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).

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）。