  Mathematical analysis on the existence of eigenvalues is essential because it
is equivalent to the occurrence of localization, which is an exceptionally
crucial property of quantum walks. We construct the method for the eigenvalue
problem via the transfer matrix for space-inhomogeneous $n$-state quantum walks
in one dimension with $n-2$ self-loops, which is an extension of the technique
in a previous study (Quantum Inf. Process 20(5), 2021). This method reveals the
necessary and sufficient condition for the eigenvalue problem of a two-phase
three-state quantum walk with one defect whose time evolution varies in the
negative part, positive part, and at the origin.
