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

We study space-inhomogeneous quantum walks (QWs) on the integer lattice, which we assign three different coin matrices to the positive part, negative part, and to the origin, respectively. We call the model the two-phase QW with one defect. It covers the one-defect and the two-phase QW, which have been intensively researched. Localization is one of the most characteristic properties of QWs, and various types of two-phase QW with one-defect occurs localization. Moreover, the existence of eigenvalues is deeply related to localization. In this paper, we obtain the 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 QW with one defect.