  For any positive real number $s$, we study the scattering theory in a unified
way for the fractional Schr\"{o}dinger operator $H=H_0+V$, where
$H_0=(-\Delta)^\frac s2$ and the real-valued potential $V$ satisfies short
range condition. We prove the existence and asymptotic completeness of the wave
operators $W_\pm=\mathrm{s-}\lim_{t\rightarrow\pm\infty}e^{itH}e^{-itH_0}$, the
discreteness and finite multiplicity of the non-zero pure point spectrum
$\sigma_\mathrm{pp}\setminus\{0\}$ of $H$, and the finite decay property of
eigenfunctions. The short range condition is sharp with respect to the allowed
decay rate of $V$, and the decay threshold for the existence and non-existence
of the wave operators is faster than $|x|^{-1}$ at the infinity in some sense.
Our approach is inspired by the theory of limiting absorption principle for
simply characteristic operators established by S. Agmon and L. H\"{o}rmander in
the 1970s.
