Absence of singular continuous spectra and embedded eigenvalues for one dimensional quantum walks with general long range coins
aa r X i v : . [ m a t h - ph ] J u l Absence of singular continuous spectra and embeddedeigenvalues for one dimensional quantum walks with generallong range coins
Masaya Maeda, Akito Suzuki, Kazuyuki WadaJuly 29, 2020
AMS Classification: 47A10, 47A40, 34L40, 81U30, 39A06
Abstract
This paper is a continuation of the paper W [29] by the third author, which studied quantumwalks with special long-range perturbations of the coin operator. In this paper, we considergeneral long-range perturbations of the coin operator and prove the non-existence of a singu-lar continuous spectrum and embedded eigenvalues. The proof relies on the construction ofgeneralized eigenfunctions (Jost solutions) which was studied in the short-range case in MSSSSdis [22].
Quantum Walks (QWs), which are usually considered to be the quantum counterpart of classicalrandom walks
ADZ, ANVW, G, M [3, 5, 12, 15] are now attracting diverse interest due to the connections with variousfields in mathematics and physics. From a numerical analysis point of view, QWs are splittingmethod of Dirac equations. Indeed, the first model of QWs which appears in Feynman’s textbookis obtained by discretization of the generator of Dirac equations in 1D FH [9]. It seems that thisconnection between Dirac equations and QWs is getting important ANF, BES, LKN, MB, MBD, MDB, MaSu, Sh, St [4, 6, 14, 16, 17, 19, 20, 25, 26].More directly, it has begun to be realized that many tools for the study of dispersive equations,such as the Dirac equations and the more well-studied Schr¨odinger equations, are transferable tothe study of QWs. More specifically, the spectral and scattering theory for QWs were studied forshort-range perturbations
ABC, ABJ, CGMV, FFSI, FFSII, HM, MoSe, MSSSSdis, RSTI, RSTII, S, W [1, 2, 7, 10, 11, 18, 21, 22, 23, 24, 27, 29]. For other directions, such asquasi-periodic perturbations and random perturbations, see
FO17JFA, K, YKE [8, 13, 28].In this paper, we study the spectral theory for long-range perturbations. In particular, weshow the absence of singular continuous spectrum and embedded eigenvalues, which are fundamentalproblems of spectral theory.To establish such result, a standard stratedy is to apply the commutator theory for unitaryoperators
ABC, ABJ, RSTI, W [1, 2, 23, 29]. However, in this paper, we construct Jost solutions with modified phases,which also called modified plane waves. This stratedy, which only applicable for one dimensionalproblems, gives the strongest result because it is based on ordinary differential (actually difference)equations. Our method also seems to be applicable to one dimensional Dirac equation with arbitrarylong range potential, which are no results up to the authors knowledge.We note that the long range perturbation was first considered by the third author for a specialclass of perturbations W [29], which generates no long range phase modification.1 .1 Set up For general C -valued map u : Z → C , we write u ( x ) = (cid:18) u ↑ ( x ) u ↓ ( x ) (cid:19) , x ∈ Z . For Banach spaces X and Y , we denote the Banach space of all bounded linear operators from X to Y by L ( X, Y ). We also set L ( X ) := L ( X, X ). For u : Z → C , we define τ u ( x ) := u ( x − x ∈ Z .Let H := l ( Z , C ). We define the shift operator S ∈ L ( H ) by (cid:18) S (cid:18) u ↑ u ↓ (cid:19)(cid:19) ( x ) := (cid:18)(cid:18) τ τ − (cid:19) (cid:18) u ↑ u ↓ (cid:19)(cid:19) ( x ) = (cid:18) u ↑ ( x − u ↓ ( x + 1) (cid:19) , x ∈ Z . (1.1) def:shift Let α, β : Z → C and θ : Z → R satisfying | α ( x ) | < | α ( x ) | + | β ( x ) | = 1 for all x ∈ Z . Wedefine the coin operator C = C α,β,θ ∈ L ( H ) by Cu ( x ) := C α ( x ) ,β ( x ) ,θ ( x ) u ( x ) , C α ( x ) ,β ( x ) ,θ ( x ) := e i θ ( x ) (cid:18) β ( x ) α ( x ) − α ( x ) β ( x ) (cid:19) ∈ U (2) , (1.2) def:coin:pre where U (2) is the set of all 2 × S and C , we can define the time evolutionoperator of a quantum walk (QW) by U u := SCu. (1.3) def:U
By gauge transformation, one can reduce the coin operator to the case β ( x ) > lem:Gauge Lemma 1.1.
There exists an unitary operator G ∈ L ( H ) such that U = G − S e CG, (cid:16) e Cu (cid:17) ( x ) = C α ( x ) , √ −| α ( x ) | ,θ ( x ) . For the proof, see Lemma 2.1 of
MSSSSdis [22].By Lemma 1.1, it suffices to consider the coin operator with β ( x ) = ρ ( x ) := p − | α ( x ) | > x ∈ Z . In the following, we write C α,θ := C α,ρ ( x ) ,θ = e i θ ( x ) (cid:18) ρ ( x ) α ( x ) − α ( x ) ρ ( x ) (cid:19) . (1.4) def:coin We are interested in the situation that the coin operator converges to a fixed unitary matrix as | x | → ∞ . In particular, we consider the following “long-range” situation: X x ∈ Z ( | α ( x + 1) − α ( x ) | + | θ ( x + 1) − θ ( x ) | ) < ∞ , | α ( x ) − α ∞ | + | θ ( x ) | → , | x | → ∞ , (1.5) ass:long where 0 = α ∞ ∈ B C (0 ,
1) := { z ∈ C | | z | < } . Remark . The former assumption of (1.5) implies that there exist α ± ∈ B C (0 ,
1) and θ ± ∈ R such that | α ( x ) − α ± | + | θ ( x ) − θ ± | → x → ±∞ . The latter assumption is added to ensure α + = α − ∈ B C (0 , \ { } and θ + = θ − = 0. We remark that there is no loss of generality assuming θ + = 0. 2 emark . We set U ∞ := SC ∞ , where C ∞ is given by C ∞ ( x ) := (cid:18) ρ ∞ α ∞ − α ∞ ρ ∞ (cid:19) , x ∈ Z , (1.6) inf:coin and ρ ∞ := p − | α ∞ | . From (1.5), C α,θ − C ∞ is compact. Therefore U − U ∞ = S ( C α,θ − C ∞ )is also compact. Since an essential spectrum of a unitary operator is invariant under a compactperturbation (Lemma 2.2 of MoSe [21]), we have σ ess ( U ) = σ ess ( U ∞ ) = { e i λ | | cos λ | ≤ ρ ∞ } , (1.7) ess where σ ess ( U ) is the set of an essential spectrum of U . The last equality in (1.7) comes from Lemma4.1 and Proposition 4.5 of RSTI [23].
Our main result is the following. main:spec
Theorem 1.4.
Assume (1.5) . Let σ sc ( U ) and σ p ( U ) are the set of a singular continuous spectrumand eigenvalues of U , respectively. Then, we have σ sc ( U ) = ∅ , (1.8) no:sing and σ p ( U ) ∩ { e i λ | | cos λ | < ρ ∞ } = ∅ . (1.9) no:emb Remark . From (1.8), the interior of essential spectrum of U is corresponds to the absolutelycontinuous spectrum.To prove Theorem 1.4, we construct modified plane waves of U and apply the limiting absorptionprinciple for unitary operators. For short-range cases, such strategy was adapted in MSSSSdis [22] (see also HM [18]). In MSSSSdis [22] the idea to construct the plane wave was to rewrite the generalized eigenvalue problemto an difference equation using the transfer matrix (TM), see Proposition 2.1. Then, consideringthe TM as the perturbation of the TM of the constant coin case, they were able to construct thesolution with the desired asymptotics because the perturbation was l .Apparently, for the construction of modified plane waves, considering the TM as a pertubationof the TM with constant coin is not sufficient. Indeed, to construct the modified phase Z (see (2.27)),we need to diagonalize the transfer matrix at each point sufficiently near infinity and subtract theremainder to make the pertubation to be in l . This simple but new strategy requires detailedanalysis for the dispersion relation of QWs and more than half of the proof of this paper is devotedto such analysis. We note that this detailed analysis of dispersion relation should be useful forstudying other properties of QWs such as scattering and inverse scattering theory. After obtainingthe esitmates of the dispersion relation, the construction of modified plane wave will be more or lesssimilar to MSSSSdis [22].This paper is organized as follows. In Sections 2.1 and 2.2, we introduce the transfer matrixfrom the generalized eigenvalue problem and study the dispersion relation induced by the transfermatrix. In Section 2.3, we construct modified plane waves which are solutions of the generalizedeigenvalue problem. In Section 3, we prove the main result by using modified plane waves and thelimiting absorption principle. 3
Generalized eigenvalue problem sec:genev
In this section, we study the following (generalized) eigenvalue problem:
U u = e i λ u, λ ∈ C / π Z , (2.1) ev:prob where u : Z → C is not necessary in H = l ( Z , C ). subsec:tm We recall several results from
MSSSSdis [22]. We set a unitary operator J VE by J VE u ( x ) := (cid:18) u ↓ ( x − u ↑ ( x ) (cid:19) . (2.2) def:jev We denote the inverse of J VE by J EV u ( x ) := J − u ( x ) = (cid:18) u ↓ ( x ) u ↑ ( x + 1) (cid:19) (2.3) def:jve For λ ∈ C / π Z and x ∈ Z , we set T λ ( x ) := ρ ( x ) − (cid:18) e i( λ − θ ( x )) α ( x ) α ( x ) e − i( λ − θ ( x )) (cid:19) . (2.4) def:tla The generalized eigenvalue problem can be rewritten using transfer matrix. prop:equiv
Proposition 2.1 (Proposition 3.2 of
MSSSSdis [22]) . For u : Z → C , the generalized eigenvalue problem (2.1) is equivalent to ( J VE u ) ( x + 1) = T λ ( x ) ( J VE u ) ( x ) , ∀ x ∈ Z . (2.5) eq:equiv As for ordinary differential equations, the Wronskian is invariant. Corollary 2.3 will be used toexclude the existence of embedded eigenvalues. prop:Wronsky
Proposition 2.2 (Proposition 3.3 of
MSSSSdis [22]) . Let v = J VE u and v = J VE u satisfy (2.5) . Then, det( v ( x ) v ( x )) is independent of x ∈ Z . cor:Wron Corollary 2.3 (Corollary 3.4 of
MSSSSdis [22]) . Let u and u satisfy (2.1) , u is bounded on Z ≥ := { x ∈ Z | x ≥ } and u ∈ H . Then, u and u are linearly dependent. The kernel of the resolvent of U can be written down using the generalized eigenfunctions. For v ∈ C (column vector), we set v ⊤ (row vector) to be the transposition of v . prop:kernel Proposition 2.4 (Proposition 3.6 of
MSSSSdis [22]) . Let v = J VE u and v = J VE u satisfy (2.5) with W λ = det( v v ) = 0 . We set K λ ( x, y ) := e − i λ W − λ (cid:18) v ( x ) v ( y ) ⊤ (cid:18)
Two eigenvalues of T λ ( x ) can be written as e ± i ξ for ξ ∈ C / π Z with Im ξ ≥ satisfyingthe relation: ρ ( x ) cos ξ = cos ( λ − θ ( x )) . (2.7) disprel Proof.
Since det T λ ( x ) = 1, we can express two eigenvalues of T λ ( x ) as e ± i ξ . Further, without loseof generality, we can assume that Im ξ ≥
0. Finally comparing the trace, we obtain (2.7).Formally, we can express λ as λ = θ + Arccos ( ρ cos ξ ). Since it is an analytic function, it can bedefined on its Riemann surface R . However, since the Riemann surface will depend on ρ ( x ) whichdepends on x in general, we restrict the domain, which will consist of two connected components.For b >
0, we set T b := { z ∈ C / π Z | < Im z < b } , T Re := { z ∈ C / π Z | Im z = 0 } , and T ± b := { ( ± , z ) | z ∈ T b } , T ± Re = { ( ± , z ) | z ∈ T Re } . (2.8)We set R b := T + b ⊔ T − b , R Re := T +Re ⊔ T − Re , R E := { ( ± , , ( ± , π ) } . (2.9) def:Rb Remark . The element of R b (closure of R b ) should be expressed as ( s , z ) where s ∈ {±} . Wewill sometimes use the notation z s := ( s , z ). Further, by abuse of notations, we express z s as z with z ∈ T b and comment on which sheet (i.e. s = + or − ) is z in if necessary. For ξ ± = ( ± , ξ ) ∈ R b , wedefine e i ξ ± := e i ξ and also define trigonometric functions in R b too.In the following, we assume 0 < | α | < < ρ < | ( − , = (cid:0) cos | (0 ,π ) (cid:1) − can be extended analytically in C \ (( −∞ , − ∪ [1 , ∞ )). Bythe formula cos ξ = cos ξ R cosh ξ I − i sin ξ R sinh ξ I , ξ = ξ R + i ξ I , (2.10) complexcos it is seen that cos( · ) maps { u + i v | < u < π, < ± v } to C ∓ := { z ∈ C | ∓ Im z > } . Thus Arccos( · )maps C ± to { u + i v | ≤ u ≤ π, < ∓ v } (See Figure 1 below). − C + C − π · ) on the complex plane.For given α ∈ B C (0 , b α > b α = ρ − = (1 − | α | ) − / . Then, for ξ ∈ T b α , we have − < Re ( ρ cos ξ ) < < b < b α , we definethe analytic function λ α,θ : R b → C / π Z by λ α,θ ( ξ ) = θ ± Arccos ( ρ cos ξ ) , ξ ∈ T ± b . (2.11) def:lambda For 0 < b < b α ∞ , we also define λ ∞ : R b → C / π Z by λ ∞ ( ξ ) := λ α ∞ , ( ξ ) = ± Arccos ( ρ ∞ cos ξ ) , ξ ∈ T ± b . (2.12) def:lambdainf Remark . R b α is a subset of the Riemann surface R of λ α,θ ( ξ ) = θ + Arccos ( ρ cos ξ ), whichconsists of two sheets. If one considers λ α,θ as an analytic extension of λ α,θ initially defined on T +Re , λ α,θ will correspond to the above definition on R b α .For 0 < b < b α , we set D α,θ,b := λ α,θ ( R b ) := { λ α,θ ( ξ ) | ξ ∈ R b } ⊂ C / π Z , E α,θ := λ α,θ ( R E ) , (2.13) def:D and D α,θ,b := D α,θ,b ∪ ∂ D α,θ,b , (2.14) def:barD where ∂ D α,θ,b := ∪ s , s ∈{±} ∂ D s , s α,θ ∪ ∂ D out α,θ,b and ∂ D s , ± α,θ := ( { λ ± i0 | θ + Arccos( ρ ) ≤ λ ≤ θ + Arccos( − ρ ) } s = + , { λ ± i0 | θ − Arccos( − ρ ) ≤ λ ≤ θ − Arccos( ρ ) } s = − ,∂ D out α,θ,b := { λ α,θ ( ξ ) | Im ξ = b } . The boundary ∂ D s , ± α,θ can be also expressed as ∂ D s , ± α,θ,b = { λ α,θ ( ξ ) | ξ ∈ T s Re , ∓ ξ ∈ [0 , π ] } . − π π i b T + b θ − π θ θ + πλ α,θ ( T + b ) − π π i b T − b θ − π θ θ + πλ α,θ ( T − b )6igure 2. Domain and image of λ α,θ . − π π i b T + b θ − π θ θ + πλ α,θ ( T + b ) − π π i b T − b θ − π θ θ + πλ α,θ ( T − b ) ∂ D + , + α,θ ∂ D + , − α,θ ∂ D − , − α,θ ∂ D − , + α,θ Figure 3. Image of ∂ D s , ± α,θ ( s = + , − ). Remark . The domain D α,θ,b consists of two connected components which are the image of T ± b .The above definition for D α,θ,b means that we distinguish λ α,θ ( ξ ) and λ α,θ ( − ξ ) for ξ ∈ R Re except ξ = 0 ± , π ± . Thus, each connected component of D α,θ,b has a slit in the interior which we extend λ α,θ from above and below. lem:bihol Lemma 2.9.
Let b ≤ b α . Then, λ α,θ : R b → D α,θ,b is a homeomorphism. Moreover, on restrictedto R b , λ α,θ : R b → D α,θ,b is a biholomorphism.Proof. By the definition of D α,θ,b , it is obvious that λ α,θ | R b is a surjection.Next, we show the injectivity. First, for ξ ∈ T b , we have 0 < Re (Arccos( ρ cos ξ )) < π . Indeed,since Re (Arccos( ρ cos ξ )) ∈ (0 , π ) for ξ ∈ T Re , if the above does not hold, there exists ξ ∈ T b suchthat Re (Arccos( ρ cos ξ )) = 0 or π . By (2.10), we see that if Re Arccos z = 0 or π , then z ∈ R , | z | ≥ ξ ∈ T b , Re ρ cos ξ ∈ ( − ,
1) so there exists no ξ ∈ T b such that Re Arccos( ρ cos ξ ) = 0or π . Thus, we have λ α,θ ( T + b ) ∩ λ α,θ ( T − b ) = ∅ . The injectivity of λ α,θ | T ± b follows from the injectivityof cos( · ) in T ± b , Arccos( · ) in { z ∈ C | Re z ∈ ( − , } and the definition of D α,θ,b , where we haveinsert cuts to have the injectivity.To show λ α,θ : R b → D α,θ,b is a biholomorphism, it suffices to show λ α,θ is analytic and thederivative do not vanish. The analyiticity is obvious and by direct computation, we have ∂ ξ λ α,θ ( ξ ) = ± ρ sin ξ p − ρ cos ξ = 0 , ξ ∈ T ± b . (2.15) dxil Thus, we see that λ α,θ is a biholomorphism in D α,θ,b .Finally, we show that λ α,θ : R b → D α,θ,b is a homeomorphism. By the above argument, itsuffices to restrict the domain to T + b . Dividing T + b = { z ∈ T + b | ≤ Re z ≤ π } ∪ { z ∈ T + b | − π ≤ Re z ≤ } , we see that the function ρ cos ξ restricted on each region are continuous up to theboundaries where the image are in { z ∈ C | Im z ≤ , − < Re z < } and { z ∈ C | Im z ≥ , − < Re z < } , respectively. Moreover, the inverse are also continuous up to the boundaries. Next, byconsidering cos λ defined on { z ∈ C / π Z | < Re z < π, Im z ≥ } and { z ∈ C / π Z | < Re z <π, Im z ≤ } , we see that it is also continuous up to the boundaries and its inverse is also continuous.Therefore, we have the conclusion. 7e will denote the inverse of λ α,θ as ξ α,θ . That is, ξ α,θ : D α,θ,b α → R b α , ξ α,θ ( λ ) := λ − α,θ ( λ ) . We also define ξ α ∞ := λ − ∞ . Recall that the definition of λ ∞ appears in (2.12). Notice that e ± i ξ α ( x ) ,θ ( x ) ( λ ) are eigenvalues of T λ ( x ).In the following, we will be expressing λ as λ = λ ∞ ( ξ ), so we have to consider the function ζ ( x, ξ ) = ξ α ( x ) ,θ ( x ) ( λ ∞ ( ξ )). The domain of ξ α ( x ) ,θ ( x ) depends on x , we would like to restrict ourselvesto a smaller region that we can define it for all x sufficiently large.We set b := b α ∞ . Then, there exists δ > | α − α ∞ | + | θ | < δ , then 2 b < b α < b , D α ∞ , , b \ ∂ D α,θ ⊂ D α,θ, b α , and ( ± θ + Arccos ρ, ± θ + Arccos( − ρ )) ⊂ (Arccos ( ρ ∞ cosh b ) , Arccos ( − ρ ∞ cosh b )) . Thus, setting e R := { ξ ∈ R b , | Im ξ > b , if Re ξ = 0 , π } , we have λ ∞ ( e R ) ⊂ D α,θ, b α for | α − α ∞ | + | θ ∞ | < δ. Therefore, we can define the analytic function ζ α,θ : e R → R b by ζ α,θ ( ξ ) := ξ α,θ ( λ ∞ ( ξ )) . Moreover, we can continuously extend ζ α,θ on R Re \ R E .We set r > | x | ≥ r , then | α ( x ) − α ∞ | + | θ ( x ) | < δ . Setting f R := e R ∪ ( R Re \ R E ),we define ζ ( x, ξ ) : Z × f R → R b by ζ ( x, ξ ) := ( ξ α ( x ) ,θ ( x ) ( λ ∞ ( ξ )) | x | ≥ r ,ξ | x | < r . (2.16) pointfreq The function ζ ( x, ξ ) converges to ξ as | x | → ∞ . lem:limzeta Lemma 2.10. ζ ( x, ξ ) → ξ as | x | → ∞ uniformly in compact sets of R b contained in f R .Proof. Let Ω ⊂ f R be a compact set of R b . We claim ξ α,θ ( λ ∞ ( ξ )) is continuous w.r.t. α, θ and ξ in { α ∈ C | | α − α ∞ | ≤ δ }× [ − δ , δ ] × Ω. Since this domain is compact, it is also uniformly continuous.Thus, for any ε >
0, there exists δ > | α − α ′ | + | δ − δ ′ | < ε , then sup ξ ∈ Ω | ξ α,θ ( λ ∞ ( ξ )) − ξ α ′ ,θ ′ ( λ ∞ ( ξ )) | < ε . Combining this fact with ξ α ( x ) ,θ ( x ) ( λ ∞ ( ξ )) → ξ α ∞ , ( λ ∞ ( ξ )) = ξ which alsocomes from the continuity of ξ α,θ we have the conclusion.It remains to show the continuity of ξ α,θ ( λ ∞ ( ξ )). Since λ ∞ is continuous, it suffices to showthe continuity of ξ α,θ ( λ ) w.r.t. the variables α, θ and λ . Let ( α n , θ n , λ n ) → ( α , θ , λ ). We set ξ n = ξ α n ,θ n ( λ n ) and ξ = ξ α ,θ ( λ ). Then, we have λ α n ,θ n ( ξ n ) = λ n → λ because λ α,θ is theinverse function of ξ α,θ . Next, from the uniform continuity, we have λ α n ,θ n ( ξ n ) − λ α ,θ ( ξ n ) → λ α ,θ ( ξ n ) − λ = λ α ,θ ( ξ n ) − λ α n ,θ n ( ξ n )+ λ α n ,θ n ( ξ n ) − λ →
0. Finally, assume ξ n ξ .Then, taking subsequence if necessary we have ξ n → ξ = ξ by the compactness of the domain.Since λ α ,θ is continuous we have λ α ,θ ( ξ n ) → λ α ,θ ( ξ ) which is different from λ because of theinjectivity of λ α ,θ proved in Lemma 2.9. Therefore, we have the conclusion.8 − b π − π i b i2 b T +2 b π − π i b i2 b Figure 4: f R consists of two sheets T ± b . Thick lines are excluded. A mark ◦ is correspond to anelement of R E .The function ζ has the following symmetry. lem:symzeta Lemma 2.11.
For ξ ∈ f R , we have ζ ( x, − ξ ) = − ζ ( x, ξ ) .Proof. We will only consider the case ξ ∈ T +2 b . If | x | < r , then the statement is trivial so we onlyconsider | x | ≥ r . First, λ ∞ ( − ξ ) = Arccos( ρ cos( − ξ )) = Arccos( ρ cos( − ξ )) = λ ∞ ( ξ ) . (2.17) sym:lam Next, since ξ α,θ is an inverse of λ α,θ , we have λ α,θ ( ξ α,θ ( λ )) = λ. On the other hand λ α,θ ( − ξ α,θ ( λ )) = θ + Arccos (cid:16) ρ cos( − ξ α,θ ( λ )) (cid:17) = θ + Arccos ( ρ cos( − ξ α,θ ( λ ))) = λ. Since λ α,θ is an injection (as an D α,θ,b function) by Lemma 2.9, we have ξ α,θ ( λ ) = − ξ α,θ ( λ ) . (2.18) sym:xi Combining (2.17) and (2.18), we have the conclusion.For ǫ >
0, we define f R ǫ := { ξ ∈ f R | | sin ξ | ≥ ǫ } . (2.19) def:Re In the following, we will encounter several (large) numbers depending on ǫ all denoted by r ǫ .Since it suffices to consider the maximum of such r ǫ ’s, we will not distinguish them. lem:zetare Lemma 2.12.
Let ǫ > . Then, there exists r ǫ > s.t. if | x | ≥ r ǫ and ξ ∈ f R ǫ ∩ R Re , then ζ ( x, ξ ) ∈ R Re .Proof. First, there exists δ = δ ǫ > λ ∞ ( f R ǫ ∩ R Re ) ⊂ ∂D α ∞ , andRe λ ∈ ( − Arccos( − ρ ) + δ, − Arccos( ρ ) − δ ) ∪ (Arccos( − ρ ) + δ, Arccos( ρ ) − δ ) . Thus, taking r ǫ > λ ∈ ( θ ( x ) − Arccos( − ρ ( x )) , θ ( x ) − Arccos( ρ ( x ))) ∪ ( θ ( x ) + Arccos( − ρ ( x )) , θ ( x ) + Arccos( ρ ( x ))) , for all | x | ≥ r ǫ . Therefore, we have λ ∞ ( ξ ) ∈ ∂ D α ( x ) ,θ ( x ) , which implies ξ α ( x ) ,θ ( x ) ( λ ∞ ( ξ )) ∈ R Re .9 rop:oddzeta Proposition 2.13.
Let ǫ > . Then, there exists r ǫ > s.t. if | x | > r ǫ and ξ ∈ f R ǫ ∩ R Re , then ζ ( x, − ξ ) = − ζ ( x, ξ ) .Proof. This is a direct consequence of Lemmas 2.11 and 2.12.In the following, we will need a bound for | ζ ( x + 1 , ξ ) − ζ ( x, ξ ) | . By the definition of ζ , it sufficesto study the derivative of ξ α,θ w.r.t. α and θ . Notice that ξ α,θ ( λ ) is analytic w.r.t. Re α, Im α and θ .By the identity λ α,θ ( ξ α,θ ( λ )) = λ, partial derivatives w.r.t. X = Re α, Im α, θ are ∂ X λ α,θ ( ξ α,θ ( λ )) + ∂ ξ λ α,θ ( ξ λ,θ ( λ )) · ∂ X ξ α,θ ( λ ) = 0 , X = Re α, Im α, θ. Thus we have ∂ X ξ α,θ ( λ ) = − ∂ X λ α,θ ( ξ α,θ ( λ )) ∂ ξ λ α,θ ( ξ α,θ ( λ )) , X = Re α, Im α, θ. (2.20) diff:xilem:zetadiff Lemma 2.14.
Let ǫ > . Then, there exists r ǫ > s.t. sup | x |≥ r ǫ ,ξ ∈ f R ǫ | ∂ X ζ α ( x ) ,θ ( x ) ( ξ ) | . ǫ − , X = Re α, Im α, θ, (2.21) diff:xi:bound2 where the implicit constant depends only on ρ ∞ .Proof. By (2.15) and (2.20), it suffices to show | sin( ξ α ( x ) ,θ ( x ) ( λ ∞ ( ξ ))) | & ǫ, | x | ≥ r ǫ , ξ ∈ f R ǫ . By Lemma 2.10, there exists r ǫ s.t. the conclusion holds.By Lemma 2.14, we have a quantitative version of Lemma 2.10. prop:zetaxi Proposition 2.15.
Let ǫ > . Then, there exists r ǫ > s.t. for ξ ∈ f R ǫ and | x | ≥ r ǫ , we have | ζ ( x, ξ ) − ξ | = | ζ α ( x ) ,θ ( x ) ( ξ ) − ζ α ∞ , ( ξ ) | . ǫ − ( | α ( x ) − α ∞ | + | θ ( x ) | ) . (2.22) eq:diffzetaxi Proof.
This is trivial from Lemma 2.14. cor:nondeg
Corollary 2.16.
Let ǫ > . Then, there exists r ǫ > s.t. if ξ ∈ f R ǫ and | x | ≥ r ǫ , then we have ζ ( x, ξ ) ∈ g R ǫ/ .Proof. This is trivial from the estimate (2.22). prop:zetal1
Proposition 2.17.
Let ǫ > . Then, there exists r ǫ > s.t. for | x | ≥ r ǫ and ξ ∈ f R ǫ , we have | ζ ( x + 1 , ξ ) − ζ ( x, ξ ) | . ǫ − ( | α ( x + 1) − α ( x ) | + | θ ( x + 1) − θ ( x ) | ) . (2.23) eq:zeta:diff In particular, for any ξ ∈ f R , we have ζ ( · + 1 , ξ ) − ζ ( · , ξ ) ∈ l ( Z ) . roof. As the proof of Proposition 2.15, by Lemma 2.14, for | x | ≥ r ǫ + 1, we have | ζ ( x + 1 , ξ ) − ζ ( x, ξ ) | = | ζ α ( x +1) ,θ ( x +1) ( ξ ) − ζ α ( x ) ,θ ( x ) ( ξ ) | . ǫ − ( | α ( x + 1) − α ( x ) | + | θ ( x + 1) − θ ( x ) | ) . Thus, we have (2.23). The fact ζ ( · + 1 , ξ ) − ζ ( · , ξ ) ∈ l ( Z ) follows from the above estimate and theassumption (1.5).By the definition of ζ , we see that if | x | ≥ r ǫ and ξ ∈ f R ǫ , e ± i ζ ( x,ξ ) are the two eigenvalues of T ( x, ξ ) := T λ ∞ ( ξ ) ( x ) = ρ ( x ) − (cid:18) e i( λ ∞ ( ξ ) − θ ( x )) α ( x ) α ( x ) e − i( λ ∞ ( ξ ) − θ ( x )) (cid:19) . (2.24) trans2 We remark that e i ζ ( x,ξ ) = e − i ζ ( x,ξ ) from Corollary 2.16.By explicit computation, setting P ( x, ξ ) := (cid:18) α ( x ) α ( x ) ρ ( x ) e i ζ ( x,ξ ) − e i( λ ∞ ( ξ ) − θ ( x )) ρ ( x ) e − i ζ ( x,ξ ) − e i( λ ∞ ( ξ ) − θ ( x )) (cid:19) , (2.25) P we have det P ( x, ξ ) = − α ( x ) ρ ( x ) sin ζ ( x, ξ ) . Corollary (2.16) shows that the right-hand side in the above equation is uniformly bounded frombelow for | x | ≥ r ǫ and ξ ∈ f R ǫ . Thus, we can diagonalize T ( x, ξ ) such as P ( x, ξ ) − T ( x, ξ ) P ( x, ξ ) = (cid:18) e i ζ ( x,ξ ) e − i ζ ( x,ξ ) (cid:19) . lem:est:diffP Proposition 2.18.
We have X | x |≥ r k P ( x + 1 , ξ ) − P ( x, ξ ) − k L ( C ) . ǫ X x ∈ Z ( | α ( x + 1) − α ( x ) | + | θ ( x + 1) − θ ( x ) | ) < ∞ . (2.26) Proof.
Since k P ( x + 1 , ξ ) k L ( C ) . ǫ
1, it suffices to show k P ( x, ξ ) − P ( x + 1 , ξ ) k L ( C ) . ǫ | α ( x + 1) − α ( x ) | + | θ ( x + 1) − θ ( x ) | . However, this is obvious from Proposition 2.17 and (2.25). subsec:modwave
We are now in the position to construct modified plain waves. We set ν ± ( ξ ) := (cid:18) α ∞ ρ ∞ e ± i ξ − e i λ ∞ ( ξ ) (cid:19) , and Z ( x, ξ ) := x − X y =0 ζ ( y, ξ ) . (2.27) eq:modphase Here we use the convention P x − y =0 = − P y = x − for x ≤ rop:main Proposition 2.19.
Let ξ ∈ e R . Then, there exist solutions of φ ± ( x + 1 , ξ ) = T ( x, ξ ) φ ± ( x, ξ ) , φ ± ( x, ξ ) − e ± i Z ( x,ξ ) ν ± ( ξ ) → , x → ±∞ . (2.28) modified Moreover, setting m ± ( x, ξ ) := e ∓ i Z ( x,ξ ) φ ± ( x, ξ ) , we have m ± ∈ C ( f R , l ∞ ( Z , C )) ∩ C ω ( f R , l ∞ ( Z , C )) . (2.29) m:reg Remark . If α, θ ∈ l ( Z ) (the short-range case), then Z ( x, ξ ) − ξx − c ± → x → ±∞ forsome constant c ± . In the long-range case, Z ( x, ξ ) represents the desired phase correction. Proof.
We only consider the + case. Since f R = ∪ ǫ> f R ǫ , it suffices to show the statement ofProposition 2.19 for f R replaced by f R ǫ with arbitrary ǫ > ξ ∈ f R ǫ , we set ψ ( x, ξ ) = e − i Z ( x,ξ ) e P ( x, ξ ) − φ + ( x, ξ ) , (2.30) def:psi where e P ( x, ξ ) = ( P ( x, ξ ) | x | ≥ r ǫ ,P ∞ ( ξ ) := (cid:16) ν + ( ξ ) ν − ( ξ ) (cid:17) | x | < r ǫ . We remark that P ∞ ( ξ ) diagonalize T ∞ ( ξ ) := lim | x |→∞ T λ ∞ ( ξ ) ( x ). Since Z ( x +1 , ξ ) = Z ( x, ξ )+ ζ ( x, ξ )and T ( x, ξ ) = e i ζ ( x,ξ ) e P ( x, ξ ) ( A ( x, ξ ) + B ( x, ξ )) e P ( x, ξ ) − , where A ( x, ξ ) := (cid:18) e − ζ ( x,ξ ) (cid:19) , and B ( x, ξ ) = ( e − i ζ ( x,ξ ) e P ( x, ξ ) − ( T ( x, ξ ) − T ∞ ( ξ )) e P ( x, ξ ) | x | < r ǫ , | x | ≥ r ǫ . By substituting (2.30) into (2.28), we obtain ψ ( x + 1 , ξ ) = ( A ( x, ξ ) + V ( x, ξ )) ψ ( x, ξ ) , ψ ( x, ξ ) − (cid:18) (cid:19) → , x → ∞ , (2.31) eq:psi where V ( x, ξ ) = (cid:16) e P ( x + 1 , ξ ) − e P ( x, ξ ) − (cid:17) A ( x, ξ ) + e P ( x + 1 , ξ ) − e P ( x, ξ ) B ( x, ξ ) . By Proposition 2.18, for x ≥ r ǫ , we havesup ξ ∈ f R ǫ k V ( · , ξ ) k l ( Z ·≥ x , L ( C )) . ǫ X x ≤ y ( | α ( y + 1) − α ( y ) | + | θ ( y + 1) − θ ( y ) | ) < ∞ .
12e will divide the x region in three parts. First, we take x ≥ r ǫ sufficiently large so thatsup ξ ∈ e R ǫ k V ( · , ξ ) k l ( Z ·≥ x , L ( C )) ≤ /
2. We consider the region x ≥ x . Rewriting (2.31) in theDuhamel form, it suffices to solve ψ ( x, ξ ) = (cid:18) (cid:19) − ( D ( ξ ) ψ ( · , ξ )) ( x ) , (2.32) eq:Neu1 where ( D ( ξ ) v ) ( x ) := ∞ X y = x (cid:18) e P yw = x ζ ( w,ξ ) (cid:19) V ( y, ξ ) v ( y ) . Since | e P yw = x ζ ( w,ξ ) | ≤
1, we have k D ( ξ ) k L ( l ∞ ( Z ·≥ x , C ) ) ≤ k V ( · , ξ ) k l ( Z ·≥ x , L ( C )) ≤ / . (2.33)Therefore, we have ψ ( x, ξ ) = (cid:18) (cid:19) + ∞ X n =1 (cid:18) D n ( ξ ) (cid:18) (cid:19)(cid:19) ( x ) , where the infinite series converges in l ∞ ( Z ·≥ x , C ). Therefore, we have the conclusion with Z replaced by Z ·≥ x .Next, we take x < ξ ∈ e R ǫ k V ( · , ξ ) k l ( Z ·≤ x , L ( C )) ≤ /
2. In the finite region ( x , x ) ∩ Z , we simply define ψ using (2.31).Finally, to define ψ in the region x < x , notice that as (2.32), it suffices to solve ψ ( x, ξ ) = (cid:18) e P x − w = x ζ ( w,ξ ) (cid:19) ψ ( x , ξ ) − ( E ( ξ ) ψ ( ξ, · )) ( x ) , where ( E ( ξ ) v ) ( x ) := x − X y = x (cid:18) e P yw = x ζ ( w,ξ ) (cid:19) V ( y, ξ ) v ( y ) . Since we can show k E ( ξ ) k L ( l ∞ ( Z ·≤ x , C )) ≤ /
2, we can express ψ ( x, ξ ) as ψ ( x, ξ ) = ψ ( x , ξ ) + ∞ X n =1 E ( ξ ) n ψ ( x , ξ ) . Therefore, we have the conclusion. sec:prmain
We first prove the non-existence of embedded eigenvalues.
Proof of (1.9) of Theorem 1.4.
We will only consider the eigenvalues e i λ in the upper half plane of C . That is, we assume λ ∈ (0 , π ) and | cos λ | < ρ ∞ . Then, there exists ξ ∈ T \ { , π } s.t. λ ∞ ( ξ ) = λ .Now, suppose that e i λ ∞ ( ξ ) is an eigenvalue of U . That is, there exists ψ ∈ H s.t. U ψ = e i λ ∞ ( ξ ) ψ .On the other hand, φ + ( · , ξ ) is another solution of (2.5) bounded on x ≥
0. Corollary 2.3 shows that φ + ( · , ξ ) and ψ are linearly dependent. However, this cannot happen so it is a contradiction.13or the proof of (1.8), we recall some basic facts which will be a direct consequence of thelimiting absorption principle.By Proposition 2.4 and Proposition 2.19, for ξ ∈ f R , we have (cid:16) ( U − e − i λ ∞ ( ξ ) ) − u (cid:17) ( x ) = X y ∈ Z K ξ ( x, y ) u ( y ) , where K ξ ( x, y ) := e − i λ ∞ ( ξ ) W − ξ (cid:18) e i P y − w = x ζ ( w,ξ ) m − ( x, ξ ) m + ( y, ξ ) ⊤ (cid:18)
Let ξ ∈ R Re \ R E . Then, lim ε → +0 ( U − e i λ ∞ ( λ +i ε ) ) − exists in L ( l σ , l − σ ) for σ > / .Proof. Since k ( U − e − i λ ∞ ( ξ ) ) − k L ( l σ ,l − σ ) ≤ k h x i − σ K ξ ( x, y ) h y i − σ k l ( Z , L ( C )) , it suffices to show that h x i − σ K ξ +i ǫ ( x, y ) h y i − σ → h x i − σ K ξ ( x, y ) h y i − σ as ǫ → +0 in l ( Z , L ( C )).However, this immediately follows from Proposition 2.19 and the fact that Im ζ ( x, ξ ) ≥ x ∈ Z and ξ ∈ e R Proof of (1.8) of Theorem 1.4.
By Stone’s formula for unitary operators and the limiting absorptionprinciple (Proposition 3.1) imply the non-existence of the singular continuous spectrum.
Acknowledgments
The first author was supported by the JSPS KAKENHI Grant Numbers JP19K03579, G19KK0066A,JP17H02851 and JP17H02853. The second author was supported by the JSPS KAKENHI GrantNumbers JP18K03327. This work was supported by the Research Institute of Mathematical Sciences,an International Joint Usage/Research Center located in Kyoto University and by 2019 IMI JointUse Research Program Short-term Joint Research “Mathematics for quantum walks as quantumsimulators”.
References [1] M. A. Astaburuaga, O. Bourget, and V. H. Cort’es. Commutation relations for unitaryoperators I, J. Funct. Anal., 268(8):21882230, 2015.[2] J. Asch, O. Bourget and A. Joye Spectral stability of unitary network models, Rev.Math.Phys, vol.27, No.7, 1530004, 2015.[3] Y. Aharonov, L. Davidovich and N. Zagury, Quantum random walks, Phys. Rev.A, 48 ,16871690, (1993). 144] P. Arrighi, V. Nesme, and M. Forets, The dirac equation as a quantum walk: higherdimensions, observational convergence, Journal of Physics A: Mathematical and Theoret-ical 47 (2014), no. 46, 465302.[5] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, One-dimensional quantumwalks, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing,ACM, New York, 2001, pp. 3749.[6] A. J. Bracken, D. Ellinas, and I. Smyrnakis, Free-dirac-particle evolution as a quantumrandom walk, Phys. Rev. A 75 (2007), 022322.[7] M. J. Cantero, F. A. Grnbaum, L. Moral, and L. Velzquez, One-dimensional quantum walkswith one defect, Rev. Math. Phys. 24(2): 1250002, 52p, 2012.[8] J. Fillman and D. C. Ong, Purely singular continuous spectrum for limit-periodic CMVoperators with applications to quantum walks, J. Funct. Anal. (2017), no. 12, 5107–5143.[9] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, emended ed., DoverPublications, Inc., Mineola, NY, 2010, Emended and with a preface by Daniel F. Styer.[10] T. Fuda, D. Funakawa, and A. Suzuki, Localization of a multi-dimensional quantum walkwith one defect, Quantum Information Processing , Vol59, No.203, 2017.[11] T. Fuda, D. Funakawa, and A. Suzuki, Localization for a one-dimensional split-step quantumwalk with bound states robust against perturbations, J. Math. Phys. 59, 082201, 2018.[12] S. P. Gudder, Quantum Probability. Probability and Mathematical Statistics, AcademicPress Inc., Boston, MA, 1988.[13] N. Konno, One-dimensional discrete-time quantum walks on random environments, Quan-tum Information Processing, Vol.8, No.5, pp.387-399 2009.[14] C. W. Lee, P. Kurzy’nski, and H. Nha, Quantum walk as a simulator of nonlinear dynamics:Nonlinear dirac equation and solitons, Phys. Rev. A 92 (2015), 052336.[15] D. A. Meyer, From quantum cellular automata to quantum lattice gases, J. Stat. Phys., 85,551574, (1996).[16] L. Mlodinow and T. A. Brun, Discrete spacetime, quantum walks, and relativistic waveequations, Phys. Rev. A 97 (2018), 042131.[17] G. di Molfetta, M. Brachet, and F. Debbasch, Quantum walks as massless dirac fermions incurved space-time, Phys. Rev. A 88 (2013), 042301[18] H. Morioka, Generalized eigenfunctions and scattering matrices for position-dependentquantum walks, Rev. Math.Phys., https://doi.org/10.1142/S0129055X19500193, 2019.[19] G. di Molfetta, F. Debbasch, and M. Brachet, Nonlinear optical galton board:Thermalization and continuous limit, Phys. Rev. E 92 (2015), 042923.[20] M. Maeda and A. Suzuki, Continuous limits of linear and nonlinear quantum walks, Rev.Math. Phys., https://doi.org/10.1142/S0129055X20500087.1521] H. Morioka and E. Segawa, Detection of edge defects by embedded eigenvalues of quantumwalks, Quantum Inf Process. 18:283, https://doi.org/10.1007/s11128-019-2398-z, 2019.[22] M. Maeda, H. Sasaki, E. Segawa, A. Suzuki, and K. Suzuki, Dispersive estimates for quantumwalks on 1d lattice, preprint (arXiv:1808.05714v2)[23] S. Richard, A. Suzuki and R. Tiedra de Aldecoa, Quantum walks with an anisotropic coinI: spectral theory, Lett. Math. Phys. vol. 108, 331-357, 2018.[24] S. Richard, A. Suzuki and R. Tiedra de Aldecoa, Quantum walks with an anisotropic coinII: scattering theory, Lett. Math. Phys. vol. 109, 2, 61-88, 2019.[25] Y. Shikano, From discrete time quantum walk to continuous time quantum walk in limitdistribution, J. Comput. Theor. Nanosci. 10 (2013), 15581570.[26] F. W. Strauch, Relativistic quantum walks, Phys. Rev. A 73 (2006), 054302.[27] A. Suzuki, Asymptotic velocity of a position-dependent quantum walk, Quantum Informa-tion Processing, 15, 103119, 2016.[28] Y. Yin, D. E. Katsanos, and S. N. Evangelou, Quantum walks on a random environment,Phys. Rev. A 77, 022302. https://doi.org/10.1103/PhysRevA.77.022302[29] K. Wada, A weak limit theorem for a class of long-range-type quantum walks in 1d, QuantumInformation Processing (2020) 19: 2. https://doi.org/10.1007/s11128-019-2491-3Department of Mathematics and Informatics, Graduate School of Science, Chiba University,Chiba 263-8522, Japan.
E-mail Address : [email protected] Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato,Nagano 380-8553, Japan.
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E-mail Address : [email protected]@hachinohe.kosen-ac.jp