**1. Introduction**

There is no doubt that a study on scattering theory is one of the most interesting topics of the Schrödinger equation. Recently, it has been revealed that the scatterings of some fundamental stationary Schrödinger equations on the real line with not only delta potentials [1–3] but also continuous potential [4] can be recovered by discrete-time quantum walks. These induced quantum walks are given by the following setting: the non-trivial quantum coins are assigned to some vertices in a finite region on the one-dimensional lattice as the impurities and the free-quantum coins are assigned at the other vertices. The initial state is given so that a quantum walker inflows into the perturbed region at every time step. It is shown that the scattering matrix of the quantum walk on the one-dimensional lattice can be explicitly described by using a path counting in [5] and this path counting method can be described by a discrete analogue of the Feynmann path integral [4]. There are some studies for the scattering theory of quantum walks under slightly general settings and related topics [6–12].

Such a setting is the special setting of [13,14] in that the regions where a quantum walker moves freely coincide with tails in [13,14], and the perturbed region can be regarded as a finite and connected graph in [13,14]. The properties of not only the scattering on the surface of the internal graph but also the stationary state in the internal graph for the Szegedy walk are characterized by [15] with a constant inflow from the tails.

**Citation:** Higuchi, K.; Komatsu, T.; Konno, N.; Morioka, H.; Segawa, E. A Discontinuity of the Energy of Quantum Walk in Impuritie. *Symmetry* **2021**, *13*, 1134. https://doi.org/10.3390/sym13071134

Academic Editor: Motoichi Ohtsu

Received: 29 May 2021 Accepted: 21 June 2021 Published: 24 June 2021

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By [14], this quantum walk converges to a stationary state. Therefore, let *ϕ*(·): Z → C<sup>2</sup> be the stationary state of the quantum walk on Z. The perturbed region is Γ*M* := {0, 1, . . . , *M* − 1} and we assign the quantum coin

$$\mathbf{C} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

to each vertex in Γ*M*. The inflow into the perturbed region at time *n* is expressed by *ω<sup>n</sup>* (|*ω*| = <sup>1</sup>). In this paper, we compute (1) the scattering on the surface of the perturbed region Γ*M* in the one-dimensional lattice; (2) the energy of the quantum walk. Here, the energy of quantum walk is defined by

$$\mathcal{E}\_M(\omega) = \sum\_{x=0}^{M-1} \left| \left| \vec{q}(x) \right| \right|\_{\mathbb{C}^2}^2.$$

This is the quantity that quantum walkers accumulate to the perturbed region Γ*M* in the long time limit. We obtain a necessary and sufficient condition for the perfect transmitting, and also obtain the energy. As a consequence of our result on the energy, we observe a discontinuity of the energy with respect to the frequency of the inflow. Moreover, our result implies that the condition for *<sup>θ</sup>*(*ω*) ∈ N is equivalent to the condition for the perfect transmitting. Then, we obtain that the situation of the perfect transmitting not only releases quantum walker to the opposite outside but also accumulates quantum walkers in the perturbed region. Note that since this quantum walk can be converted to a quantum walk with absorption walls, the problem is reduced to analysis on a finite matrix *EM*, which is obtained by picking up from the total unitary time evolution operator with respect to the perturbed region Γ*M*. See [16] for a precise spectral results on *EM*.

This paper is organized as follows. In Section 2, we explain the setting of this model and give some related works. In Section 3, an explicit expression for the stationary state is computed using the Chebyshev polynomials. From this expression, we obtain the transmitting and reflecting rates and a necessary and sufficient condition for the perfect transmitting. We also give the energy in the perturbed region. In Section 4, we estimate the asymptotics of the energy to see the discontinuity with respect to the incident inflow.

### **2. The Setting of our Quantum Walk**

The total Hilbert space is denoted by H := <sup>2</sup>(<sup>Z</sup>; C<sup>2</sup>) ∼= <sup>2</sup>(*A*). Here *A* is the set of arcs of one-dimensional lattice whose elements are labeled by {(*x*; *<sup>R</sup>*),(*<sup>x</sup>*; *L*) | *x* ∈ <sup>Z</sup>}, where (*x*; *R*) and (*x*; *L*) represents the arcs "from *x* − 1 to *x*", and "from *x* + 1 to *x*", respectively. We assign a 2 × 2 unitary matrix to each *x* ∈ Z so-called local quantum coin

$$\mathbf{C}\_{\mathfrak{X}} = \begin{bmatrix} a\_{\mathfrak{X}} & b\_{\mathfrak{X}} \\ c\_{\mathfrak{X}} & d\_{\mathfrak{X}} \end{bmatrix}.$$

Putting |*L* := [1, <sup>0</sup>]&, |*R* := [0, 1]& and *L*| = [1, 0], *R*| = [0, 1], we define the following matrix valued weights associated with the motion from *x* to left and right by

$$P\_{\mathfrak{x}} = |L\rangle\langle L|\mathbb{C}\_{\mathfrak{x}\_{\prime}} \ \ Q\_{\mathfrak{x}} = |R\rangle\langle R|\mathbb{C}\_{\mathfrak{x}\_{\prime}}$$

respectively. Then, the time evolution operator on <sup>2</sup>(<sup>Z</sup>; C<sup>2</sup>) is described by

$$(\mathcal{U}\psi)(\mathfrak{x}) = P\_{\mathfrak{x}+1}\psi(\mathfrak{x}+1) + Q\_{\mathfrak{x}-1}\psi(\mathfrak{x}-1)$$

for any *ψ* ∈ <sup>2</sup>(<sup>Z</sup>; <sup>C</sup><sup>2</sup>). Its equivalent expression on <sup>2</sup>(*A*) is described by

$$\begin{aligned} (\mathsf{U}'\phi)(\mathsf{x};\mathsf{L}) &= a\_{\mathsf{x}+1}\phi(\mathsf{x}+1;\mathsf{L}) + b\_{\mathsf{x}+1}\phi(\mathsf{x}+1;\mathsf{R}),\\ (\mathsf{U}'\phi)(\mathsf{x};\mathsf{R}) &= c\_{\mathsf{x}-1}\phi(\mathsf{x}-1;\mathsf{L}) + d\_{\mathsf{x}-1}\phi(\mathsf{x}-1;\mathsf{R}) \end{aligned} \tag{1}$$

for any *ψ* ∈ <sup>2</sup>(*A*). We call *ax* and *dx* the transmitting amplitudes, and *bx* and *cx* the reflection amplitudes at *x*, respectively. If we put *ax* = *dx* = 1 and *bx* = *cx* = √ −1 = *i*, then the primitive form of QW in [17] is reproduced. Remark that *U* and *U* are unitarily equivalent such that letting *η* : <sup>2</sup>(<sup>Z</sup>; C<sup>2</sup>) → <sup>2</sup>(*A*) be

$$(\eta\psi)(\mathfrak{x};\mathbb{R}) = \langle \mathbb{R}|\psi\rangle,\ (\eta\psi)(\mathfrak{x};L) = \langle L|\psi\rangle$$

then we have *U* = *η*<sup>−</sup>1*U η*. The free quantum walk is the quantum walk where all local quantum coins are described by the identity matrix, i.e.,

$$(\mathcal{U}\_0 \psi)(\mathbf{x}) = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \psi(\mathbf{x} + \mathbf{1}) + \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \psi(\mathbf{x} - \mathbf{1}).$$

Then, the walker runs through one-dimensional lattices without any reflections in the free case.

In this paper, we set "impurities" on

$$
\Gamma\_M := \{0, 1, \ldots, M - 1\}
$$

in the free quantum walk on one-dimensional lattice; that is,

$$\mathbb{C}\_{\mathbf{x}} = \begin{cases} \begin{bmatrix} a & b \\ c & d \end{bmatrix} & \colon \mathbf{x} \in \Gamma\_{M\prime} \\\\ I\_2 & \colon \mathbf{x} \notin \Gamma\_M . \end{cases} \tag{2}$$

We consider the initial state Ψ0 as follows.

$$\Psi\_0(\mathbf{x}) = \begin{cases} e^{i\frac{\pi}{6}\mathbf{x}}|R\rangle & : \mathbf{x} \le 0;\\ 0 & : \text{otherwise}. \end{cases}$$

where *ξ* ∈ R/2*π* Z. Note that this initial state belongs to no longer 2 category. The region Γ*M* is obtained a time dependent inflow *e*<sup>−</sup>*iξ<sup>n</sup>* from the negative outside. On the other hand, if a quantum walker goes out side of Γ*M*, it never come back again to Γ*M*. We can regard such a quantum walker as an outflow from Γ*M*. Roughly speaking, in the long time limit, the inflow and outflow are balanced and obtain the stationary state with some modification. Indeed, the following statement holds.
