  In amorphous materials, groups of particles can rearrange locally into a new
stable configuration. Such elementary excitations are key as they determine the
response to external stresses, as well as to thermal and quantum fluctuations.
Yet, understanding what controls their geometry remains a challenge. Here we
build a scaling description of the geometry and energy of low-energy
excitations in terms of the distance to an instability, as predicted for
instance at the dynamical transition in mean field approaches of supercooled
liquids. We successfully test our predictions in ultrastable computer glasses,
with a gapped and ungapped (regular) spectrum. Overall, our approach explains
why excitations become less extended, with a higher energy and displacement
scale upon cooling.
