The scattering of fractional Schrödinger operators with short range potentials (Q2020101)

The authors study existence and asymptotic completeness of the wave operators. One of the main theorems of the paper states that the wave operators \[W_{\pm}u=\lim\limits_{t\to\pm\infty}e^{itH}e^{-itH_0}u,\] exist and are isometric. Here \(u\in L^2(\mathbb{R}^n)\), \(H=H_0+V\), and \(H_0=(-\Delta)^{\frac{s}{2}}\), where \(s\) is a positive real number and \(V\) is a real-valued multiplication operator that is a potential satisfying the short range condition. Another theorem proves that \(W_{\pm}\) are asymptotically complete and that the scattering operator \(S=W^*_+W_-\) is unitary. The authors also establish the discreteness of the non-zero pure point spectrum of \(H\) as well as the finite decay property for the eigenfunctions.