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

The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices