  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.
