Quantum walks with an anisotropic coin II : scattering theory

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Quantum walks with an anisotropic coin II : scattering

theory

Serge Richard, A Suzuki, R Tiedra de Aldecoa

To cite this version:

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Quantum walks with an anisotropic coin II: scattering theory

S. Richard

1

_{, A. Suzuki}

2y

_{, R. Tiedra de Aldecoa}

3z

1 _{Graduate school of mathematics, Nagoya University, Chikusa-ku,}

Nagoya 464-8602, Japan

2 _{Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato,}

Nagano 380-8553, Japan

3 _{Facultad de Matemáticas, Ponticia Universidad Católica de Chile,}

Av. Vicuña Mackenna 4860, Santiago, Chile

E-mails: [email protected], [email protected], [email protected] Abstract

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest.

2010 Mathematics Subject Classication: 46N50, 47A40, 47B47, 60F05.

Keywords: Quantum walks, unitary operators, scattering theory, weak limit theorem

1 Introduction

This paper is motivated by recent works on topological phenomena for quantum walks [3,6,7,13,14,15,

16,23]. It is the second part of a series of papers on one-dimensional quantum walks with an anisotropic behaviour at innity. In our rst paper [25], we performed the spectral analysis of the quantum walks and we developed abstract commutator methods for unitary operators in a two-Hilbert spaces setting. Here we pursue our study by investigating the scattering theory of the quantum walks and establishing a weak limit theorem [17,18]. We also present a suitable abstract framework for the proof of the existence and completeness of wave operators for unitary operators in a two-Hilbert spaces setting.

The one-dimensional anisotropic quantum walks that we consider are described by a unitary operator U := SC in the Hilbert space H := `2_{(Z; C}2_{), where S is a shift operator and C is a coin operator acting}

by multiplication by unitary matrices C(x) 2 U(2), x 2 Z, with short-range asymptotics at innity:

C(x) =(C`+ O(jxj 1 "`) as x ! 1

Cr+ O(jxj 1 "r) as x ! +1 with C`; Cr2 U(2); "`; "r> 0: (1.1)

The assumption (1.1) covers a wide range of quantum walks such as homogeneous (or translation-invariant) quantum walks [1, 12, 17, 18], one-defect models [5, 19, 20, 29], and two-phase quantum

_{Supported by the grant Topological invariants through scattering theory and noncommutative geometry from Nagoya}

University, and on leave of absence from Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France.

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walks [8, 9, 10]. Some classes of inhomogeneous (or position-dependent) quantum walks [4, 28] and split-step quantum walks [14] are also covered by our assumption. We refer to the introduction of [25] for additional references on earlier works.

A weak limit theorem for quantum walks is a result of the following type: If Xn denotes the random

variable for the position of a quantum walker at time n 2 Z, then Xn=n converges in law to a random

variable V as n ! 1. Since Xn=n is the average velocity of the quantum walker, the random variable V can

be interpreted as the asymptotic velocity of the walker. It is therefore of particular interest to determine the density function of the probability distribution V of V. As was put into evidence in [28], where the

second author considered the case C` = Cr in (1.1), the key ingredients for the proof of the weak limit

theorem are the following:

(i) absence of singular continuous spectrum for U, (ii) existence of an asymptotic velocity operator Vacfor U.

Once these assertions are proved, a weak limit theorem can be established in a way similar to [28] and the distribution V can be expressed as

V= EpU in _H2 0+ EVac()E_acU in

2_H; (1.2) with 0 the Dirac measure for the point 0, in 2 H the initial state of the walker, EpU and EacU the

projections onto the pure point and absolutely continuous subspaces of U, and EVac the spectral measure

of Vac. In our rst paper [25], we proved the assertion (i) and provided information on the eigenvalues of

U by constructing a conjugate operator A for U under the assumption (1.1). Here, we build on the results of [25] to prove the assertion (ii) and to establish a detailed formula for the distribution (1.2).

The organisation of the paper is the following. In Section 2, we develop our framework for the scattering theory for unitary operators in a two-Hilbert spaces setting. Given two unitary operators U and U0 acting in Hilbert spaces H and H0, and a bounded operator J : H0 ! H, we establish in Theorem

2.5and Corollary2.6criteria for the existence and completeness under smooth perturbations of the local wave operators

W(U; U0; J; ) := s-lim_n!1U nJU0nEU0(); (1.3)

where EU0 is the spectral measure of U

0 and T := fz 2 C j jzj = 1g an open set. These results for

the scattering theory of unitary operators in two Hilbert spaces are new in such a generality. They are a natural analogue of similar results for the scattering theory of self-adjoint operators in two Hilbert spaces, which can be found for example in [26, 30].

In Section3, we apply our results on scattering theory to anisotropic quantum walks with full evolution operator U = SC and free evolution operator U0:= U` Ur, where U`:= SC` and Ur:= SCr describe the

behaviour of the quantum walker as x ! 1 and x ! +1. We prove in Theorem3.3the existence and completeness of the wave operators for the pair fU0; Ug, and in Proposition 3.4we give a description of

the initial sets of the wave operators in terms of the velocity operators V`; Vr for the evolution operators

U`; Ur.

Section 4 is dedicated to the proof of the weak limit theorem for the anisotropic quantum walks. First, we prove in Proposition4.1the assertion (ii) above, that is, the existence of an asymptotic velocity operator Vacfor the full evolution operator U. We show that Vacis given by

Vac= W+(U; U0; J; )V0W+(U; U0; J; );

with V0:= V` Vrthe asymptotic velocity operator for the free evolution operator U0. Then, in Theorem

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set a` := j(C`)1;1j and ar:= j(Cr)1;1j, let fK: R (0; 1] ! [0; 1) be the Konno function, and write B

for the characteristic function for a set B, then we prove that V has distribution V(d) = 00(d) + ` 1(d) + r1(d)

+ [ a`;0)()w`()12fK(; a`)d + (0;ar]()wr()12fK(; ar)d; (1.4)

with 0 := kEUp ink2H, `; r 0 and w`; wr : R ! [0; 1). In addition, we show that ` is nontrivial

when a` = 1, r is nontrivial when ar= 1, w` is nontrivial and has support in [ a`; 0) when a` 2 (0; 1),

and wris nontrivial and has support in (0; ar] when ar2 (0; 1). See Theorem4.8for the explicit formulas

of `; r and w`; wr. We also show that the decomposition (1.4) of V is unique.

An interpretation of the formula (1.4) detailed after Theorem4.3and in Example4.10is the following. Localisation occurs if the probability that the asymptotic velocity vanishes is positive, i.e., P(V = 0) > 0. Since (1.4) implies that P(V = 0) = 0= kEpU ink2H, localisation occurs if and only if the initial state in

has an overlap with the pure point subspace of U. Furthermore, the quantum walker moves asymptotically to the left at speed 2 [ a`; 0) if a` 2 (0; 1) and at speed = 1 if a` = 1. Similarly, the quantum

walker moves asymptotically to the right at speed 2 (0; ar] if ar2 (0; 1) and at speed = 1 if ar= 1.

In particular, if a` > ar, then the quantum walker can move faster on the left-hand side than on the

right-hand side.

Finally, in Example4.11at the end of Section4, we explain how our formula (1.4) for the distribution V generalises several formulas already available in the literature. For example, it generalises a similar

formula for isotropic quantum walks where C(x) = C1+ O(jxj 1 ") [28], which include one-defect

models [20] and homogeneous quantum walks [17,18,12]. The formula (1.4) also generalises the formula obtained in [9] for two-phase quantum walks where C(x) = C for x 1 and C(x) = C+ for x 1,

and (C )1;1= (C+)1;1.

2 Scattering theory in a two-Hilbert spaces setting

We discuss in this section the existence and the completeness under smooth perturbations of the local wave operators for unitary operators in a two-Hilbert spaces setting. Namely, given two unitary operators U0; U in Hilbert spaces H0; H with spectral measures EU0; EU, a bounded operator J : H0! H, and an

open set T := fz 2 C j jzj = 1g, we give criteria for the existence and the completeness of the strong limits

W(U; U0; J; ) := s-lim_n!1U nJU0nEU0()

under the assumption that the dierence JU0 UJ factorises as a product of a locally U-smooth operator

on and a locally U0-smooth operator on . We start with a standard result on the intertwining property

of wave operators. Note that we use the notation B(H1; H2) (resp. K (H1; H2)) for the set of bounded

(resp. compact) operators between Hilbert spaces H1 and H2, and we set B(H1) := B(H1; H1) and

K (H1) := K (H1; H1).

Lemma 2.1 (Intertwining property). Let U0; U be unitary operators in Hilbert spaces H0; H with spectral

measures EU0, EU, let J 2 B(H₀; H), and let T be an open set. Assume that W(U; U₀; J; ) exist.

Then, we have for each bounded Borel function : T ! C the intertwining property

W(U; U0; J; )(U0) = (U)W(U; U0; J; ): (2.1)

Proof. A direct calculation implies the equality W(U; U0; J; )U0k = UkW(U; U0; J; ) for each k 2 Z.

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Next, we dene the closed subspaces of H N(U; J; ) := ' 2 H j lim n!1 JUnEU()' _H 0 = 0 ;

and note that EU_{(T n )H N}_{(U; J; ), that U is reduced by N}_{(U; J; ), and that}

Ran W(U; U0; J; ) ? N(U; J; );

this last fact being shown as in the self-adjoint case, see [30, Lemma 3.2.1]. In particular, one has the inclusion

Ran W(U; U0; J; ) EU()H N(U; J; );

which motivates the following denition:

Denition 2.2 (J-completeness). Assume that W(U; U0; J; ) exist. The operators W(U; U0; J; ) are

J-complete on if

Ran W(U; U0; J; ) = EU()H N(U; J; ):

Remark 2.3. In the particular case H0 = H and J = 1H, the J-completeness on reduces to the

completeness of W(U; U0; J; ) on in the usual sense. Namely, Ran W(U; U0; 1H; ) = EU()H,

and the operators W(U; U0; 1H; ) are unitary from EU0()H to EU()H.

The following criterion for J-completeness is shown as in the self-adjoint case, see for example [30, Thm. 3.2.4]:

Lemma 2.4. If W(U; U0; J; ) and W(U0; U; J; ) exist, then W(U; U0; J; ) are J-complete on .

Proof. The intertwining property and the existence of the operators W(U0; U; J; ) imply that for any

' 2 H and 2 H0 _W (U; U0; J; )'; _H₀ = '; EU()W(U; U0; J; ) _H = lim n!1 EU_{()'; U} n_JUn 0EU0() H = lim n!1 EU0_()U n 0 JUnEU()'; iH0 = W(U0; U; J; )'; _H₀:

Thus, W(U0; U; J; ) is the adjoint of W(U; U0; J; ). Since ker W(U0; U; J; ) = N(U; J; ) and

EU_{(T n )H N}

(U; J; ), it follows that

Ran W(U; U0; J; ) = H ker W(U; U0; J; )

= H N(U; J; )

= EU_{()H N}

(U; J; );

which proves the claim.

For the next theorem, we recall that the spectral support suppU(') of a vector ' 2 H with respect

to U is the smallest closed set T such that EU_{()' = '. We also recall that if G is an auxiliary}

Hilbert space, then an operator T 2 B(H; G) is locally U-smooth on an open set T if for each closed set 0_{there exists c}₀ _{0 such that}

X n2Z T UnEU(0)' 2 G c0k'k2H for each ' 2 H; (2.2)

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Theorem 2.5. Let U0; U be unitary operators in Hilbert spaces H0; H with spectral measures EU0; EU,

J 2 B(H0; H), and T be an open set. Let G be an auxiliary Hilbert space, T02 B(H0; G) a locally U0

-smooth operator on and T 2 B(H; G) a locally U--smooth operator on such that JU0 UJ = TT0.

Then, the wave operators

W(U; U0; J; ) = s-lim_n!1U nJU0nEU0() (2.3)

exist, are J-complete on , and satisfy the relations

W(U; U0; J; )= W(U0; U; J; ) and W(U; U0; J; )(U0) = (U)W(U; U0; J; )

for each bounded Borel function : T ! C.

Proof. We adapt the proof of [2, Thm. 7.1.4] to the case of unitary operators in a two-Hilbert spaces setting. The existence of the limits (2.3) is a direct consequence of the following assertion: For each '0 2 H0 such that 0 := suppU0('0) , and for each 2 C1c (; R) such that () = 1 on a

neighbourhood of 0

s-lim

n!1(U)U n_JUn

0'0exist and _n!1lim 1 (U)U nJU₀n'0

H = 0: (2.4)

To prove the rst claim in (2.4), we take ' 2 H and observe that Wn := (U)U nJU0n satises for

m n 1 '; (Wn Wm)'0_H = n X j=m+1 '; (U)U j_(JU 0 UJ)U0j 1'0_H = n X j=m+1 T Uj_{(U)'; T} 0U0j 1'0_G   n X j=m+1 T Uj(U)' 2_G   1=2  n X j=m+1 T0U₀j 1'0 2_G   1=2 c1=2₁ k'kH   n X j=m+1 T0U₀j 1)'0 2 G   1=2 ;

with 1:= supp() and c1 the constant appearing in the denition (2.2) of a locally U-smooth operator.

Since T0is locally U0-smooth on , it follows that k(Wn Wm)'0kH! 0 as m ! 1 or n ! 1. This

proves the rst claim in (2.4).

To prove the second claim in (2.4), we take 02 Cc1(; R) such that 0 1 on 0and 0= 0.

Then, we have '0= 0(U0)'0 and

1 (U)J0(U0) = 1 (U) J0(U0) 0(U)J;

and thus the second claim in (2.4) follows from lim n!1 J0(U0) 0(U)JU₀n'0 _H= 0:

Since the set of monomials zk _{with z 2 T and k 2 Z is total in C(T) for the sup norm, it is sucient to}

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for all k 2 Z. For k 1, the result follows from the formula JUk

0 UkJ = Pkj=1Uj 1(JU0 UJ)U0k j

and the local U0-smoothness of T0since

lim n!1 JU₀k UkJU₀n'0 H lim_n!1 k X j=1 (JU0 UJ)U₀k jU₀n'0 H Const:k lim m!1 T0U₀m0(U0)'0 _G = 0;

and for k 0 the result follows from what precedes since JU₀jkj U jkj_{J = U} jkj _JUjkj

0 UjkjJU0jkj.

So, the existence of the limits (2.3) has been established. Similar arguments, using the relation U

0J JU= T0T instead of JU0 UJ = TT0, show that W(U0; U; J; ) exist too. This, together

with standard arguments in scattering theory, implies the claims that follow (2.3).

To present the last result of this section, we need to recall some basic denitions of the conjugate operator theory borrowed from [2, Chap. 5]: Let S 2 B(H) and let A be a self-adjoint operator in H with domain D(A). For k 2 N, we say that S 2 Ck_{(A) if the map R 3 t 7! e} itA_{S e}itA_{2 B(H) is strongly of}

class Ck_{. In the case k = 1, one has S 2 C}1_{(A) if and only if the quadratic form}

D(A) 3 ' 7! A'; S'_H

'; SA'_H 2 C

is continuous for the topology induced by H on D(A). The operator associated to the continuous extension of the form is denoted by [A; S] 2 B(H). Three regularity conditions slightly stronger than S 2 C1_(A)

are dened as follows: S 2 C1;1_{(A) if}

Z 1

0

e itAS eitA+ eitAS e itA 2S _B(H) dt t2 < 1:

S 2 C1+0_{(A) if S 2 C}1_{(A) and}

Z 1

0

e itA[A; S] eitA [A; S] _B(H) dt

t < 1: S 2 C1+"_{(A) for some " 2 (0; 1) if S 2 C}1_{(A) and}

e itA[A; S] eitA [A; S] _B(H) Const:t" for all t 2 (0; 1).

As banachisable topological vector spaces, these sets satisfy the continuous inclusions [2, Sec. 5.2.4] C2_{(A) C}1+"_{(A) C}1+0_{(A) C}1;1_{(A) C}1_{(A) C}0_(A):

Let us also recall from [25, Sec. 3.1] that if U is unitary operator in H with U 2 C1_{(A), then the}

function%eAU : T ! ( 1; 1] is dened by

e %A

U() := sup a 2 R j 9" > 0 such that EU(; ")U 1[A; U]EU(; ") & aEU(; ") ; 2 T;

where EU_{(; ") := E}U _{(; "), (; ") := f}0_{2 T j j arg(}0_{)j < "g, and for S; T 2 B(H) the notation}

T & S means that there exists K 2 K (H) such that T + K S. By analogy with the self-adjoint case, we say that A is conjugate to U at a point 2 T if%eAU() > 0, and we write

e

A_{(U) := 2 T j}

e %A

U() > 0

for the open subset of T where A is conjugate to U. The sete

A_{(U) is open because the function}

e %A

U()

is lower semicontinuous. Finally, we denote by p(U0) and p(U) the pure point spectra of U0and U.

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Corollary 2.6. Let U0; U be unitary operators in Hilbert spaces H0; H with spectral measures EU0; EUand

A0; A self-adjoint operators in H0; H. Assume either that U0; U have a spectral gap and U02 C1;1(A0); U 2

C1;1_{(A), or that U}

02 C1+0(A0); U 2 C1+0(A). Let

:= eA0_(U

0) n p(U0) \ eA(U) n p(U) ;

J 2 B(H0; H), G be an auxiliary Hilbert space, and assume there exist T02 B(H0; G) and T 2 B(H; G)

with JU0 UJ = TT0and such that T0 extends continuously to an element of B D(hA0is); G and T

extends continuously to an element of B D(hAis₎_{; G for some s > 1=2. Then, the strong limits}

W(U; U0; J; ) := s-lim_n!1U nJU0nEU0()

exist, are J-complete on , and satisfy the relations

for each bounded Borel function : T ! C.

3 Scattering theory for quantum walks with an anisotropic coin

In this section, we present our results on the scattering theory for the pair fU0; Ug when U is the evolution

operator of a one-dimensional quantum walk with an anisotropic coin and U0 is the corresponding free

evolution operator. We start by recalling from [25, Sec. 4] the needed denitions and facts on the operators U and U0.

Let H be the Hilbert space of square-summable C2_{-valued sequences}

H := `2_{(Z; C}2_{) =} ( : Z ! C2_jX x2Z k (x)k2 2< 1 ) ;

where k k2is the usual norm on C2. Then, the evolution operator of the one-dimensional quantum walk

in H that we consider is given by U := SC, with S a shift operator dened by

(S )(x) := (0)_{(x + 1)} (1)_(x ₁₎ ; = (0) (1) 2 H; x 2 Z;

and C a coin operator dened by

(C )(x) := C(x) (x); 2 H; x 2 Z; C(x) 2 U(2):

In particular, the evolution operator U is unitary in H since both S and C are unitary in H.

We assume that the coin operator C has an anisotropic behaviour at innity. More precisely, we assume that C converges with short-range rate to two asymptotic coin operators, one on the left and one on the right in the following way:

Assumption 3.1 (Short-range assumption). There exist C`; Cr 2 U(2), `; r > 0, and "`; "r > 0 such

that C(x) C` _B(C₂₎ `jxj 1 "` if x < 0 C(x) Cr B(C2₎ rjxj 1 "r if x > 0;

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This assumption provides us with two new unitary operators U`:= SC` and Ur:= SCr describing the

asymptotic behaviour of U on the left and on the right.

From now on, we shall use the symbol ? to denote either the index ` or the index r, and we dene the space

Hn:= [ n2N

2 H j (x) = 0 if jxj n H;

the Hilbert space K := L2 _{[0; 2);}dk 2; C2

, and the unitary Fourier transform F : H ! K which corresponds to the unique continuous extension of the operator

(F )(k) :=X

x2Z

e ikx_{(x); 2 H}

n; k 2 [0; 2):

The operator U? is decomposable in the Fourier representation, namely, for all f 2 K and almost every

k 2 [0; 2) we have (F U?Ff )(k) = cU?(k)f (k) with cU?(k) := eik ₀ 0 e ik C?2 U(2):

Also, since cU?(k) 2 U(2), the spectral theorem implies that

c U?(k) = 2 X j=1 ?;j(k)?;j(k);

with ?;j(k) the eigenvalues of cU?(k) and ?;j(k) the corresponding orthogonal projections. Furthermore,

for j 2 f1; 2g we let v?;j: [0; 2) ! R be the bounded function given by

v?;j(k) := i 0?;j(k) ?;j(k) 1; (3.1)

where ()0_{means the derivative with respect to k. The function v}

?;jis real valued because ?;jtakes values

in T. Then, we dene for all f 2 K and almost every k 2 [0; 2) the decomposable operator bV?2 B(K),

b V?f (k) := bV?(k)f (k) where bV?(k) := 2 X j=1 v?;j(k)?;j(k) 2 B(C2); (3.2)

and we call asymptotic velocity operator the operator V?:= FVb?F .

We can now start studying the scattering theory for the operator U. As free evolution operator, we use the unitary operator U0:= U` Ur in the Hilbert space H0:= H H. In [25, Sec. 4.2], it has been

shown that the spectrum of U0 coincides with the essential spectrum of U, namely,

ess(U) = (U`) [ (Ur) = (U0):

As identication operator between the Hilbert spaces H0 and H, we use the operator J 2 B(H0; H)

dened by

J( `; r) := j` `+ jr r; ( `; r) 2 H0;

where

jr(x) :=(1 if x 0_{0 if x 1} and j` := 1 jr:

The rst lemma of the section consists in a simple observation related to the J-completeness of the wave operators for the pair fU0; Ug :

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Proof. We know from [25, Lemma 4.7] that JJ_{= 1}

H. Therefore, we have for any n 2 Z and 2 H

kJ_Un_k2

H0 = JUn ; JUn H0= U

n_{; JJ}_Un

H= kUn k2H= k k2H;

which implies the claim.

A direct consequence of this lemma is that the abstract spaces N(U; J; ) dened in Section 2are

trivial in our case:

N(U; J; ) = EU(T n )H: (3.3) Now, in order to prove with the help of Corollary 2.6 the existence and the completeness of the wave operators for the pair fU0; Ug, we need to recall some facts about conjugate operators A0 and A

introduced in [25]. In the proof of [25, Thm. 4.5(c)], it has been shown that there exists for ? = `; r an operator A?, dened in terms of the velocity operator V? and essentially self-adjoint on Hn, such

that U? 2 C2(A?). In addition, the operator A? is conjugate to the operator U? outside the set @(U?)

of boundary points of (U?) in T. As a consequence, the operator A0 := A` Ar is well-dened and

conjugate to U0on the set (U0) n (U), with

(U) := @(U`) [ @(Ur):

The set (U), which contains at most 8 values, is called for this reason the set of thresholds of U. In [25, Lemma 4.9], it has also been shown that the operator JA0Jis essentially self-adjoint on Hn, with

self-adjoint extension denoted by A, and that A is conjugate to U on (U0) n (U).

We also recall a relation between the conjugate operator A and the position operator Q given by Q (x) := x (x); x 2 Z; 2 D(Q) := 2 H j kQ kH< 1 :

This relation has already been used in the proof of [25, Lemma 4.13], but we make it more explicit now. As mentioned in that proof, the operator hQi 1_A

?dened on Hnextends continuously to an element of

B(H). This implies that D(hQi) D(A?), and thus that D(hQi) D(A) due to the equality

A = JA0J= j`A`j`+ jrArjr on Hn: (3.4)

Therefore, we obtain by real interpolation the inclusions

D(hQi)s _D(hAi)s _{and B D(hQi)} s_{; H B D(hAi)} s_{; H} _(3.5)

for each s > 0.

We can now state our theorem on the J-completeness of the wave operators for the pair fU0; Ug,

with the notation EU

acfor the orthogonal projection on the absolutely continuous subspace of U.

Theorem 3.3. Let := f(U`) [ (Ur)g n f(U) [ p(U)g. Then, the operators

W(U; U0; J; ) = s-lim_n!1U nJU0nEU0() (3.6)

exist and satisfy Ran W(U; U0; J; ) = EacUH. In addition, the relations

hold for each bounded Borel function : T ! C.

Before the proof, it is convenient to highlight some properties of the projection EU0(). First, let

the matrices C?2 U(2) be parameterised as

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with a?; b?2 [0; 1] satisfying a2?+ b?2= 1, and ?; ?; ?2 ( ; ]. Then, recall from [25, Lemma 4.1 &

Prop. 4.5] that the operator U?has pure point spectrum with (U?) (U) if a?= 0 and purely absolutely

continuous spectrum with @(U?) (U) if a? 2 (0; 1]. Since we also know from [25, Thm. 2.4] that

the number of eigenvalues of U in any closed set 0_{T n (U) is nite, we infer that}

EU0_{() = E}U0 ac =            1H 1H if a`; ar2 (0; 1], 1H 0H if a`2 (0; 1] and ar= 0, 0H 1H if a`= 0 and ar2 (0; 1], 0H 0H if a`= ar= 0. (3.8)

Thus, in the generic case a`; ar2 (0; 1], the projection EU0() appearing in (3.6) can simply be replaced

by 1H0 = 1H 1H.

Proof of Theorem3.3. All the claims except the equality Ran W(U; U0; J; ) = EacUH follow from

Corollary2.6whose assumptions are checked now.

The proof of [25, Prop. 4.5(c)] implies that U02 C2(A0), [25, Lemma 4.13] implies that U 2 C1+"(A)

for each " 2 (0; 1) with " minf"`; "rg, and [25, Prop. 4.11] implies that

f(U`) [ (Ur)g n f(U) [ p(U)g eA0(U0) n p(U0) \ eA(U) n p(U) :

Thus, in order to apply Corollary2.6, it is sucient to prove the existence of operators T02 B(H0; G)

and T 2 B(H; G) with JU0 UJ = TT0 and such that T0 extends continuously to an element of

B D(hA0is); G and T extends continuously to an element of B D(hAis); G for some s > 1=2. For

that purpose, we set s := (1 + ")=2 and dene the sesquilinear form D : H0 H ! C by

D ( `; r); := * hQis_; X ?2f`;rg [j?; S]C? S(C C?)j?hQis ? + H

for each ( `; r) 2 Hn Hn and 2 Hn. With arguments similar to the ones used in the proofs

of [25, Lemmas 4.12 & 4.13], one shows that the form D extends continuously to a bounded form on H0 H. Thus, there exists an operator D 2 B(H0; H) (the same notation is used on purpose) such that

D ( `; r); = ; D( `; r)_H; ( `; r) 2 H0; 2 H:

Also, we dene the operators T0 := hQi s hQi s 2 B(H0) and T := DhQi s 2 B(H0; H), and

observe that JU0 UJ = TT0 due to the denition of D and Equation [25, Eq. (4.6)]:

JU0 UJ =

X

?2f`;rg

[j?; S]C? S(C C?)j?:

Finally, we note that the second inclusion in (3.5) implies that

hQi s _{2 B D(hQi)} s_{; H B D(hAi)} s_{; H;}

and thus that T 2 B D(hAi) s_{; H}

0. Similarly, since

hQi s _hQi s _{2 B D(hQi} s_hQi s_{); H}

0 B D(hA0i) s; H0;

we have that T02 B D(hA0i) s; H0, and thus all the assumptions of Corollary2.6are veried.

Therefore, it only remains to show that Ran W(U; U0; J; ) = EacUH. For this, we recall from (3.3)

that N(U; J; ) = EU(T n )H. This, together with the J-completeness of the wave operators and [25,

Thm. 2.4], implies that

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In the last proposition of the section, we determine explicitly the kernels of the wave operators W(U; U0; J; ). We use the notation for the characteristic function of a set R and for the

characteristic functions of the sets (0; 1) and ( 1; 0), respectively.

Proposition 3.4. Let := f(U`)[(Ur)gnf(U)[p(U)g. Then, the wave operators W(U; U0; J; ) :

H0! H are partial isometries with initial sets

H

0 := (V`)H (Vr)H: (3.9)

Let us make two observations before giving the proof. Firstly, if a` 6= 0 and ar 6= 0, then [25,

Lemma 4.6] implies that the value 0 is not an atom of the spectral measures of V` and Vr. Therefore, one

has the following orthogonal decomposition of H0:

H0= H H = (V`) + +(V`)H +(Vr) + (Vr)H = H+0 H0:

Secondly, if a`= 0, then [25, Lemma 4.2(a)] implies that V`= 0. Thus (V`) = 0, and (3.9) implies that

W(U; U0; J; ) are isometric only on vectors 0 r with r2 (Vr)H. Such a result is not surprising

since we know from [25, Lemma 4.1(a)] that in this case one has (U`) = p(U`) and

W(U; U0; J; ) = W(U; U0; J; )EU0() = W(U; U0; J; ) 0 EUr():

A similar result holds if ar= 0.

Proof of proposition3.4. We give the proof for W+(U; U0; J; ), treating separately the cases

correspond-ing to the dierent values of a` and ar. The proof for W (U; U0; J; ) is similar.

(i) If a` = ar= 0, then we know from (3.8) that EU0() = 0, and

W+(U; U0; J; ) = W+(U; U0; J; )EU0() = 0:

Thus, the wave operator W+(U; U0; J; ) is isometric only on the subspace f0g f0g. But, we also have

H+

0 = f0g f0g since V` = 0 = Vr. So, W+(U; U0; J; ) is a partial isometry with (trivial) initial set H+0.

(ii) If a`; ar2 (0; 1], we know from (3.8) that EU0() = 1H0. To show that H0 ker W+(U; U0; J; ),

take ( `; r) 2 H0 such that [";1)(V`) ` = ` and ( 1; "](Vr) r= r for some " > 0. Then, one

has W+(U; U0; J; )( `; r) H = s-lim_n!1U n_JUn 0( `; r) H = lim n!1 X ?2f`;rg U n_j ?U?n ? H X ?2f`;rg lim n!1 U nj?U_?n ? _H = X ?2f`;rg lim n!1 U_?nj?U_?n ? _H: Now, if `; r2 C(R; [0; 1]) satisfy `(s) :=(1 if s < 0_{0 if s "} and r(s) :=(0 if s "_{1 if s > 0,}

one obtains for each n 2 N_{the inequality}

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Furthermore, since

`(V`) `= `(V`)[";1)(V`) ` = 0 and r(Vr) r= r(Vr)( 1; "](Vr) r= 0;

one infers from [28, Thm. 4,1] and a standard result on the convergence in the strong resolvent sense [24, Thm. VIII.20(b)] that lim n!1 U_?n?(Q=n)U_?n ? _H= ?(V?) ? _H = 0: Putting together what precedes, one obtains that

W+(U; U0; J; )( `; r) _H X ?2f`;rg lim n!1 U_?nj?U_?n ? _H= 0; meaning that ( `; r) 2 ker W+(U; U0; J; ). Since

( `; r) = [";1)(V`) `; ( 1; "](Vr) r;

a density argument taking into account the fact that the value 0 is not an atom of the spectral measures of V` and Vr then shows that H0 ker W+(U; U0; J; ).

To show that W+(U; U0; J; ) is an isometry on H+0, take ( `; r) 2 H+0 such that ( 1; "](V`) ` =

` and [";1)(Vr) r= rfor some " > 0, and let `; r2 C(R; [0; 1]) satisfy

`(s) :=(0 if s "

1 if s > 0 and r(s) :=

(1 if s < 0 0 if s ". Then, using successively the identity EU0() = 1_H

0, the identity JJ = j` jr of [25, Lemma 4.7], the

denition of the asymptotic velocity V?, and the assumption on the support of ?, one gets

W+(U; U0; J; )( `; r) 2_H ( `; r) 2_H 0 =_n!1lim U nJU₀n( `; r) 2_H ( `; r) 2_H 0 = lim n!1 _Un 0( `; r); (JJ 1)U0n( `; r)_H₀ = lim n!1 Un 0( `; r); (1 j` jr)U0n( `; r)_H₀ X ?2f`;rg lim n!1 ?; U?n?(Q=n)U?n ?_H = X ?2f`;rg ?; ?(V?) ?_H = 0: Thus, W+(U; U0; J; ) is isometric on ( `; r). Since

( `; r) = ( 1; "](V`) `; [";1)(Vr) r;

a density argument taking into account the fact that the value 0 is not an atom of the spectral measures of V` and Vr then shows that W+(U; U0; J; ) is an isometry on whole of H0+.

(iii) If a` = 0 and ar2 (0; 1] or if a` 2 (0; 1] and ar= 0, then the claim can be shown as in point (ii).

We leave the details to the reader.

Remark 3.5. Let = ( `; r) 2 H0. Then, we have

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with

W(U; U?; j?; ) := s-lim_n!1U nj?U?nEU?():

That is, the wave operators W(U; U0; J; ) act as the sum of the operators W(U; U?; j?; ) :

W(U; U0; J; ) = W(U; U`; j`; ) `+ W(U; Ur; jr; ) r:

This simple observation will be used in the following section.

4 Weak limit theorem

We prove in this section a weak limit theorem for quantum walks with an anisotropic coin, and we give an interpretation of this weak limit theorem by comparing it with its classical analogue, the central limit theorem for classical random walks. Our rst proposition gives a description of the asymptotic velocity operator associated to the the full evolution operator U. To state it, we introduce the Heisenberg evolution Q(n) := U n_QUn_{, n 2 Z, of the position operator Q, and the velocity operator V}₀_{:= V}_`_V_r _{for the free}

evolution operator U0= U` Ur.

Proposition 4.1. Let := f(U`)[(Ur)gnf(U)[p(U)g, Vac:= W+(U; U0; J; )V0W+(U; U0; J; ),

and 2 R. Then, one has

s-lim

n!1e

iQ(n)=n_{= E}U

p + eiVacEUac:

Proof. The niteness of (U) and [25, Thm. 2.4] imply that U has no singular continuous spectrum, and [28, Thm. 4.2] implies the equality s-limn!1eiQ(n)=nEpU= EpU. Therefore,

s-lim n!1e iQ(n)=n_{= s-lim} n!1e iQ(n)=n _EU p + EacU= EpU+ s-lim_n!1eiQ(n)=nEacU;

and thus it is sucient to show that s-lim

n!1e

iQ(n)=n_EU

ac= eiVacEacU: (4.1)

Now, we know from Theorem3.3and Proposition3.4that Ran W+(U; U0; J; ) = EacUH and that

W+(U; U0; J; ) is a partial isometry with initial set H+0. Furthermore, we have the inclusion V0H+0 H+0

due to the denition of H+

0 (see (3.9)). So, we obtain for each k 2 N the identity

W+(U; U0; J; )V0W+(U; U0; J; )kEacU = W+(U; U0; J; )V0kW+(U; U0; J; )EacU;

and thus the r.h.s. of (4.1) satises eiVacEU ac= X k0 1 k!W+(U; U0; J; ) iV0 k_W +(U; U0; J; )EacU

= W+(U; U0; J; ) eiV0W+(U; U0; J; )EacU: (4.2)

On another hand, if we set Wn := U nJU0n and Q0(n) := U0n(Q Q)U0n for n 2 Z, then a direct

calculation using the denition of the operator J implies that eiQ(n)=n_{= W}

neiQ0(n)=nWn:

Therefore, we obtain that eiQ(n)=n_EU

ac eiVacEacU = WneiQ0(n)=nWnEacU W+(U; U0; J; ) eiV0W+(U; U0; J; )EacU

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with

I1(n) := WneiQ0(n)=n Wn W+(U; U0; J; )EacU;

I2(n) := Wn eiQ0(n)=n eiV0W+(U; U0; J; )EacU;

I3(n) := Wn W+(U; U0; J; ) eiV0W+(U; U0; J; )EacU:

But, Theorem3.3and the identity EU_{() = E}U

acimply that s-limn!1I1(n) = 0. Theorem3.3, the identity

EU0() = EU0

ac, the inclusion Ran W+(U; U0; J; ) EacU0H, and the fact that eiV0EacU0 = EacU0eiV0

(which follows from (3.8)) imply that s-limn!1I3(n) = 0. Finally, [28, Thm. 4,1] and [24, Thm. VIII.20(b)]

imply that s-limn!1I2(n) = 0. Therefore, we obtain that

s-lim

n!1 e

iQ(n)=n_EU

ac eiVacEacU= 0;

which proves (4.1).

Remark 4.2. If we dene the asymptotic velocity operator V 2 B(H) for the full evolution operator U as

V :=(0 if 2 EpUH

Vac if 2 EacUH;

then the result of Proposition4.1can be rephrased in the more compact form s-lim

n!1e

iQ(n)=n_{= e}iV_{; 2 R:}

We are now in a position to state the weak limit theorem. For that purpose, we denote by Xn the

random variable for the position of a quantum walker with evolution U and initial normalised state in2 H

at time n 2 Z. The probability distribution of Xn is given by

P(Xn= x) := Un in(x) 2_C2; x 2 Z;

and the characteristic function of the average velocity Xn=n of the quantum walker is given by

E(eiXn=n_{) := U}n

in; eiQ=nUn in_H; = in; eiQ(n)=n in_H; 2 R;

We also use the notation 0 for the Dirac measure on R for the point 0 :

Theorem 4.3 (Weak limit theorem). Let in2 H with k inkH = 1, let := f(U`) [ (Ur)g n f(U) [

p(U)g, and let V be the random variable with probability distribution

V:= EpU in 2_H0+ EV`()W+(U; U`; j`; ) in

2_H+ EVr()W+(U; Ur; jr; ) in

2_H (4.3) with the operators W(U; U?; j?; ) dened in Remark3.5. Then, the average velocity Xn=n converges in

law to V as n ! 1, namely,

lim

n!1E(e

iXn=n) = E(eiV); 2 R: (4.4)

Since the average velocity Xn=n of the quantum walker converges in law to V as n ! 1, V can

be interpreted as the asymptotic velocity of the quantum walker, with V its probability distribution.

Therefore, Theorem4.3implies that the probability that the quantum walker has velocity 0 is P(V = 0) = V(f0g) EpU in 2_H:

Accordingly, we say that localisation occurs if P(V = 0) > 0, and (4.3) tells us that this happens if the initial state in has an overlap with the pure point subspace of U. Later in this section, we will see that

P(V = 0) = EU p in

2 H

and that localisation occurs in fact if and only if the initial state in has an overlap with the pure point

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Proof. Using Proposition4.1, Equation (4.2), and the identity W+(U; U0; J; )EacU = W+(U; U0; J; ), one obtains lim n!1E(e iXn=n) = lim n!1 in; eiQ(n)=n in_H = in; EpU+ eiVacEacU in_H = EU p in 2

H+ in; W+(U; U0; J; ) eiV0W+(U; U0; J; )EacU in_H

= EU p in

2

H+ W+(U; U0; J; )EacU in; eiV0W+(U; U0; J; )EUac in_H₀

=Z Re i E_pU in 2_H0(d) + EV0(d)W+(U; U0; J; )E_acU in 2_H 0 :

Therefore, to prove the claim, it only remains to observe that EV0()W+(U; U0; J; )E_acU in 2_H 0 = EV`_{() E}Vr_()W +(U; U0; J; ) in 2 H0 = EV`()W

+(U; U`; j`; ) in 2_H+ EVr()W+(U; Ur; jr; ) in 2_H:

In order to give a better description of the probability distribution V, we recall from Section3that

we have for k 2 [0; 2) c U?(k) = 2 X j=1 ?;j(k)?;j(k); Vb?(k) = 2 X j=1 v?;j(k)?;j(k) and v?;j(k) = i 0?;j(k) ?;j(k) 1:

We also recall from [25, Lemma 4.2] the following properties of the functions v?;j given in terms of the

parameters a?; ?; ? of (3.7):

(i) If a?= 0, then v?;j= 0 for j 2 f1; 2g.

(ii) If a?2 (0; 1), then v?;j(k) = ( 1)_?j_(k)&?(k) for j 2 f1; 2g, k 2 [0; 2) and

?(k) := a?cos(k + ? ?=2); ?(k) :=p1 ?(k)2; &?(k) := a?sin(k + ? ?=2):

(iii) If a?= 1, then v?;j(k) = ( 1)j for j 2 f1; 2g and k 2 [0; 2).

With this done, we can start our study of the probability distribution Vby collecting some information

on the operators W+(U; U?; j?; ) appearing in (4.3):

Lemma 4.4. Let := f(U`) [ (Ur)g n f(U) [ p(U)g.

(a) If a?= 0, then W+(U; U?; j?; ) = 0.

(b) If a` 2 (0; 1), then W+(U; U`; j`; ) is a partial isometry with initial subspace (V`)H, and if

ar2 (0; 1), then W+(U; Ur; jr; ) is a partial isometry with initial subspace +(Vr)H.

(c) If a`= 1, then W+(U; U`; j`; ) is a partial isometry with initial subspace `2 Z; C0, and if ar= 1,

then W+(U; Ur; jr; ) is a partial isometry with initial subspace `2 Z; C0.

Proof. The claim (a) is a direct consequence of Remark 3.5 and Equation (3.8). The claim (b) is a direct consequence of Remark3.5, Equation (3.8) and Proposition3.4. For the claim (c), we recall from Proposition 3.4and the point (iii) above that in the case a` = 1 the initial subspace of W+(U; U`; j`; )

coincides with the eigenspace of V` for the eigenvalue 1 (see the case j = 1 in the point (iii) above).

Then, the formula for u?;1(k) in [25, Sec. 4.1] directly implies that this eigenspace is equal to the space

`2 _Z; C

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Now, to pursue our study of the probability distribution V, we recall the denition of the Konno function fK: R (0; 1] ! [0; 1); (; r) 7! ( p_{1 r}2 (1 2₎p_r2 2 if jj < r, 0 otherwise. With this denition at hand, we can establish the following:

Proposition 4.5. Let in 2 H with k inkH= 1, and let := f(U`) [ (Ur)g n f(U) [ p(U)g.

(a) If a?= 0, then EV?()W+(U; U?; j?; ) in 2

H= 0. (b) If a?2 (0; 1) and 2 R, then d EV? ( 1; ]W₊(U; U_?; j_?; ) _in 2 H d = (₁ 2[ a`;0)()!`()fK(; a`) if ? = ` 1 2(0;ar]()!r()fK(; ar) if ? = r

for some nonnegative functions !` 2 L1 [ a`; 0);1₂fK(; a`)d and !r2 L1 (0; ar];1₂fK(; ar)d.

(c) If a?= 1, then EV?()W+(U; U?; j?; ) in 2_H = ( W+(U; U`; j`; ) in 2 H 1 if ? = ` W+(U; Ur; jr; ) in 2_H1 if ? = r with 1 the Dirac measures on R for the points 1.

Proof of (a) and (c). In the case a` = 0, the claim (a) follows directly from Lemma 4.4(a). In the

case of a` = 1, we know from Lemma 4.4(c) that W+(U; U`; j`; ) has initial subspace `2 Z; C0,

so that W+(U; U`; j`; ) in 2 `2 Z; C0. On the other hand, we have shown in [25, Sec. 4.1] that

(V`) = p(V`) = f 1; 1g, with `2 Z; C0

the eigenspace associated to the eigenvalue 1. Thus, we obtain for any Borel set B R that

EV`(B)W+(U; U`; j`; ) in 2_H= ( W+(U; U`; j`; ) in 2_H if 1 2 B, 0 otherwise. Since a similar result holds for the operator W+(U; Ur; jr; ), the eigenspace `2 Z; C0

and the eigenvalue +1, we infer the result of the claim (c).

In order to prove the claim (b) of Proposition4.5, we need two preparatory lemmas. The rst one is the following:

Lemma 4.6. Dene for ? 2 f`; rg, j 2 f1; 2g, m 2 f0; 1g and a?2 (0; 1) the function

k?;j;m : [ a?; a?] ! Im; 7! ₂? ?+ m + arcsin ( 1)j+mb ? a?p1 2 ; where I0:= =2 ?+ ?=2; =2 ?+ ?=2 and I1:= I0+ :

Then, the function k?;j;m is dierentiable on ( a?; a?) with

k0

?;j;m() = ( 1)j+m fK(; a?); 2 ( a?; a?); (4.5)

and k?;j;m is a bijection with inverse v?;jjIm, that is,

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Proof. Set f?() := _a_?pb₁? 2 for 2 [ a?; a?]. Since f?0() = _a_?₍₁b?2₎3=2 > 0 on ( a?; a?), the function

f?is strictly increasing on ( a?; a?) with f?([ a?; a?]) = [ 1; 1]. It follows that k?;j;m is dierentiable on

( a?; a?) with derivative k0 ?;j;m() = d arcsin ( 1) j+m_f ? d () = ( 1)j+m_f0 ?() p 1 f?()2 = ( 1)j+mp 1 a2 ? p a2 ? 2(1 2) : This implies (4.5).

Now, the fact that k0

?;j;m 6= 0 on ( a?; a?) implies that k?;j;m is invertible. Since &? k?;j;m() =

( 1)j_a

?f?() and ? k?;j;m() = pb2?+ a?2f?2() for 2 [ a?; a?], one also obtains that

v?;j k?;j;m() = ( 1) j_& ? k?;j;m() ? k?;j;m() = a?f?() p b2 ?+ a2?f?()2 = :

On the other hand, since f? v?;j(k) = ( 1)jsin(k + ? ?=2) for k 2 [0; 2), one obtains that

k?;j;m v?;j(k) = ₂? ?+ m + arcsin ( 1)msin(k + ? ?=2):

Therefore, if m = 0 and k 2 I0, then k + ? ?=2 2 [ =2; =2] and thus k?;j;m(v?;j(k)) = k. And if

m = 1 and k 2 I1, then k +? ?=2 2 [=2; 3=2] and ( 1)msin(k +? ?=2) = sin(k +? ?=2 ).

Thus k + ? ?=2 2 [ =2; =2], and one obtains once again k?;j;m v?;j(k) = k, which concludes

the proof.

To state our second preparatory lemma, we need to introduce some notations. For k 2 [0; 2), we denote by u?;j(k) 2 C2a normalised eigenvector of the operator cU?(k) for the eigenvalue ?;j(k). In [25,

Sec. 4], we have shown that u?;j(k) can be chosen C1 in the variable k. For ? 2 f`; rg, we set

G?:= L2 [ a?; a?];12fK(; a?)d:

Finally, for j 2 f1; 2g and m 2 f0; 1g, we dene the operator K?;j;m: H ! G?by

K?;j;m () := u?;j k?;j;m(); (F ) k?;j;m()_C2; 2 H; 2 [ a?; a?]:

Lemma 4.7. The operator K?;j;m is bounded and satises the following:

(a) For g 2 G?, one has (K?;j;m)g = FImg v?;j()u?;j

. (b) For j; j0_{2 f1; 2g and m; m}0_{2 f0; 1g, one has K}

?;j;m(K?;j0_;m0)= _j;j0_m;m0id_G_?.

(c) P_{j2f1;2g;m2f0;1g}(K?;j;m)K?;j;m = idH.

(d) For any Borel function F : [ a?; a?] ! C, the multiplication operator by F in G? (denoted by the

same symbol) satises the equation X

j2f1;2g;m2f0;1g

(K?;j;m)F K?;j;m = F (V?):

Proof. Using the change of variables k := k?;j;m(), Equation (4.5), and the normalisation of u?;j(k), we

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which proves that K?;j;mis bounded. Furthermore, using the change of variables k := k?;j;m() and (4.6),

we obtain for any g 2 G?

g; K?;j;m _G_? =12 Z a? a? g()u?;j k?;j;m(); (F ) k?;j;m()_C2fK(; a?)d = 1 2 Z Im g v?;j(k)u?;j(k); (F )(k)_C2dk = F Img v?;j()u?;j ; _H;

which proves (a). To prove (b), we observe that points (a) and (4.6) imply for any g 2 G?and 2 [ a?; a?]

K?;j;m(K?;j0_;m0)g() = u_?;j k_?;j;m(); F (K_?;j0_;m0)g k_?;j;m()

C2

= Im0 (k?;j;m()g v?;j0 k?;j;m()u?;j k?;j;m(); u?;j0 k?;j;m()C2

= j;j0_m;m0g():

On the other hand, for any 2 H and k 2 [0; 2) we have

F (K?;j;m)K?;j;m (k) = Im(k) K?;j;m v?;j(k)u?;j(k) = Im(k)u?;j k?;j;m v?;j(k); (F ) k?;j;m v?;j(k)_C2u?;j(k) = Im(k)u?;j(k); (F )(k)_C2u?;j(k): Hence, we obtain (K?;j;m)K?;j;m = FIm u?;j(); (F )()_C2u?;j = F Im?;j()F ; and thus X j2f1;2g;m2f0;1g (K?;j;m)K?;j;m = X j2f1;2g;m2f0;1g F Im?;j()F = ;

which shows (c). Finally, for any 2 H and k 2 [0; 2) we have

F (K?;j;m)F K?;j;m (k) = Im(k) F K?;j;m v?;j(k)u?;j(k)

= Im(k)F v?;j(k)?;j(k)F

which implies that X j2f1;2g;m2f0;1g (K?;j;m)F K?;j;m = X j2f1;2g F_{F v} ?;j()?;j()F = FF ( bV?)F = F (V?); as stated in (d).

We can now provide the proof of the claim (b) of Proposition4.5:

Proof of Proposition4.5(b). Take a Borel set B R. Then, Lemma 4.4(b), Lemma 4.7(d) and [25, Lemma 4.2(b)] imply that

EV`_(B)W

+(U; U`; j`; ) in= B(V`) (V`)W+(U; U`; j`; ) in

= X

j2f1;2g;m2f0;1g

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Thus, it follows by Lemma4.7(b) that EV`(B)W+(U; U`; j`; ) in 2_H = X j2f1;2g;m2f0;1g B\[ a_`;0)K`;j;mW+(U; U`; j`; ) in 2_G ` =1 2 X j2f1;2g;m2f0;1g Z B\[ a`;0) K`;j;mW+(U; U`; j`; ) in ()2fK(; a`)d;

which means that the density function of EV`(B)W₊(U; U_`; j_`; ) _in 2 H is given by d EV` ( 1; ]W₊(U; U_`; j_`; ) _in 2 H d =1 2 X j2f1;2g;m2f0;1g [ a`;0)() K`;j;mW+(U; U`; j`; ) in ₍₎2_f K(; a`):

Since a similar argument leads to

d EVr ( 1; ]W₊(U; U_r; j_r; ) _in 2 H d = 1 2 X j2f1;2g;m2f0;1g (0;ar]() Kr;j;mW+(U; Ur; jr; ) in ()2fK(; ar);

the claim follows by setting !?() :=

X

j2f1;2g;m2f0;1g

(K?;j;mW+(U; U?; j?; ) in)()2: (4.7)

In our nal theorem, we gather the results obtained so far on the probability distribution V and we

prove a uniqueness result.

Theorem 4.8. Let in2 H with k inkH= 1, and let V be the random variable dened by (4.4). Then,

V has probability distribution

V(d) = 00(d) + ` 1(d) + r1(d) + [ a`;0)()w`()12fK(; a`)d + (0;ar]()wr()12fK(; ar)d; 2 R; (4.8) with 0; ? 0 given by 0:= EpU in 2 H and ?:= ( W+(U; U?; j?; ) in 2 H if a?= 1 0 otherwise; and with w`2 L1 [ a`; 0);1₂fK(; a`)d and wr2 L1 (0; ar];1₂fK(; ar)d given by

w?() := (_P j2f1;2g;m2f0;1g K?;j;mW+(U; U?; j?; ) in ()2 if a?2 (0; 1) 0 otherwise:

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Proof. In view of Theorem4.3and Proposition4.5, only the uniqueness of the decomposition of V has

to be established. For that purpose, we observe that V is the sum of a pure point measure located at the

three distinct points 1; 0; 1 and an absolutely continuous measure. As a consequence, the coecients `; 0; r are unique. In addition, since the absolutely continuous measure is the sum of two absolutely

continuous measures with disjoint supports [ a`; 0) and (0; ar], each of these measures is unique in the

L1_{-sense given by their density functions in L}1 _{[ a}

`; 0);1₂fK(; a`)d and L1 (0; ar];1₂fK(; ar)d.

Remark 4.9. In the case of the one-dimensional random walk where the walker moves to the left with probability p and to the right with probability q, the classical central limit theorem implies that the random variable Xn n(q p)

2pnpq converges in law as n ! 1 to a random variable Z with standard normal

distribution N(0; 1). Therefore, the weak limit theorem4.3can be interpreted as a quantum analogue of the classical central limit theorem. On the other hand, the classical central limit theorem implies that the average velocity Xn=n always converges in law as n ! 1 to a Gaussian random variable with distribution

N(q p; 4pq=n), whereas the weak limit theorem leads to a variety of limit distributions V depending on

the outgoing states W+(U; U?; j?; ) in.

To conclude, we discuss in more detail some particular cases of Theorem4.8: Example 4.10. (a) In the case a` = 0 and ar2 (0; 1), the formula (4.8) reduces to

V(d) = 00(d) +1₂(0;ar]()wr()fK(; ar)d:

Thus, the density function of the absolutely continuous part of V is supported in (0; ar], and the quantum

walker can asymptotically move only to the right at a speed belonging to the interval (0; ar].

(b) In the case a` = 1 and ar2 (0; 1), the formula (4.8) reduces to

V(d) = 00(d) + ` 1(d) +₂1(0;ar]()wr()fK(; ar)d:

Thus, the quantum walker can asymptotically move to the left at speed 1 or to the right at a speed belonging to the interval (0; ar].

(c) In the case a`; ar2 (0; 1), the formula (4.8) reduces to

V(d) = 00(d) +₂1[ a`;0)()w`()fK(; a`)d +12(0;ar]()wr()fK(; ar)d: (4.9)

Thus, the quantum walker can asymptotically move to the left at a speed belonging to the interval (0; a`]

or to the right at a speed belonging to the interval (0; ar]. In particular, if a` > ar, then the quantum

walker can move faster on the left-hand side than on the right-hand side.

Finally, we note that the formula (4.9) covers various previously known results: Example 4.11. (a) In the case a` = ar= 1=p2, the formula (4.9) reduces to

V(d) = 00(d) +1₂ [ 1=p2;0)()w`() + (0;1=p2]()wr()fK(; 1=p2)d: (4.10)

This generalises the formula obtained by Endo et al. in the case of a two-phase quantum walk with one defect [9, Thm. 2.1]. Indeed, in their work Endo et al. consider a coin operator given by

C(x) =          C+=p1₂ _e1i+ ei+₁ if x 1 C0= 1 00 1 if x = 0 C = _p1 2 1 e i e i ₁ if x 1;

for some 2 [0; 2), and they obtain the formula

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where c := P_x2Z limN!1N1PN 1n=0 P(Xn = x) and w : R ! R is some particular function. However,

by applying a discrete analogue of the RAGE theorem, or Wiener's theorem, or in a way similar to [27, Appendix], one can prove that c = 0 = EpU in 2_H, thus showing that (4.10) is a generalisation of (4.11). Moreover, the uniqueness of the decomposition (4.9) leads to the explicit formula for w:

w() = 1

2 [ 1=p2;0)()w`() + (0;1=p2]()wr();

Endo et al. obtained an explicit expression for w only in the case the initial state satises in(x) = 0

everywhere except for x = 0.

(b) The isotropic case where C(x) = C1+ O(jxj 1 "), C1 2 U(2), and a1 := j(C1)1;1j 2 (0; 1),

was studied in [28] by the second author. In our setup, this corresponds to setting C` = Cr = C1 and

"` = "r= " in Assumption 3.1and having a` = ar= a12 (0; 1) in (3.7). In [28], it was shown that

V(d) = 00(d) + w()fK(; a1)d;

where w 2 L1 _{[ a}₁_{; a}₁_];1

2fK(; a1)d is a function similar to (4.7) but dened in terms of the wave

operator W+(U; U1; 1H; ), with the identity 1H instead of the identication operator J and the evolution

operator U1 := SC1 in H instead of the evolution operator U0 in H0. Thus, by using once again the

uniqueness of the decomposition of (4.9), we obtain the explicit formula for w: w() = 1

2[ a1;0)()w`() + 12(0;a1]()wr():

In the one-defect case, where C(x) = C1everywhere except for x = 0, Konno et al. obtained in [19] an

explicit expression of w only in the case the initial state satises in(x) = 0 everywhere except for x = 0.

(c) The homogeneous case, where C(x) = C1and a12 (0; 1), has been studied by several authors

[12,17,18]. It is a particular case of the isotropic case (b) above. Because U = SC1 has no eigenvalue

whenever a12 (0; 1), (4.9) reduces to

V(d) = w()fK(; a1)d

with

w() = 1

2[ a1;0)()w`() + 1₂(0;a1]()wr():

As noted in Remark4.9, the outgoing states determines w?uniquely. However, in the homogeneous case,

the initial state in and the coin matrix C1 determine the function w 2 L1 [ a1; a1];12fK( ; a1) d

uniquely. For a special initial state in and coin matrix C1, w can be computed explicitly. We refer to

the works [21,22] of Machida for more information on the relation between the limit distribution and the initial state.

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