The scattering of fractional Schrödinger operators with short range potentials

Abstract: 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.