Quantum walks with an anisotropic coin I: spectral theory › ~richard › papers › unitaries_spectral.pdf · Quantum walks with an anisotropic coin are also related to Kitagawa’s

Lett Math Phys (2018) 108:331–357https://doi.org/10.1007/s11005-017-1008-1

Quantum walks with an anisotropic coin I:spectral theory

S. Richard1 · A. Suzuki2 · R. Tiedra de Aldecoa3

Received: 29 March 2017 / Revised: 24 July 2017 / Accepted: 21 September 2017 /Published online: 27 September 2017© Springer Science+Business Media B.V. 2017

Abstract We perform the spectral analysis of the evolution operator U of quantumwalks with an anisotropic coin, which include one-defect models, two-phase quan-tum walks, and topological phase quantum walks as special cases. In particular, wedetermine the essential spectrum of U , we show the existence of locally U -smoothoperators, we prove the discreteness of the eigenvalues ofU outside the thresholds, andwe prove the absence of singular continuous spectrum forU . Our analysis is based onnew commutator methods for unitary operators in a two-Hilbert spaces setting, whichare of independent interest.

S. Richard was supported by JSPS Grant-in-Aid for Young Scientists A no 26707005, and on leave ofabsence 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.A. Suzuki was supported by JSPS Grant-in-Aid for Young Scientists B no 26800054.R. Tiedra de Aldecoa was supported by the Chilean Fondecyt Grant 1170008.

B R. Tiedra de [email protected]

S. [email protected]

A. [email protected]

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, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860,Santiago, Chile

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http://crossmark.crossref.org/dialog/?doi=10.1007/s11005-017-1008-1&domain=pdf

332 S. Richard et al.

Keywords Quantum walks · Spectral theory · Commutator methods · Unitaryoperators

Mathematics Subject Classification 81Q10 · 47A10 · 47B47 · 46N50

1 Introduction

Discrete-time quantumwalks appear in numerous contexts [1,2,20,21,33,46]. Amongthem, Gudder [21], Meyer [33], and Ambainis et al. [2] introduced one-dimensionalquantum walks as a quantum mechanical counterpart of classical random walks.Nowadays, these quantumwalks and their generalisations have been physically imple-mented in various ways [31]. Versatile applications of quantum walks can be found in[12,22,35,45] and references therein.

Recently, because of the controllability of their parameters, discrete-time quantumwalks have attracted attention as promising candidates to realise topological insula-tors. In [25,26], Kitagawa et al. have shown that one- and two-dimensional quantumwalks possess topological phases, and they experimentally observed a topologicallyprotected bound state between two distinct phases. We refer for example to [24] for anintroductory review on topological phenomena in quantumwalks, see also [11,19,23].Motivated by these studies, Endo et al. [15] (see also [13,14]) have performed athorough analysis of the asymptotic behaviour of two-phase quantum walks, whoseevolution is given by unitary operatorsUTP = SC with S a shift operator and C a coinoperator defined as a multiplication by unitary matrices C(x) ∈ U(2), x ∈ Z. WhenC(x) is given by

C(x) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1√2

(1 eiσ+

e−iσ+ −1

)

if x ≥ 0

1√2

(1 eiσ−

e−iσ− −1

)

if x ≤ −1

(1.1)

with σ± ∈ [0, 2π), the two-phase quantum walk with evolution operatorUTP is calledcomplete two-phase quantum walk, and when C(x) satisfies the alternative conditionat 0

C(0) =(1 00 −1

)

, (1.2)

the quantum walk is called two-phase quantum walk with one defect. In [14,15],Endo et al. have proved a weak limit theorem [27,28] similar to the de Moivre–Laplace theorem (or the central limit theorem) for random walks, which describes theasymptotic behaviour of the two-phase quantum walk.

In the present paper and the companion paper [37], we consider one-dimensionalquantum walks U = SC with a coin operator C exhibiting an anisotropic behaviourat infinity, with short-range convergence to the asymptotics. Namely, we assume thatthere exist matrices C�,Cr ∈ U(2) and constants ε�, εr > 0 such that

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Quantum walks with an anisotropic coin I: spectral theory 333

C(x) ={C� + O

(|x |−1−ε�)

as x → −∞Cr + O

(|x |−1−εr)

as x → ∞.(1.3)

We call this type of quantum walks quantum walks with an anisotropic coin or simplyanisotropic quantumwalks. They include two-phase quantumwalkswith coins definedby (1.1) and (1.2) and one-defect models [10,29,30,48] as special cases. In the caseC0 := C� = Cr and ε0 := ε� = εr, quantum walks with an anisotropic coin reduce toone-dimensional quantum walks with a position dependent coin

C(x) = C0 + O(|x |−1−ε0

), |x | → ∞,

for which the absence of the singular continuous spectrum was proved in [4] and forwhich a weak limit theorem was derived in [43].

Quantum walks with an anisotropic coin are also related to Kitagawa’s topologicalquantum walk model called split-step quantum walk [24–26]. Indeed, if R(θ) ∈ U(2)is a rotation matrix with rotation angle θ/2, R(� j ) the multiplication operator byR(θ j ( · )

) ∈ U(2) with θ j : Z → [0, 2π), j = 1, 2, and T↓, T↑ shift operatorssatisfying S = T↓T↑ = T↑T↓, then the evolution operator of the split-step quantumwalk is defined as

USS(θ1, θ2) := T↓ R(�2)T↑ R(�1).

Now, as mentioned in [24], USS(θ1, θ2) is unitarily equivalent to T↑R(�1)T↓R(�2).Thus, our evolution operator U describes a quantum walk unitarily equivalent to theone described by USS(θ1, θ2) if θ1 ≡ 0 and C( · ) = R

(θ2( · )) (see [34,42] for the

definition of unitary equivalence between two quantumwalks). In [24], Kitagawa dealtwith the case

θ2(x) := 12 (θ2− + θ2+) + 1

2 (θ2+ − θ2−) tanh(x/3), θ2−, θ2+ ∈ [0, 2π), x ∈ Z,

which corresponds to taking the anisotropic coin (1.3) with C� = R(θ2−) and Cr =R(θ2+), and which cannot be covered by two-phase models.

The main goal of the present paper and [37] is to establish a weak limit theoremfor the evolution operator U of the quantum walk with an anisotropic coin satisfying(1.3). As put into evidence in [43], in order to establish a weak limit theorem one hasto prove along the way the following two important results: (i) absence of singularcontinuous spectrum and (ii) existence of the asymptotic velocity.

In the present paper, we perform the spectral analysis of the evolution operator Uof quantum walks with an anisotropic coin. We determine the essential spectrum ofU , we show the existence of locally U -smooth operators, we prove the discretenessof the eigenvalues of U outside the thresholds, and we prove the absence of singularcontinuous spectrum forU . In the companion paper [37],wewill develop the scatteringtheory for the evolution operatorU . We will prove the existence and the completenessof wave operators for U and a free evolution operator U0, we will show the existenceof the asymptotic velocity forU , and we will finally establish a weak limit theorem for

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U . Other interesting related topics such as the existence and the robustness of a boundstate localised around the phase boundary or a weak limit theorem for the split-stepquantum walk with θ1 �= 0 are considered in [18] and [17], respectively.

The rest of this paper is structured as follows. In Sect. 2, we give the precise defi-nition of the evolution operator U for the quantum walk with an anisotropic coin andwe state our main results on the essential spectrum of U (Theorem 2.2), the locallyU -smooth operators (Theorem 2.3), and the eigenvalues and singular continuous spec-trum of U (Theorem 2.4). Section 3 is devoted to mathematical preliminaries. Here,we develop new commutator methods for unitary operators in a two-Hilbert spacessetting, which are a key ingredient for our analysis and are of independent interest.In Sect. 4, we prove our main theorems as an application of the commutator methodsdeveloped in Sect. 3. In Sect. 4.2, we prove Theorem 2.2 and we define in Lemma 4.9a conjugate operator A for the evolution operatorU built from conjugate operators forthe asymptotic evolution operators U� := SC� and Ur := SCr, where C� and Cr arethe constant coin matrices given in (1.3). Finally, in Sect. 4.3 we prove Theorems 2.3and 2.4.

2 Model and main results

In this section, we give the definition of the model of anisotropic quantum walks thatwe consider, we state ourmain results on quantumwalks, andwe present themain toolswe use for the proofs. These tools are results of independent interest on commutatormethods for unitary operators in a two-Hilbert spaces setting. The proofs of our resultson commutator methods are given in Sect. 3, and the proofs of our results on quantumwalks are given in Sect. 4.

Let H be the Hilbert space of square-summable C2-valued sequences

H := �2(Z, C2) = {

� : Z → C2 | ∑x∈Z ‖�(x)‖22 < ∞}

,

where ‖ · ‖2 is the usual norm on C2. The evolution operator of the one-dimensional

quantum walk inH that we consider is given byU := SC , with S a shift operator andC a coin operator defined by

(S�)(x) :=(

�(0)(x + 1)�(1)(x − 1)

)

, � =(

�(0)

�(1)

)

∈ H, x ∈ Z,

(C�)(x) := C(x)�(x), � ∈ H, x ∈ Z, C(x) ∈ U(2).

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

Throughout the paper, we assume that the coin operator C exhibits an anisotropicbehaviour at infinity. More precisely, we assume that C converges with short-rangerate to two asymptotic coin operators, one on the left and one on the right in thefollowing way:

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Assumption 2.1 (Short-range) There exist C�,Cr ∈ U(2), κ�, κr > 0, and ε�, εr > 0such that

∥∥C(x) − C�

∥∥B(C2)

≤ κ� |x |−1−ε� if x < 0∥∥C(x) − Cr

∥∥B(C2)

≤ κr |x |−1−εr if x > 0,

where the indexes � and r stand for “left” and “right”.

This assumption provides us two new unitary operators

U� := SC� and Ur := SCr (2.1)

describing the asymptotic behaviour of U on the left and on the right. The precisesense (from the scattering point of view) in which the operators U�,Ur describe theasymptotic behaviour of U on the left and on the right will be given in [37], and thespectral properties of U�,Ur are determined in Sect. 4.1. Here, we just introduce theset

τ(U ) := ∂σ(U�) ∪ ∂σ(Ur),

where ∂σ(U�), ∂σ (Ur) denote the boundaries in the unit circleT := {z ∈ C | |z| = 1}of the spectra σ(U�), σ (Ur) of U�,Ur. In Sect. 4.1, we show that τ(U ) is finite andcan be interpreted as the set of thresholds in the spectrum of U .

Ourmain results onU , proved in Sects. 4.2 and 4.3, are the following three theoremson locallyU -smooth operators and on the structure of the spectrum ofU . The symbolsσess(U ), σp(U ) and Q stand for the essential spectrum of U , the pure point spectrumof U , and the position operator in H, respectively (see (4.9) for precise definition ofQ).

Theorem 2.2 (Essential spectrum of U ) One has σess(U ) = σ(U�) ∪ σ(Ur).

Theorem 2.3 (U -smooth operators) Let G be an auxiliary Hilbert space, and let� ⊂T be an open set with closure� ⊂ T\τ(U ). Then, each operator T ∈ B(H,G)whichextends continuously to an element of B

(D(〈Q〉−s),G)for some s > 1/2 is locally

U-smooth on �\σp(U ).

Theorem 2.4 (Spectrum of U ) For any closed set � ⊂ T\τ(U ), the operator U hasat most finitely many eigenvalues in �, each one of finite multiplicity, and U has nosingular continuous spectrum in �.

The content of Theorem 2.2 could be inferred from [9, Thm. 3.1], but we provide analternative proof. To prove these theorems, we develop in Sect. 3 commutator meth-ods for unitary operators in a two-Hilbert spaces setting: Given a triple (H,U, A)

consisting in a Hilbert space H, a unitary operator U , and a self-adjoint operator A,we determine how to obtain commutator results for (H,U, A) in terms of commu-tator results for a second triple (H0,U0, A0) also consisting in a Hilbert space, aunitary operator, and a self-adjoint operator. In the process, an identification operator

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J : H0 → H must also be chosen. The intuition behind this approach comes fromscattering theory which tells us that given a unitary operatorU describing some quan-tum system in a Hilbert space H there often exists a simpler unitary operator U0 ina second Hilbert space H0 describing the same quantum system in some asymptoticregime.

Our main results in this context are the following. First, we present in Theorem3.6 conditions guaranteeing that U and A satisfy a Mourre estimate on a Borel set� ⊂ T as soon asU0 and A0 satisfy a Mourre estimate on� (equivalently, we presentconditions guaranteeing that A is a conjugate operator for U on � as soon as A0 isa conjugate operator for U0 on �). Next, we present in Proposition 3.7 conditionsguaranteeing thatU is regular with respect to A (that is,U ∈ C1(A)) as soon asU0 isregular with respect to A0 (that is,U0 ∈ C1(A0)). Finally, we give in Assumption 3.9and Corollaries 3.10–3.11 conditions guaranteeing that the most natural choice for theoperator A, namely A = J A0 J ∗, is indeed a conjugate operator for U as soon as A0is a conjugate operator for U0.

3 Unitary operators in a two-Hilbert spaces setting

In this section, we start by recalling some facts on the spectral family of unitary oper-ators, on locally smooth operators for unitary operators, and on commutator methodsfor unitary operators in one Hilbert space. In particular, we introduce in (3.2)–(3.3)the functions � and � which will play an essential role in the two-Hilbert space settingand which have never been used before for unitary operators. Then, we develop theabstract theory of commutator methods for unitary operators in a two-Hilbert spacessetting. Note that the theory in one Hilbert space has also been introduced in [5,6],but without the �-functions mentioned above.

3.1 Commutator methods in one Hilbert space

Let H be a Hilbert space with norm ‖ · ‖H and scalar product 〈 · , · 〉H linear in thesecond argument,B(H) the set of bounded linear operators inHwith norm ‖·‖B(H),and K (H) the set of compact linear operators in H. A unitary operator U in H isan element U ∈ B(H) satisfying U∗U = UU∗ = 1. Since U∗U = UU∗, thespectral theorem for normal operators implies that U admits exactly one complexspectral family EU , with support supp(EU ) ⊂ T, such that U = ∫

Cz EU (dz). The

support supp(EU ) is the set of points of non-constancy of EU , which coincides withthe spectrum σ(U ) of U [47, Thm. 7.34(a)]. In addition, the measure EU admits adecomposition into a pure point, a singular continuous and an absolutely continuouscomponents, and the corresponding orthogonal decomposition

H = Hp(U ) ⊕ Hsc(U ) ⊕ Hac(U )

reduces the operatorU . The sets σp(U ) := σ(U |Hp(U )

), σsc(U ) := σ

(U |Hsc(U )

), and

σac(U ) := σ(U |Hac(U )

)are called pure point spectrum, singular continuous spectrum,

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and absolutely continuous spectrum ofU , respectively, and the set σc(U ) := σsc(U )∪σac(U ) is called the continuous spectrum of U . Finally, if G is an auxiliary Hilbertspace, then an operator T ∈ B(H,G) is locally U -smooth on an open set � ⊂ T iffor each closed set �′ ⊂ � there exists c�′ ≥ 0 such that

∑

n∈Z

∥∥T UnEU (�′)ϕ

∥∥2G ≤ c�′ ‖ϕ‖2H for each ϕ ∈ H, (3.1)

and T is (globally) U -smooth if (3.1) is satisfied with �′ = T. The condition (3.1) isinvariant under rotation by ω ∈ T in the sense that if T is U -smooth on �, then T is(ωU )-smooth on ω� since

∥∥T (ωU )n EωU (ω�′)ϕ

∥∥G = ∥

∥T UnEU (�′)ϕ∥∥G

for each closed set�′ ⊂ � and eachϕ ∈ H.An important consequenceof the existenceof a locallyU -smooth operator T on � is the inclusion EU (�)T ∗G∗ ⊂ Hac(U ), withG∗ the adjoint space of G (see [7, Thm. 2.1] for a proof).

Now, we present some results on commutator methods for unitary operators inone Hilbert space, starting with definitions and results borrowed from [3,16,41]. LetS ∈ B(H) and let A be a self-adjoint operator in H with domain D(A). For k ∈ N,we say that S belongs to Ck(A), with notation S ∈ Ck(A), if the map R � t �→e−i t A S ei t A ∈ B(H) is strongly of class Ck . In the case k = 1, one has S ∈ C1(A) ifand only if the quadratic form

D(A) � ϕ �→ ⟨Aϕ, Sϕ

⟩

H − ⟨ϕ, SAϕ

⟩

H ∈ C

is continuous for the topology induced byH on D(A). The operator associated to thecontinuous extension of the form is denoted by [A, S] ∈ B(H), and it verifies

[A, S] = s-limτ→0

[Aτ , S] with Aτ := (iτ)−1( eiτ A −1) ∈ B(H), τ ∈ R\{0}.

Three regularity conditions slightly stronger than S ∈ C1(A) are defined as follows:S belongs to C1,1(A), with notation S ∈ C1,1(A), if

∫ 1

0

∥∥ e−i t A S ei t A + ei t A S e−i t A −2S

∥∥B(H)

dt

t2< ∞.

S belongs to C1+0(A), with notation S ∈ C1+0(A), if S ∈ C1(A) and

∫ 1

0

∥∥ e−i t A[A, S] ei t A −[A, S]∥∥B(H)

dt

t< ∞.

S belongs to C1+ε(A) for some ε ∈ (0, 1), with notation S ∈ C1+ε(A), if S ∈ C1(A)

and

∥∥ e−i t A[A, S] ei t A −[A, S]∥∥B(H)

≤ Const. tε for all t ∈ (0, 1).

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As banachisable topological vector spaces, the sets C2(A), C1+ε(A), C1+0(A),C1,1(A),C1(A), andC0(A) = B(H), satisfy the continuous inclusions [3, Sec. 5.2.4]

C2(A) ⊂ C1+ε(A) ⊂ C1+0(A) ⊂ C1,1(A) ⊂ C1(A) ⊂ C0(A).

Now, we adapt to the unitary framework the definition of two functions introducedin [3, Sec. 7.2] in the self-adjoint set-up. For that purpose, we let U be a unitaryoperator with U ∈ C1(A), for S, T ∈ B(H) we write T � S if there exists anoperator K ∈ K (H) such that T + K ≥ S, and for θ ∈ T and ε > 0 we set

�(θ; ε) := {θ ′ ∈ T | | arg(θ − θ ′)| < ε

}and EU (θ; ε) := EU (

�(θ; ε)).

With these notations at hand, we define the functions �AU : T → (−∞,∞] and

�AU : T → (−∞,∞] by

�AU (θ) :=sup

{a ∈ R | ∃ε>0 such that EU (θ; ε)U−1[A,U ]EU (θ; ε)≥a EU (θ; ε)

}

(3.2)and

�AU (θ) :=sup

{a ∈ R | ∃ε>0 such that EU (θ; ε)U−1[A,U ]EU (θ; ε)�a EU (θ; ε)

}.

(3.3)In applications, the function �A

U is more convenient than the function �AU since it is

defined in terms of a weaker positivity condition (positivity up to compact terms). Asimple argument shows that �A

U (θ) can be defined in an equivalent way by

�AU (θ) = sup

{a ∈ R | ∃η ∈ C∞(T, R) such that η(θ) �= 0

and η(U )U−1[A,U ]η(U ) � a η(U )2}. (3.4)

Further properties of the functions �AU and �A

U are collected in the following lemmas,with first lemma corresponding to [16, Prop. 2.3].

Lemma 3.1 (Virial Theorem forU ) Let U be a unitary operator inH, and let A be aself-adjoint operator in H with U ∈ C1(A). Then, EU ({θ})U−1[A,U ]EU ({θ}) = 0for each θ ∈ T. In particular, one has

⟨ϕ,U−1[A,U ]ϕ⟩

H = 0 for each eigenvectorϕ ∈ H of U.

Lemma 3.2 Let U be a unitary operator in H, and let A be a self-adjoint operatorin H with U ∈ C1(A). Assume there exist an open set � ⊂ T and a ∈ R such thatEU (�)U−1[A,U ]EU (�) � a EU (�). Then, for each θ ∈ � and η > 0 there existε > 0 and a finite rank orthogonal projection F with EU ({θ}) ≥ F such that

EU (θ; ε)U−1[A,U ]EU (θ; ε) ≥ (a − η)(EU (θ; ε) − F

) − ηF.

In particular, if θ is not an eigenvalue of U, then

EU (θ; ε)U−1[A,U ]EU (θ; ε) ≥ (a − η)EU (θ; ε),

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while if θ is an eigenvalue of U, one has only

EU (θ; ε)U−1[A,U ]EU (θ; ε) ≥ min{a − η,−η}EU (θ; ε).

Proof The proof relies on the Virial Theorem forU and is analogous to the proof of [3,Lemma 7.2.12] in the self-adjoint case. One just needs to replace in that proof [i H, A]byU−1[A,U ], E(J ) by EU (�), E({λ}) by EU ({θ}), and E(λ; 1/k) by EU (θ; 1/k).��Lemma 3.3 Let U be a unitary operator inH, and let A be a self-adjoint operator inH with U ∈ C1(A).

(a) The function �AU : T → (−∞,∞] is lower semicontinuous, and �A

U (θ) < ∞ ifand only if θ ∈ σ(U ).

(b) The function �AU : T → (−∞,∞] is lower semicontinuous, and �A

U (θ) < ∞ ifand only if θ ∈ σess(U ).

(c) �AU ≥ �A

U .(d) If θ ∈ T is an eigenvalue of U and �A

U (θ) > 0, then �AU (θ) = 0. Otherwise,

�AU (θ) = �A

U (θ).

Proof The claims are shown as in the proofs of Lemma 7.2.1, Proposition 7.2.3(a),Proposition 7.2.6, and Theorem 7.2.13 of [3] in the self-adjoint case. ��

By analogy with the self-adjoint case, we say that A is conjugate to U at a pointθ ∈ T if �A

U (θ) > 0, and that A is strictly conjugate to U at θ if �AU (θ) > 0. Since

�AU (θ) ≥ �A

U (θ) for each θ ∈ T by Lemma 3.3(c), strict conjugation is a propertystronger than conjugation.

Theorem 3.4 (U -smooth operators) Let U be a unitary operator in H, let A be aself-adjoint operator inH, and let G be an auxiliary Hilbert space. Assume either thatU has a spectral gap and U ∈ C1,1(A), or that U ∈ C1+0(A). Suppose also thereexist an open set � ⊂ T, a number a > 0 and an operator K ∈ K (H) such that

EU (�)U−1[A,U ]EU (�) ≥ aEU (�) + K .

Then, each operator T ∈ B(H,G) which extends continuously to an element ofB

(D(〈A〉s)∗,G)for some s > 1/2 is locally U-smooth on �\σp(U ).

Proof The claim follows by adapting the proof of [16, Prop. 2.9] to locallyU -smoothoperators T with values in the auxiliary Hilbert space G. ��

The last theorem of this section corresponds to [16, Thm. 2.7]:

Theorem 3.5 (Spectrum of U ) Let U be a unitary operator in H, and let A be aself-adjoint operator inH. Assume either that U has a spectral gap andU ∈ C1,1(A),or that U ∈ C1+0(A). Suppose also there exist an open set � ⊂ T, a number a > 0and an operator K ∈ K (H) such that

EU (�)U−1[A,U ]EU (�) ≥ aEU (�) + K .

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Then, U has at most finitely many eigenvalues in�, each one of finite multiplicity, andU has no singular continuous spectrum in �.

3.2 Commutator methods in a two-Hilbert spaces setting

From now on, in addition to the triple (H,U, A), we consider a second triple(H0,U0, A0) with H0 a Hilbert space, U0 a unitary operator in H0, and A0 a self-adjoint operator in H0. We also consider an identification operator J ∈ B(H0,H).The existence of two such triples with an identification operator is quite standard inscattering theory of unitary operators, at least for the pairs (H,U ) and (H0,U0) (seefor instance [8,49]). Part of our goal in this section is to show that the existence of theconjugate operators A and A0 is also natural, in the same way it is in the self-adjointcase [38].

In the one-Hilbert space setting, the unitary operator U is usually a multiplicativeperturbation of the unitary operatorU0. In this case, ifU −U0 is compact, the stabilityof the function �

A0U0

under compact perturbations allows one to infer information onU from similar information on U0 (see [16, Cor. 2.10]). In the two-Hilbert spacessetting, we are not aware of any general result relating the functions �A

U and �A0U0. The

obvious reason for this being the impossibility to consider U as a direct perturbationofU0 since these operators do not act in the same Hilbert space. Nonetheless, the nexttheorem provides a result in that direction. For Hilbert spaces H1,H2 and operatorsS, T ∈ B(H1,H2), we use the notation T ≈ S if (T − S) ∈ K (H1,H2).

Theorem 3.6 Let (H0,U0, A0) and (H,U, A) be as above, let J ∈ B(H0,H), andassume that

(i) U0 ∈ C1(A0) and U ∈ C1(A),(ii) JU−1

0 [A0,U0]J ∗ −U−1[A,U ] ∈ K (H),(iii) JU0 −U J ∈ K (H0,H),(iv) For each η ∈ C(C, R), η(U )(J J ∗ − 1)η(U ) ∈ K (H).

Then, one has �AU ≥ �

A0U0.

An induction argument together with a Stone–Weierstrass density argument showsthat (iii) is equivalent to the apparently stronger condition

(iii’) For each η ∈ C(C, R), Jη(U0) − η(U )J ∈ K (H0,H).

Therefore, in the sequel, we will sometimes use the condition (iii’) instead of (iii).

Proof For each η ∈ C(C, R), we have

η(U )U−1[A,U ]η(U ) ≈ η(U )JU−10 [A0,U0]J ∗η(U )

≈ Jη(U0)U−10 [A0,U0]η(U0)J

∗ (3.5)

due to Assumption (i)–(iii). Furthermore, if there exists a ∈ R such that

η(U0)U−10 [A0,U0]η(U0) � a η(U0)

2,

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then Assumptions (iii)–(iv) imply that

Jη(U0)U−10 [A0,U0]η(U0)J

∗ � a Jη(U0)2 J ∗ ≈ a η(U )J J ∗η(U ) ≈ a η(U )2.

(3.6)Thus, we obtain η(U )U−1[A,U ]η(U ) � a η(U )2 by combining (3.5) and (3.6). Thislast estimate, together with the definition (3.4) of the functions �

A0U0

and �AU , implies

the claim. ��The regularity ofU0 with respect to A0 is usually easy to check, while the regularity

ofU with respect to A is in general difficult to establish. For that purpose, various per-turbative criteria have been developed for self-adjoint operators in one Hilbert space,and often a distinction is made between short-range and long-range perturbations.Roughly speaking, the two terms of the formal commutator [A,U ] = AU − U Aare treated separately in the short-range case, while [A,U ] is really computed in thelong-range case. In the sequel, we discuss short-range type perturbations for unitaryoperators in a two-Hilbert spaces setting. The results we obtain are analogous to theones obtained in [38, Sec. 3.1] for self-adjoint operators in a two-Hilbert spaces setting.

We start by showing how the conditionU ∈ C1(A) and the assumptions (ii)-(iii) ofTheorem 3.6 can be verified for a class of short-range type perturbations. Our approachis to infer the desired information onU from equivalent information onU0, which areusually easier to obtain. Accordingly, our results exhibit some perturbative flavour.The price one has to pay is to impose some compatibility conditions between A0 andA. For brevity, we set

B := JU0 −U J ∈ B(H0,H) and B∗ := JU∗0 −U∗ J ∈ B(H0,H).

Proposition 3.7 Let U0 ∈ C1(A0), assume that D ⊂ H is a core for A such thatJ ∗D ⊂ D(A0), and suppose that

BA0 � D(A0) ∈ B(H0,H), B∗A0 � D(A0) ∈ B(H0,H)

and (J A0 J ∗ − A) � D ∈ B(H). (3.7)

Then, U ∈ C1(A).

Proof For ϕ ∈ D , a direct calculation gives

⟨Aϕ,Uϕ

⟩

H − ⟨ϕ,U Aϕ

⟩

H = ⟨Aϕ,Uϕ

⟩

H − ⟨ϕ,U Aϕ

⟩

H − ⟨ϕ, J [A0,U0]J ∗ϕ

⟩

H+ ⟨

ϕ, J [A0,U0]J ∗ϕ⟩

H= ⟨

ϕ, BA0 J∗ϕ

⟩

H − ⟨B∗A0 J

∗ϕ, ϕ⟩

H+ ⟨

U∗ϕ, (J A0 J∗ − A)ϕ〉H

− ⟨(J A0 J

∗ − A)ϕ,Uϕ⟩

H + ⟨ϕ, J [A0,U0]J ∗ϕ

⟩

H.

Furthermore, we have

∣∣⟨ϕ, BA0 J

∗ϕ⟩

H − ⟨B∗A0 J

∗ϕ, ϕ⟩

H∣∣ ≤ Const.‖ϕ‖2H

123


due to the first two conditions in (3.7), and we have

∣∣⟨U∗ϕ, (J A0 J

∗ − A)ϕ⟩

H − ⟨(J A0 J

∗ − A)ϕ,Uϕ⟩

H∣∣ ≤ Const.‖ϕ‖2H

due to the third condition in (3.7). Finally, since U0 ∈ C1(A0) and J ∈ B(H0,H)

we also have

∣∣⟨ϕ, J [A0,U0]J ∗ϕ

⟩

H∣∣ ≤ Const.‖ϕ‖2H.

Since D is a core for A, this implies that U ∈ C1(A). ��In the next proposition, we show how the assumption (ii) of Theorem 3.6 is verified

for short-range type perturbations. Since the hypotheses are slightly stronger than theones of Proposition 3.7, U automatically belongs to C1(A).

Proposition 3.8 Let U0 ∈ C1(A0), assume that D ⊂ H is a core for A such thatJ ∗D ⊂ D(A0), and suppose that

BA0 � D(A0) ∈ B(H0,H), B∗A0 � D(A0) ∈ K (H0,H)

and (J A0 J ∗ − A) � D ∈ K (H). (3.8)

Then, the difference of bounded operators JU−10 [A0,U0]J ∗ −U−1[A,U ] belongs to

K (H).

Proof The facts that U0 ∈ C1(A0) and J ∗D ⊂ D(A0) imply the inclusions

U0 J∗D ⊂ U0D(A0) ⊂ D(A0).

Using this and the last two conditions of (3.8), we obtain for ϕ ∈ D and ψ ∈ U−1Dthat

⟨ψ,

(JU−1

0 [A0,U0]J ∗ −U−1[A,U ])ϕ⟩

H= ⟨

ψ, B∗A0U0 J∗ϕ

⟩

H + ⟨B∗A0 J

∗Uψ, ϕ⟩

H + ⟨(J A0 J

∗ − A)Uψ,Uϕ⟩

H− ⟨

ψ, (J A0 J∗ − A)ϕ

⟩

H= ⟨

ψ, K1U0 J∗ϕ

⟩

H + ⟨K1 J

∗Uψ, ϕ⟩

H + ⟨K2Uψ,Uϕ

⟩

H − ⟨ψ, K2ϕ

⟩

H

with K1 ∈ K (H0,H) and K2 ∈ K (H). Since D and U−1D are dense in H, itfollows that the operator JU−1

0 [A0,U0]J ∗ −U−1[A,U ] belongs toK (H). ��In the rest of the section, we particularise the previous results to the case A =

J A0 J ∗. This case deserves a special attention since it represents the most natu-ral choice of conjugate operator A for U when a conjugate operator A0 for U0 isgiven. However, one needs in this case the following assumption to guarantee theself-adjointness of the operator A:

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Assumption 3.9 There exists a set D ⊂ D(A0 J ∗) ⊂ H such that J A0 J ∗ � D isessentially self-adjoint, with corresponding self-adjoint extension denoted by A.

Assumption 3.9 might be difficult to check in general, but in concrete situationsthe choice of the set D can be quite natural (see for example Lemma 4.9 for the caseof quantum walks or [39, Rem. 4.3] for the case of manifolds with asymptoticallycylindrical ends). The following two corollaries follow directly from Propositions3.7-3.8 in the case Assumption 3.9 is satisfied.

Corollary 3.10 Let U0 ∈ C1(A0), suppose that Assumption 3.9 holds for someset D ⊂ H, and assume that BA0 � D(A0) ∈ B(H0,H) and B∗A0 � D(A0) ∈B(H0,H). Then, U belongs to C1(A).

Corollary 3.11 Let U0 ∈ C1(A0), suppose that Assumption 3.9 holds for someset D ⊂ H, and assume that BA0 � D(A0) ∈ B(H0,H) and B∗A0 � D(A0) ∈K (H0,H). Then, the difference of boundedoperators JU−1

0 [A0,U0]J ∗−U−1[A,U ]belongs toK (H).

4 Quantum walks with an anisotropic coin

In this section, we apply the abstract theory of Sect. 3 to prove our results on thespectrum of the evolution operator U of the quantum walk with an anisotropic coindefined in Sect. 2. For this, we first determine in Sect. 4.1 the spectral properties andprove a Mourre estimate for the asymptotic operators U� and Ur. Then, in Sect. 4.2,we use the Mourre estimate for U� and Ur to derive a Mourre estimate for U . Finally,in Sect. 4.3, we use the Mourre estimate for U to prove our results on U . We recallthat the behaviour of the coin operator C at infinity is determined by Assumption 2.1.

4.1 Asymptotic operators U� and Ur

For the study of the asymptotic operators U� and Ur, we use the symbol � to denoteeither the index � or the index r. Also, we introduce the subspaceHfin ⊂ H of elementswith finite support

Hfin := ⋃n∈N

{� ∈ H | �(x) = 0 if |x | ≥ n

},

the Hilbert space K := L2([0, 2π), dk

2π , C2), and the discrete Fourier transform F :

H → K, which is the unitary operator defined as the unique continuous extension ofthe operator

(F�)(k) :=∑

x∈Ze−ikx �(x), � ∈ Hfin, k ∈ [0, 2π).

123


A direct computation shows that the operator U� is decomposable in the Fourierrepresentation, namely for all f ∈ K and almost every k ∈ [0, 2π) we have

(F U�F∗ f )(k) = U�(k) f (k) with U�(k) :=

(eik 00 e−ik

)

C� ∈ U(2).

Moreover, since U�(k) ∈ U(2) the spectral theorem implies that U�(k) can be writtenas

U�(k) =2∑

j=1

λ�, j (k)��, j (k),

with λ�, j (k) the eigenvalues of U�(k) and ��, j (k) the corresponding orthogonal pro-jections.

The next lemma furnishes some information on the spectrum of U�. To state it, weuse the following parametrisation for the matrices C� :

C� = eiδ�/2(

a� ei(α�−δ�/2) b� ei(β�−δ�/2)

−b� e−i(β�−δ�/2) a� e−i(α�−δ�/2)

)

(4.1)

with a�, b� ∈ [0, 1] satisfying a2� +b2� = 1, and α�, β�, δ� ∈ (−π, π ]. The determinantdet(C�) of C� is equal to eiδ� . For brevity, we also set

τ�(k) := a� cos(k + α� − δ�/2),

η�(k) :=√1 − τ�(k)2,

ς�(k) := a� sin(k + α� − δ�/2),

θ� := arccos(a�).

Lemma 4.1 (Spectrum of U�)

(a) If a� = 0, then U� has pure point spectrum

σ(U�) = σp(U�) = {i eiδ�/2,−i eiδ�/2

}

with each point an eigenvalue of U� of infinite multiplicity.(b) If a� ∈ (0, 1), then σp(U�) = ∅ and

σ(U�) = σc(U�) = {eiγ | γ ∈ [δ�/2 + θ�, π + δ�/2 − θ�]

∪ [π + δ�/2 + θ�, 2π + δ�/2 − θ�]}.

(c) If a� = 1, then σp(U�) = ∅ and σ(U�) = σc(U�) = T.

123


Proof Using the parametrisation (4.1), one gets U�(k) = eiδ�/2(

a�(k) b�(k)−b�(k) a�(k)

)with

the coefficients a�(k) := a� ei(k+α�−δ�/2) and b�(k) := b� ei(k+β�−δ�/2). Therefore,the spectrum of U� is given by

σ(U�) = {λ�, j (k) | j = 1, 2, k ∈ [0, 2π)

},

with λ�, j (k) the solution of the characteristic equation det(U�(k) − λ�, j (k)

) = 0,j = 1, 2 and k ∈ [0, 2π). ��We now exhibit normalised eigenvectors u�, j (k) of U�(k) for the eigenvalues

λ�, j (k) which are C∞ in the variable k :⎧⎪⎨

⎪⎩

u�, j (k) :=√

η�(k)+(−1) j−1ς�(k)b�

√2η�(k)

(ib�(k)

ς�(k) + (−1) jη�(k)

)

if a� ∈ [0, 1)u�,1(k) := (

10

)and u�,2(k) := (

01

)if a� = 1.

We leave the reader check that u�, j (k) are indeed normalised eigenvectors of U�(k)with eigenvalues λ�, j (k). In addition, since for a� ∈ [0, 1) one has η�(k) > 0 andη�(k) + (−1) j−1ς�(k) > 0, the 2π -periodic map R � k �→ u�, j (k) ∈ C

2 is of classC∞.

Our next goal is to construct a conjugate operator for the operator U�. For this, afew preliminaries are necessary. First, we equip the interval [0, 2π) with the additionmodulo 2π , and for any n ∈ N we define the space Cn

([0, 2π), C2) ⊂ K as the set of

functions [0, 2π) → C2 of class Cn . In particular, we have u�, j ∈ C∞([0, 2π), C

2),

and the spaceFHfin ⊂ C∞([0, 2π), C2)is the set of C

2-valued trigonometric poly-nomials.

Next, we define the asymptotic velocity operator for the operatorU�. For j = 1, 2,we let v�, j : [0, 2π) → R be the bounded function given by

v�, j (k) := i λ′�, j (k)

(λ�, j (k)

)−1, k ∈ [0, 2π). (4.2)

Here, ( · )′ stands for the derivative with respect to k, and v�, j is real valued becauseλ�, j takes values in T. Finally, for all f ∈ K and almost every k ∈ [0, 2π), we definethe decomposable operator V� ∈ B(K) by

(V� f

)(k) := V�(k) f (k) where V�(k) :=

2∑

j=1

v�, j (k)��, j (k) ∈ B(C2), (4.3)

and we call asymptotic velocity operator the operator V� := F ∗ V� F . The basicspectral properties of V� are collected in the following lemma.

Lemma 4.2 (Spectrum of V�) Let C� be parameterised as in (4.1).

(a) If a� = 0, then v�, j = 0 for j = 1, 2, and V� = 0.

123


(b) If a� ∈ (0, 1), then v�, j (k) = (−1) jς�(k)η�(k)

for j = 1, 2 and k ∈ [0, 2π), σp(V�) = ∅

and

σ(V�) = σc(V�) = [−a�, a�].

(c) If a� = 1, then v�, j = (−1) j for j = 1, 2, and V� has pure point spectrum

σ(V�) = σp(V�) = {−1, 1}

with each point an eigenvalue of V� of infinite multiplicity.

Proof The claims follow from simple calculations using the formulas for λ�, j (k) inthe proof of Lemma 4.1 and the definition (4.2) of v�, j (k). ��

For any ξ, ζ ∈ C([0, 2π), C

2), we define the operator |ξ 〉〈ζ | : C([0, 2π), C

2) →

C([0, 2π), C

2)by

(|ξ 〉〈ζ | f )(k) := ⟨ζ(k), f (k)

⟩

2 ξ(k), f ∈ C([0, 2π), C

2), k ∈ [0, 2π),

where 〈 · , · 〉2 is the usual scalar product on C2. This operator extends continuously

to an element of B(K), with norm satisfying the bound

∥∥|ξ 〉〈ζ |∥∥B(K)

≤ ‖ξ‖L∞([0,2π), dk2π ,C2)‖ζ‖L∞([0,2π), dk2π ,C2)

. (4.4)

We also define the self-adjoint operator P in K

P f := −i f ′, f ∈ D(P)

D(P) := {f ∈ K | f is absolutely continuous, f ′ ∈ K, and f (0) = f (2π)

}.

With these definitions at hand, we can prove the self-adjointness of an operator usefulfor the definition of our future the conjugate operator for U :

Lemma 4.3 The operator

X� f := −2∑

j=1

(∣∣u�, j

⟩⟨u�, j

∣∣P − i

∣∣u�, j

⟩⟨u′

�, j

∣∣)f, f ∈ FHfin,

is essentially self-adjoint inK, with closure denoted by the same symbol. In particular,the Fourier transform X� := F ∗ X�F of X� is essentially self-adjoint on Hfin inH.

Proof The proof simply consists in checking the assumptions of Nelson’s commutatortheorem [36, Thm. X.37] applied with the comparison operator N := P2 + 1. ��

The main relations between the operators introduced so far are summarised inthe following proposition whose proof is left to the reader. To state it, we need one

123


more decomposable operator H� ∈ B(K) defined for all f ∈ K and almost everyk ∈ [0, 2π) by

(H� f

)(k) := H�(k) f (k) where H�(k) := −

2∑

j=1

v′�, j (k)��, j (k) ∈ B(C2).

We also need the inverse Fourier transform H� := F ∗ H�F of H�.

Proposition 4.4 (a) One has the equality [i X�, V�] = H� in the form sense onHfin.(b) U�, V� and H� are mutually commuting.(c) One has the equality [X�,U�] = U�V� in the form sense onHfin.

Since X� is essentially self-adjoint on Hfin, Proposition 4.4(a) implies that V� ∈C1(X�). Therefore,

A�� := 12

(X�V� + V�X�

)�, � ∈ D(A�) := {

� ∈ H | V�� ∈ D(X�)},

is self-adjoint in H, and essentially self-adjoint on Hfin (see [44, Lemma 2.4]). Wecan now state and prove the main results of this section. The symbols Int(�) and ∂�

denote the interior and the boundary of a set � ⊂ T.

Proposition 4.5 (a) U� ∈ C1(A�) with U−1� [A�,U�] = V 2

� .(b) �

A�

U�= �

A�

U�, and

(i) if a� = 0, then �A�

U�(θ) = 0 for θ ∈ {

i eiδ�/2,−i eiδ�/2}and �

A�

U�(θ) = ∞

otherwise,(ii) if a� ∈ (0, 1), then �

A�

U�(θ) > 0 for θ ∈ Int

(σ(U�)

), �

A�

U�(θ) = 0 for θ ∈

∂σ(U�), and �A�

U�(θ) = ∞ otherwise,

(iii) if a� = 1, then �A�

U�(θ) = 1 for all θ ∈ T.

(c) (i) If a� ∈ (0, 1), then U� has purely absolutely continuous spectrum

σ(U�) = σac(U�) = {eiγ | γ ∈ [δ�/2 + θ�, π + δ�/2 − θ�]

∪ [π + δ�/2 + θ�, 2π + δ�/2 − θ�]}.

(ii) If a� = 1, then U� has purely absolutely continuous spectrum σ(U�) =σac(U�) = T.

Proof (a) A calculation in the forme sense on Hfin using points (b) and (c) of Propo-sition 4.4 gives

[A�,U�] = 12

(V�[X�,U�] + [X�,U�]V�

) = U�V2� .

Since A� is essentially self-adjoint on Hfin, this implies that U� ∈ C1(A�) withU−1

� [A�,U�] = V 2� .

123


(b) Take θ ∈ T and ε > 0. Then, the result of point (a) and (4.3) imply for almostevery k ∈ [0, 2π)

(F EU� (θ; ε)U−1

� [A�,U�]EU� (θ; ε)F ∗)(k) = (F EU� (θ; ε)V 2

� EU� (θ; ε)F ∗)(k)

= EU�(k)(θ; ε) V�(k)2EU�(k)(θ; ε)

≥ min{v�,1(k)

2, v�,2(k)2}EU�(k)(θ; ε).

Then, the definition (4.2) of v�, j (k) shows that v�, j (k) = 0 if and only if λ′�, j (k) = 0,

which occurs when λ�, j (k) ∈ ∂σ(U�). Therefore, one gets �A�

U�= �

A�

U�by Lemma

3.3(d), and to conclude one just has to take into account the form of the boundary setsσ(U�) given in Lemma 4.1.

(c) We know from point (a) that U� ∈ C1(A�) with U−1� [A�,U�] = V 2

� , andProposition 4.4(a) implies that V� ∈ C1(A�). Thus, U� ∈ C2(A�). Therefore, ifa� ∈ (0, 1), we infer from point (b.ii) and Theorem 3.5 that U� has no singularcontinuous spectrum in Int

(σ(U�)

). This, together with Lemma 4.1(b), implies the

claim in the case a� ∈ (0, 1). The claim in the case a� = 1 is proved in a similar way.��

4.2 Mourre estimate for U

In this section, we use the Mourre estimate for the asymptotic operators U�,Ur toderive a Mourre estimate for U . To achieve this, we apply the abstract constructionintroduced in Sect. 3.2, starting by choosing H0 := H ⊕ H as second Hilbert spaceand U0 := U� ⊕Ur as second unitary operator in H0.

The spectral properties of U0 are obtained as a consequence of Lemma 4.1(a),Proposition 4.5(c) and the direct sum decomposition of U0 :Lemma 4.6 (Spectrum of U0) One has σ(U0) = σ(U�) ∪ σ(Ur) and σsc(U0) = ∅.Furthermore,

(a) if a� = ar = 0, then U0 has pure point spectrum

σ(U0) = σp(U0) = σp(U�) ∪ σp(Ur) = {i eiδ�/2,−i eiδ�/2, i eiδr/2,−i eiδr/2

}

with each point an eigenvalue of U0 of infinite multiplicity,(b) if a� = 0 and ar ∈ (0, 1], then σac(U0) = σac(Ur) with σac(Ur) as in Proposition

4.5(c), and

σp(U0) = σp(U�) = {i eiδ�/2,−i eiδ�/2

}

with each point an eigenvalue of U0 of infinite multiplicity,(c) if a� ∈ (0, 1] and ar = 0, then σac(U0) = σac(U�) with σac(U�) as in Proposition

4.5(c), and

σp(U0) = σp(Ur) = {i eiδr/2,−i eiδr/2

}

with each point an eigenvalue of U0 of infinite multiplicity,

123


(d) if a�, ar ∈ (0, 1], then U0 has purely absolutely continuous spectrum

σ(U0) = σac(U0) = σac(U�) ∪ σac(Ur)

with σac(U�) and σac(Ur) as in Proposition 4.5(c).

Also, as intuition suggests and as already stated in Theorem 2.2, the spectrum ofU0 coincides with the essential spectrum of U , namely σess(U ) = σ(U�) ∪ σ(Ur) =σ(U0).

Proof of Theorem 2.2 The proof is based on an argument using crossed product C∗-algebras inspired from [32]. LetA be the algebra of functions Z → B(C2) admittinglimits at ±∞, and let A0 be the ideal of A consisting in functions Z → B(C2)

vanishing at ±∞. Since A is equipped with an action of Z by translation, namely

(Tyϕ

)(x) := ϕ(x + y), x, y ∈ Z, ϕ ∈ A,

we can consider the crossed product algebraA�Z, and the functoriality of the crossedproduct implies the identities

(A � Z)/(A0 � Z) ∼= (A/A0) � Z = (B(C2) ⊕ B(C2)

)� Z

= (B(C2) � Z

) ⊕ (B(C2) � Z

), (4.5)

where the equalityA/A0 = B(C2)⊕B(C2) is obtained by evaluation of the functionsϕ ∈ A at ±∞.

Now, the algebrasA�Z andA0�Z can be faithfully represented inH by mappingthe elements of A and A0 to multiplication operators in H and the elements of Z tothe shifts Tz . Writing A and A0 for these representations of A � Z and A0 � Z inH,we can note three facts. First, A0 is equal to the ideal of compact operators K (H).Secondly, the operator U belongs to A, since

U = SC = T1

(1 00 0

)

C + T−1

(0 00 1

)

C

with T1, T−1 shifts and(1 00 0

)C,

(0 00 1

)C multiplication operators in H. Thirdly,

the essential spectrum of U in A is equal to the spectrum of the image of U in thequotient algebra A/K (H) = A/A0. These facts, together with (4.5) and Lemma 4.6,imply the equalities

σess(U ) = σ(SC(−∞) ⊕ SC(+∞)

) = σ(SC� ⊕ SCr

) = σ(U�) ∪ σ(Ur) = σ(U0),

which prove the claim. ��Next, we define the identification operator J ∈ B(H0,H) by

J (��,�r) := j� �� + jr �r, (��,�r) ∈ H0,

123


where

jr(x) :={1 if x ≥ 0

0 if x ≤ −1and j� := 1 − jr.

The adjoint operator J ∗ ∈ B(H,H0) satisfies J ∗� = ( j� �, jr �) for � ∈ H. Usingthe same notation for the functions j�, jr and the associated multiplication operatorsinH, one directly gets:

Lemma 4.7 J ∗ J = j� ⊕ jr is an orthogonal projection, and J J ∗ = 1H.

The first result of the next lemma is an analogue of Proposition 4.5(a) in the Hilbertspace H0. To state it, we need to introduce the operator A0 := A� ⊕ Ar (which willbe used as a conjugate operator for U0) and the operator V0 := V� ⊕ Vr.

Lemma 4.8 (a) U0 ∈ C1(A0) with U−10 [A0,U0] = V 2

0 .(b) B := JU0 −U J ∈ K (H0,H) and B∗ := JU∗

0 −U∗ J ∈ K (H0,H).

Proof The proof of point (a) is similar to the proof of Proposition 4.5(a); one just hasto replace the operators U�, A�, V� inH by the operators U0, A0, V0 inH0. For point(b), a direct computation with (��,�r) ∈ H0 gives

B(��,�r) = (j�U�� + jrUr�r

) −U(j�� + jr�r

)

= ([ j�,U�] − (U −U�) j�)�� + ([ jr,Ur] − (U −Ur) jr

)�r

= ([ j�, S]C� − S(C − C�) j�)�� + ([ jr, S]Cr − S(C − Cr) jr

)�r.

(4.6)

Since we have [ j�, S] ∈ K (H) and (C − C�) j� ∈ K (H) as a consequence ofAssumption 2.1, it follows that B ∈ K (H0,H). The inclusion B∗ ∈ K (H0,H) isproved in a similar way. ��

The next step is to define a conjugate operator A for U by using the conjugateoperator A0 for U0. For this, we consider the operator J A0 J ∗ which is well-definedand symmetric on Hfin. We have the equality

J A0 J∗ = j� A� j� + jr Ar jr on Hfin, (4.7)

and J A0 J ∗ is essentially self-adjoint onHfin :Lemma 4.9 (Conjugate operator for U ) The operator J A0 J ∗ is essentially self-adjoint on Hfin, with corresponding self-adjoint extension denoted by A.

Proof The operator j� := F j�F ∗ ∈ B(K) satisfies j�D(P) ⊂ D(P) and [ j�, P] =0 on D(P). Therefore, we have the following equalities onFHfin

F j�A� j�F∗ = 1

2F j�(X�V� + V�X�

)j�F

∗

= 12 j�

(X�V� + V� X�

)j�

123


= j�(V� X� − i

2 H�

)j�

= −2∑

j=1

(j�∣∣v�, j u�, j

⟩⟨u�, j

∣∣ j� P − i j�

∣∣v�, j u�, j

⟩⟨u′

�, j

∣∣ j�

)− i

2 j� H� j�.

which give onFHfin

F J A0 J∗F ∗ = −

2∑

j=1

∑

�∈{�,r}j�∣∣v�, j u�, j

⟩⟨u�, j

∣∣ j� P

+i2∑

j=1

∑

�∈{�,r}j�∣∣v�, j u�, j

⟩⟨u′

�, j

∣∣ j� − i

2

∑

�∈{�,r}j� H� j�.

The rest of the proof consists in an application of Nelson’s commutator theorem[36, Thm. X.37] with the comparison operator N := P2 + 1. As a consequence, itfollows that F J A0 J ∗F ∗ is essentially self-adjoint on FHfin, and thus that J A0 J ∗is essentially self-adjoint on Hfin. ��

We are thus in the set-up of Assumption 3.9 with D = Hfin. So, the next step isto show the inclusion U ∈ C1(A). For this, we use Corollary 3.10. Using Corollary3.11, we also get an additional compacity result:

Lemma 4.10 U ∈ C1(A) and JU−10 [A0,U0]J ∗ −U−1[A,U ] ∈ K (H).

Proof First, we recall that U0 ∈ C1(A0) due to Lemma 4.8(a), and that Assumption3.9 holds with D = Hfin. Next, we note that the expression for B(��,�r) with(��,�r) ∈ H0 is given in (4.6), and that

B∗(��,�r) = (C∗[ j�, S∗] − (C∗ − C∗

� ) j�S∗)�� + (

C∗[ jr, S∗] − (C∗ − C∗r ) jrS

∗)�r.

Furthermore, we know from Lemma 4.8(b) that B, B∗ ∈ K (H0,H). In consequence,due to Corollaries 3.10–3.11, the claims will follow if we show that BA0 � D(A0) ∈B(H0,H) and B∗A0 � D(A0) ∈ K (H0,H). For this, we first note that computationsas in the proof of Lemma 4.9 imply on Hfin the equalities

A� = −F ∗{

P2∑

j=1

(∣∣u�, j

⟩⟨v�, j u�, j

∣∣ + i

∣∣u′

�, j

⟩⟨v�, j u�, j

∣∣)}

F + i2H�

= QF ∗{ 2∑

j=1

(∣∣u�, j

⟩⟨v�, j u�, j

∣∣ + i

∣∣u′

�, j

⟩⟨v�, j u�, j

∣∣)}

F + i2H� (4.8)

with Q the self-adjoint multiplication operator defined by

(Q�

)(x) = x�(x), x ∈ Z, � ∈ D(Q) := {

� ∈ H | ‖Q�‖H < ∞}. (4.9)

123


Therefore, since all the operators on the right of Q in (4.8) are bounded, it is sufficientto show that

B(Q ⊕ Q) � D(Q) ⊕ D(Q) ∈ B(H0,H)

and B∗(Q ⊕ Q) � D(Q) ⊕ D(Q) ∈ K (H0,H).

But, this can be deduced from the Assumption 2.1 once the following observationsare made:

[j�, S

] = Sm� with m� : Z → B(C2) a function with compact support,[ j�, S∗] = S∗n� with n� : Z → B(C2) a function with compact support, and S∗Q =QS∗ + b with b ∈ L∞ (

Z,B(C2)). ��

Recall that the set τ(U ) = ∂σ(U�) ∪ ∂σ(Ur) has been introduced in Sect. 2. Dueto Lemma 4.1, τ(U ) contains at most 8 values. Moreover, since we show in the nextproposition that a Mourre estimate holds outside τ(U ), it is natural to interpret τ(U )

as the set of thresholds in the spectrum of U .

Proposition 4.11 (Mourre estimate for U) We have �AU ≥ �

A0U0

with �A0U0

=min

{�A�

U�, �

ArUr

}and �

A�

U�, �

ArUr

given in Proposition 4.5. In particular, �A0U0

(θ) > 0if θ ∈ {σ(U�) ∪ σ(Ur)}\τ(U ).

Proof The first claim follows from Theorem 3.6, with the assumptions of this theoremverified in Lemmas 4.7–4.10. The second claim follows from Proposition 4.5 andstandard results on the function �

A0U0

when A0 andU0 are direct sums of operators (see[3, Prop. 8.3.5] for a proof in the case of direct sums of self-adjoint operators). ��

4.3 Spectral properties of U

In order to go one step further in the study ofU , a regularity property ofU with respectto A stronger than U ∈ C1(A) has to be established. This regularity property will beobtained by considering first the operator JU0 J ∗, and then by analysing the differenceU − JU0 J ∗. We note that JU0 J ∗ and U − JU0 J ∗ satisfy the equalities

JU0 J∗ = j�U� j� + jrUr jr (4.10)

and

U − JU0 J∗ = j�(U −U�) j� + jr(U −Ur) jr + j�U jr + jrU j�. (4.11)

Lemma 4.12 JU0 J ∗ ∈ C2(A).

Proof The proof is based on standard results from toroidal pseudodifferential calculus,as presented for example in [40, Chap. 4]. The normalisation we use for the Fouriertransform differs from the one used in [40], but the difference is harmless.

(i) First, we note that j� is a toroidal pseudodifferential operator on FHfin withsymbol in S0ρ,0(T × Z) for each ρ > 0 (see the definitions 4.1.7 and 4.1.9 of [40]).

Similarly, Eq. (4.8) shows that A� is a first-order differential operator onFHfin with

123


matrix coefficients in M(2,C∞(T)

) ⊂ M(2, S0ρ,0(T × Z)

)for each ρ > 0. In con-

sequence, it follows from [40, Thm. 4.7.10] that the commutator[j�, A�

]on FHfin

is well-defined and equal to a toroidal pseudodifferential operator with matrix coeffi-cients in M

(2, S1−ρ

ρ,0 (T × Z))for each ρ > 0. This implies that

[j�, A�

]is bounded

on FHfin, and thus that j� ∈ C1( A�) since FHfin is dense in D( A�). By Fouriertransform, it follows that j� ∈ C1(A�).

(ii) A calculation in the form sense on Hfin using Eqs. (4.7) and (4.10) and theidentities j� jr = 0 = jr j� gives

[JU0 J

∗, A] = [

j�U� j�, j� A� j�] + [

jrUr jr, jr Ar jr]

=∑

�∈{�,r}j�(U� j�A� − A� j�U�

)j�

=∑

�∈{�,r}j�([U�, j�

]A� + [

j�U�, A�

])j�. (4.12)

Since j�U� ∈ C1(A�) by Proposition 4.5(a), point (i) and [3, Prop. 5.1.5], the secondterm on the r.h.s. of (4.12) belongs to B(H). Furthermore, a calculation using thedefinition of the shift operator S shows that

[U�, j�

] = [S, j�

]C� = B�m� with

B� ∈ B(H) and m� : Z → B(C2) a function with compact support. It followsfrom (4.8) that

[U�, j�

]A� is bounded on Hfin. Therefore, both terms on the r.h.s. of

(4.12) are bounded on Hfin, and thus we infer from the density of Hfin in D(A) thatJU0 J ∗ ∈ C1(A).

(iii) To show that JU0 J ∗ ∈ C2(A), one has to commute the r.h.s. of (4.12) oncemore with A. Doing this in the form sense onHfin with the notation

∑�∈{�,r} j�D� j�

with D� := [U�, j�]A� + [ j�U�, A�] for the r.h.s. of (4.12), one gets that JU0 J ∗ ∈C2(A) if the operators [D�, A�], [D�, j�]A� and A�[D�, j�] defined in the form senseonHfin extend continuously to elements ofB(H).

For the first operator, we have in the form sense on Hfin the equalities

[D�, A�] = [[U�, j�]A� + j�[U�, A�] + [ j�, A�]U�, A�

]

= [[U�, j�]A�, A�

] + j�[[U�, A�], A�

] + [ j�, A�][U�, A�]+ [ j�, A�][U�, A�] + [[ j�, A�], A�

]U� . (4.13)

Then, simple adaptations of the arguments presented in points (i) and (ii) show thatthe operators [ j�, A�], [U�, j�] ∈ B(H) can be multiplied in the form sense on Hfinby several operators A� on the left and/or on the right and that the resultant operatorsextend continuously to elements ofB(H). Therefore, the first, the third, the fourth, andthe fifth terms in (4.13) extend continuously to elements ofB(H). For the second term,we note from Propositions 4.4(a) and 4.5(a) that U�, V� ∈ C1(A�) with [U�, A�] =−U�V 2

� . In consequence, we have U�V 2� ∈ C1(A�) by [3, Prop. 5.1.5] and

j�[[U�, A�], A�

] = − j�[U�V

2� , A�

] ∈ B(H).

123


The proof that the operators [D�, j�]A� and A�[D�, j�] defined in the form senseon Hfin extend continuously to elements of B(H) is similar. The only noticeabledifference is the appearance of terms [U�V 2

� , j�]A� and A�[U�V 2� , j�]. However, by

observing that V 2� ∈ C1(A�) and that [V 2

� , j�] is a toroidal pseudodifferential oper-ator with matrix coefficients in M

(2, S−ρ

ρ,0(T × Z))for each ρ > 0, one infers that

[U�V 2� , j�]A� and A�[U�V 2

� , j�] extend continuously to elements ofB(H). ��In the next lemma, we prove that U satisfies sufficient regularity with respect to

A, namely that U ∈ C1+ε(A) for some ε ∈ (0, 1). We recall from Sect. 3.1 thatthe sets C2(A), C1+ε(A), C1+0(A) and C1,1(A) satisfy the continuous inclusionsC2(A) ⊂ C1+ε(A) ⊂ C1+0(A) ⊂ C1,1(A).

Lemma 4.13 U ∈ C1+ε(A) for each ε ∈ (0, 1) with ε ≤ min{ε�, εr}.Proof (i) Since JU0 J ∗ ∈ C2(A) by Lemma 4.12 and C2(A) ⊂ C1+ε(A), it is suf-ficient to show that U − JU0 J ∗ ∈ C1+ε(A), with U − JU0 J ∗ given by (4.11).Moreover, calculations as in the proof of Lemma 4.12 show that the last two termsj� U jr and jr U j� of (4.11) belong to C2(A). So, it only remains to show thatj�(U −U�) j� + jr(U −Ur) jr ∈ C1+ε(A).(ii) In order to show this inclusion, we first observe from (2.1) and (4.7) that we

have in the form sense on Hfin the equalities

[j�(U −U�) j� + jr(U −Ur) jr, A

] =∑

�∈{�,r}

[j�(U −U�) j�, j�A� j�

]

=∑

�∈{�,r}

(j�S(C − C�) j�A� j�

− j�A� j�S(C − C�) j�). (4.14)

Then, using Assumption 2.1, the formula (4.8) for A� on Hfin, and a similar formulawith the operator Q on the right (recall that Q is the position operator defined in (4.9)),one obtains that the operator on the r.h.s. of (4.14) defined as

D� := j�S(C − C�) j�A� j� − j�A� j�S(C − C�) j�

extends continuously to an element ofB(H). SinceHfin is dense inD(A), this impliesthat j�(U −U�) j� + jr(U −Ur) jr ∈ C1(A).

(iii) To show that j�(U − U�) j� + jr(U − Ur) jr ∈ C1+ε(A), it remains to checkthat

∥∥ e−i t A D� e

i t A −D�

∥∥B(H)

≤ Const. tε for all t ∈ (0, 1).

But, algebraic manipulations as presented in [3, pp. 325–326] show that for all t ∈(0, 1)

∥∥ e−i t A D� e

i t A −D�

∥∥B(H)

≤ Const.(‖ sin(t A)D�‖B(H) + ‖ sin(t A)(D�)

∗‖B(H)

)

123


≤ Const.(‖t A (t A + i)−1D�‖B(H)

+ ‖t A (t A + i)−1(D�)∗‖B(H)

).

Furthermore, if we set At := t A (t A + i)−1 and �t := t〈Q〉(t〈Q〉 + i)−1, we obtainthat

At = (At + i(t A + i)−1A 〈Q〉−1)�t

with A〈Q〉−1 ∈ B(H)due to (4.7)–(4.8). Thus, since∥∥At+i(t A+i)−1A 〈Q〉−1

∥∥B(H)

is bounded by a constant independent of t ∈ (0, 1), it is sufficient to prove that

‖�t D�‖B(H) + ‖�t (D�)∗‖B(H) ≤ Const. tε for all t ∈ (0, 1).

Now, this estimate will hold if we show that the operators 〈Q〉εD� and 〈Q〉ε(D�)∗

defined in the form sense on Hfin extend continuously to elements of B(H). Forthis, we fix ε ∈ (0, 1) with ε ≤ min{ε�, εr}, and note that 〈Q〉1+ε(C − C�) j� ∈B(H). With this inclusion and the fact that 〈Q〉−1A� defined in the form sense onHfin extend continuously to elements of B(H), one readily obtains that 〈Q〉εD� and〈Q〉ε(D�)

∗ defined in the form sense on Hfin extend continuously to elements ofB(H), as desired. ��

With what precedes, we can now prove our last two main results on U which havebeen stated in Sect. 2.

Proof of Theorem 2.3 Theorem 3.4, whose assumptions are verified in Proposition4.11 and Lemma 4.13, implies that each T ∈ B(H,G) which extends continuously toan element of B

(D(〈A〉s)∗,G)for some s > 1/2 is locally U -smooth on �\σp(U ).

Moreover, we know from the proof of Lemma 4.13 thatD(Q) ⊂ D(A). Therefore, wehaveD(〈Q〉s) ⊂ D(〈A〉s) for each s > 1/2, and it follows by duality thatD(〈A〉s)∗ ⊂D(〈Q〉s)∗ ≡ D(〈Q〉−s) for each s > 1/2. In consequence, any operator T ∈ B(H,G)

which extends continuously to an element of B(D(〈Q〉−s),G)

some s > 1/2 alsoextends continuously to an element of B

(D(〈A〉s)∗,G). This concludes the proof. ��

Proof of Theorem 2.4 The claim follows from Theorem 3.5, whose hypotheses areverified in Lemma 4.13 and Proposition 4.11. ��Acknowledgements R. Tiedra de Aldecoa thanks the Graduate School of Mathematics of Nagoya Uni-versity for its warm hospitality in January–February 2017. The authors also thank the anonymous refereefor the valuable comments and for pointing out missing references which have been added.

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Quantum walks with an anisotropic coin I: spectral theory › ~richard › papers › unitaries_spectral.pdf · Quantum walks with an anisotropic coin are also related to Kitagawa’s

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