  We study the spectral analysis and the scattering theory for time evolution
operators of position-dependent quantum walks. Our main purpose of this paper
is construction of generalized eigenfunctions of the time evolution operator.
Roughly speaking, the generalized eigenfunctions are not square summable but
belong to $\ell^{\infty}$-space on ${\bf Z}$. Moreover, we derive a
characterization of the set of generalized eigenfunctions in view of the
time-harmonic scattering theory. Thus we show that the S-matrix associated with
the quantum walk appears in the singularity expansion of generalized
eigenfunctions.
