  Localization is a characteristic phenomenon of space-inhomogeneous quantum
walks in one dimension, where particles remain localized around their initial
position. The existence of eigenvalues of time evolution operators is a
necessary and sufficient condition for the occurrence of localization, and
their associated eigenvectors are deeply related to the amount of localization,
i.e., the probability that the walker stays around the starting position in the
long-time limit. In a previous study by authors, the eigenvalues of two-phase
quantum walks with one defect were studied using a transfer matrix, which
focused on the occurrence of localization (Quantum Inf. Process 20(5), 2021).
In this paper, we introduce the analytical method to calculate eigenvectors
using the transfer matrix and also extend our results to characterize
eigenvalues not only for two-phase quantum walks with one defect but also for a
more general space-inhomogeneous model.
  With these results, we quantitatively evaluate localization and study the
strong trapping property by deriving the time-averaged limit distributions of
five models studied previously.
