**5. Conclusions**

We considered the quantum walk on the line with the perturbed region {0, 1, ... , *<sup>M</sup>*}; that is, an non-trivial quantum coin is assigned at the perturbed region and the free quantum coin is assigned at the other region. We set an ∞ initial state so that free quantum walkers are inputted at each time step to the perturbed region. A closed form of the stationary state of this dynamical system was obtained and we computed the energy of the quantum walk in the perturbed region. This energy represents how quantum walker feels "comfortable" in the perturbed region. We showed that the "feeling" of quantum walk depends on the frequency of the initial state. We can divide the region of the frequency into three parts to classify the asymptotics of the energy for large *M*; *Bin*, *Bout*, *δB*. The region *Bin* coincides with the continuous spectrum of the quantum walk with *M* → ∞ [5]. We showed that quantum walkers prefer to the initial state whose frequency corresponds to the continuous spectrum in the infinite system. More precisely, the energy of the quantum walk in the perturbed region is estimated by *O*(1) if *θ* ∈ *Bout*, while one is estimated by *O*(*M*) if

*θ* ∈ *δB* and *almost all* pseudo momentum *θ* gives *<sup>O</sup>*(*M*)-energy, but some momentum gives *O*(*M*<sup>3</sup>) if *θ* ∈ *Bin* (Theorem 2). Such an initial state exactly exists but it is quite rare from the view point of the Lebesgue measure. The most comfortable initial state for quantum walkers has the frequency whose pseudo momentum *θ* lives in some neighborhood of the boundary *∂B* and accomplishes the perfect transmitting. If the momentum of the initial state exceeds the boundary *∂B* from the internal region *Bin*, then the energy is immediately reduced to *<sup>O</sup>*(1). It suggests that the control of the frequency of the initial state to give the maximal energy in the perturbed region is quite sensitive from the view point of an implementation.

The spectrum of the boundary *∂B* for *M* → ∞ produces the two singular points of the density function of the Konno limit distribution and is characterized by the Airy functions. In [16], details of the spectrum behavior around *∂B* is discussed. Indeed, a kind of "speciality" also appears as the non-diagonalizability of *T* when *θ* ∈ *∂B* in our work (Lemma 2). Note that the infinite system does not have any *edges*, which means every node is "impurity", while our quantum walker feels the *edges* of the impurities; nodes 0 and *M*. Therefore, to see the effect of such a finiteness on the behavior of the quantum walker comparing with the infinite system, computing how a quantum walker is distributed in the perturbed region is interesting which may be possible from the explicit expression of the stationary state in Theorem 1. Moreover, to consider the escaping time from the perturbed region seems to be useful to estimate the finesse as the interferometer motivated by quantum walk and it would be possible to extract some information from (3) and (4). This remains one of the interesting problems for the future.

**Author Contributions:** Conceptualization, E.S.; Formal analysis, E.S., H.M., T.K., K.H. and N.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** HM was supported by the grant-in-aid for young scientists No. 16K17630, JSPS. ES acknowledges financial supports from the Grant-in-Aid of Scientific Research (C) No. JP19K03616, Japan Society for the Promotion of Science and Research Origin for Dressed Photon.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
