  We treat a position dependent quantum walk (QW) on the line which we assign
two different time-evolution operators to positive and negative parts
respectively. We call the model "the two-phase QW" here, which has been
expected to be a mathematical model of the topological insulator. We obtain the
stationary and time-averaged limit measures related to localization for the
two-phase QW with one defect. This is the first result on localization for the
two-phase QW. The analytical methods are mainly based on the splitted
generating function of the solution for the eigenvalue problem, and the
generating function of the weight of the passages of the model. In this paper,
we call the methods "the splitted generating function method" and "the
generating function method", respectively. The explicit expression of the
stationary measure is asymmetric for the origin, and depends on the initial
state and the choice of the parameters of the model. On the other hand, the
time-averaged limit measure has a starting point symmetry and localization
effect heavily depends on the initial state and the parameters of the model.
Regardless of the strong effect of the initial state and the parameters, the
time-averaged limit measure also suggests that localization can be always
observed for our two-phase QW. Furthermore, our results imply that there is an
interesting relation between the stationary and time-averaged limit measures
when the parameters of the model have specific periodicities, which suggests
that there is a possibility that we can analyze localization of the two-phase
QW with one defect from the stationary measure.
