A one-dimensional Hadamard walk with one defect

We consider a one-dimensional space-inhomogeneous discrete time quantum walk. This model is the Hadamard walk with one defect at the origin which is different from the model introduced by Wojcik et al. [14]. We obtain a stationary measure of the model by solving the eigenvalue problem and an asymptotic behaviour of the return probability by the path counting approach. Moreover, we get the time-averaged limit measure using the space-time generating function method. The measure is symmetric for the origin and independent of the initial coin state at the starting point. So localization depends only on the parameter which determines the model.