  We numerically study the vibrations of jammed packings of particles
interacting with finite-range, repulsive potentials at zero temperature. As the
packing fraction $\phi$ is lowered towards the onset of unjamming at
$\phi_{c}$, the density of vibrational states approaches a non-zero value in
the limit of zero frequency. For $\phi>\phi_{c}$, there is a crossover
frequency, $\omega^{*}$ below which the density of states drops towards zero.
This crossover frequency obeys power-law scaling with $\phi-\phi_{c}$.
Characteristic length scales, determined from the dominant wavevector
contributing to the eigenmode at $\omega^{*}$, diverge as power-laws at the
unjamming transition.
