  We study a general class of functionals providing an analytic handle on the
conformal bootstrap equations in one dimension. We explicitly identify the
extremal functionals, corresponding to theories saturating conformal bootstrap
bounds, in two regimes. The first corresponds to functionals that annihilate
the generalized free fermion spectrum. In this case, we analytically find both
OPE and gap maximization functionals proving the extremality of the generalized
free fermion solution to crossing. Secondly, we consider a scaling limit where
all conformal dimensions become large, equivalent to the large $AdS$ radius
limit of gapped theories in $AdS_2$. In this regime we demonstrate analytically
that optimal bounds on OPE coefficients lead to extremal solutions to crossing
arising from integrable field theories placed in large $AdS_2$. In the process,
we uncover a close connection between asymptotic extremal functionals and
S-matrices of integrable field theories in flat space and explain how 2D
S-matrix bootstrap results can be derived from the 1D conformal bootstrap
equations. These points illustrate that our formalism is capable of capturing
non-trivial solutions of CFT crossing.
