  For expansions in one-dimensional conformal blocks, we provide a rigorous
link between the asymptotics of the spectral density of exchanged primaries and
the leading singularity in the crossed channel. Our result has a direct
application to systems of SL(2,R)-invariant correlators (also known as 1d
CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis
of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In
addition, a similar argument controls the spectral density asymptotics in large
N gauge theories.
