Mellin amplitudes for 1d CFT

We define a Mellin amplitude for CFT$_1$ four-point functions. Its analytical properties are inferred from physical requirements on the correlator. discuss analytic continuation that is necessary fully nonperturbative definition of transform. The resulting bounded, meromorphic function single complex variable used to derive an infinite set sum rules CFT data exchanged operators, which we test known examples. then consider perturbative setup produced by quartic interactions with arbitrary number derivatives in bulk AdS$_2$ field theory. With our formalism, obtain closed-form expression transform tree-level contact and first correction scaling dimension "two-particle" operators generalized free theory