aa r X i v : . [ m a t h . SP ] N ov BALLISTIC TRANSPORT FOR PERIODICJACOBI OPERATORS ON Z d JAKE FILLMAN
Abstract.
In this expository work, we collect some background results and givea short proof of the following theorem: periodic Jacobi matrices on Z d exhibitstrong ballistic motion. Introduction
This expository note is concerned with the properties of Jacobi operators on Z d .Concretely, we fix the dimension d ∈ N and consider linear operators J = J a,b : ℓ ( Z d ) → ℓ ( Z d ) given by[ J u ] x = X y ∈ Z d k x − y k =1 a x , y u y + b x u x , u ∈ ℓ ( Z d ) , x ∈ Z d , where a and b are bounded, b is real-valued, and a y , x = a ∗ x , y = 0 for all x and y in Z d for which k x − y k = 1. For convenience, we will write x ∼ y for x , y ∈ Z d tomean k x − y k = 1.In the case d = 1, we simply write a x := a x,x +1 and one obtains the familiar Jacobimatrix on ℓ ( Z ): J = . . . . . . . . . a ∗ b a a ∗ b a . . . . . . . . . . Jacobi matrices have inspired intense study over the years owing to their closeconnections with orthogonal polynomials, integrable systems, and mathematicalphysics; see, e.g., [3, 10, 12, 14] and references therein.We will be interested in the case in which J is periodic , i.e., there exists a full-ranksubgroup L ⊆ Z d such that U ℓ J U − ℓ = J for all ℓ ∈ L , where U ℓ denotes the shift δ x δ x + ℓ . Equivalently,(1.1) a x + ℓ , y + ℓ = a x , y and b x + ℓ = b x for all x ∼ y ∈ Z d and all ℓ ∈ L . Of course, if { ℓ , . . . , ℓ d } are linearly independent elements of Z d generating a lattice L for which (1.1) holds, then it is a straightforward calculation to show that (1.1)also holds for the lattice L ′ = r Z d = { r n : n ∈ Z d } , where r = | det( ℓ | · · · | ℓ d ) | isthe volume of R d / L ; consequently, no real generality is lost in considering latticesgenerated by multiples of the standard basis vectors, so we shall consider precisely J. F. was supported in part by Simons Foundation Collaboration Grant this scenario in the present note. To that end, given q ∈ N d , say J is q -periodic if U mq J U − mq = J for all m ∈ Z d , where mq = ( m q , m q , . . . , m d q d ) . Equivalently, J is q periodic if (1.1) holds for the lattice L = L dj =1 q j Z .The goal of the note is to discuss quantum dynamics associated with such periodicoperators. In particular, we focus on the growth of the position observables. For alinear operator O , we denote by O ( t ) = e itJ Oe − itJ the corresponding time evolutionwith respect to J . For 1 ≤ j ≤ d , the j th position operator X j is given by X j δ x = x j δ x , where D ( X j ) = (cid:8) ψ ∈ ℓ ( Z d ) : X j ψ ∈ ℓ ( Z d ) (cid:9) . The vector position operator X : D ( X ) = T dj =1 D ( X j ) → ℓ ( Z d ) ⊗ C d given by X ψ = ( X ψ, . . . , X d ψ ) . The primary phenomenon that we will discuss is that of ballistic motion , i.e.,linear growth of the position observable(s). This was established for continuumSchr¨odinger operators by Asch–Knauf [1] and was later extended to Jacobi matri-ces in d = 1 by Damanik–Lukic–Yessen [2]. The result we want to discuss is thegeneralization of [2] to the case of general d ≥ Theorem 1.1. If J is periodic, then it exhibits ballistic motion in the followingsense. There are bounded, self-adjoint operators Q k , ≤ k ≤ d , such that lim t →∞ X k ( t ) t = Q k in the strong sense, and ker( Q k ) = { } . In particular, Q := lim t →∞ X ( t ) t in the strong sense and ker( Q ) = { } . Naturally, since Asch–Knauf already worked in higher dimension, the result is notsurprising and could indeed be considered a folklore result, since it is simply a convexcombination of [1] and [2]. Nevertheless, we felt it would be worthwhile to have anessentially self-contained exposition of the proof somewhere in the literature.In recent years, there has also been substantial interest in studying the phenom-enon of ballistic motion in specific aperiodic models. For instance, this has beenestablished for limit-periodic and quasi-periodic models [4, 5, 6, 7, 8, 15].In Section 2 we discuss a direct integral decomposition of J , and then explainin Section 3 how to use this to prove Theorem 1.1. Since the paper is expositoryin nature, we aim to supply proofs so that the article is self-contained, modulobackground facts from functional analysis and analytic perturbation theory. Acknowledgements.
J.F. is grateful to Ilya Kachkovskiy and Milivoje Lukic forhelpful conversations. Work supported in part by Simons Foundation CollaborationGrant
ALLISTIC TRANSPORT FOR PERIODIC OPERATORS 3 Decomposition of J We first explain how to decompose J as a direct integral of operators on thefundamental domain with suitable self-adjoint boundary conditions. The reader isreferred to [11] for additional background about direct integrals. More precisely, let T d = R d / Z d , putΓ = Z d ∩ d Y j =1 [0 , q j ) = { , , . . . , q − } × · · · × { , , . . . , q d − } , and consider H ( θ ) = H ( θ , q ) ⊂ ℓ ∞ ( Z d ) comprising all those ψ : Z d → C suchthat(2.1) ψ x + nq = e πi h θ , n i ψ x . With the inner product h ψ, ϕ i H ( θ ) = X x ∈ Γ ψ x ϕ x , H ( θ ) becomes a Hilbert space of dimension ¯ q = Q dj =1 q j . We will use d θ todenote the Lebesgue measure on T d . Lemma 2.1. If J is q -periodic, then J maps H ( θ , q ) into itself for all θ .Proof. This is a short calculation. If ψ ∈ H ( θ ), then[ J ψ ] x + nq = X x ∼ y a x + nq , y + nq ψ y + nq + b x + nq ψ x + nq = e πi h θ , n i X x ∼ y a x , y ψ y + e πi h θ , n i b x ψ x = e πi h θ , n i [ J ψ ] x , whence J ψ ∈ H ( θ ). (cid:3) In view of the lemma, we may define J ( θ ) = J | H ( θ ) for each θ ∈ T d . Writing C Γ for the space of functions Γ → C , one can view H ( θ ) ∼ = C Γ via the identification(2.2) C Γ ∋ δ x X n ∈ Z d e πi h θ , n i δ x + nq ∈ H ( θ ) , so we also freely consider J ( θ ) as a linear operator on C Γ .To describe the decomposition of J , define H = Z ⊕ T d H ( θ ) d θ , which consists of measurable functions f mapping T d into S θ ∈ T d H ( θ ) such that f ( θ ) ∈ H ( θ ) for all θ and k f k H := Z T d k f ( θ ) k H ( θ ) d θ < ∞ . Equipped with the inner product h f, g i H = Z T d h f ( θ ) , g ( θ ) i H ( θ ) d θ , J. FILLMAN H is a Hilbert space; see [11] for details. Write f ( θ , x ) for the x th coordinate of f ( θ ). Identifying the fibers of H with C Γ as in (2.2), we can also view H simplyas the collection of square-integrable maps T d → C Γ , which we shall do freely whenit is convenient to do so.For ψ ∈ ℓ ( Z d ), define[ F ψ ]( θ , x ) = X m ∈ Z d ψ x + mq e πi h θ , m i . Lemma 2.2.
For every ψ ∈ ℓ ( Z d ) , F ψ ∈ H , k F ψ k H = k ψ k ℓ ( Z d ) , and theimage of ℓ ( Z d ) is dense in H . In particular, F extends to a unitary operator F : ℓ ( Z d ) → H .Proof. For x ∈ Γ and n ∈ Z d , denote F δ x + nq = ϕ x , n and note that(2.3) ϕ x , n ( θ ) = X m ∈ Z d e πi h θ , n − m i δ x + mq . Since (cid:8) e πi h· , n i : n ∈ Z d (cid:9) is an orthonormal basis of L ( T d ), one can check that (cid:8) ϕ x , n : x ∈ Γ , n ∈ Z d (cid:9) is an orthonormal basis of H , so the lemma follows imme-diately. (cid:3) The unitary operator F “diagonalizes” J in the sense that it transforms J to a(matrix) multiplication operator given by pointwise multiplication by J ( θ ) on H .Concretely, define a linear operator b J : H → H by(2.4) [ b J g ]( θ ) = J ( θ ) g ( θ ) . It is convenient to use the direct integral notation for operators enjoying a decom-position as in (2.4); for instance, we will write b J = Z ⊕ T d J ( θ ) d θ . Theorem 2.3. b J = F J F ∗ .Proof. This follows from a direct calculation. Recall ϕ x , n = F δ x + nq for x ∈ Γ and n ∈ Z d . Since ϕ x , n ( θ ) ∈ H ( θ ) for each θ ∈ T d , (2.3) yields J ( θ ) ϕ x , n ( θ ) = J ϕ x , n ( θ )= X m ∈ Z d e πi h θ , n − m i J δ x + mq = X m ∈ Z d e πi h θ , n − m i X z ∼ x + mq a x + mq , z δ z + b x + mq δ x + mq ! = X m ∈ Z d e πi h θ , n − m i X y ∼ x a x , y δ y + mq + b x δ x + mq ! = X y ∼ x a x , y ϕ y,n ( θ ) + b x ϕ x , n ( θ ) . ALLISTIC TRANSPORT FOR PERIODIC OPERATORS 5
On the other hand, a direct calculation from the definitions yields[ F J F ∗ ϕ x , n ]( θ ) = [ F J δ x + nq ]( θ )= " F X y ∼ x a x , y δ y + nq + b x δ x + nq ! ( θ )= X y ∼ x a x , y ϕ y , n ( θ ) + b x ϕ x , n ( θ ) . Thus b J ϕ x , n = F J F ∗ ϕ x , n for all x and n . Since { ϕ x , n : x ∈ Γ , n ∈ Z d } is a basisof H , the theorem is proved. (cid:3) Corollary 2.4.
Let ¯ q = Q dj =1 q j , let E ( θ ) ≤ · · · ≤ E ¯ q ( θ ) denote the eigenvalues of J ( θ ) , and define I k = { E k ( θ ) : θ ∈ T d } . Then σ ( H ) = ¯ q [ k =1 I k = [ θ ∈ T d σ ( J ( θ )) . Proof.
Since E k ( θ ) depends continuously on θ for every 1 ≤ k ≤ ¯ q , this is animmediate consequence of Theorem 2.3. (cid:3) For later use, we note the following:
Lemma 2.5.
For a.e. θ ∈ T d and each ≤ k ≤ d , ∂E j ∂θ k ( θ ) exists and is nonzero.Proof. This follows from analytic eigenvalue perturbation theory [9]. We describethe broad strokes and leave the details to the reader.Let 1 ≤ k ≤ d , choose and fix θ j ∈ T for j = k . For s ∈ T , define θ ( k ) ( s ) ∈ T d by θ ( k ) j ( s ) = θ j for j = k and θ ( k ) k ( s ) = s . Extending into the complex plane, we havethat A ( s ) := J ( θ ( k ) ( s )) , s ∈ C is an analytic family of matrices. As such, one can choose branches of the eigen-values λ ( s ) , . . . , λ t ( s ) and the associated eigenprojections which are real-analyticfunctions of s . In fact, these will be holomorphic functions of s ∈ C away froma discrete set of points. We note that the multiplicity m j of λ j is constant, againaway from a discrete set. We refer the reader to Kato [9] for details about analyticperturbation theory. See especially [9, Theorem II.6.1]; see also [13, Theorem 1.4.1and Corollary 1.4.5].Since λ r is non-constant for each 1 ≤ r ≤ t , it follows that ∂E j /∂θ k exists and isnonzero for each 1 ≤ j ≤ ¯ q and a.e. θ ∈ T d . (cid:3) Note that we use the different notation λ to emphasize that in general, the analytic enumeration λ j ( θ ) need not necessarily coincide with the ordered enumeration E j ( θ ). J. FILLMAN Ballistic Motion
Let us make a few observations. Formally, for each 1 ≤ k ≤ d , ddt X k ( t ) = iJ e itJ X k e − itJ − ie itJ X k J e − itJ = P k ( t ) , where P k = i [ J, X k ] = i ( J X k − X k J ). Thus,(3.1) X k ( t ) t = X k t + 1 t Z s P k ( s ) ds, so one wants to understand the time averages of P k . Since the integral equation(3.1) involves the unbounded operator X k , some care is needed, so let us make thisprecise. For each N ∈ N , let X k,N denote the bounded operator given by X k,N δ x = ( x k δ x | x k | ≤ NN δ x otherwise , and define P k,N = i [ J, X k,N ]. Proposition 3.1.
Let J be a bounded Jacobi matrix on ℓ ( Z d ) . For all k and t , D ( X k ( t )) = D ( X k ) and one has (3.2) X k ( t ) ψ = X k ψ + Z t P k ( s ) ψ ds. The following representation of P k in the standard basis will be helpful. Proposition 3.2.
For x ∈ Z d , P k δ x = ia x , x − e k δ x − e k − ia x , x + e k δ x + e k . In particular, k P k k ≤ k a k ∞ .Proof. For x ∈ Z d , observe that J δ x = d X j =1 (cid:0) a x , x + e j δ x + e j + a x , x − e j δ x − e j (cid:1) + b x δ x , so X k J δ x = d X j =1 (cid:0) ( x k + δ j,k ) a x , x + e j δ x + e j + ( x k − δ j,k ) a x , x − e j δ x − e j (cid:1) + x k b x δ x . Subtracting this from
J X k δ x = x k J δ x , we get P k δ x = ia x , x − e k δ x − e k − ia x , x + e k δ x + e k , as desired. (cid:3) Proof of Proposition 3.1.
This follows from the same argument as [2, Theorem 2.1].Since X k,N is bounded, a direct calculation shows that (3.2) holds with X k replacedby X k,N , that is,(3.3) X k,N ( t ) ψ = X k,N ψ + Z t P k,N ( s ) ψ ds. ALLISTIC TRANSPORT FOR PERIODIC OPERATORS 7
Since X k,N → X k and P k,N → P k strongly, (3.2) follows immediately from (3.3), theuniform bound k P k k ≤ k a k ∞ , and dominated convergence. (cid:3) From the representation of P k in Proposition 3.2, we can see that P k is q -periodicwhenever J is q -periodic, so it too defines operators P k ( θ ) := P k | H ( θ ) for each θ ∈ T d . Theorem 3.3.
We have F P k F ∗ = b P k , where P k = R ⊕ T d P k ( θ ) d θ , that is, [ b P k g ]( θ ) = P k ( θ ) g ( θ ) . Proof.
This is essentially the same calculation as in the proof of Theorem 2.3. (cid:3)
We now have all the necessary pieces in place in order to prove the main result.
Proof of Theorem 1.1.
For ψ ∈ C Γ , x ∈ Γ, and θ ∈ T d , define x / q = ( x /q , . . . , x d /q d )and define the multiplication operator M : H → H by [ M g ]( θ ) = M ( θ ) g ( θ ),where [ M ( θ ) ψ ] x = e πi h θ , x / q i ψ x Write e J ( θ ) = M ( θ ) − J ( θ ) M ( θ ) and likewise for e P k ( θ ). A direct calculation shows e P k ( θ ) = q k π ∂∂θ k e J ( θ ) , and thus, denoting the projection onto the eigenspace of E j ( θ ) by Π j ( θ ), we haveΠ j ( θ ) P k ( θ )Π j ( θ ) = q k π ∂E j ∂θ k ( θ )Π j ( θ )for a.e. θ (by Lemma 2.5).Thus, we have X k ( t ) t = X k t + 1 t Z t P k ( s ) ds = X k t + 1 t Z t Z ⊕ T d e isJ ( θ ) P k ( θ ) e − isJ ( θ ) d θ ds → q k π Z ⊕ T d ¯ q X j =1 ∂E j ∂θ k ( θ )Π j ( θ ) d θ , where we applied dominated convergence to deduce the final line. By Lemma 2.5, ∂E j /∂θ k = 0 a.e., so ker( Q k ) = { } . (cid:3) J. FILLMAN
References
1. J. Asch and A. Knauf,
Motion in periodic potentials , Nonlinearity (1998), no. 1, 175–200.MR 14929562. D. Damanik, M. Lukic, and W. Yessen, Quantum dynamics of periodic and limit-periodicJacobi and block Jacobi matrices with applications to some quantum many body problems ,Comm. Math. Phys. (2015), no. 3, 1535–1561. MR 33391853. David Damanik,
Schr¨odinger operators with dynamically defined potentials , Ergodic TheoryDynam. Systems (2017), no. 6, 1681–1764. MR 36819834. J. Fillman, Ballistic transport for limit-periodic Jacobi matrices with applications to quantummany-body problems , Comm. Math. Phys. (2017), no. 3, 1275–1297. MR 36074755. Lingrui Ge and Ilya Kachkovskiy,
Ballistic transport for one-dimensional quasiperiodicSchr¨odinger operators , arxiv:2009.02896 (2020).6. Ilya Kachkovskiy,
On transport properties of isotropic quasiperiodic XY spin chains , Comm.Math. Phys. (2016), no. 2, 659–673. MR 35149557. , On the relation between strong ballistic transport and exponential dynamical localiza-tion , arxiv:2001.01314 (2020).8. Yulia Karpeshina, Young-Ran Lee, Roman Shterenberg, and G¨unter Stolz,
Ballistic transportfor the Schr¨odinger operator with limit-periodic or quasi-periodic potential in dimension two ,Comm. Math. Phys. (2017), no. 1, 85–113. MR 36565139. T. Kato,
Perturbation theory for linear operators , Classics in Mathematics, Springer-Verlag,Berlin, 1995, Reprint of the 1980 edition. MR 133545210. C. A. Marx and S. Jitomirskaya,
Dynamics and spectral theory of quasi-periodic Schr¨odinger-type operators , Ergodic Theory Dynam. Systems (2017), no. 8, 2353–2393. MR 371926411. M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of oper-ators , Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.MR 049342112. B. Simon,
Szeg˝o’s theorem and its descendants , M. B. Porter Lectures, Princeton UniversityPress, Princeton, NJ, 2011, Spectral theory for L perturbations of orthogonal polynomials.MR 274305813. , Operator theory , A Comprehensive Course in Analysis, Part 4, American Mathemat-ical Society, Providence, RI, 2015. MR 336449414. G. Teschl,
Jacobi operators and completely integrable nonlinear lattices , Mathematical Surveysand Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 171153615. Zhiyuan Zhang and Zhiyan Zhao,
Ballistic transport and absolute continuity of one-frequencySchr¨odinger operators , Comm. Math. Phys.351