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Remarks on Scattering Matrices for Schrödinger Operators with Critically Long-Range Perturbations
Annales Henri Poincaré ( IF 1.4 ) Pub Date : 2020-08-01 , DOI: 10.1007/s00023-020-00943-z
Shu Nakamura

We consider scattering matrix for Schrödinger-type operators on \(\mathbb {R}^d\) with perturbation \(V(x)=O(\langle x \rangle ^{-1})\) as \(|x|\rightarrow \infty \). We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator and analyze its spectrum. We present examples of which the spectrum of the scattering matrices has dense point spectrum, and absolutely continuous spectrum, respectively. These give a partial answer to an open question posed by Yafaev (Scattering theory: some old and new problems. Springer Lecture Notes in Mathematical, vol 1735, 2000).



中文翻译:

关于具有严重长范围扰动的Schrödinger算子的散射矩阵的评论

我们考虑在薛定谔型运营商散射矩阵\(\ mathbb {R} ^ d \)与扰动\(V(X)= O(\ langle X \ rangle ^ { - 1})\)作为\(| X | \ rightarrow \ infty \)。我们证明了散射矩阵(具有与时间无关的修饰符)是伪微分算子,并分析了其光谱。我们给出了散射矩阵光谱分别具有密点光谱和绝对连续光谱的示例。这些部分回答了Yafaev提出的一个悬而未决的问题(散射理论:一些新老问题。施普林格数学讲义,第1735卷,2000年)。

更新日期:2020-08-01
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