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We study space-inhomogeneous quantum walks (QWs) on the integer lattice whichwe assign three different coin matrices to the positive part, the negativepart, and the origin, respectively. We call them two-phase QWs with one defect.They cover one-defect and two-phase QWs, which have been intensivelyresearched. 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. Inthis paper, we obtain a necessary and sufficient condition for the existence ofeigenvalues. Our analytical methods are mainly based on the transfer matrix, auseful tool to generate the generalized eigenfunctions. Furthermore, weexplicitly derive eigenvalues for some classes of two-phase QWs with onedefect, 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).

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