Eigenvalues of two-phase quantum walks with one defect in one dimension

We study space-inhomogeneous quantum walks (QWs) on the integer lattice which we assign three different coin matrices to the positive part, the negative part, and the origin, respectively. We call them two-phase QWs with one defect. They cover one-defect and two-phase QWs, which have been intensively researched. Localization is one of the most characteristic properties of QWs, and various types of two-phase QWs with one defect exhibit localization. Moreover, the existence of eigenvalues is deeply related to localization. In this paper, we obtain a necessary and sufficient condition for the existence of eigenvalues. Our analytical methods are mainly based on the transfer matrix, a useful tool to generate the generalized eigenfunctions. Furthermore, we explicitly derive eigenvalues for some classes of two-phase QWs with one defect, and illustrate the range of eigenvalues on unit circles with figures. Our results include some results in previous studies, e.g., Endo et al. (Entropy 22(1):127, 2020).

Automated summary

This paper investigates space-inhomogeneous quantum walks on the integer lattice, assigning three different coin matrices to different parts. The study focuses on two-phase QWs with one defect, uncovering the relationship between localization and the existence of eigenvalues.Analytical methods involving transfer matrices are used to determine necessary and sufficient conditions for eigenvalues and derive them for specific classes of QWs, with results extending previous studies.