*Article* **A Discontinuity of the Energy of Quantum Walk in Impurities**

**Kenta Higuchi 1, Takashi Komatsu 2, Norio Konno 3, Hisashi Morioka 4 and Etsuo Segawa 5,\***


**Abstract:** We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix *C* at the perturbed region {0, 1, ... , *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 *ω<sup>n</sup>* at time *n* (|*ω*| = <sup>1</sup>). 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. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

**Keywords:** quantum walk; scattering theory; energy
