  Using heterogeneous-elasticity theory (HET) and a generalisation of HET
theory (GHET), obtained by applying a newly developed procedure for obtaining
the continuum limit of the glass's Hessian, we investigate the nature of
vibrational excitations, which are present in small systems, which do not allow
for low-frequency phonons. We identify two types of such non-phononic
excitations. In marginally stable systems, which can be prepared by quenching
from a rather high parental temperature, the low-frequency regime is dominated
by random-matrix vibrational wavefunctions (type-I) which in macroscopic
samples gives rise to the boson peak. They show a density of states (DOS),
which scales as $g(\omega)\sim \omega^2$. In more stable systems (reached by a
somewhat lower parental temperature) a gap appears in the type-I spectrum. This
gap is filled with other type-II non-phononic excitations, which are not
described by the previous version of HET, and have a DOS, which scales as
$g(\omega)\sim\omega^s$ with $3<s<5$. Using GHET we demonstrate that the
type-II excitations are due to local non-irrotational oscillations associated
with the stress field. The frequency scaling exponent $s$ turns out to be
non-universal, depending on the details of the interaction potential.
Specifically, we demonstrate that $s$ depends on the statistics of the small
values of the local frozen-in stresses, which are, in turn, governed by the
shape of the pair potential close to values where the potential or its first
derivative vanishes. All these findings are verified by extensive numerical
simulations of small soft-sphere glasses. Further, using level-distance
statistics, we demonstrate that both types of non-phononic excitations obey the
Gaussian-Orthogonal-Ensemble random-matrix statistics, which means that they
are extended and not localized.
