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We consider the time-independent scattering theory for time evolutionoperators of one-dimensional two-state quantum walks. The scattering matrixassociated with the position-dependent quantum walk naturally appears in theasymptotic behavior at spatial infinity of generalized eigenfunctions. Theasymptotic behavior of generalized eigenfunctions is a consequence of anexplicit expression of the Green function associated with the free quantumwalk. When the position-dependent quantum walk is a finite rank perturbation ofthe free quantum walk, we derive a kind of combinatorial constructions of thescattering matrix by counting paths of quantum walkers. We also mention someremarks on the tunneling effect.

Generalized eigenfunctions for quantum walks via path counting approach