We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.

中文翻译：

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幺正算子的稳态散射理论在量子行走中的应用

我们对二希尔伯特空间设置中酉算子的平稳散射理论进行了一般说明。对于希尔伯特空间 ${\cal H}_0,{\cal H}$ 中的酉算子 $U_0,U$ 和识别算子 $J:{\cal H}_0\to{\cal H}$，我们给出三元组 $(U,U_0,J)$ 的驻波算子、强波算子、散射算子和散射矩阵的定义和集合属性。特别地，我们展示了驻波算符和强波算符存在并重合的条件，并推导出驻波算符和散射矩阵的表示公式。作为一个应用，我们表明这些表示公式对于最近在文献中引入的一类各向异性量子游走是满足的。