自伴算子的piecewise continuous函数的Spectral theory
Spectral theory of piecewise continuous functions of self-adjoint operators
摘要：
令$H_0$，$H$是一对自伴算子，对于它们，散射理论的光滑版本的标准假设成立。本文给出了分段连续函数$θ$的算子$\mathcal{D}_θ=θ(H)-θ(H_0)$的绝对连续谱的一个显式描述。本说明涉及在$θ$的不连续处评估的对$H_0$、$H$的散射矩阵。我们还证明了$\mathcal{D}_θ$的奇异连续谱是空的，该算子的特征值具有有限的重数，只能累积到$\mathcal{D}_θ$的绝对连续谱的“阈值”。我们的方法依赖于为函数$θ$的每个跳转构造“模型”运算符。这些模型算子被定义为某些允许显式谱分析的对称Hankel算子。我们发展了模型算符和算符$θ(H)-θ(H_0)$的多通道散射理论。作为我们的方法的副产品，我们还构造了一般对称Hankel算子的散射理论与分段连续符号。
英文摘要：
Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous spectrum of the operator $\mathcal{D}_θ=θ(H)-θ(H_0)$ for piecewise continuous functions $θ$. This description involves the scattering matrix for the pair $H_0$, $H$, evaluated at the discontinuities of $θ$. We also prove that the singular continuous spectrum of $\mathcal{D}_θ$ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the "thresholds" of the absolutely continuous spectrum of $\mathcal{D}_θ$. Our approach relies on the construction of "model" operators for each jump of the function $θ$. These model operators are defined as certain symmetrised Hankel operators which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator $θ(H)-θ(H_0)$. As a by-product of our approach, we also construct the scattering theory for general symmetrised Hankel operators with piecewise continuous symbols.
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