We develop scattering theory for non-local Schrödinger operators defined by functions of the Laplacian that include its fractional power (−Δ) with 0 < ρ⩽ 1. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions for perturbational potentials. ρ