Threshold between short and long-range potentials for non-local Schrödinger operators

We develop scattering theory for non-local Schr\"odinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. 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.