Strongly trapped space-inhomogeneous quantum walks in one dimension

Localization is a characteristic phenomenon of discrete-time space-inhomogeneous quantum walks on the one-dimensional integer lattice, 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 (Kiumi and Saito in Quantum Inf Process 20(5): 1-11, 2021). In this paper, we introduce the analytical method to calculate eigenvectors using the transfer matrix and also extend and simplify 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 a finite number of defects. 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.