Abstract
We consider a position-dependent coined quantum walk on \(\mathbb {Z}\) and assume that the coin operator C(x) satisfies
with positive \(c_1\) and \(\epsilon \) and \(C_0 \in U(2)\). We show that the Heisenberg operator \(\hat{x}(t)\) of the position operator converges to the asymptotic velocity operator \(\hat{v}_+\) so that
provided that U has no singular continuous spectrum. Here \(\Pi _\mathrm{p}(U)\) (resp., \(\Pi _\mathrm{ac}(U)\)) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable \(X_t\) denoting the position of a quantum walker at time \(t \in \mathbb {N}\), \(X_t/t\) converges in law to a random variable V with the probability distribution
where \(\Psi _0\) is the initial state, \(\delta _0\) the Dirac measure at zero, and \(E_{\hat{v}_+}\) the spectral measure of \(\hat{v}_+\).
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1 Introduction
The weak limit theorems for discrete time quantum walks have been studied in various models (for reviews, see [7, 12]). In his papers [5, 6], Konno first proved the weak limit theorem for a position-independent quantum walk on \(\mathbb {Z}\). Grimmett et al. [4] simplified the proof and extended the result to higher dimensions. For position-dependent qunatum walks on \(\mathbb {Z}\), the weak limit theorems were obtained by Konno et al. [9], Endo and Konno [2], and Endo et al. [3].
We consider a position-dependent quantum walk on \(\mathbb {Z}\) given by a unitary evolution operator U:
where \(\Psi \) is a state vector in the Hilbert space \(\mathcal {H} = \ell ^2(\mathbb {Z};\mathbb {C}^2)\) of states and
Let \(C(x) = P(x) + Q(x) \in U(2)\) and S be a shift operator such that \(U = SC\). Suppose that there exists a unitary matrix \(C_0 = P_0 + Q_0 \in U(2)\) such that
with positive \(c_1\) and \(\epsilon \) independent of x. Here \(\Vert M\Vert \) stands for the operator norm of a matrix \(M \in M_2(\mathbb {C})\). A typical example is the quantum walks with one defect [1, 8, 9, 13], which clearly satisfies (1.1). We note that the condition (1.1) allows not only finite but also infinite defects, whereas the models introduced in [2, 3] do not satisfy (1.1). The unitary operator \(U_0 = SC_0\) also defines an evolution of a position-independent quantum walk on \(\mathbb {Z}\) and satisfies
with \(C_0 = P_0 + Q_0\). Let \(\hat{x}\) be the position operator defined by \((\hat{x} \Psi )(x) = x \Psi (x), \quad x \in \mathbb {Z}\). and \(\hat{x}_0(t) = U_0^{-t} \hat{x} U_0^t\) the Heisenberg operator of \(\hat{x}\) at time \(t \in \mathbb {N}\) with the evolution \(U_0\). In Grimmett et al. [4] essentially proved that the operator \(\hat{x}_0(t)/t\) weakly converges to the asymptotic velocity operator \(\hat{v}_0\) so that
Let \(X^{(0)}_t\) be the random variable denoting the position of a quantum walker at time \(t \in \mathbb {N}\) with the evolution operator \(U_0\). Then, the characteristic function of \(X^{(0)}_t/t\) is given by
where \(\Psi _0\) is the initial state of the quantum walker. Hence, (1.2) means that the random variable \(X^{(0)}_t/t\) converges in law to a random variable \(V_0\), which represents the linear spreading of the quantum walk: \(X^{(0)}_t \sim t V_0\).
In this paper, we derive the asymptotic velocity \(\hat{v}_+\) for the Heisenberg operator \(\hat{x}(t) = U^{-t} \hat{x} U^t\) with the evolution U of the position-dependent quantum walk. The decaying condition (1.1) implies that \(U-U_0\) is a trace class operator and allows us to prove the existence and completeness of the wave operator
using a discrete analogue of the Kato–Rosenblum Theorem (see [11] for details), where \(\Pi _\mathrm{ac}(U_0)\) is the orthogonal projection onto the subspace of absolute continuity of \(U_0\). We also prove that
under a reasonable condition, which is essentially the same as that of [4]. Furthermore, we assume that U has no singular continuous spectrum. Then, we prove that
where \(\Pi _\mathrm{p}(U)\) is the orthogonal projection onto the direct sum of all eigenspaces of U and \(\hat{v}_+ = W_+ \hat{v}_0 W_+^*\). We believe that the absence of a singular continuous spectrum can be checked with a concrete example such as the one-defect model. As a consequence of (1.3), we have the following weak limit theorem. Let \(X_t\) be the random variable denoting the position of a quantum walker at time \(t \in \mathbb {N}\) with the evolution operator U and the initial state \(\Psi _0\). We prove that \(X_t/t\) converges in law to a random variable V with a probability distribution
where \(\delta _0\) is the Dirac measure at zero and \(E_{\hat{v}_+}\) the spectral measure of \(\hat{v}_+\).
The remainder of this paper is organized as follows. In Sect. 2, we present the precise definition of the model and our results. Section 3 is devoted to the proof of the existence and completeness of the wave operator. In Sect. 4, we construct the asymptotic velocity.
2 Definition of the model
Let \(\mathcal {H} = \ell ^2(\mathbb {Z};\mathbb {C}^2)\) be the Hilbert space of the square summable functions \(\Psi :\mathbb {Z} \rightarrow \mathbb {C}^2\). We define a shift operator S and a coin operator C on \(\mathcal {H}\) as follows. For a vector \(\Psi = \begin{pmatrix} \Psi ^{(0)} \\ \Psi ^{(1)} \end{pmatrix} \in \mathcal {H}\), \(S\Psi \) is given by
Let \(\{C(x)\}_{x \in \mathbb {Z}} \subset U(2)\) be a family of unitary matrices with
\(C\Psi \) is given by
We define an evolution operator as \(U = SC\). U satisfies
with
For a matrix \(M \in M(2,\mathbb {C})\), we use \(\Vert M\Vert \) to denote the operator norm in \(\mathbb {C}^2\): \(\Vert M\Vert = \sup _{\Vert {\varvec{x}}\Vert _{\mathbb {C}^2}=1} \Vert M{\varvec{x}}\Vert _{\mathbb {C}^2}\). We suppose that:
-
(A.1)
There exists a unitary matrix \(C_0 = \begin{pmatrix} a_0 &{} b_0 \\ c_0 &{} d_0 \end{pmatrix}\in U(2)\) such that
$$\begin{aligned} \Vert C(x) - C_0\Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z} \setminus \{0\} \end{aligned}$$with some positive \(c_1\) and \(\epsilon \) independent of x.
We denote by \(\mathscr {T}_1\) the set of trace class operators.
Lemma 2.1
Let U satisfy (A.1) and set \(U_0 = SC_0\). Then, \(U - U_0 \in \mathscr {T}_1\).
Proof
Let \(T = U-U_0\) and \(T(x) = C(x) - C_0\). Then,
is the multiplication operator by the matrix-valued function \(T(x)^*T(x)\). Let \(t_i(x)\) (\(i=1,2\)) be the eigenvalues of the Hermitian matrix \(T(x)^*T(x) \in M(2,\mathbb {C})\) and take an orthonormal basis (ONB) \(\{ \tau _i(x)\}_{i=1,2}\) of corresponding eigenvectors for all \(x \in \mathbb {Z}\). We use \(|\xi \rangle \langle \eta |\) to denote the operator on \(\mathcal {H}\) defined by \(|\xi \rangle \langle \eta |\Psi = \langle \eta , \Psi \rangle \xi \). Then, we have
where \(\{\tau _{i,x}\}\) is the ONB given by
Since \(T^*(x)T(x) \ge 0\), we have \(t_i(x) \ge 0\). By (A.1), we know that
Hence, we have
which means that \(T \in \mathscr {T}_1\). Since \(\mathscr {T}_1\) is an ideal, \(U-U_0 = ST \in \mathscr {T}_1\). \(\square \)
Example 2.1
(one-defect model) Let \(C_0, C_0^\prime \in U(2)\) be unitary matrices with \(C_0 \not = C_0^\prime \) and set
\(U = SC\) satisfies (A.1), because \(C(x) - C_0 = 0\) if \(x \not = 0\).
Example 2.2
Let \(C_0 \in U(2)\) be a unitary matrix and \(\{C(x)\} \subset U(2)\) a family of unitary matrices. Assume that
where \(M_{ij}\) denotes the ij-component of a matrix M. Then, \(U = SC\) satisfies (A.1), because all norms on a finite-dimensional vector space are equivalent.
We prove the following theorem in Sect. 3 using a discrete analogue of the Kato–Rosenblum theorem.
Theorem 2.1
Let U and \(U_0\) be as above and assume that (A.1) holds. Then,
exists and is complete.
In what follows, we introduce the asymptotic velocity \(\hat{v}_0\), obtained first in [4], of the quantum walk with the evolution \(U_0\) as follows. Let
Since \(\hat{U}_0(k) \in U(2)\), \(\hat{U}_0(k)\) is represented as
where \(\lambda _j(k)\) is an eigenvalue of \(\hat{U}_0(k)\) and \(u_j(k)\) is the corresponding eigenvector with \(\Vert u_j(k)\Vert =1\). The function \(k \mapsto e^{ik}\) is analytic, and so is \(\lambda _j(k)\). We need the following assumption on \(u_j(k)\):
-
(A.2)
The functions \(k \mapsto u_j(k)\) are continuously differentiable in k with
$$\begin{aligned} \sup _{k \in [0,2\pi )} \left\| \frac{\hbox {d}}{\hbox {d}k} u_j(k) \right\| _{\mathbb {C}^2} < \infty . \end{aligned}$$
Let \(\mathcal {K}\) be the Hilbert space of square integrable functions \(f{:} [0,2\pi ) \rightarrow \mathbb {C}^2\) with norm
Let \(\mathcal {F}_0{:} \mathcal {H} \rightarrow \mathcal {K}\) be the discrete Fourier transform given by
We also use \(\hat{\Psi }(k) = \begin{pmatrix} \hat{\Psi }^{(0)}(k) \\ \hat{\Psi }^{(1)}(k) \end{pmatrix}\) to denote the Fourier transform of \(\Psi \). The asymptotic velocity \(\hat{v}_0\) is the self-adjoint operator defined by
The position operator \(\hat{x}\) is a self-adjoint operator defined by
with domain
Let \(\hat{x}_0(t) = U_0^{-t} \hat{x} U_0^t\) be the Heisenberg operator of \(\hat{x}\) for the evolution \(U_0\).
Theorem 2.2
Let \(\hat{v}_0\) and \(\hat{x}_0\) be as above. Suppose that (A.2) holds. Then,
Proof
By [10, TheoremVIII.21], (2.3) holds if and only if
which is proved in Sect. 4.1. \(\square \)
Example 2.3
-
(i)
Let \(C_0 = \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix}\). Then, \(\hat{U}_0(k)\) has eigenvalues 1 and \(-1\), which are independent of k. By definition, \(\hat{v}_0\) = 0. Hence, the random variable \(X^{(0)}_t/t\) converges in law to a random variable \(V_0\) with a probability distribution \(\delta _0\).
-
(ii)
Let \(C_0 = \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix}\). \(\hat{U}_0(k)\) has eigenvalues \(e^{ik}\) and \(-e^{-ik}\). Hence, \(\hat{v}_0\) has eigenvalues \(-1\) and 1. The random variable \(X^{(0)}_t/t\) converges in law to a random variable \(V_0\) with a probability distribution \(\Vert \Psi ^{(0)}\Vert ^2 \delta _{-1} + \Vert \Psi ^{(1)}\Vert ^2 \delta _1\).
-
(iii)
Let \(C_0\) be the Hadamard matrix. The eigenvalues of \(\hat{U}_0(k)\) are given by \(\lambda _j(k) = ((-1)^j w(k) + i \sin k)/\sqrt{2}\) (\(j=1,2\)), where \(w(k) = \sqrt{1+\cos ^2 k}\). Hence, \(\hat{v}_0\) has no eigenvalue. The corresponding eigenvectors
$$\begin{aligned} u_j(k) = \sqrt{\frac{w(k)+(-1)^j \cos k}{2w(k)}} \begin{pmatrix} e^{ik} \\ (-1)^j w(k) - \cos k \end{pmatrix} \end{aligned}$$form an ONB of \(\mathbb {C}^2\) and satisfy (A.2). The random variable \(X^{(0)}_t/t\) converges in law to a random variable \(V_0\) with a probability distribution \(\Vert E_{\hat{v}_0}(\cdot )\Psi _0\Vert ^2\), where \(E_{\hat{v}_0}\) is the spectral measure of \(\hat{v}_0\). Let us consider the Hadmard walk starting from the origin. Let the initial state \(\Psi _0\) satisfy \(\Psi _0(0)= \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\) (\(|\alpha |^2 + |\beta |^2=1\)) and \(\Psi (x) = 0\) if \(x\not =0\). Then,
$$\begin{aligned} d \Vert E_{\hat{v}_0}(v)\Psi _0 \Vert ^2 = (1 - c_{\alpha ,\beta } v) f_K\left( v;\frac{1}{\sqrt{2}}\right) \hbox {d}v, \end{aligned}$$where \(c_{\alpha , \beta } = |\alpha |^2-|\beta |^2+\alpha \bar{\beta } + \bar{\alpha } \beta \),
$$\begin{aligned} f_K(v;r) = \frac{\sqrt{1-r^2}}{\pi (1-v^2)\sqrt{r^2-v^2}}I_{(-r,r)}(v) \end{aligned}$$is the Konno function, and \(I_A\) is the indicator function of a set A. For more details, the reader can consult [4, 7].
Let \(\hat{x}(t) = U^{-t} \hat{x} U\) be the Heisenberg operator of \(\hat{x}\) and define the asymptotic velocity \(\hat{v}_+\) for the evolution U by
We need the following assumption:
-
(A.3)
The singular continuous spectrum of U is empty.
We are now in a psition to state our main result, which is proved in Sect. 4.2.
Theorem 2.3
Let \(\hat{x}(t)\) and \(\hat{v}_+\) be as above. Suppose that (A.1)–(A.3) hold. Then,
Let \(X_t\) be the random variable denoting the position of the walker at time \(t \in \mathbb {N}\) with the initial state \(\Psi _0\). We use \(\Pi _\mathrm{p}(U)\) to denote the orthogonal projection onto the direct sum of all eigenspaces of U and \(E_{A}\) to denote the spectral projection of a self-adjoint operator A.
Corollary 2.4
Let \(X_t\) be as above. Suppose that (A.1)–(A.3) hold. Then, \(X_t/t\) converges in law to a random variable V with a probability distribution
where \(\delta _0\) is the Dirac measure at zero.
Proof
From Theorem 2.1, \(\text{ s- }\lim _{t \rightarrow \infty } U_0^{-t} U^t ~\Pi _\mathrm{ac}(U)\) exists and is equal to \(W_+^*\). Then, \(W_+\) is unitary from \(\mathrm{Ran}W_+^* = \mathrm{Ran}\Pi _\mathrm{ac}(U_0)\) to \(\mathrm{Ran}W_+ = \mathrm{Ran}\Pi _\mathrm{ac}(U)\). Since, by Lemma 4.1, \(U_0\) is strongly commuting with \(\hat{v}_0\), we know, from the intertwining property \(UW_+ = W_+ U_0\), that U is also strongly commuting with \(\hat{v}_+\). Hence, \(\hat{v}_+\) is strongly commuting with \(\Pi _\mathrm{ac}(U)\) and \(e^{i \xi \hat{v}_+} \Pi _\mathrm{ac}(U) = \Pi _\mathrm{ac}(U) e^{i \xi \hat{v}_+}\). Hence, by Theorem 2.3, \(\mathrm{exp}(i\xi \hat{x}(t)/t) \Psi _0\) converges strongly to \(\Pi _\mathrm{p}(U)\Psi _0 + e^{i \xi \hat{v}_+} \Pi _\mathrm{ac}(U)\Psi _0\) and
which proves the corollary. \(\square \)
Example 2.4
Let \(C_0\) be the Hadmard matrix and C(x) satisfy (A.1). As seen in Example 2.3 (iii), (A.2) is satisfied and the spectrum of \(U_0\) is purely absolutely continuous. Let \(\Psi _+ \in \mathcal {H}\) satisfy \(\Psi _+(0) = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\) (\(|\alpha |^2 + |\beta |^2 = 1\)) and \(\Psi _+(x) = 0\) if \(x\not =0\). By Example 2.3,
Let \(\Psi _\mathrm{p} \in \mathrm{Ran} \Pi _\mathrm{p}(U_0)\) be a unit vector and take the initial state \(\Psi _0\) as \(\Psi _0 = C_1 \Psi _\mathrm{p} + C_2 W_+ \Psi _+\) (\(|C_1|^2 + |C_2|^2 = 1\)). Suppose that \(U = SC\) satisfies (A.3). By Corollary 2.4, \(X_t/t\) converges in law to V with a probability distribution \(\mu _V\) and
3 Wave operator
To prove Theorem 2.1, we use the following general proposition:
Proposition 3.1
Let U and \(U_0\) be unitary operators on a Hilbert space \(\mathcal {H}\) and suppose that \(U-U_0 \in \mathscr {T}_1\). The following limit exists:
Proof of Theorem 2.1
Since, by Lemma 2.1, \(U-U_0 \in \mathscr {T}_1\), the wave operator \(W_+\) exists. If we interchange the roles of U and \(U_0\), then the proposition says that the limit \(\text{ s- }\lim _{t \rightarrow \infty } U_0^{-t} U^t \Pi _\mathrm{ac}(U)\) also exists, which implies that \(W_+\) is complete. This completes the proof. \(\square \)
In the remainder of this section, we suppose that \(U-U_0 \in \mathscr {T}_1\) and prove Proposition 3.1. This is done by a discrete analogue of [11, Theorem 6.2]. We use \(\mathcal {H}_\mathrm{ac}\) and \(\mathcal {H}_\mathrm{p}\) to denote the subspaces of absolute continuity and the direct sum of all eigenspaces of \(U_0\). Let \(E_0\) be the spectral measure of \(U_0\) with \(E_0([0,2\pi )) = I\). Let
where \(L^2 = L^2([0,2\pi ))\) and \(L^\infty = L^\infty ([0,2\pi ))\). Although the following lemma may be well known, we give proofs for completeness.
Lemma 3.1
\(\mathcal {H}_\mathrm{ac,0}\) is dense in \(\mathcal {H}_\mathrm{ac}\).
Proof
For all \(\psi \in \mathcal {H}_\mathrm{ac}\), there exists a positive function \(F \in L^1\) such that \(d\Vert E_0(\lambda )\psi \Vert ^2 = F (\lambda ) d\lambda \). Let \(B_n = F^{-1}([0,n])\), and let \(\chi _{B_n}\) be the characteristic function of \(B_n\). We set \(G_n = \sqrt{F} \chi _{B_n}\) and \(\psi _n = E_0(B_n)\Psi \). Then, \(G_n \in L^2 \cap L^\infty \) and \(\Vert E_0(B) \psi _n\Vert ^2 = \int _B G_n(\lambda )^2 d\lambda \). Hence, \(\psi _n \in \mathscr {H}_\mathrm{ac,0}\) and \(\psi = \lim _n \psi _n\). This completes the proof. \(\square \)
Lemma 3.2
Let \(\phi \in \mathcal {H}\) and \(\psi \in \mathcal {H}_\mathrm{ac,0}\). Then,
Proof
Let \(\psi \in \mathcal {H}_\mathrm{ac,0}\) and \(\mathcal {L} = L^2([0,2\pi ), G^2_\psi (\lambda ) d\lambda )\). Let \(H_0\) be the self-adjoint operator defined by \(\langle \xi , H_0 \eta \rangle = \int _0^{2\pi } \lambda d\langle \xi , E_0(\lambda ) \eta \rangle \) (\(\xi , \eta \in \mathcal {H}\)). Let \(\mathscr {U}{:} \mathcal {L} \rightarrow \mathcal {H}\) be an injection defined by \(\mathscr {U}f = f(H_0)\psi \) (\(f \in \mathcal {L}\)). Then \(\mathscr {U} 1 = \psi \) and \(\mathscr {U} e^{it\lambda } = U_0^t\psi \) (\(t \in \mathbb {N}\)). We use \(\Pi \) to denote the orthogonal projection onto \(U\mathcal {L}\). Let \(\phi \in \mathcal {H}\) and \(F = \mathscr {U}^{-1}\Pi \phi \in \mathcal {L}\). Then, we have
Hence, by Parseval’s identity, we obtain
This completes the proof. \(\square \)
Let \(W_t = U^{-t} U_0^t \).
Lemma 3.3
Let \(t, s \in \mathbb {N}\) (\(s\not =t\)). Then, \(\text{ s- }\lim _{r \rightarrow \infty } (W_t - W_s)U_0^r\Pi _\mathrm{ac}(U_0) = 0\).
Proof
For \(t, s \in \mathbb {N}\) (\(t>s\)), we have \(W_t = \sum _{k=s+1}^t (W_k - W_{k-1}) + W_s\) and \(W_k - W_{k-1} = U^{-k}(-T)U_0^{k-1}\), where \(T = U-U_0 \in \mathscr {T}_1\). Since \(\mathscr {T}_1\) is an ideal, we know that
In particular, \(W_t-W_s\) is compact. Let \(H_0\) be the self-adjoint operator defined in the proof of Lemma 3.2. Since \(\text{ w- }\lim _{r \rightarrow \infty } e^{irH_0}\Pi _\mathrm{ac}(H_0) = 0\), we have
This completes the proof. \(\square \)
Proof of Proposition 3.1
By Lemma 3.1, it suffices to prove that, for \(\psi \in \mathcal {H}_\mathrm{ac,0}\),
Because
we need only to prove that
By direct calculation, we have, for \(r > 1\),
Since
we obtain
Since, by Lemma 3.3, \(\text{ s- }\lim _{r \rightarrow \infty }U_0^{-r} W_t^* (W_t - W_s) U_0^r \psi = 0\), we have
where
By Lemma 3.4 below, we know that
This completes the proof. \(\square \)
Lemma 3.4
Let \(Y \in \mathscr {T}_1\) and \(\{Q(t,s)\}\) be a family of bounded operators with \(\sup _{t,s}\Vert Q(t,s)\Vert < \infty \). Then, for all \(\psi \in \mathcal {H}_\mathrm{ac,0}\),
-
(1)
\(\lim _{t, s \rightarrow \infty } \left\langle \psi , Z_{t,s}(Y Q(t,s) )\psi \right\rangle = 0\);
-
(2)
\(\lim _{t, s \rightarrow \infty } \left\langle \psi , Z_{t,s}(Q(t,s) Y)\psi \right\rangle = 0\).
Proof
Let \(Y = \sum _{n=1}^\infty \lambda _n |\psi _n \rangle \langle \phi _n |\) be the canonical expansion of the compact operator Y. Since \(Y \in \mathscr {T}_1\), \(\sum _n \lambda _n < \infty \). Then, by the Cauchy–Schwartz inequality, we have
where
By Lemma 3.2, we have
where we have used the fact that \(\phi _n\) is a normalized vector. Let \(u_k= \sum _{n=1}^{\infty } \lambda _n |\left\langle \psi _n,\right. \left. U_0^k \psi \right\rangle |^2\). Then, similar to the above, we observe that \(\{u_k\} \in \ell ^1(\mathbb {Z})\). Hence, we have
This proves (i). The same proof works for (ii). \(\square \)
4 Asymptotic velocity
4.1 Proof of Theorem 2.2
Let
We use \(\mathcal {D}\) to denote a subspace of vectors \(\Psi \in \mathcal {H}\) whose Fourier transform \(\hat{\Psi }\) is differentiable in k with
Note that \(\mathcal {H}_0\) is a core for \(\hat{x}\), and so is \(\mathcal {D}\). Let \(D = \mathscr {F} \hat{x} \mathscr {F}^{-1}\). Then, by direct calculation, we know that \((D \hat{\Psi })(k) = i\frac{\hbox {d}}{\hbox {d}k} \hat{\Psi }(k)\) for \(\Psi \in \mathcal {D}\). We prove the following theorem:
Theorem 4.1
Suppose that (A.2) holds. Then,
Proof
For all \(\Psi \in \mathcal {H}\) and \(\epsilon > 0\), there exists a vector \( \Psi _\epsilon \in \mathcal {D}\) such that \(\Vert \Psi - \Psi _\epsilon \Vert \le \epsilon \). Because, by the second resolvent identity,
it suffices to prove that
Note that
Since \(\lambda _j(k)\) is analytic and \(|\lambda _j(k)|=1\), we observe from (A.2) that \((\hat{v}_0 - z)^{-1}\) leaves \(\mathcal {D}\) invariant. Hence, we only need to prove that
By direct calculation, we have
By the definition of \(\mathcal {D}\) and (A.2), we know that
Hence, we have
which completes the proof. \(\square \)
4.2 Proof of Theorem 2.3
The proof falls naturally into two parts:
Theorem 4.2
Let U be a unitary operator on \(\mathcal {H}\). \(\hat{x}(t) = U^{-t} \hat{x} U^t\) satisfies
Theorem 4.3
Let \(U=SC\) and \(U_0 = SC_0\) satisfy (A.1) and (A.2). Then,
Proof of Theorem 2.3
By (A.3), we have
This prove the theorem. \(\square \)
It remains to prove Theorems 4.2 and 4.3.
Proof of Theorem 4.2
Let \(\mathcal {H}_\mathrm{p}(U)\) be the direct sum of all eigenspaces of U. It suffices to prove that, for \(\Psi \in \mathcal {H}_\mathrm{p}(U)\),
Let \(\lambda _n\) be the eigenvalues of U and take an ONB \(\{\eta _n\}_{n=1}^\infty \) of \(\mathcal {H}_\mathrm{p}\) such that \(U\eta _n = \lambda _n \eta _n\). We have \(\Pi _\mathrm{p}(U) = \sum _n |\eta _n \rangle \langle \eta _n|\). Let \(\epsilon >0\). For sufficiently large N, \(\Psi _N = \sum _{n=1}^N \langle \eta _n, \Psi \rangle \eta _n\) satisfies \(\Vert \Psi - \Psi _N\Vert \le \epsilon \). Then,
By direct calculation, we have
Since \(\lim _{t \rightarrow \infty } |1-e^{i \xi x/t}|=0\), \(|1-e^{i \xi x/t}| \le 2\) and \(\sum _{x} \Vert \eta _n(x)\Vert _{\mathbb {C}^2}^2 = \Vert \eta _n\Vert ^2 < \infty \), we have
which, combined with (4.2), completes the proof. \(\square \)
Lemma 4.1
\([U_0, \mathrm{exp}(i \xi \hat{v}_0)] = 0\).
Proof
By direct calculation, we have
Proof of Theorem 4.3
By (A.1) and (A.2), Theorems 2.1 and 2.2 hold. Then, \(W_+\) is a unitary operator from \(\mathcal {H}_\mathrm{ac}(U_0)\) to \(\mathcal {H}_\mathrm{ac}(U)\). Hence, we have
By direct calculation, we observe that
where
Because \(W_t\) and \(\mathrm{exp}(i \xi \hat{x}_0(t)/t)\) are uniformly bounded, we know from Theorems 2.1 and 2.2 that \(\text{ s- }\lim _{t \rightarrow \infty }I_1(t) = \text{ s- }\lim _{t \rightarrow \infty }I_2(t) = 0\). Hence, we have
where we have used the fact that \(\mathrm{Ran} W_+^* =\mathcal {H}_\mathrm{ac}(U_0)\). Since, by Lemma 4.1, \([ \mathrm{exp}(i \xi \hat{v}_0), \Pi _\mathrm{ac}(U_0)]=0\), we obtain from Theorem 2.1, that \(\text{ s- }\lim _{t \rightarrow \infty } I(t)=0\). This completes the proof. \(\square \)
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This work was supported by Grant-in-Aid for Young Scientists (B) (No. 26800054).
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Suzuki, A. Asymptotic velocity of a position-dependent quantum walk. Quantum Inf Process 15, 103–119 (2016). https://doi.org/10.1007/s11128-015-1183-x
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DOI: https://doi.org/10.1007/s11128-015-1183-x