  We calculate numerically the normal modes of vibrations in 3D jammed packings
of soft spheres as a function of the packing fraction and obtain the energy
diffusivity, a spectral measure of transport that controls sound propagation
and thermal conductivity. The crossover frequency between weak and strong
phonon scattering is controlled by the coordination and shifts to zero as the
system is decompressed towards the critical packing fraction at which rigidity
is lost. Below the crossover, the diffusivity displays a power-law divergence
with inverse frequency, which suggests that the vibrational modes are primarily
transverse waves, weakly scattered by disorder. Above it, a large number of
modes appear whose diffusivity plateaus at a nearly constant value independent
of the inter-particle potential, before dropping to zero above the Anderson
localization frequency. The thermal conductivity of a marginally jammed solid
just above the rigidity threshold is calculated and related to the one measured
experimentally at room temperature for most glasses.
