带piecewise continuous符号的Toeplitz算子的多信道散射理论
Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols
摘要：
自伴Toeplitz算子具有纯绝对连续谱。对于具有分段连续符号的Toeplitz运算符$T$，我们建议进一步的谱分类，该分类由运算符$T$的传播特性确定，即由$t\to\pm\infty$的$\exp(-iTt) f$的行为确定。结果表明，光谱被自然地划分为三个不相交的子集：厚光谱、薄光谱和混合光谱。在厚频谱上，传播特性由符号的连续部分来建模，而在薄频谱上，模型算子由符号的跳跃来确定。在混合谱上，这两种类型的渐近演化$\exp(-iTt) f$共存。这种分类是合理的散射理论的框架。我们证明了波算子的存在性，将模型算子与Toeplitz算子$T$联系起来。这些波算子的值域是成对正交的，它们的正交和耗尽了整个空间，即，这些波算子的集合是渐近完备的。
英文摘要：
Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of $\exp(-iTt) f$ coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator $T$. The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space, i.e., the set of these wave operators is asymptotically complete.
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