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