Instabilities of jammed packings of frictionless spheres under compression

We consider the contribution to the density of vibrational states and the distribution of energy barrier heights of incipient instabilities in a glass modeled by a jammed packing of spheres. On approaching an instability, the frequency of a normal mode and the height of the energy barrier to cross into a new ground state both vanish. These instabilities produce a contribution to the density of vibrational states that scales as $\omega^3$ at low frequencies $\omega$, but which vanishes in the thermodynamic limit. In addition, they affect an anharmonic property, the distribution of energy barriers $\Delta H$, giving a contribution that scales as $\Delta H^{-1/3}$ at low barrier heights, which should be universal for jammed and glassy systems.