  We obtain exact expressions for a general class of correlation functions in
the 1D quantum mechanical model described by the Schwarzian action, that arises
as the low energy limit of the SYK model. The answer takes the form of an
integral of a momentum space amplitude obtained via a simple set of
diagrammatic rules. The derivation relies on the precise equivalence between
the 1D Schwarzian theory and a suitable large $c$ limit of 2D Virasoro CFT. The
mapping from the 1D to the 2D theory is similar to the construction of
kinematic space. We also compute the out-of-time ordered four point function.
The momentum space amplitude in this case contains an extra factor in the form
of a crossing kernel, or R-matrix, given by a 6j-symbol of SU(1,1). We argue
that the R-matrix describes the gravitational scattering amplitude near the
horizon of an AdS${}_2$ black hole. Finally, we discuss the generalization of
some of our results to ${\cal N}=1$ and ${\cal N}=2$ supersymmetric Schwarzian
QM.
