Detection of edge defects by embedded eigenvalues of quantum walks
DDETECTION OF EDGE DEFECTS BY EMBEDDEDEIGENVALUES OF QUANTUM WALKS
HISASHI MORIOKA AND ETSUO SEGAWA
Abstract.
We consider a position-dependent quantum walk on Z . In partic-ular, we derive a detection method for edge defects by embedded eigenvaluesof its time evolution operator. In the present paper, an edge defect is a set { y − , y } for y ∈ Z on which the coin operator is an anti-diagonal matrix.In fact, under some suitable assumptions, the existence of a finite number ofedge defects is equivalent to the existence of embedded eigenvalues of the timeevolution operator. In view of applications, by checking the spectrum, we candetect the existence of disconnecting edge (in the sense of edge defects above)on the line without directly watching it. Introduction
Quantum walks have been studied in various kinds of research fields (see [1],[17], [21] et al. and its references). Recently, there is an abundance of studies onposition-dependent quantum walks in view of the spectral theory of unitary opera-tors. Some results of the weak limit theorem for position-dependent quantum walkswere proved by Konno-Luczak-Segawa [9], Endo-Konno [4] and Endo et al. [5]. Inview of the scattering theory, the wave operators associated with the time evolutionoperator were introduced by Suzuki [18] under the short-range type condition, aswell as the asymptotic velocity of the quantum walker and the weak limit theoremwere considered as applications. We also mention about Richard-Suzuki-Tiedra deAldecoa [15]. A Mourre theory for unitary operators is given and its application tothe spectral theory of the quantum walk is derived.In some models of quantum walks, localization occurs depending on its initialstates, and eigenvalues of the time evolution operator have a crucial role in thelocalization. If U is a unitary time evolution operator for one-dimensional, two-state quantum walks, eigenvalues and eigenspaces are defined as follows. If thereexists a non-trivial solution ψ ∈ (cid:96) ( Z ; C ) to the equation U ψ = e iθ ψ for θ ∈ [0 , π ),we call e iθ an eigenvalue of U . Thus the associated eigenspace E ( θ ) is a subspace of (cid:96) ( Z ; C ). As has been shown by Cantero et al. [3], and Suzuki [18], if the initialstate has an overlap with E ( θ ) i.e. the initial state is not in E ( θ ) ⊥ in the sense Date : August 2, 2019.2000
Mathematics Subject Classification.
Primary 47A75, Secondary 47A40.
Key words and phrases.
Quantum walk, Eigenvalue, Edge defect.H. Morioka was supported by the Grant-in-aid for young scientists (B) No. 16K17630, JSPS.E. Segawa was supported by the Grant-in-aid of Scientific Research (C) No. 19K036116, JSPS,and Research Origin for Dressed Photon. a r X i v : . [ m a t h . SP ] A ug H. MORIOKA AND E. SEGAWA of (cid:96) ( Z ; C ), the localization occurs in the associated quantum walk. Examples oflocalizations with one-defect model are in Cantero et al. [3], Konno-Luczak-Segawa[9] and Fuda-Funakawa-Suzuki [6]. More generally, we can see a similar result forlocalizations for quantum walks on graphs (see Segawa-Suzuki [16]).In this paper, we consider an approach of detection of edge defects by usingembedded eigenvalues of the time evolution operator of the one-dimensional, two-state quantum walk. The rigorous meaning of edge defects will be defined below.Let H = (cid:96) ( Z ; C ) be the space of states. The unitary operator U is given by( U ψ )( x ) = P ( x + 1) ψ ( x + 1) + Q ( x − ψ ( x − , x ∈ Z , for every ψ ∈ H and P ( x ) = (cid:20) a ( x ) b ( x )0 0 (cid:21) , Q ( x ) = (cid:20) c ( x ) d ( x ) (cid:21) . Here we assume C ( x ) := P ( x ) + Q ( x ) ∈ U (2) for every x ∈ Z and U is rewrittenby U = SC where S is the shift operator defined by( Sψ )( x ) = (cid:20) ψ (0) ( x + 1) ψ (1) ( x − (cid:21) , ψ ∈ H , x ∈ Z . Taking an initial state ψ ∈ H , we put ψ ( t, · ) := U t ψ for t ∈ { , , , . . . } . Sincethe operator U depends on the position, we call this discrete time evolution onedimensional position-dependent quantum walk . Thus we call C the coin operator of the operator U . The corresponding position-independent quantum walk is givenby U = SC where C := P + Q ∈ U (2) and P = (cid:20) a b (cid:21) , Q = (cid:20) c d (cid:21) . We adopt the representation of C which is introduced in [15]. Precisely, we put a = pe iα , b = qe iβ , c = − qe − i ( β − γ ) and d = pe − i ( α − γ ) for α, β, γ ∈ [0 , π ) and p, q ∈ [0 ,
1] with p + q = 1 :(1.1) C = e iγ/ (cid:20) pe i ( α − γ/ qe i ( β − γ/ − qe − i ( β − γ/ pe − i ( α − γ/ (cid:21) . Throughout of the paper, we assume that there exist constants ρ, M > (cid:107) C ( x ) − C (cid:107) ∞ ≤ M e − ρ (cid:104) x (cid:105) , x ∈ Z , where (cid:107) · (cid:107) ∞ is the norm of 2 × (cid:107) A (cid:107) ∞ = max ≤ j,k ≤ | a jk | , A = [ a jk ] ≤ j,k ≤ , and (cid:104) x (cid:105) = √ x .In the present paper, we consider the existence or the non-existence of edgedefects on Z . Here we define edge defects as follows. ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 3
Definition 1.1.
We call the set e y = { y − , y } for y ∈ Z an edge defect if C ( x ) = C for x ∈ e y where(1.3) C = e iγ (cid:48) / (cid:20) e i ( β (cid:48) − γ (cid:48) / − e − i ( β (cid:48) − γ (cid:48) / (cid:21) , for β (cid:48) , γ (cid:48) ∈ [0 , π ).Let us make a remark on Definition 1.1 in view of applications. If the edgedefect occurs, then there is a disconnection between { y − , y } in the network bythe definition. So in this paper we propose a detection way of the existence ofa disconnecting part without directly watching it. Turning our mind to quantumsearch algorithms driven by quantum walks, we notice that the quantum coins at thetarget vertices are also perfect reflection operators. Then it is possible to interpretthat the setting of the edge defect is an infinite system’s analogue of quantumsearch algorithms whose target vertices are e.g., { , } ; in this “algorithm”, we canfind how the defects occurs at the targets checking the spectrum of this system (seeFigs. 1-4 in § U embedded in the interior of the continuousspectrum σ ess ( U ). The first result of the present paper is as follows. Theorem 1.2.
Let p ∈ (0 , . We assume that there is no edge defect i.e. thereexists a constant δ > such that | a ( x ) | ≥ δ for all x ∈ Z . Moreover, supposethat C and C satisfy the condition (1.2). Then the continuous spectrum of U is σ ess ( U ) = { e iθ ; θ ∈ J γ } where J γ = J γ, ∪ J γ, with J γ, = [arccos p + γ/ , π − arccos p + γ/ ,J γ, = [ π + arccos p + γ/ , π − arccos p + γ/ . Moreover, there is no eigenvalue in σ ess ( U ) \ T where T = { e iθ ∈ σ ess ( U ) ; θ ∈ J γ, T } with J γ, T = (cid:40) arccos p + γ/ , π − arccos p + γ/ ,π + arccos p + γ/ , π − arccos p + γ/ (cid:41) . If there are some edge defects, the operator U is given as follows. Let C bedefined by (1.3). For a positive integer N >
0, we take y , · · · , y N ∈ Z , and put e = N (cid:91) j =1 e y j , e y j = { y j − , y j } . For any subset A ⊂ Z , let the operator F A on H be defined by ( F A ψ )( x ) = ψ ( x )for x ∈ A and ( F A ψ )( x ) = 0 for x ∈ Z \ A . Then we put(1.4) C = N (cid:88) j =1 F e yj C + (1 − F e ) C = F e C + (1 − F e ) C , H. MORIOKA AND E. SEGAWA where the coin operator C given by C ( x ) = (cid:20) a ( x ) b ( x ) c ( x ) d ( x ) (cid:21) ∈ U (2) , x ∈ Z , satisfies the assumption (1.2) and there exists a constant δ > | a ( x ) | ≥ δ for all x ∈ Z . In this case, the situation of U and U is same as Theorem 1.3 in Z \ e . However, there exists an embedded eigenvalue as follows. Theorem 1.3.
Let p ∈ (0 , and C be given by (1.4).(1) The continuous spectrum of U is σ ess ( U ) = { e iθ ; θ ∈ J γ } .(2) For any γ (cid:48) ∈ [0 , π ) , we have ± ie iγ (cid:48) / ∈ σ p ( U ) , and we can take associatedeigenfunctions Ψ ± ∈ H such that suppΨ ± ⊂ e .(3) If ( γ (cid:48) + π ) / ∈ J γ \ J γ, T , we have ± ie iγ (cid:48) / ∈ σ p ( U ) ∩ ( σ ess ( U ) \T ) . Any associatedeigenfunctions Ψ ± vanish in { x ∈ Z ; x > x ∗ or x < x ∗ } where x ∗ = max { x ∈ e } and x ∗ = min { x ∈ e } . As a consequence of Theorems 1.2 and 1.3, we can state the conclusion of thispaper.
Corollary 1.4.
Let p ∈ (0 , and ( γ (cid:48) + π ) / ∈ J γ \ J γ, T . Suppose C is given by(1.4). There is no edge defect i.e. e = ∅ if and only if U has no eigenvalue in σ ess ( U ) \ T . Theorems 1.2 and 1.3 are analogues of the Rellich type uniqueness theorem forthe Helmholtz equation ( − ∆ − λ ) u = 0 on the Euclidean space. Originally it wasintroduced by Rellich [14] and Vekoua [20]. This theorem has been generalizedto a broad class of partial differential equations, since it plays important roles inthe spectral theory ([19], [10], [11], [7], [12] and [13]). Recently, this theorem wasgeneralized for the discrete Schr¨odinger operator on perturbed periodic graphs ([8],[22] and [2]). Note that the Rellich type uniqueness theorem holds in a Banach spacelarger than L -space or (cid:96) -space. However, it is sufficient to prove in (cid:96) ( Z ; C ) forour purpose of the paper. For the proof, we use a Paley-Wiener theorem and thetheory of complex variable.The plan of this paper is as follows. In §
2, we recall basic properties of spectra ofunitary operators. The proof of Theorem 1.2 is given in §
3. The precise constructionof embedded eigenvalues and the associated eigenfunctions are given in §
4. Wesummarize our arguments in §
5, using some numerical examples.Throughout of this paper, we use the following basic notations. We denote theflat torus by T = R / (2 π Z ) and the complex torus by T C = C / (2 π Z ). For any s ∈ R , we put (cid:104) s (cid:105) = √ s . The unit circle on the complex plane C is denotedby S . ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 5 Continuous spectrum
Spectral decomposition of unitary operators.
Here let us recall somegeneral properties of spectra of unitary operators. Let H be a Hilbert space. Wedenote by ( · , · ) H the inner product of H and by (cid:107) · (cid:107) H the associated norm.Let U be a unitary operator on H . It is well-known that there exists a spectraldecomposition E U ( θ ) for θ ∈ R such that U = (cid:90) π e iθ dE U ( θ ) , where E U ( θ ) is extended to be zero for θ ∈ ( −∞ ,
0) and to be 1 for θ ∈ [2 π, ∞ ). Itis well-known that σ ( U ) ⊂ S . Since E U ( θ ) is a measure on R , applying Radon-Nikod´ym theorem, it provides the orthogonal decomposition of H associated with U as H = H p ( U ) ⊕ H sc ( U ) ⊕ H ac ( U ) , where H p ( U ) is spanned by eigenfunctions of U , H sc ( U ) and H ac ( U ) are orthogonalprojections on the pure point, the singular continuous and the absolutely continuoussubspace of H , respectively. Then we put σ p ( U ) = the set of eigenvalues of U in H ,σ sc ( U ) = σ ( U | H sc ( U ) ) , σ ac ( U ) = σ ( U | H ac ( U ) ) , and we call them the point spectrum, the singular continuous spectrum and theabsolutely continuous spectrum of U , respectively.We also define the discrete spectrum and the essential spectrum of U . Thediscrete spectrum σ d ( U ) is the set of isolated eigenvalues of U with finite multiplic-ities. The essential spectrum σ ess ( U ) is defined by σ ess ( U ) = σ ( U ) \ σ d ( U ). Thenif λ ∈ σ ess ( U ), λ is either an eigenvalue of infinite multiplicity or an accumulationpoint of σ ( U ).As in the case of self-adjoint operators, the essential spectrum of U is character-ized by singular sequences as follows. Lemma 2.1.
We have e iθ ∈ σ ess ( U ) for θ ∈ [0 , π ) if and only if there exists asequence { ψ n } ∞ n =1 in H such that (cid:107) ψ n (cid:107) H = 1 , ψ n → weakly in H and (cid:107) ( U − e iθ ) ψ n (cid:107) H → as n → ∞ . Proof. Suppose e iθ ∈ σ ess ( U ). When e iθ is an eigenvalue of infinite multiplici-ties, we can take an orthonormal basis { ψ n } ∞ n =1 in Ker( U − e iθ ). When e iθ is anaccumulation point of σ ( U ), we can take a sequence { θ n } ∞ n =1 such that e iθ n ∈ σ ( U )and θ n → θ . We take sufficiently small (cid:15) n > I n = ( θ n − (cid:15) n , θ n + (cid:15) n )satisfies I n ∩ I m = ∅ for m (cid:54) = n . By choosing ψ n ∈ Ran( E U ( I n )) with (cid:107) ψ n (cid:107) H = 1,we have an orthonormal basis { ψ n } ∞ n =1 . Moreover, we obtain (cid:107) ( U − e iθ ) ψ n (cid:107) H = (cid:90) I n | e is − e iθ | d ( E U ( s ) ψ n , ψ n ) H ≤ C(cid:15) n → . H. MORIOKA AND E. SEGAWA
Suppose that there exists a sequence { ψ n } ∞ n =1 such that ψ n satisfies the conditionin the statement of the lemma. If e iθ (cid:54)∈ σ ( U ), there exists a constant δ > E U (( θ − δ, θ + δ )) = 0 and (cid:107) ( U − e iθ ) ψ (cid:107) H ≥ δ for any ψ ∈ H . This is a contradiction.If e iθ ∈ σ d ( U ), there exists a constant (cid:15) > E U (( θ − (cid:15), θ + (cid:15) )) = E U ( { θ } )for e iθ (cid:54) = 1 or E U (( − (cid:15), (cid:15) )) + E U ((2 π − (cid:15), π + (cid:15) )) = E U ( { } ) + E U ( { π } ) for e iθ = 1.In the following, we shall prove the case e iθ (cid:54) = 1. For e iθ = 1, the proof is similar.We can take an orthonormal basis { φ j } mj =1 of Ker( U − e iθ ) for a positive integer m . Applying the Gram-Schmidt orthonormalization to { φ j } mj =1 ∪ { ψ k } ∞ k =1 , we putthe resulting sequence { φ (cid:48) j } ∞ j =1 . Note that φ (cid:48) j = φ j for j = 1 , · · · , m . Hence wehave (cid:107) ( U − e iθ ) φ (cid:48) j (cid:107) H → j → ∞ . On the other hand, we have (cid:107) ( U − e iθ ) φ (cid:48) j (cid:107) H = (cid:90) | s − θ |≥ (cid:15) | e i ( s − θ ) − | d ( E U ( s ) φ (cid:48) j , φ (cid:48) j ) H ≥ (cid:15) , for j > m . This is a contradiction. (cid:3) As a consequence, we can see that compact perturbations of U do not change itsessential spectrum. Lemma 2.2.
Let U (cid:48) and U be unitary operators on H . If U (cid:48) − U is compact on H , we have σ ess ( U (cid:48) ) = σ ess ( U ) . Proof. Let e iθ ∈ σ ess ( U ). In view of Lemma 2.1, there exists a sequence { ψ n } ∞ n =1 in H such that (cid:107) ψ n (cid:107) H = 1, ψ n → H and (cid:107) ( U − e iθ ) ψ n (cid:107) H → n → ∞ .Since U (cid:48) − U is compact, we have ( U (cid:48) − U ) ψ n → H . Then we have (cid:107) ( U (cid:48) − e iθ ) ψ n (cid:107) H ≤ (cid:107) ( U − e iθ ) ψ n (cid:107) H + (cid:107) ( U (cid:48) − U ) ψ n (cid:107) H → . Applying Lemma 2.1 to U (cid:48) , we obtain e iθ ∈ σ ess ( U (cid:48) ). This implies σ ess ( U ) ⊂ σ ess ( U (cid:48) ). We can prove σ ess ( U (cid:48) ) ⊂ σ ess ( U ) by the same way. (cid:3) Essential spectrum.
We turn to the quantum walk. In the following, thenotations U and U are used in order to represent the unitary operators of timeevolution for the quantum walk, and H = (cid:96) ( Z ; C ). Let F : H → (cid:98) H := L ( T ; C )be the unitary operator defined by( F ψ )( ξ ) = (cid:34) (cid:98) ψ (0) ( ξ ) (cid:98) ψ (1) ( ξ ) (cid:35) , (cid:98) ψ ( j ) ( ξ ) = 1 √ π (cid:88) x ∈ Z e − ixξ ψ ( j ) ( x ) , for ξ ∈ T , j = 0 ,
1, and every ψ ∈ H . Then the adjoint operator F ∗ : (cid:98) H → H isgiven by ( F ∗ (cid:98) φ )( x ) = (cid:20) φ (0) ( x ) φ (1) ( x ) (cid:21) , φ ( j ) ( x ) = 1 √ π (cid:90) T e ixξ (cid:98) φ ( j ) ( ξ ) dξ, for x ∈ Z , j = 0 ,
1, and every (cid:98) φ ∈ (cid:98) H .Letting (cid:98) U = F U F ∗ = F SC F ∗ , ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 7 we have that (cid:98) U is the operator of multiplication by the unitary matrix(2.1) (cid:98) U ( ξ ) = (cid:20) a e iξ b e iξ c e − iξ d e − iξ (cid:21) . In view of (1.1), we have(2.2) (cid:98) U ( ξ ) = e iγ/ (cid:20) pe i ( α − γ/ e iξ qe i ( β − γ/ e iξ − qe − i ( β − γ/ e − iξ pe − i ( α − γ/ e − iξ (cid:21) . Moreover, we obtain for any λ ∈ C (2.3) det( (cid:98) U ( ξ ) − λ ) = λ − λpe iγ/ cos (cid:16) ξ + α − γ (cid:17) + e iγ . In view of (2.3), we can see the following fact. For the proof, see Lemma 4.1 in [15].
Lemma 2.3. (1) If p = 0 , we have σ ( U ) = σ p ( U ) = {± ie iγ/ } .(2) If p ∈ (0 , , we have σ ( U ) = σ ac ( U ) = { e iθ ; θ ∈ J γ } .(3) If p = 1 , we have σ ( U ) = σ ac ( U ) = S . In view of the assumption (1.2), the operator U − U is compact on H . ApplyingLemma 2.2, we obtain the following lemma. Lemma 2.4. (1) If p ∈ (0 , , we have σ ess ( U ) = σ ess ( U ) = { e iθ ; θ ∈ J γ } .(2) If p = 1 , we have σ ess ( U ) = σ ess ( U ) = S . Absence of embedded eigenvalues
Thresholds.
Let M ( θ ) = { ξ ∈ T ; p ( ξ, θ ) = 0 } , (3.1) M reg ( θ ) = { ξ ∈ T ; p ( ξ, θ ) = 0 , ∂ ξ p ( ξ, θ ) (cid:54) = 0 } , (3.2) M sng ( θ ) = { ξ ∈ T ; p ( ξ, θ ) = 0 , ∂ ξ p ( ξ, θ ) = 0 } , (3.3)where p ( ξ, θ ) = det( (cid:98) U ( ξ ) − e iθ ). Note that p ( ξ, θ ) is a trigonometric polynomial in ξ (see (2.3)). Lemma 3.1.
Suppose p ∈ (0 , . If θ ∈ J γ \ J γ, T , we have M ( θ ) = M reg ( θ ) and M sng ( θ ) = ∅ . If θ ∈ J γ, T , we have M ( θ ) = M sng ( θ ) and M reg ( θ ) = ∅ . Proof. Note that ∂ ξ p ( ξ, θ ) = 2 pe iγ/ e iθ sin (cid:16) ξ + α − γ (cid:17) . Then ∂ ξ p ( ξ, θ ) = 0 if and only if ξ + α − γ/ π . If p ( ξ, θ ) = ∂ ξ p ( ξ, θ ) = 0,we have that e iθ must be equal to one of the following values : e iγ/ (cid:16) p ± i (cid:112) − p (cid:17) , e iγ/ (cid:16) − p ± i (cid:112) − p (cid:17) . The lemma follows from these observations. (cid:3)
H. MORIOKA AND E. SEGAWA
Absence of embedded eigenvalues. In § σ p ( U ) ∩ ( σ ess ( U ) \ T ) andwe show a contradiction.Let us recall the assumptions which we adopt in § p ∈ (0 ,
1] and there exists a constant δ > | a ( x ) | ≥ δ for all x ∈ Z .(2) There exist constants ρ, M > (cid:107) C ( x ) − C (cid:107) ∞ ≤ M e − ρ (cid:104) x (cid:105) for any x ∈ Z .We assume e iθ ∈ σ p ( U ) ∩ ( σ ess ( U ) \ T ) and let ψ ∈ H be the associated eigen-function. Putting f = − ( U − U ) ψ ∈ H , the equation ( U − e iθ ) ψ = 0 is rewrittenas ( U − e iθ ) ψ = f on Z . In view of the assumption (2), we have e r (cid:104)·(cid:105) f ∈ H for any r ∈ (0 , ρ ). Passing to theFourier series, we have(3.4) ( (cid:98) U ( ξ ) − e iθ ) (cid:98) ψ = (cid:98) f on T . Moreover, we multiply the equation (3.4) by the cofactor matrix of (cid:98) U ( ξ ) − e iθ . Notethat each component of the cofactor matrix is trigonometric polynomials. Then thematrix (cid:98) U ( ξ ) − e iθ is diagonalized and it is sufficient to consider the equation of theform(3.5) p ( ξ, θ ) (cid:98) u = (cid:98) g on T , where (cid:98) u, (cid:98) g ∈ L ( T ).Here we need a Paley-Wiener type theorem. The following one is Theorem 6.1in [22]. Theorem 3.2.
Let k > be a constant. For a function φ ∈ (cid:96) ( Z ) , e k (cid:104)·(cid:105) φ ∈ (cid:96) ( Z ) for any k ∈ (0 , k ) if and only if the function (cid:98) φ extends to analytic function in { z ∈ T C ; | Im z | < k / (2 π ) } . As a direct consequence, we have the following fact.
Lemma 3.3.
The function (cid:98) g in (3.5) extends to an analytic function in { z ∈ T C ; | Im z | < ρ/ (2 π ) } . Proof. Since we have e r (cid:104)·(cid:105) f ∈ H for any r ∈ (0 , ρ ), we apply Theorem 3.2 to f sothat (cid:98) f is analytic in { z ∈ T C ; | Im z | < ρ/ (2 π ) } . Each component of the cofactormatrix is trigonometric polynomials. Then (cid:98) g is also analytic in { z ∈ T C ; | Im z | <ρ/ (2 π ) } . (cid:3) Next we discuss about the regularity of (cid:98) u . Lemma 3.4.
Let (cid:98) u ∈ L ( T ) satisfy the equation (3.5). Then (cid:98) u ∈ C ∞ ( T ) . Inparticular, we have (cid:98) g ( ξ ( θ )) = 0 for ξ ( θ ) ∈ M ( θ ) . ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 9
Proof. We take ξ ( θ ) ∈ M ( θ ). Note that M ( θ ) = M reg ( θ ) from e iθ ∈ σ p ( U ) ∩ ( σ ess ( U ) \ T ). Let χ ∈ C ∞ ( T ) satisfy χ ( ξ ( θ )) = 1 with small support. In view of ξ ( θ ) ∈ M reg ( θ ), we have ∂ ξ p ( ξ ( θ ) , θ ) (cid:54) = 0. Thus we can make the change of variable η = cos (cid:16) ξ + α − γ (cid:17) − cos (cid:16) ξ ( θ ) + α − γ (cid:17) , in a small neighborhood of ξ ( θ ). Letting (cid:98) u χ = χ (cid:98) u and (cid:98) g χ = χ (cid:98) g , we rewrite theequation (3.5) as(3.6) η (cid:98) u χ = − p e − i ( θ + γ/ (cid:98) g χ on T . Now let us define the Fourier transformation by (cid:102) u χ ( t ) = 1 √ π (cid:90) ∞−∞ e − itη (cid:98) u χ ( η ) dη, t ∈ R . We define (cid:102) g χ ( t ) by the same way. Then the equation (3.6) is reduced to the differ-ential equation(3.7) ∂ t (cid:102) u χ = i p e − i ( θ + γ/ (cid:102) g χ . Integrating this equation, we have (cid:102) u χ ( t ) = i p e − i ( θ + γ/ (cid:90) t (cid:102) g χ ( s ) ds + (cid:102) u χ (0) . In view of Lemma 3.3, (cid:98) g χ is smooth. Hence (cid:102) g χ is rapidly decreasing at infinity.From (cid:98) u χ ∈ L ( T ), we have (cid:102) u χ ( t ) → | t | → ∞ . Then the limitlim t →∞ (cid:102) u χ ( t ) = i p e − i ( θ + γ/ (cid:90) ∞ (cid:102) g χ ( s ) ds + (cid:102) u χ (0) , exists and we obtain (cid:102) u χ (0) = − i p e − i ( θ + γ/ (cid:90) ∞ (cid:102) g χ ( s ) ds. Therefore, (cid:102) u χ is represented by the rapidly decreasing function(3.8) (cid:102) u χ ( t ) = − i p e − i ( θ + γ/ (cid:90) ∞ t (cid:102) g χ ( s ) ds, t ≥ . Similarly, we have as t → −∞ lim t →−∞ (cid:102) u χ ( t ) = − i p e − i ( θ + γ/ (cid:90) −∞ (cid:102) g χ ( s ) ds + (cid:102) u χ (0) , and (cid:102) u χ (0) = i p e − i ( θ + γ/ (cid:90) −∞ (cid:102) g χ ( s ) ds. Hence we obtain(3.9) (cid:102) u χ ( t ) = i p e − i ( θ + γ/ (cid:90) t −∞ (cid:102) g χ ( s ) ds, t ≤ . Then (cid:102) u χ ( t ) is rapidly decreasing as | t | → ∞ and this implies that (cid:98) u χ ∈ C ∞ ( T ).Obviously, (cid:98) u is smooth outside any small neighborhood of ξ ( θ ). Then we have (cid:98) u ∈ C ∞ ( T ). It follows from the equation (3.5) that (cid:98) g vanishes at ξ ( θ ). (cid:3) Lemma 3.5.
The meromorphic function (cid:98) g ( z ) /p ( z, θ ) is analytic in { z ∈ T C ; | Im z | <ρ/ (2 π ) } . Proof. If p ( z, θ ) = 0 for e iθ ∈ σ ess ( U ) \ T , we havecos (cid:16) z + α − γ (cid:17) = 1 p cos (cid:16) θ − γ (cid:17) . This implies Im z = 0 if p ( z, θ ) = 0 for e iθ ∈ σ ess ( U ) \ T . Therefore, in order toshow the analyticity of (cid:98) g ( z ) /p ( z, θ ), it is sufficient to consider a neighborhood of ξ ( θ ) ∈ M ( θ ). We expand p ( z, θ ) and (cid:98) g ( z ) into Taylor series at ξ ( θ ) ∈ M ( θ ) : p ( z, θ ) = ∞ (cid:88) n =0 p n ( z − ξ ( θ )) n , (cid:98) g ( z ) = ∞ (cid:88) n =0 g n ( z − ξ ( θ )) n , for p n , g n ∈ C . In view of M ( θ ) = M reg ( θ ), we have p = 0 and p (cid:54) = 0. ThenLemma 3.4 implies g = 0 and (cid:98) g ( z ) /p ( z, θ ) is analytic in a neighborhood of ξ ( θ ).The Lemma follows from Lemma 3.3. (cid:3) In the next step, we show that the eigenfunction ψ decays super-exponentiallyas | x | → ∞ . Lemma 3.6.
For any k > , we have e k (cid:104)·(cid:105) ψ ∈ H . Proof. It follows from Lemma 3.5 that the function u ( x ) := 1 √ π (cid:90) T e ixξ (cid:98) u ( ξ ) dξ, satisfies e r (cid:104)·(cid:105) u ∈ (cid:96) ( Z ) for r ∈ (0 , ρ ) so that e r (cid:104)·(cid:105) ψ ∈ H . The assumption (2) impliesthat the function f = ( U − U ) ψ satisfies e r (cid:104)·(cid:105) f ∈ H for any r ∈ (0 , ρ ). Repeatingthe arguments in the proofs of Lemmas 3.3-3.5, we can see e r (cid:104)·(cid:105) ψ ∈ H . We canrepeat this procedure any number of times. Therefore, we have e mr (cid:104)·(cid:105) ψ ∈ H forany m > (cid:3) Proof of Theorem 1.2.
Plugging Lemmas 3.3-3.6, the eigenfunction ψ satisfies e k (cid:104)·(cid:105) ψ ∈ H for any k >
0. The equation ( U − e iθ ) ψ = 0 is rewritten as a ( x + 1) ψ (0) ( x + 1) + b ( x + 1) ψ (1) ( x + 1) = e iθ ψ (0) ( x ) , (3.10) c ( x − ψ (0) ( x −
1) + d ( x − ψ (1) ( x −
1) = e iθ ψ (1) ( x ) . (3.11)Recalling the assumptions (1) and (2), we put K = max (cid:26) , sup x ∈ Z (cid:107) C ( x ) (cid:107) ∞ (cid:27) , K = max (cid:8) , δ − (cid:9) . From the equations (3.10) and (3.11), we have a ( x ) ψ (0) ( x ) = (cid:0) − e − iθ b ( x ) c ( x −
1) + e iθ (cid:1) ψ (0) ( x − − e − iθ b ( x ) d ( x − ψ (1) ( x − , and then | ψ (0) ( x ) | ≤ K K (cid:16) | ψ (0) ( x − | + | ψ (1) ( x − | (cid:17) . ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 11
Repeating the same estimate on the right-hand side, we can see for any y > | ψ (0) ( x ) | ≤ y − K y K y (cid:16) | ψ (0) ( x − y ) | + | ψ (1) ( x − y ) | (cid:17) . In view of Lemma 3.6, we obtain | ψ (0) ( x ) | ≤ y K y K y e − k (cid:104) x − y (cid:105) , for any k >
0. Taking a sufficiently large k and tending y → ∞ , we see | ψ (0) ( x ) | = 0.Since x ∈ Z is arbitrary, ψ (0) vanishes on Z .Let us go back the equation (3.11). The equation is rewritten as d ( x − ψ (1) ( x −
1) = e iθ ψ (0) ( x ) , so that | ψ (1) ( x ) | ≤ K | ψ (1) ( x − | ≤ · · · ≤ K y | ψ (1) ( x − y ) | , for any y >
0. Hence we also have | ψ (1) ( x ) | ≤ K y e − k (cid:104) x − y (cid:105) , for any k >
0. Taking a sufficiently large k > y → ∞ , we obtain ψ (1) ( x ) = 0 for any x ∈ Z . (cid:3) Existence of embedded eigenvalues
Finite support of eigenfunctions.
In this section, we turn to the coinoperator C given by (1.4). Since C ( x ) − C satisfies the assumption (1.2), Lemma2.4 also holds for this case i.e. σ ess ( U ) = σ ac ( U ). The set of thresholds T is alsodefined by the same manner of Theorem 1.2. Thus the assertion (1) of Theorem1.3 holds. On the other hand, the assertion of Theorem 1.2 does not hold for thiscase. However, we can prove the assertion (3) of Therem 1.3 which is weaker thanTheorem 1.2. Proof of (3) of Theorem 1.3.
We can apply Lemmas 3.3-3.6 to U . Then we have e k (cid:104)·(cid:105) ψ ∈ H for any k >
0. Since we have a ( x ) = pe iα (cid:54) = 0 for x < x ∗ , we can usethe estimate which is derived in the proof of Theorem 1.2. Then we have ψ = 0 for x < x ∗ . In view of the equations (3.10) and (3.11), we have d ( x ) ψ (1) ( x ) = − e iθ a ( x + 1) c ( x ) ψ (0) ( x + 1)+ (cid:0) e iθ − e − iθ b ( x + 1) c ( x ) (cid:1) ψ (1) ( x + 1) . Note that d ( x ) = pe iα e iγ (cid:54) = 0 for x > x ∗ . Then we have | ψ (1) ( x ) | ≤ y − K y K y e − k (cid:104) x + y (cid:105) , for any large k > y >
0. We obtain ψ (0) ( x ) = 0 for x > x ∗ tending y → ∞ .From the equation (3.10), we have | ψ (0) ( x ) | ≤ K y | ψ (0) ( x + y ) | ≤ K y e − k (cid:104) x + y (cid:105) , for any large k > y >
0. Hence we also obtain ψ (1) ( x ) = 0 for x > x ∗ tending y → ∞ . (cid:3) Embedded eigenvalues.
In order to construct eigenfunctions precisely, weconsider the auxiliary operator U = SC . Note that σ ( U ) = σ p ( U ) = {± ie iγ (cid:48) / } (see Lemma 2.3). Lemma 4.1.
Let δ ( x ) = δ x for x ∈ Z . Then the function (4.1) ψ ± ( x ) = 1 √ (cid:20) ∓ ie i ( β (cid:48) − γ (cid:48) / δ ( x + 1) δ ( x ) (cid:21) , β (cid:48) , γ (cid:48) ∈ [0 , π ) , are normalized eigenfunctions of U with eigenvalues ± ie iγ (cid:48) / , respectively. Proof. The equation ( U − ( ± ie iγ (cid:48) / )) ψ ± = 0 is equivalent to (cid:20) ∓ ie iγ (cid:48) / e iβ (cid:48) e iξ − e − iβ (cid:48) e iγ (cid:48) e − iξ ∓ ie iγ (cid:48) / (cid:21) (cid:34) (cid:98) ψ (0) ± ( ξ ) (cid:98) ψ (1) ± ( ξ ) (cid:35) = 0 , ξ ∈ T . By a direct computation, we have (cid:34) (cid:98) ψ (0) ± ( ξ ) (cid:98) ψ (1) ± ( ξ ) (cid:35) = s ( ξ ) (cid:20) ∓ ie i ( β (cid:48) − γ (cid:48) / e iξ (cid:21) , for any scalar functions s ( ξ ). Taking s ( ξ ) = (2 √ π ) − , we obtain the lemma. (cid:3) The operator of translation T y for y ∈ Z is defined by(4.2) ( T y ψ )( x ) = ψ ( x − y ) , x ∈ Z , for ψ ∈ H . Obviously, T y ψ ± are also eigenfunctions of U with eigenvalues ± ie iγ (cid:48) / ,respectively. Moreover, we have supp T y ψ (0) ± = { y − } and supp T y ψ (1) ± = { y } . Proof of (2) of Theorem 1.3.
We putΨ ± = κ T y ψ ± + · · · + κ N T y N ψ ± , for any κ , · · · , κ N ∈ C , where ψ ± is given by (4.1). Then we have suppΨ (0) ± = { y − , · · · , y N − } and suppΨ (1) ± = { y , · · · , y N } . Since we have ( F e yj C ) (cid:12)(cid:12) e yj = C for each j = 1 , · · · , N , Ψ ± satisfies the equation U Ψ ± = ± ie iγ (cid:48) / Ψ ± . Then ± ie iγ (cid:48) / ∈ σ p ( U ) for any γ (cid:48) ∈ [0 , π ).In view of the assertion (3) of Theorem 1.3, if ± ie iγ (cid:48) / ∈ σ p ( U ) ∩ ( σ ess ( U ) \ T ),associated eigenfunctions vanish for x > x ∗ and x < x ∗ . (cid:3) Summary and discussion
Finally, we summarize our results of the present paper as a conclusive remarkby using typical numerical examples. We consider two typical cases. We put e = e ∪ e = {− , , } . Let U v = SC v and U e = SC e be defined by C v = F e (cid:20) (cid:21) + (1 − F e ) (cid:20) / √ / √ − / √ / √ (cid:21) , (5.1) C e = F e (cid:20) − (cid:21) + (1 − F e ) (cid:20) / √ / √ − / √ / √ (cid:21) . (5.2) ETECTION OF EDGE DEFECTS BY EMBEDDED EIGENVALUES OF QW 13
Figure 1.
The distri-bution of P v ( X t = x ) at t = 100. Figure 2.
The distri-bution of P e ( X t = x ) at t = 100. Figure 3.
The distri-bution of σ ( U v ). Figure 4.
The distri-bution of σ ( U e ).For U e , e and e are edge defects. On the other hand, U v does not have edgedefects but are perturbed on e . From Lemma 2.4, we have σ ess ( U v ) = σ ess ( U e ) = { e iθ ; θ ∈ J } with J = [ π/ , π/ ∪ [5 π/ , π/ . Taking the initial state ψ given by ψ ( x ) = (cid:20) / √ i/ √ (cid:21) , x ∈ e , ψ (cid:12)(cid:12) Z \ e = 0 , we put ψ v ( t, · ) := U tv ψ and ψ e ( t, · ) := U te ψ for t ≥
0. Then we compute theprobability P ∗ ( X t = x ) = | ψ ∗ ( t, x ) | where ∗ = v or e and X t is the position of thequantum walker at time t . For the numerical results at t = 100, see Figures 1 and2. Localization occurs near x = 0 for both of P v ( X t = x ) and P e ( X t = x ). Herelocalization means lim sup t →∞ P ∗ ( X t = x ) > x ∈ Z . Thus we cannotdetect edge defects by the existence of localization. If the initial state ψ has an overlap with an eigenvector of U ∗ , then localizationoccurs (see [16]). For the locations of σ ( U v ) and σ ( U e ), see Figures 3 and 4. σ ess ( U ∗ )is approximated by eigenvalues of the finite rank operator U ∗ (cid:12)(cid:12) {− ≤ x ≤ } . Theoperator U v has discrete eigenvalues. On the other hand, U e has eigenvalues ± i which are embedded in the interior of σ ess ( U e ). Localizations of U v and U e occurdue to eigenvectors of these eigenvalues. Thus the existence of edge defects isdistinguished by the location of eigenvalues. Precisely, if there exist eigenvaluesembedded in the interior of the continuous spectrum, there are some edge defects.These examples are typical situations to which our main results are applicable(see Theorems 1.2 and 1.3 and Corollary 1.4). References [1] A. Ambainis,
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E-mail address : [email protected] (E. Segawa) Graduate School Educational Promotion Center, Yokohama NationalUniversity, Tokiwadai 79-7, Hodogaya-ku, Yokohama 240-8501, Japan
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