We develop scattering theory for non-local Schrodinger 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.

中文翻译：

---

非局部薛定谔算子的短程和长程势之间的阈值

我们为由拉普拉斯算子函数定义的非局部薛定谔算子开发散射理论，其中包括其分数幂 $(-\Delta)^\rho$ 和 $0<\rho\leqslant1$。特别是，我们的函数属于比伯恩斯坦函数集更广泛的类。通过显示波算符的存在和不存在，我们阐明了扰动势的短程和长程衰减条件之间的阈值。