论文标题
具有短距离电位的分数Schrödinger操作员的散射
The scattering of fractional Schrödinger operators with short range potentials
论文作者
论文摘要
对于任何正实数$ s $，我们以统一的方式研究散射理论，用于分数schrödinger$ h = h_0+v $，其中$ h_0 =（ - δ）^\ frac s2 $和现实价值的潜在$ v $满足短范围条件。我们证明了Wave Operator的存在和渐近完整性$ W_ \ PM = \ Mathrm {S - } \ Lim_ { $σ_\ mathrm {pp} \ setMinus \ {0 \} $ $ h $的$和特征functions的有限衰减属性。相对于$ v $的允许衰减速率，短距离条件是鲜明的，并且在某种意义上，在无限的情况下，波浪运算符的存在和不存在的衰减阈值要快。我们的方法的灵感来自于S. Agmon和L.Hörmander在1970年代建立的简单特征操作员的限制理论。
For any positive real number $s$, we study the scattering theory in a unified way for the fractional Schrödinger operator $H=H_0+V$, where $H_0=(-Δ)^\frac s2$ 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 $σ_\mathrm{pp}\setminus\{0\}$ 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. Hörmander in the 1970s.
