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

Weak limit theorem of a two-phase quantum walk with one defect