  It is now well established that structural glasses possess disorder- and
frustration-induced soft quasilocalized excitations, which play key roles in
various glassy phenomena. Recent work has established that in model
glass-formers in three dimensions, these non-phononic soft excitations may
assume the form of quasilocalized, harmonic vibrational modes whose frequency
follows a universal density of states $D(\omega)\!\sim\!\omega^4$,
independently of microscopic details, and for a broad range of glass
preparation protocols. Here we further establish the universality of the
non-phononic density of vibrational modes by direct measurements in model
structural glasses in two dimensions and four dimensions. We also investigate
their degree of localization, which is generally weaker in lower spatial
dimensions, giving rise to a pronounced system-size dependence of the
non-phononic density of states in two dimensions, but not in higher dimensions.
Finally, we identify a fundamental glassy frequency scale $\omega_c$ above
which the universal $\omega^4$ law breaks down.
