A discontinuity of the energy of quantum walk in impurities

We consider the discrete-time quantum walk whose local dynamics is denoted by $C$ at the perturbed region ${0,1,dots,M-1}$ and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow $omega^n$ at time $n$ $(|omega|=1)$. From this expression, we compute the scattering on the surface of $-1$ and $M$ and also compute the quantity how quantum walker accumulates in the perturbed region; namely the energy of the quantum walk, in the long time limit. We find a discontinuity of the energy with respect to the frequency of the inflow.