Glasses have a large excess of low-frequency vibrational modes in comparison
with crystalline solids. We show that such a feature is a necessary consequence
of the geometry generic to weakly connected solids. In particular, we analyze
the density of states of a recently simulated system, comprised of weakly
compressed spheres at zero temperature. We account for the observed a)
constancy of the density of modes with frequency, b) appearance of a
low-frequency cutoff, and c) power-law increase of this cutoff with
compression. We predict a length scale below which vibrations are very
different from those of a continuous elastic body.Comment: 4 pages, 2 figures. Argument rewritten, identical result
