  We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW)
with one defect at the origin, which has two different quantum coins in
positive and negative parts. We call the QW "the two-phase QW", which we
treated concerning localization theorems [10]. The two-phase QW has been
expected to be a mathematical model of the topological insulator [16] which is
an intense issue both theoretically and experimentally [3,5,11]. In this paper,
we derive the weak limit theorem describing the ballistic spreading, and as a
result, we obtain the mathematical expression of the whole picture of the
asymptotic behavior. Our approach is based mainly on the generating function of
the weight of the passages. We emphasize that the time-averaged limit measure
is symmetric for the origin [10], however, the weak limit measure is
asymmetric, which implies that the weak limit theorem represents the asymmetry
of the probability distribution.
