  In this work we apply the lightcone bootstrap to a four-point function of
scalars in two-dimensional conformal field theory. We include the entire
Virasoro symmetry and consider non-rational theories with a gap in the spectrum
from the vacuum and no conserved currents. For those theories, we compute the
large dimension limit (h/c>>1) of the OPE spectral decomposition of the
Virasoro vacuum. We then propose a kernel ansatz that generalizes the spectral
decomposition beyond h/c>>1. Finally, we estimate the corrections to the OPE
spectral densities from the inclusion of the lightest operator in the spectrum.
