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.


Introduction
There is no doubt that studies on scattering theory is one of the interesting topic of the Schrödinger equation. Recently, it is revealed that the scatterings of some fundamental stationary Schrödinger equations on the real line with not only delta potentials [26,7,18] but also continuous potential [6] 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 freequantum coins are assigned at the other vertices. The initial state is given so that quantum walkers 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 [11] and this path counting method can be described by a discrete analogue of the Feynmann path integral [6]. There are some studies for the scattering theory of quantum walks under slightly general settings and related topics [25,23,24,20,21,27,12].
Such a setting is the special setting of [3,8] in that the regions where a quantum walker moves freely coincide with tails in [3,8], and the perturbed region can be regarded as a finite and connected graph in [3,8]. 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 [19] with a constant inflow from the tails.
By [8], this quantum walk converges to a stationary state. So let ϕ(·) : Z → C 2 be the stationary state of the quantum walk on Z. to each vertex in [M]. The inflow into the perturbed region at time n is expressed by ω n (|ω| = 1). In this paper, we compute (1) the scattering on the surface of the perturbed region [M] in the one-dimensional lattice; (2) the energy of the quantum walk. Here the energy of quantum walk is defined by This is the quantity how much quantum walkers accumulate to the perturbed region [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 θ(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 E M which is obtained by picking up from the total unitary time evolution operator with respect to the perturbed region [M]. See [10] for a precise spectral results on E M . 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 neseccory and sufficient condition for the perfect transmitting. We also gives 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.

The setting of our quantum walk
The total Hilbert space is denoted by H := ℓ 2 (Z; C 2 ) ∼ = ℓ 2 (A). Here A is the set of arcs of one-dimensional lattice whose elements are labeled by {(x; R), (x; L) | x ∈ Z}, 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 Putting respectively. Then the time evolution operator on ℓ 2 (Z; C 2 ) is described by for any ψ ∈ ℓ 2 (Z; C 2 ). Its equivalent expression on ℓ 2 (A) is described by for any ψ ∈ ℓ 2 (A). We call a x and d x the transmitting amplitudes, and b x and c x the reflection amplitudes at x, respectively * . Remark that U and U ′ are unitarily equivalent such that letting η : The free quantum walk is the quantum walk where all local quantum coins are described by the identity matrix i.e.
Then the walker runs through one-dimensional lattices without any reflections in the free case.
In this paper we set "impurities" on in the free quantum walk on one-dimensional lattice; that is, (2.2) * If we put a x = d x = 1 and b x = c x = √ −1 = i, then the primitive form of QW in [5] is reproduced.
We consider the initial state Ψ 0 as follows.
where ξ ∈ R/2πZ. Note that this initial state belongs to no longer ℓ 2 category. The region Γ M is obtain a time dependent inflow e −iξn 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.
(1) This quantum walk converges to a stationary state in the following meaning: (2) This stationary state is a generalized eigenfunction satisfying Relation to an absorption problem Let the reflection amplitude at time n beγ n (z) := L|Φ n (−1) with z = e iξ . We can see thatγ n (z) is rewritten by using U ′ as follows: The first term is the amplitude that the inflow at time n cannot penetrate into Γ M ; the m-th term is the amplitude that the inflow at time n − (m − 1) penetrates into Γ M and escapes Γ M from 0 side at time n. Therefore each term corresponds to the "absorption" amplitude to −1 with the absorption walls −1 and M with the initial state δ (0;R) . Then Put γ n := | δ (−1;L) , U ′ n δ (0;R) | 2 and τ n := | δ (M ;R) , U ′ n δ (0;R) | 2 which are the absorption/ first hitting probabilities at positions −1 and M, respectively starting from (0 : R). From the above observation, for example, we can express the m-th moments of the absorption/hitting times to −1 and M as follows:

Relation to Scattering of quantum walk
The stationary state Φ ∞ is the generalized eigenfunction of U in ℓ ∞ (Z; C 2 ). The scattering matrix naturally appears in Φ ∞ (see [11]). In the time independent scattering theory, the inflow can be considered as the incident "plane wave", and the impurities cause the scattered wave by transmissions and reflections. Thus we can see the transmission coefficient and the For studies of a general theory of scattering, we also mention the recent work by Tiedra de Aldecoa [27].
3 Computation of stationary state

Preliminary
Recall that |L and |R represent the standard basis of C 2 ; that is, We put the principal submatrix of U with respect to the impurities by E M := χUχ * . The matrix form of E M with the computational basis χδ 0 |L , χδ 0 |R , . . . , χδ M −1 |L , χδ M −1 |R is expressed by the following 2M × 2M matrix: Putting ψ n := χΨ n , we have Then, putting φ n := e i(n+1)ξ ψ n , we have From [8], ϕ := ∃ lim n→∞ φ n . Then the stationary state restricted to Γ M satisfies About the uniqueness of this solution is ensured by the following Lemma because the existence of the inverse of (e −iξ − E M ) is included by this Lemma.
Proof. Let ψ ∈ ℓ 2 (Γ M , C 2 ) be an eigenvector of eigenvalue λ ∈ σ(E M ). Then Here for the inequality, we used the fact that χ * χ is the projection operator onto while for the final equality, we used the fact that χχ * is the identity operator on ℓ 2 (Γ M ; C 2 ). If the equality in (3.6) holds, then χ * χUχ * ψ = Uχ * ψ holds. Then we have the eigenequation Uχ * ψ = λχ * ψ by taking χ * to both sides of the original eigenequation χUχ * ψ = λψ. However there are no eigenvectors having finite supports in a position independent quantum walk on Z with a = 0 since its spectrum is described by only a continuous spectrum in general. Thus |λ| 2 < 1.
Now let us solve this equation (3.5). The matrix representation of E M with the permutation of the labeling such that (

Then the equation (3.5) is expressed by
Here we changed the way of blockwise of E M and we put z = e −iξ . Putting where ϕ(x) = [ϕ(x; R), ϕ(x; L)] ⊤ for any x ∈ Γ M . The inverse matrix of B z exists since z = 0. Then we have where Here ∆ = det(P + Q). For the boundaries, there exists κ such that (3.9) By (3.8) and (3.9), κ satisfies which is equivalent to Now the problem is reduced to consider the n-th power of T because the eigengector is expressed by ϕ(n) = T n ϕ(0). Since T is a just 2 × 2 matrix, we can prepare the following lemma.
Remark 3.1. The condition of "(α−δ) 2 +4βγ = 0 and A = ǫI" is a necessary and sufficient condition of the non-diagonalizability of A.

Remark 3.2.
For A = T case, the condition of (1) is reduced to Moreover if ω = e ik , the Chebyshev polynomial is described by U −1 (·) = 0, Here λ ± in RHS are the solutions of the quadratic equation

Transmitting and reflecting rates
Let us divide the unit circle in the complex plain as follows: By the unitarity of a b c d and using the Chebyshev recursion; U n+1 (x) = 2xU n (x)−U n−1 (x), we insert (1) and (2) in Lemma 3.2 into (3.8), and we have an explicit expression for the stationary state as follows.
: ω ∈ ∂B Here ǫ R = sgn(Re(ω)) and ǫ I = sgn(Im(ω)) Since the transmitting and reflecting rates are computed by we obtain explicit expressions for them as follows.
Corollary 3.1. Assume abcd = 0. For any ω ∈ R/(2πZ), we have Note that the unitarity of the time evolution can be confirmed by T + R = 1. By Corollary 3.1, we can find a necessary and sufficient conditions for the perfect transmitting; that is , T = 1. On the other hand, the perfect reflection never occurs.
Remark that if ω / ∈ B in , then the perfect transmitting never happens.

Energy in the perturbed region
Taking the square modulus to ϕ(n) in Theorem 3.1, the relative probability at position n ∈ {0, . . . , M − 1} can be computed as follows.
Proposition 3.1. Assume abcd = 0. Then the relative probability is described by Proof. Let us consider the case for ω / ∈ ∂B. Using the property of the Chebyshev polynomial, Then we can see how much quantum walkers accumulate in the perturbed region {0, . . . , M− 1} by We call it the energy of quantum walk.
In particular, E M (·) is continuous at every ω * ∈ ∂B and Proof. Using the properties of the Chebyshev polynomial for example, U 2 n − U n+1 U n−1 = 1, T n = (U n − U n−2 )/2, we have Then we have Here we used (3.17) in the last equality.
If ω ∈ ∂B, then by directly computation taking summation of (3.15) over n ∈ {0, 1, . . . , M − 1}, we obtain the conclusion. Let us see E M (·) is continuous at ∂B. We put x := (1/|a|) cos k and ζ ′ m (x) := ζ ′ m . Remark that ω → ω * implies |x| → 1. In the following, we consider x → 1 case. The Taylor expansion of ζ ′ m (x) around x = 1 is The reason for obtaining the expansion until ǫ 1 order is Then inserting all of them into (3.16), we obtain

Asymptotics of Energy
If ω ∈ ∂B, then by Corollary 3.3, it is immediately obtained that Let us consider the case of ω ∈ B in ∪ B out as follows. Note that λ ± = sgn(cos k)e ±θ : ω ∈ B out , e ±iθ : ω ∈ B in , where (1/|a|) cos k = cosh θ (ω ∈ B out ), while (1/|a|) cos k = cos θ (ω ∈ B in ) such that sin θ > 0 and sinh θ > 0. To observe the asymptotics of E M (ω) for ω / ∈ ∂B, we rewrite E M (ω) as follows: : ω ∈ B out  In the following, we regard E M (ω) as a function of θ, M; that is E(M, θ) because θ can be expressed by ω and consider the asymptotics for large M.

ω ∈ B out
Let us see that (4.20) Note that sinh Mθ ∼ e M θ /2 ≫ M. Then by (4.19), we have By (4.20), if ω → ω * ∈ ∂B, then E M (ω) ∼ 1/θ → ∞. To connect it to the limit for the case of ω * ∈ ∂B described by (4.18) continuously, we consider M → ∞ and θ → 0 simultaneously, so that Mθ ∼ θ * ∈ (0, ∞). Let us see that Noting that sinh mθ = sinh mθ * = 0, for m = 1, 2 and sinh θ ∼ θ * /M, we have Therefore if we design the parameter θ * so that then the energy of B out continuously closes to that of ∂B in the sufficient large system size M.
Then from now on, let us consider the following three cases having a magnitude relation between θ and M; This is consistent with (4.18).