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arXiv 2001.01962math-phmath.MPmath.SP
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分数阶薛定谔算子与短程势的散射

The scattering of fractional Schrödinger operators with short range potentials

Rui Zhang, Tianxiao Huang, Quan Zheng

备注：
We relax the range of $ε_j$ in Proposition 1.5
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摘要：

对任意正实数$s$，统一研究了分数阶薛定谔算子$H=H_0+V$的散射理论，其中$H_0=(-Δ)^\frac s2$和实势$V$满足短程条件.证明了波算子$W_\pm=\mathrm{s-}\lim_{t\rightarrow\pm\infty}e^{itH}e^{-itH_0}$的存在性和渐近完备性，$H$的非零纯点谱$σ_\mathrm{pp}\setminus\{0\}$的离散性和有限重数，以及本征函数的有限衰减性.短程条件相对于$V$的允许衰变率是尖锐的，并且波算子存在和不存在的衰变阈值在无穷远处在某种意义上比$|x|^{-1}$快。我们的方法是受S. Agmon和L. 20世纪70年代的霍曼德。

英文摘要：

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.

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