  We introduce spectral functions that capture the distribution of OPE
coefficients and density of states in two-dimensional conformal field theories,
and show that nontrivial upper and lower bounds on the spectral function can be
obtained from semidefinite programming. We find substantial numerical evidence
indicating that OPEs involving only scalar Virasoro primaries in a c>1 CFT are
necessarily governed by the structure constants of Liouville theory. Combining
this with analytic results in modular bootstrap, we conjecture that Liouville
theory is the unique unitary c>1 CFT whose primaries have bounded spins. We
also use the spectral function method to study modular constraints on CFT
spectra, and discuss some implications of our results on CFTs of large c and
large gap, in particular, to what extent the BTZ spectral density is universal.
