We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

中文翻译：

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通过路径计数方法的量子游走的广义特征函数

我们考虑一维二维量子行走的时间演化算子的​​与时间无关的散射理论。与位置相关的量子行走相关的散射矩阵自然地出现在广义本征函数空间无穷大处的渐近行为中。广义特征函数的渐近行为是与自由量子游走相关的格林函数的显式表达的结果。当位置相关的量子游走是自由量子游走的有限秩扰动时，我们通过计算量子游走的路径推导出一种散射矩阵的组合构造。我们还提到了一些关于隧道效应的评论。