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

TL;DR

This paper analyzes eigenvalues of two-phase one-dimensional quantum walks with a defect, providing necessary and sufficient conditions for their existence and illustrating their range, which relates to localization phenomena.

Contribution

It offers a complete characterization of eigenvalues for two-phase quantum walks with a defect, extending previous results and using transfer matrix methods.

Findings

Derived necessary and sufficient conditions for eigenvalues.

Explicitly calculated eigenvalues for specific classes.

Illustrated eigenvalue ranges on the complex unit circle.

Abstract

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. (2020).