  We study analytically the constraints of the conformal bootstrap on the
low-lying spectrum of operators in field theories with global conformal
symmetry in one and two spacetime dimensions. We introduce a new class of
linear functionals acting on the conformal bootstrap equation. In 1D, we use
the new basis to construct extremal functionals leading to the optimal upper
bound on the gap above identity in the OPE of two identical primary operators
of integer or half-integer scaling dimension. We also prove an upper bound on
the twist gap in 2D theories with global conformal symmetry. When the external
scaling dimensions are large, our functionals provide a direct point of contact
between crossing in a 1D CFT and scattering of massive particles in large
$\textrm{AdS}_2$. In particular, CFT crossing can be shown to imply that
appropriate OPE coefficients exhibit an exponential suppression characteristic
of massive bound states, and that the 2D flat-space S-matrix should be analytic
away from the real axis.
