  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).
