamma_{P}$, displays a collapse when scaled by $\sqrt{\gamma_P -
\gamma}$, and with the number of particles as $N^{-0.22}$. Notably, at
low-frequencies, this scaled distribution displays a robust $D(\omega_{\min})
\sim \omega^{6}_{\min}$ power-law regime, which survives in the large $N$
limit. Finally, we probe the universal properties of this ensemble through a
characterization of the second and third eigenvalues of the Hessian matrix near
a plastic event.
