Lieb-Robinson Bounds for the Toda Lattice

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Authors Islambekov, Umar

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Link to Item http://hdl.handle.net/10150/294026 Lieb-Robinson Bounds for the Toda Lattice

by Umar Islambekov

A Dissertation Submitted to the Faculty of the Department of Mathematics In Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy In the Graduate College The University of Arizona

2 0 1 3 2

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Umar Islambekov entitled Lieb-Robinson Bounds for the Toda Lattice and recommend that it be accepted as fulﬁlling the dissertation requirement for the Degree of Doctor of Philosophy.

Date: May 9, 2013 Jan Wehr

Date: May 9, 2013 Kenneth McLaughlin

Date: May 9, 2013 Robert Sims

Date: May 9, 2013 Thomas Kennedy

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Date: May 9, 2013 Robert Sims 3

Statement by Author

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to bor- rowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

Signed: Umar Islambekov 4

Acknowledgments

I would like to show my sincere gratitude to my advisor Professor Robert Sims for supervising my PhD research. His continuous patience, support and motivation will never be forgotten. Also, I extend my thanks to our co-author Professor Gerald Teschl for his participation and contributions in our joint research. Special gratitude also goes to my dissertation committee members Professors Thomas Kennedy, Kenneth McLaughlin and Jan Wehr for their help, productive discussions and useful comments. 5

Table of Contents

Abstract ...... 6

Chapter 1. Introduction ...... 7

Chapter 2. Present Study ...... 12

References ...... 15

Appendix A – Lieb-Robinson Bounds for the Toda Lattice ..... 16

Appendix B – An Estimate for Infinite Range Perturbations ... 57

Appendix C – Supplemental material to Section 2.4 ...... 60 6

Abstract

We study locality properties of the Toda lattice in terms of Lieb-Robinson bounds. The estimates we prove produce a ﬁnite Lieb-Robinson velocity depending on the initial condition. Then we establish analogous results for certain perturbations of the Toda system. Finally, we obtain generalizations of our main results in the setting of the Toda hierarchy. 7

Chapter 1 Introduction

A concept of locality is inherent in many physical systems. In relativistic systems, the support of a time-evolved local observable stays confined to a light cone, i.e. locality is preserved under the time evolution. In non-relativistic systems, the effect of a small perturbation at some site is immediately felt at all sites of the system. In other words, strictly speaking locality is not preserved by the dynamics for such systems. In 1972, E. Lieb and D. Robinson showed that for non-relativistic quantum spin systems with finite range interactions there exists nevertheless an approximate form of locality, called quasi-locality [6]. The estimates they proved, commonly referred to as Lieb-Robinson bounds, are quantitative locality bounds which show that except for exponentially small corrections, the support of a local observable evolving under the dynamics remains essentially contained in a ball with radius growing linearly in time. In later years, works on proving analogues of Lieb-Robinson bounds for certain classical oscillator systems appeared in the literature [2], [7], [8]. We will now give a brief summary of these results because of their close relevance to the main topic of the present dissertation: to investigate quasi-locality in terms of Lieb-Robinson bounds for the Toda system and its perturbations. We begin by setting the general framework following [5] and describe how quasi-locality is studied in the context of classical oscillator systems.

Consider a system of particles situated at the sites of a ﬁnite lattice Λ ⊂ Zd. Suppose the position and the momentum of a particle at site x ∈ Λ are equal to real

numbers qx and px respectively. The set of all sequences x = {(qx, px)}x∈Λ, which describe all the states of the system, is called phase space and denoted by XΛ. Time 8

evolution of the system is governed by Hamilton’s equations of motion: for each x ∈ Λ, ∂H ∂H q˙x(t) = andp ˙x(t) = − , (1.0.1) ∂px ∂qx where H is the corresponding Hamiltonian for the system. In cases when this system can be solved, the ﬂow generated by H and associated with an initial condition x is deﬁned as Φt(x) = {(qx(t), px(t))}x∈Λ. To study the properties of the solutions, it is convenient to introduce observables

(complex-valued functions deﬁned on phase space). For example, observables Qx and

Px deﬁned by Qx(x) = qx and Px(x) = px, respectively give the position and the momentum of a particle located at site x ∈ Λ. The set of all observables over XΛ is denoted by AΛ. In such systems, quasi-locality is studied in terms of the Poisson bracket – the generator of the dynamics, i.e.

d α (A) = α ({A, H}) = {α (A),H} , (1.0.2) dt t t t where the Poisson bracket of observables A and B is deﬁned as

X ∂A ∂B ∂A ∂B {A, B} = · − · . (1.0.3) ∂qx ∂px ∂px ∂qx x∈Λ A Lieb-Robinson bound is described as follows in terms of the Poisson bracket.

Fix two disjoint ﬁnite sets X and Y of Zd. Let Λ ⊂ Zd be a set containing the union of X and Y . Consider observables A and B supported in X and Y respectively, i.e.,

e.g. A does not depend on qx and px whenever x is outside X. Since X and Y are disjoint, the Poisson bracket of A and B is zero. A Lieb-Robinson bound is said to hold for a system with Hamiltonian H, if for some initial condition x there exist constants µ, C, and v such that an estimate

−µ(d(X,Y )−v|t|) |{αt(A),B} (x)| ≤ Ce , (1.0.4) 9

holds for all t sufficiently small. Here αt is the dynamics (the map A 7→ A ◦ Φt) corresponding to H and d(X,Y ) is the distance between the supports of A and B. This bound shows that the supports of B and the time-evolved A remain essentially apart from each other for times |t| ≤ d(X,Y )/v up to exponentially small errors in d(X,Y ), making Lieb-Robinson bounds useful only for short times, in contrast with more common long-time decay estimates. The number v is called a Lieb-Robinson velocity and it provides an upper bound on how fast the disturbances propagate through the system. One is also interested in what the constants C and v depend on. Most importantly, these constants must be independent of the size of the lattice whenever it is finite. Shortly after Lieb and Robinson published their above mentioned result, Mar- chioro et al. explored the problem of estimating the velocity of disturbances in classical oscillator systems [7]. In their paper, first the authors show that for a system of harmonic oscillators there is a cone in spacetime with slope v such that if an observable A is supported in a ball of radius less than vt and an observable B is supported at √ the origin, then the Poisson bracket {αt(A),B} decays as 1/ t and is exponentially small in time if the support of A has radius greater than vt (Proposition 3.1). The number v is the system’s Lieb-Robinson velocity and it is the slope of the cone which separates these two asymptotic behaviors. Then the authors investigate two different anharmonic models with the aim of finding a dependence of the distance between

the supports of A and B on time for which {αt(A),B} is exponentially small. The first model they consider is a plane rotator model with compact configuration space. Adapting the approach employed for quantum lattice models by Lieb and Robinson in [6], they establish the existence of a finite Lieb-Robinson velocity for such a system (Theorem 4.1). However, this result is not accompanied by any explicit estimates on the Poisson bracket. Next, they deal with a model having a quartic on-site potential and prove a nontrivial quasi-locality result (Theorem 5.2). They show that whenever the distance between the supports of the local observables grows faster than t4/3, 10

the relevant Poisson bracket is exponentially small in time for almost every initial condition with respect to a suitably chosen measure. In [2], Marchioro with other coauthors manage to essentially improve this result by replacing t4/3 with t logα(t) for appropriate α > 0 (Theorem 2.2). Unfortunately, these results do not guarantee the existence of a Lieb-Robinson velocity. In [8], H. Raz and R. Sims studied a system of harmonic oscillators defined on a finite volume lattice, deriving Lieb-Robinson bounds for general observables (Theorem 1, see also Theorem 3). They show that the corresponding Lieb-Robinson velocity is independent of an initial condition and the volume size. Their analysis relies on the explicit form of the solutions to the equations of motion. Perturbations of the harmonic system were also studied there using an interpolation technique, also used in the present dissertation. In this case, no explicit solutions are available and the proof is carried out only for a class of observables – Weyl observables – enjoying the property that the dynamics leaves them invariant. First, they tackle on-site perturbations where potentials are generated by a suitably chosen smooth function having appropriate decay (Theorem 2, see also Theorem 7). What is interesting here is that due to the employed interpolation technique, the corresponding anharmonic Lieb-Robinson velocity can be explicitly expressed in terms of that of the harmonic (see (3.7)). Then they generalize this result to cover multi-site perturbations with sufficiently fast decay (Theorem 8). It is worth mentioning that even though their first result (Theorem 1) also applies to Weyl observables, the authors provide an alternative proof of that using the specific properties of Weyl observables (Theorem 6). The authors remark that these perturbation results do not apply to the strong quartic on-site perturbation considered in [7] and [2]. The present dissertation is written according to published paper format. It is based on the paper [5] included in Appendix A. Chapter 2 summarizes the results of the paper as well as describes the methods used therein. Appendix B is dedicated to extending the results of Section 3.1 of the paper to infinite range perturbations of 11 the Toda system. Appendix C provides a visual illustration of the conclusions made in Section 2.4. We conclude the Introduction by outlining the dissertation author’s contributions to the paper. The author was actively involved in obtaining most of the results of the paper. A method suggested by the author helped improve the estimates covered in Section 2. This method was later adopted in other relevant sections. In the case of perturbed Toda systems, our initial locality estimate (see Section 6) was not sufficient to prove the existence of the Lieb-Robinson velocity. Numerical simulations performed by the author in the perturbed setting for special class of initial conditions, strongly suggested the existence of the Lieb-Robinson velocity when we restrict ourselves to a smaller class of initial conditions. A helpful result provided by our third coauthor G. Teschl, who later joined our research project, allowed us to rigorously justify and prove these numerical findings in Section 3. After the paper was published, the author has obtained a result analogous to Theorem 3.1 in the context of infinite range perturbations. This result is found in Appendix B. The present work serves as a positive step towards gaining the understanding of locality in the context of nonlinear oscillator systems. 12

Chapter 2 Present Study

In this chapter we give a brief summary of the main results of the dissertation. The details of these results can be found in the appended paper [5] (see Appendix A). The present work, motivated by the results discussed in the Introduction, aims at establishing locality estimates in the form of Lieb-Robinson bounds for the Toda lattice and certain class of its (natural) perturbations. The Toda lattice is a one- dimensional lattice model which describes the motion of particles or oscillators with nonlinear interaction potential. Rather than working with the natural variables of position and momentum, from the very beginning we make a change of variables (commonly known as Flaschka variables [3], [4]) and from then on concern ourselves mostly with the study of the properties the solutions of the Toda equations of motion in these variables. One of the useful known facts in this regard is that when the initial conditions are bounded then the Toda equations have unique global solutions which remain bounded for all times (see e.g. Theorem 12.6 in [9]). As we will discuss below, this property of the solutions plays a crucial role in proving our main results. An essential step in proving Lieb-Robinson bounds in the context of classical oscillator systems is to have a bound for the dependence of the solutions on changes in the initial data. Such an estimate is obtained in Section 2.1 (see Theorem 2.1), where we prove a bound with a finite Lieb-Robinson velocity which depends however on the initial condition in contrast to that of the harmonic model studied in [8]. The basic idea of the proof is as follows. Using the equations of motion, we write the solutions at some site, say n, in integral form and then differentiate them with respect to the initial condition at another site, say m. Applying uniform boundedness (in space and time) of the solutions we get a simple iterative inequality. Iteration and the use of Stirling’s 13 formula readily yield the desired bound. A Lieb-Robinson bound for the Toda lattice straightforwardly follows from this estimate as the relevant Poisson bracket, up to factors involving the norms of the observables, can be estimated in terms of the partial derivatives of the solutions with respect to the initial data (Theorem 2.3). For the general class of bounded initial conditions, the corresponding Lieb-Robinson velocity, though can be easily estimated, is hard to compute. Remarkably, in the case of one-soliton initial conditions it can be explicitly computed and compared with the actual velocity of the system. This is done in Section 2.4 where these two quantities are shown to be comparable. More interesting results are in Sections 3 and 6, where we study perturbations of the Toda system. The class of perturbations consist of adding an additional interaction term induced by a twice differentiable function with bounded second derivative. By closely following the methods used in the unperturbed case, we obtain a locality bound which however does not produce a Lieb-Robinson velocity (see Theorem 6.1). This is not surprising, since in this case we do not have uniformly bounded solutions for the general class of initial conditions. Our next challenge is to obtain a Lieb-Robinson bound for the perturbed systems, perhaps at the expense of considering a more restricted class of initial conditions for which the solutions remain globally bounded. One can show that for a suitable class of perturbations (which includes the ones we consider) and initial conditions for which the perturbed Hamiltonian is finite, the system has unique uniformly bounded global solutions (see Theorem A.1). This yields two types of estimates: first we prove a Lieb- Robinson bound for the perturbed Toda system (uniform boundedness of the solutions essentially allows the arguments used for the unperturbed case to go through in this setting) (see Theorem 3.1). Unfortunately, the resulting Lieb-Robinson velocity is not comparable to the unperturbed one corresponding to the same initial condition. Our second estimate addresses this problem: by interpolating between the Toda and the perturbed Toda dynamics we show that the Lieb-Robinson velocity for the perturbed system can be estimated in terms of that of the unperturbed (see Theorem 3.3). In 14 proving this estimate one needs to have bounds for second order partial derivatives of the solutions with respect to the initial condition. Such bounds are obtained in Lemma 3.2. Sections 4 and 5 are devoted to extending the previously obtained results to the so-called Toda hierarchy (see Theorems 4.2, 4,4, 5.1 and 5.3). The Toda hierarchy is a set of infinitely many systems of coupled differential equations parametrized by a non-negative integer r. The equations corresponding to r = 0 are exactly previously studied Toda equations. The key fact that allows for the extensions to be made is that the entire Toda hierarchy enjoys uniformly bounded solutions. Since in this setting we have to deal with infinitely many systems simultaneously, the main challenge in obtaining these generalizations is of computational nature while the methods remain essentially the same. 15

References

[1] R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications. 2nd edition, Springer, New York, 1983.

[2] P. Butt`a,E. Caglioti, S. Di Ruzza, and C. Marchioro. On the propagation of a perturbation in an anharmonic system. J. Stat. Phys. 127, 313 (2007).

[3] H. Flaschka. The Toda lattice. I. Phys. Rev. B 9, 1924–1925 (1974).

[4] H. Flaschka. The Toda lattice. II. Progr. Theoret. Phys. 51, 703–716 (1974).

[5] U. Islambekov, R. Sims, G. Teschl. Lieb-Robinson bounds for the Toda lattice. J. Stat. Phys. 148, 440-479 (2012).

[6] E.H. Lieb and D.W. Robinson. The ﬁnite group velocity of quantum spin systems. Comm. Math. Phys. 28, 251–257 (1972).

[7] C. Marchioro, A. Pellegrinotti, M. Pulvirenti, and L. Triolo. Velocity of a perturbation in inﬁnite lattice systems. J. Statist. Phys. 19, 499–510 (1978).

[8] H. Raz and R. Sims. Estimating the Lieb–Robinson velocity for classical anharmonic lattice systems. J. Stat. Phys. 137, 79–108 (2009).

[9] G. Teschl. Jacobi Operators and Completely Integrable Nonlinear Lattices. Math- ematical Surveys and Monographs 72, Amer. Math. Soc, Providence, 2000. 16

Appendix A – Lieb-Robinson Bounds for the Toda Lattice J Stat Phys (2012) 148:440–479 DOI 10.1007/s10955-012-0554-2

Lieb–Robinson Bounds for the Toda Lattice

Umar Islambekov · Robert Sims · Gerald Teschl

Received: 2 May 2012 / Accepted: 21 July 2012 / Published online: 3 August 2012 © Springer Science+Business Media, LLC 2012

Abstract We establish locality estimates, known as Lieb–Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb–Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a ﬁnite velocity for certain perturbations of these systems.

Keywords Toda lattice · Lieb–Robinson bound

1 Introduction

Analyzing the dynamics of certain non-relativistic systems is crucial to understanding a number of important problems in statistical mechanics. For example, there is much interest in rigorously justifying the emergence of macroscopic non-equilibrium phenomena, like Fourier’s Law or other forms of heat conduction, directly from a many-body Hamiltonian

R. S. and U. I. were supported, in part, by NSF grants DMS-0757424 and DMS-1101345. G. T. was supported by the Austrian Science Fund (FWF) under Grant No. Y330. U. Islambekov · R. Sims () Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, AZ 85721, USA e-mail: [email protected] url: http://math.arizona.edu/~rsims/ U. Islambekov e-mail: [email protected] url: http://math.arizona.edu/~uislambekov/

G. Teschl Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria e-mail: [email protected] url: http://www.mat.univie.ac.at/~gerald/

G. Teschl International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria Lieb–Robinson Bounds for the Toda Lattice 441 dynamics [4]. Future progress on fundamental questions such as these will require detailed information on the structure of the underlying dynamics. One of the goals of the present 18 work is to investigate an approximate form of locality, often described by a Lieb–Robinson bound, for the dynamics corresponding to a class of integrable systems. Recognizing approximate forms of locality, despite the lack of a relativistic framework, has been essential in solving many intriguing open problems. One of the first mathematical formulations of a useful locality estimate was given by Lieb and Robinson in 1972 [24]. In this work, they demonstrated that the dynamics corresponding to quantum spin systems, with e.g. finite-range interactions, remains effectively confined to a “light” cone, up to corrections which decay at least exponentially away from the light cone. Recently there have been a number of improvements and generalizations of the original result [2, 10, 20, 28, 29, 31, 34, 36, 37, 39], and these new techniques have led to some interesting applications [3, 5–7, 16–18, 21, 30, 32]. For a review of these results, we refer the interested reader to [19, 33]. Shortly after the original result of Lieb and Robinson, it was shown in [26], see also [9, 38] for more recent developments, that this notion of quasi-locality also applies to the dynamics of classical oscillator systems. Since it is this work that more closely pertains to the topic of the present article, we will discuss it briefly as follows. Consider a system of particles confined to a large but finite set Λ ⊂ Zd . To each site ∈ ∈ ∈ x Λ associate a particle, or oscillator, with position qx R and momentum px R.The state of the system in Λ is described by a sequence x ={(qx ,px )}x∈Λ, and the set of all such sequences, Λ, is called phase space. A Hamiltonian, H , is a real-valued function on phase X space. Given a Hamiltonian and a sequence x ∈ Λ, Hamilton’s equations of motion are: for each x ∈ Λ, X ∂H ∂H q˙x (t) = and p˙x (t) =− (1.1) ∂px ∂qx solved with initial condition {(qx (0), px (0))}x∈Λ = x. For many Hamiltonians, this system of coupled differential equations can be solved for all time (and any initial condition). In this case, we denote by Φt , the Hamiltonian flow, i.e., the mapping that associates to initial conditions, the solution at time t: Φt (x) ={(qx (t), px (t))}x∈Λ. Measurements of the system under consideration correspond to observables, where an observable A is a complex-valued function on phase space. For example, the position of the particle at site x ∈ Λ corresponds to an observable Qx for which Qx (x) = qx .Letus denote by Λ the set of all observables over Λ. Given a Hamiltonian for which (1.1) can A X be solved, the Hamiltonian dynamics αt : Λ → Λ is defined by αt (A) = A ◦ Φt .Inthis A A case, we can make time-dependent observations such as [αt (Qx )](x) = qx (t). It is well known that the Hamiltonian dynamics is generated by the Poisson bracket, i.e.,

d α (A) = α {A,H} = α (A), H , (1.2) dt t t t and here the Poisson bracket of two observables is the observable ∂A ∂B ∂A ∂B {A,B}= · − · . (1.3) ∂q ∂p ∂p ∂q x∈Λ x x x x

In terms of this Poisson bracket, we can now describe the Lieb–Robinson bound. Let us fix finite sets X, Y ⊂ Zd with X ∩ Y =∅.TakeΛ ⊂ Zd finite, but large enough so that X ∪ Y ⊂ Λ. Consider two observables A,B ∈ Λ with supports in X, Y respectively, A 442 U. Islambekov et al. i.e., e.g. A depends only on those qx and px with x ∈ X. It is clear that {A,B}=0since A and B have disjoint supports. A Hamiltonian H satisfies a Lieb–Robinson bound if for 19 some initial condition x there exist numbers μ, C,andv for which −μ(d(X,Y)−v|t|) αt (A), B (x) ≤ Ce , (1.4) where d(X,Y) is the distance between X and Y . In words, this estimate shows that for times t with v|t|≤d(X,Y) the support of αt (A) remains essentially disjoint from the support of B, up to exponentially small corrections in d(X,Y). The number v is often called the Lieb–Robinson velocity corresponding to H , and it represents a bound on the rate at which disturbances can propagate through the system. We will now briefly discuss what is known concerning Lieb–Robinson bounds for classical systems. The works in [9, 26, 38] prove Lieb–Robinson type bounds for a variety of harmonic systems, and it is shown that analogous results also hold for certain anharmonic perturbations. The strongest such result is in [9] and demonstrates that for an anharmonic model with a quartic on-site perturbation the relevant Poisson bracket decays to zero whenever the distance between the supports of the local observables grows faster than t logα(t) for suitable α>0. In [38], where explicit estimates on the Lieb–Robinson velocity were obtained, it is shown that the number v is independent of the initial condition x. Since the system is linear, this is not so surprising. Of course when it comes to anharmonic lattice oscillations one of the central objects is the famous Fermi–Pasta–Ulam–Tsingou (FPU) problem. It was first demonstrated by Zabusky and Kruskal [44] that one key ingredient to resolve the FPU paradox is the relation with solitons. Moreover, subsequently Toda [43] presented an anharmonic lattice which possess soliton solutions and was later on shown to be integrable by Flaschka [12, 13] and Manakov [25]. Clearly this naturally raises the question about Lieb–Robinson type locality bounds for the Toda lattice and the main goal of this work is to establish such bounds. Our estimates produce a Lieb–Robinson velocity that depends on the initial condition, see e.g. Theorem 2.3. We discuss this fact in the context of one-soliton solutions in Section 2.4.For a restricted class of initial conditions, we can prove the existence of a finite Lieb–Robinson velocity for a class of perturbations of the Toda lattice. We have two bounds of this type. The first, in Section 3.1, establishes a result by directly mimicking the methods for the unperturbed system. The next, in Section 3.2, uses interpolation and shows that the velocity of the perturbed system can be estimated in terms of the velocity of the unperturbed system. Analogous results are shown to also hold for solutions of the Toda hierarchy, see Sects. 4 and 5. For a much larger class of initial conditions, we can prove a locality bound for perturbed systems, however, we do not establish the existence of a finite Lieb–Robinson velocity. This is discussed in Section 6. The final section, Appendix, describes a set of perturbations and corresponding initial conditions for which solutions remain globally bounded. This shows that the results obtained in Sections 3.1 and 3.2 are not vacuous.

2 Locality Estimates for the Toda Lattice

In this section, we discuss the Toda Lattice and prove a Lieb–Robinson bound. We introduce the model in Section 2.1. The crucial solution estimate, Theorem 2.1, is contained in Section 2.2 as well as some useful remarks. A Lieb–Robinson bound, see Theorem 2.3,in terms of a large class of observables is proven in Section 2.3. We end this section by com- paring the velocity estimates in our Lieb–Robinson bound to known results for one-soliton solutions of the Toda Lattice. This is done in Section 2.4. Lieb–Robinson Bounds for the Toda Lattice 443

2.1 The Toda Lattice 20 The Toda Lattice is a well-studied physical model and one of the prototypical discrete integrable wave equations. We refer to the monographs [11, 40, 43] or the review articles [22, 41] for further information. Like those discussed in the introduction, the Toda Lattice corresponds to a specific system of coupled oscillators on Z. We present this model immediately in the infinite volume setting. ∈ ∈ 2 For each site n Z, we associate an ordered pair (qn,pn) R and denote by the set of X ={ } ∈ all sequences x (qn,pn) n Z. The system is said to be comprised of an infinite collection ∈ ∈ ∈ of oscillators, each situated at a site n Z with position qn R and momentum pn R. Here the state of the system is described by a sequence x ={(qn,pn)}n∈ ∈ . Z X The oscillators evolve in time according to the following coupled system of differential equations. For each n ∈ Z and any t ∈ R,

−(qn(t)−qn−1(t)) −(qn+1(t)−qn(t)) q˙n(t) = pn(t) and p˙n(t) = e − e , (2.1) with initial condition given by some state of the system x ∈ . We note that the differential X system (2.1) corresponds to the following (formal) Hamiltonian H : → R ∪{∞}given by X 2 pn H(x) = + V(qn+1 − qn), (2.2) ∈ 2 n Z where V(r)= e−r + r − 1. For many sequences, the formal Hamiltonian may be inﬁnite, however, it does formally generate the system of differential equations in (2.1) through Hamilton’s equations. For classes of solutions with prescribed decay properties we refer to [42]. Existence and uniqueness of global solutions to (2.1) on certain subsets of is well- X known. Rather than address this directly, we begin by changing variables. Suppose that (2.1) has a solution. For each n ∈ Z and t ∈ R,set

1 − − 1 a (t) = e (qn+1(t) qn(t))/2 and b (t) =− p (t), (2.3) n 2 n 2 n in terms of this given solution. This choice is commonly referred to as Flaschka variables as they were initially introduced in [12], see also [13]. Using (2.1), it is clear that these new variables satisfy the following equations of motion ˙ = − ˙ = 2 − 2 an(t) an(t) bn+1(t) bn(t) and bn(t) 2 an(t) an−1(t) , (2.4) and correspond to the following formal Hamiltonian = 2 + − 2 = 2 + 2 − − H(x) 2bn V ln 4an 2bn 4an 2ln(2an) 1. (2.5) ∈ ∈ n Z n Z In this work, we will concern ourselves mainly with the properties of the solutions of (2.4). ∞ ∞ Consider the vector space M = (Z, R) × (Z, R) of pairs of bounded, real-valued sequences. We will write each x ∈ M as = { } { } = x an n∈ , bn n∈ (an,bn) ∈ . (2.6) Z Z n Z 444 U. Islambekov et al.

M is a Banach space with respect to the norm 21 x M = max a ∞, b ∞ , c ∞ = sup |cn|. (2.7) n Given any initial condition x ∈ M, global solvability of the system (2.4) is well-known (see e.g. [40, Theorem 12.6]). We will denote the Toda ﬂow on M by Φt , i.e., Φt : M → M is the function that associates an initial condition x ∈ M to the solution of (2.4) at time t: ={ } ∈ = Φt (x) (an(t), bn(t)) n Z with Φ0(x) x.

2.2 Estimating Toda Solutions

More is known about the solutions of (2.4)onM. In fact, let us ﬁx x ∈ M and introduce operators L(x), P(x) : 2(Z) → 2(Z) by setting = + + − − + L(x)f n anfn 1 an 1fn 1 bnfn (2.8) and = + − − − P(x)f n anfn 1 an 1fn 1 . (2.9)

For each x ∈ M, the existence of global solutions imply that L(Φt (x)) and P(Φt (x)) are well-deﬁned for all t ∈ R. Whenever the initial condition x is ﬁxed, we will write L(t) = L(Φt (x)) and P(t)= P(Φt (x)) to spare notation. Observe that for each x ∈ M, Φt (x) ∈ M and so L(t) is a bounded self-adjoint operator with the operator norm satisfying ≤ ≤ + max a(t) ∞, b(t) ∞ L(t) 2 2 a(t) ∞ b(t) ∞. (2.10) Similarly P(t) is a bounded skew-adjoint operator. Since P(t) is also differentiable with respect to t, it generates a two-parameter family of unitary propagators U(t,s) satisfying d U(t,s)= P(t)U(t,s) with U(t,t)= 1, (2.11) dt for all pairs t,s ∈ R, see e.g. [40], Theorem 12.4 for more details. Moreover, a short calculation shows that P(t)and L(t) are a Lax pair associated to (2.4), i.e., d L(t) = P(t),L(t) , (2.12) dt and therefore,

L(t) = U(t,s)L(s)U(t,s)−1.

From this fact, each x ∈ M satisﬁes the a-priori estimate ≤ = ∈ Φt (x) M L(t) 2 L(0) 2 for all t R. (2.13) It is important to emphasize the fact that we work with real-valued solutions. In fact, while a local existence result can be established in the complex case, even periodic complex initial conditions can blow up in ﬁnite time (see e.g. [14]or[15]). Our proof of the Lieb–Robinson bound for the Toda Lattice, see Theorem 2.3, makes crucial use of the following estimate describing the sensitivity of solutions to changes in the initial condition. Lieb–Robinson Bounds for the Toda Lattice 445

Theorem 2.1 Let x ∈ M and μ>0. There exists a number v = v(μ,x) for which given any n, m ∈ Z, the bound 22 ∂ ∂ 8 −μ(|n−m|−v|t|) max an(t), bn(t) ≤ √ e , (2.14) ∂z ∂z 17 ∈ ∈{ } holds for all t R and each z am,bm . In fact, one may take √ 1 v = (1 + 17 )L(0) eμ+1 + . (2.15) 2 μ

Proof Fix x ∈ M. Global existence of solutions on M guarantees that for each x ∈ M and ∈ : → 2 n Z, the function Fn R R given by an(t) Fn(t; x) = (2.16) bn(t) is well-deﬁned. It is differentiable with respect to each z ∈{am,bm}, e.g. as a consequence of Lemma 4.1.9 in [1]. When convenient, we will suppress the dependence of Fn on x. Using the equations of motion, i.e. (2.4), it is clear that t an(s)(bn+1(s) − bn(s)) F (t) = F (0) + ds. (2.17) n n 2 − 2 0 2(an(s) an−1(s)) The relation ∂ ∂ t ∂ F (t) = F (0) + D (s) F + (s) ds, (2.18) ∂z n ∂z n n,e ∂z n e |e|≤1 0 with (bn+1(s) − bn(s))δ0(e) an(s)(−δ0(e) + δ1(e)) Dn,e(s) = (2.19) 4(an(s)δ0(e) − an−1(s)δ−1(e)) 0 then follows immediately from (2.17). To complete our estimate, we introduce the following notation. For each v ∈ R2,wewill denote by x |x| v = and |v|= . (2.20) y |y| Moreover, we will write x u ≤ if and only if |x|≤|u| and |y|≤|v|. (2.21) y v

With this understanding, the uniform solution estimate (2.13), and (2.19), it is clear that |t| ∂ ∂ ∂ Fn(t) ≤ Fn(0) + L(0) De Fn+e(s) ds, (2.22) ∂z ∂z 2 ∂z |e|≤1 0 where 2δ0(e) δ0(e) + δ1(e) De = . (2.23) 4(δ0(e) + δ−1(e)) 0 446 U. Islambekov et al.

Let us now consider the case that z = am, i.e., 23 ∂ 1 Fn(0) = δm(n). (2.24) ∂am 0

In this case, iteration of (2.22) yields ∞ | | k ∂ ≤ ( L(0) 2 t ) ··· ··· 1 Fn(t) δm+e1+···+ek (n)De1 Dek ∂am k! 0 k=0 |e1|≤1 |ek|≤1 ∞ k ( L(0) 2|t|) 1 ≤ Dk , (2.25) k! 0 k=|n−m| wherewehaveset 22 D = D = . (2.26) e 80 |e|≤1 Moreover, note that the remainder term in ﬁnite iterations of (2.22) converges to zero since ∂ F (t) ∂z n is continuous and thus bounded√ on compact time intervals. The eigenvalues of D are λ± = 1 ± 17, and in terms of the eigenvectors v± given by λ± 1 1 1 v± = it is clear that = √ v+ − √ v−. (2.27) 8 0 2 17 2 17

Taking the inﬁnity norm, in R2, of both sides of (2.25)showsthat ∞ | | k ∂ 8 (λ+ L(0) 2 t ) Fn(t) ≤ √ . (2.28) ∂a k! m ∞ 17 k=|n−m|

−(μ+1) Now, let μ>0beﬁxedandsetc = λ+ L(0) 2.Ifc|t|≤|n − m|e , then by Stirling ∞ k |n−m| |n−m| (c|t|) (c|t|) | | c|t| | − | | | − | − |− c | | ≤ ec t ≤ e n m ec t ≤ e μ( n m μ t ). (2.29) k! (n − m)! |n − m| k=|n−m|

Otherwise 0 ≤−μ(|n − m|−ceμ+1|t|), and so by (2.28)wehave ∂ 8 | | 8 − | − |− μ+1+ 1 | | c t μ( n m c(e μ ) t ) Fn(t) ≤ √ e ≤ √ e . (2.30) ∂am ∞ 17 17

In the case that z = bm, it is clear that 0 λ− λ+ =− v+ + v−. (2.31) 1 8(λ+ − λ−) 8(λ+ − λ−)

A short calculation leads to a prefactor of √λ+ in the analogue of (2.28),andthisislessthan 17 the one above. We have proven the result. Lieb–Robinson Bounds for the Toda Lattice 447

Remarks 24 1. The number v in our estimate (2.14) depends on the initial condition x only through the quantity L(0) 2. In fact, one can replace L(0) 2 with any uniform estimate on the solutions (uniform in n and t) as is clear in the bound from (2.18)–(2.22). 2. If |n − m| > 0, the left hand side of (2.14) is zero when t = 0.Theestimatein(2.14)then shows that the solutions of the Toda Lattice at site n are insensitive (i.e. exponentially small) to changes in the initial condition at site m for times t satisfying v|t|≤|n − m|. For this reason, the number v is often called a bound on the velocity (i.e. rate) at which disturbances propagate through the Toda Lattice. Moreover, if |n − m|≥1, then the bound in (2.28) is at most linear in |t|,forsmallt. 3. In general, the fact that solutions of the Toda Lattice have partial derivatives that decay exponentially in |n − m| will be more important than any particular decay rate μ.For this reason, given an initial condition x ∈ M, one can optimize the quantity v over all = μ+1 + 1 possible μ>0. The graph of f(μ) e μ is given below.

It is clear that the optimal μ is achieved when d 1 eμ+1 + = 0 (2.32) dμ μ

and moreover, this value is independent of the initial condition x. The unique solution of (2.32) can be expressed in terms of the Lambert W -function [35, §4.13] and is given by √ μ0 = 2W 1/(2 e) ≈ 0.47767 and f(μ0) ≈ 6.47622. (2.33)

2.3 A Lieb–Robinson Bound for Toda

We can now formulate a Lieb–Robinson bound for the Toda Lattice. In general, an observable A is a function A : M → C. Let us denote by the set of all observables with A well-deﬁned ﬁrst order partial derivatives. We will say that an observable A is pointwise bounded if A x = sup A Φt (x) < ∞∀x ∈ M. (2.34) ∈ t R 448 U. Islambekov et al.

Take (1) to be the set of those observables in for which all of the ﬁrst order partial A A derivatives are bounded and 25 ∂A ∂A A 1,x = + < ∞∀x ∈ M. (2.35) ∈ ∂an x ∂bn x n Z

∂A ∂A An observable A ∈ is said to be supported on X ⊂ Z if the observables and are A ∂an ∂bn identically zero for all n ∈ Z \ X.Thesupport of an observable A is the minimal set on which A is supported, and we will denote this set by supp(A).Set 0 ⊂ to be the set of A A all observables with compact support. For any t ∈ R and A ∈ ,theToda dynamics,which A we denote by αt , is the observable-valued mapping given by

αt (A) = A ◦ Φt , (2.36) where Φt is the Toda ﬂow described at the end of Section 2.1 above. In other words, for each x ∈ M, t ∈ R,andA ∈ , A αt (A) (x) = A Φt (x) , (2.37) which, since solutions are global on M, is a well-deﬁned quantity.

Example 2.2 The two most basic observables correspond to evaluation maps, i.e., to each ∈ n Z, we associate the functions An and Bn given by

An(x) = an and Bn(x) = bn for each x ∈ M. (2.38)

Each of these observables have support X ={n}. Moreover, the dynamically evolved observables αt (An) (x) = an(t) and αt (Bn) (x) = bn(t) (2.39) correspond to observations over time.

The modified Poisson bracket of two observables A and B is formally defined as the observable { } = 1 · ∂A · ∂B − ∂A · ∂B A,B (x) an ˜ ˜ (x) (2.40) 4 ∈ ∂an ∂bn ∂bn ∂an n Z ∂ ∂ ∂ wherewehavedenoteby ˜ = − . If, for example, either A or B has compact ∂bn ∂bn+1 ∂bn support, then the corresponding modified Poisson bracket is a well-defined observable. This quantity is of particular interest since it generates the Toda dynamics, i.e., for any A ∈ 0 A d α (A) = α {A,H} = α (A), H , (2.41) dt t t t where H is the Hamiltonian (2.5). Now, for any n, m ∈ Z, t ∈ R,andx∈ M, it is easy to see that 1 ∂ 1 ∂ αt (An), Bm (x) = am−1 an(t) − am an(t) (2.42) 4 ∂am−1 4 ∂am Lieb–Robinson Bounds for the Toda Lattice 449 using the basic observables defined in Example 2.2. In this simple case, Theorem 2.1 gives the bound 26 2 a ∞ −μ(|n−m+1|−v|t|) −μ(|n−m|−v|t|) αt (An), Bm (x) ≤ √ e + e 17

2 a ∞ ≤ √ 1 + eμ e−μ(|n−m|−v|t|). (2.43) 17

In general, we have the following Lieb–Robinson bound for the Toda Lattice.

Theorem 2.3 Let x ∈ M and μ>0. There exist numbers C and v for which given any observables A,B ∈ (1), the estimate A ∂A ∂A ∂B ∂B −μ(|n−m|−v|t|) αt (A), B (x) ≤ C a ∞ + + e ∈ ∂am x ∂bm x ∂an x ∂bn x n,m Z (2.44) holds for all t ∈ . Here C = √2 (1 + eμ) and v is as in Theorem 2.1. R 17

Proof Substituting into (2.40), we ﬁnd that = 1 · ∂ · ∂B − ∂ · ∂B αt (A), B (x) an αt (A) ˜ ˜ αt (A) (x). (2.45) 4 ∈ ∂an ∂bn ∂bn ∂an n Z ˜ Using the chain rule, it is clear that for any z ∈{an, bn}, ∂ ∂A ∂ ∂A ∂ αt (A) (x) = αt · αt (Am) + αt · αt (Bm) (x) (2.46) ∂z ∈ ∂am ∂z ∂bm ∂z m Z where we are using the notation from Example 2.2. Applying the triangle inequality and then Theorem 2.1, the bound a ∞ ∂A ∂ ∂A ∂ ∂B αt (A), B (x) ≤ am(t) + bm(t) 4 ∂a ∂a ∂b ∂a ˜ n,m m x n m x n ∂bn x a ∞ ∂A ∂ ∂A ∂ ∂B + am(t) + bm(t) 4 ∂a ˜ ∂b ˜ ∂a n,m m x ∂bn m x ∂bn n x 2 a ∞ ∂A ∂A ∂B −μ(|n−m|−v|t|) ≤ √ + e ∂a ∂b ˜ 17 n,m m x m x ∂bn x 2 a ∞ μ ∂A ∂A ∂B −μ(|n−m|−v|t|) + √ e + e (2.47) ∂a ∂b ∂a 17 n,m m x m x n x

∂ readily follows. Note that the bound on e.g. | ˜ am(t)| follows from the argument in Theo- ∂bn rem 2.1. Another triangle inequality proves (2.44) as claimed.

For many applications, the observables of interest will have disjoint supports. In this case, the bound in (2.44) can be stated as follows. For any x ∈ M and μ>0, there exist numbers 450 U. Islambekov et al.

C and v, as above, such that given any two disjoint subsets X and Y of Z the bound 27 −μ(d(X,Y)−v|t|) αt (A), B (x) ≤ C a ∞ A 1,x B 1,xe , (2.48) holds for all t ∈ R and any observables A,B ∈ (1) with supports in X, Y , respectively. Here A d(X,Y) = inf |x − y|:x ∈ X and y ∈ Y > 0, (2.49) and (2.48) corresponds with the bound (1.4) claimed in the introduction. Other norms on the observables under considerations are often useful. For example, consider ∂A ∂A ∂A x = sup max , (2.50) ∈ m Z ∂am x ∂bm x for any A ∈ (1). Theorem 2.3 then shows that: For any x ∈ M and μ>0, there exist A numbers C and v, as above, such that given any two disjoint subsets X and Y of Z, at least one of which being finite, the bound −μ(|m−n|−v|t|) αt (A), B (x) ≤ 4C a ∞ ∂A x ∂B x e , (2.51) m∈X,n∈Y holds for all t ∈ R and any observables A,B ∈ (1) with supports in X, Y , respectively. In any case, the bound in Theorem 2.3 demonstratesA that each x ∈ M propagates no faster than some finite rate. We say that the Lieb–Robinson velocity corresponding to x ∈ M is v(x,μ0) where μ0 is the infimum defined in the remarks after Theorem 2.1.

2.4 On One-Soliton Solutions

Perhaps the most important collection of solutions to the Toda Lattice are the solitons, i.e. the solitary waves, see e.g. [40, 41, 43]. In fact, solitons can be considered as the stable part of any short-range initial condition since every such solution eventually splits into a number of stable solitons plus a decaying dispersive tail [22]. Since all involved quantities can be computed explicitly for the one-soliton solution we will use it as a test case and for our Lieb–Robinson bounds. Fix κ>0. A one-soliton solution of the Toda lattice is given by 1 + exp[−2κn± 2sinh(κ)t + δ] q (t;±) = q − ln , (2.52) n 1 + exp[−2κ(n− 1) ± 2sinh(κ)t + δ] where q and δ are real constants. Here qn(t;±) represents the position of the traveling wave. ± sinh(κ) It describes a single bump traveling with speed κ and width proportional to 1/κ.In other words, the smaller the soliton the faster it propagates. Changing δ amounts to a shift of the solution and we will set δ = 0 for simplicity. From this explicit formula it is clear that, in contrast to harmonic models, the velocity of a solution to the Toda Lattice may indeed depend on the initial condition. In terms of the function f±(x, t) = 1 + exp −2κx ± 2sinh(κ)t (2.53) it is clear that f±(n, t) qn(t;±) = q − ln , (2.54) f±(n − 1,t) Lieb–Robinson Bounds for the Toda Lattice 451 and from Hamilton’s equations, we also know that 28 d f±(n − 1,t) f±(n, t) pn(t;±) = qn(t;±) = − . (2.55) dt f±(n − 1,t) f±(n, t)

In Flaschka’s variables, we have that √ 1 f±(n − 1,t)f±(n + 1,t) an(t;±) = and 2 f±(n, t) (2.56) 1 f±(n, t) f±(n − 1,t) bn(t;±) = − . 2 f±(n, t) f±(n − 1,t)

A short calculation shows that

;± = ;± = cosh(κ) supn an(0 ) a0(0 ) and 2 (2.57) sinh(κ) tanh(κ) sup b (0;±) = b (0;±) = . n n 0 2

With xκ denoting the initial condition corresponding to this one-soliton, we have proven that cosh(κ) cosh(κ) sinh(κ) tanh(κ) = max , ≤ L(x ) 2 2 2 κ 2 sinh(κ) tanh(κ) ≤ cosh(κ) + , (2.58) 2 using e.g. (2.10). Since L(xκ ) is self-adjoint its norm is equal to the spectral radius. More- over, the spectrum is given by an absolutely continuous part [−1, 1] plus the single eigenvalue ± cosh(κ) implying = L(xκ ) 2 cosh(κ). (2.59) To see this last claim note that the one-soliton solution can be computed from the inverse scattering transform by choosing one eigenvalue λ =±cosh(κ) plus the corresponding norming constant γ = (1 − e−2κ )eδ and zero reﬂection coefﬁcient (cf. [43, Sect. 3.6] or [40, Sect. 13.4]) or by using the double commutation method to add one eigenvalue λ with norming constant γ to the trivial solution (cf. [40, Sect. 14.5]). As is clear from this calculation, the Lieb–Robinson velocity does provide a reasonable estimate on the actual velocity, at least for one-soliton solutions.

3 Estimates on Perturbed Toda Systems

In this section, we consider a class of perturbations of the Toda system for which locality results analogous to Theorem 2.3 still hold. We introduce these perturbations as follows. Let W : R →[0, ∞) satisfy W ∈ C2(R). Consider the formal Hamiltonian w H = H + Wn, (3.1) ∈ n Z

={ } ∈ ∈ where H is the Toda Hamiltonian, see (2.5), and for any initial condition x (an,bn) n Z = ∞ \{ } × ∞ = 2 M0 (Z, R 0 ) (Z, R), the observable Wn(x) W(ln(4an)). Our choice of 452 U. Islambekov et al. parametrization is motivated by the fact that the potential V for the unperturbed Toda Lat- − =− 2 = 29 tice, see (2.2), is a function of qn+1 qn ln(4an). Since the choice an 0 corresponds to taking the positions of the particles at sites n and n + 1 inﬁnitely far apart, we will exclude it from our considerations. The formal Hamiltonian (3.1) corresponds to the following system of coupled differential equations w w w w a˙ (t) = a (t) b + (t) − b (t) , n n n 1 n (3.2) ˙w = w 2 − w 2 + bn (t) 2 an (t) an−1(t) Rn(t), where 1 w 2 w 2 R (t) = W ln 4a (t) − W ln 4a − (t) . (3.3) n 2 n n 1 Local existence and uniqueness of solutions of (3.2) corresponding to initial conditions ∈ w x M0 follows from standard results, [1,Thm.4.1.5].LetusdenotebyΦt the per- ∈ w = turbed Toda ﬂow, i.e., the function that associates initial conditions x M0 with Φt (x) { w w } (an (t), bn (t)) , the solution of (3.2) at time t. Consider the set of initial conditions Mb = Mb(W) ⊂ M0 for which there exist numbers C1,C2 < ∞ with 1 w ≤ ≤ sup Φt (x) M C1 and sup sup C2. (3.4) ∈ ∈ ∈ | w | t R t R n Z an (t)

It will be shown in Appendix that this Mb contains at least all initial conditions whose energy is ﬁnite under appropriate assumptions on W . For initial conditions x ∈ Mb, we have two estimates on the corresponding Lieb– Robinson velocity. The ﬁrst is obtained in Section 3.1 by simply arguing as we did in Theorem 2.1. The other, in Section 3.2, achieves an estimate of the perturbed velocity (corresponding to x) in terms of the unperturbed velocity (corresponding to x) via interpolation.

3.1 Direct Bounds

Following closely the arguments in Theorem 2.1, we obtain the next result.

∈ 2 ∈ ∞ ∈ Theorem 3.1 Fix W C (R) with W L (R), μ>0, and let x Mb. There exist numbers Cw = Cw(x,W)and vw = vw(μ, x) for which given any n, m ∈ Z, the bound ∂ w ∂ w w −μ(|n−m|−vw|t|) max a (t), b (t) ≤ C e , (3.5) ∂z n ∂z n ∈ ∈{ } holds for all t R and each z am,bm . In fact, one may take w 4C2 W ∞ μ+1 1 v = 1 + 17 + C1 e + (3.6) C1 μ with C1,C2 from (3.4).

∈ ∈ Proof Fix x Mb. Global existence again guarantees that for each n Z, the function w w = an (t) Fn (t) w (3.7) bn (t) Lieb–Robinson Bounds for the Toda Lattice 453 satisﬁes t w w w 30 a (s)(b + (s) − b (s)) F w(t) = F w(0) + n n 1 n ds. (3.8) n n w 2 − w 2 + 0 2(an (s) an−1(s) ) Rn(s) w Since W is sufﬁciently smooth, the components of Fn (t) are differentiable with respect to each z ∈{am,bm}, and the bound |t| ∂ w ∂ w w ∂ w F (t) ≤ F (0) + D F + (s) ds, (3.9) ∂z n ∂z n e ∂z n e |e|≤1 0 follows as in (2.22) with 2C δ (e) C (δ (e) + δ (e)) w = 1 0 1 0 1 De . (3.10) (4C1 + C2 W ∞)(δ0(e) + δ−1(e)) 0

Following the previous scheme, iteration (with z = am) yields ∞ | | k ∂ w (2C1 t ) w k 1 F (t) ≤ D (3.11) ∂a n k! 0 m k=|n−m| with 11 C2 W ∞ Dw = for α = 4 + . (3.12) α 0 C1 Eigenvalues and eigenvectors of Dw are √ 1 ± 1 + 4α λ± λ± = and v± = (3.13) 2 α and arguing as before, it is now clear that ∂ 2α w w ≤ √ −μ(|n−m|−v |t|) Fn (t) e (3.14) ∂am ∞ 1 + 4α with w μ+1 1 v = 2λ+C e + . (3.15) 1 μ

For z = bm, one similarly ﬁnds ∂ 2λ+ w w ≤ √ −μ(|n−m|−v |t|) Fn (t) e . (3.16) ∂bm ∞ 1 + 4α

This completes the proof.

3.2 Bounds via Interpolation

The goal of this section is to prove a different bound on the Lieb–Robinson velocity corresponding to an initial condition x ∈ Mb, see Theorem 3.3 below. The novel feature of this estimate is that it is explicit in the unperturbed Lieb–Robinson velocity of x. 454 U. Islambekov et al.

Before we state the main result of this section, we ﬁrst indicate some further estimates on the unperturbed system which will be useful in proving Theorem 3.3. To start with, we 31 prove another Lieb–Robinson type estimate for the Toda system, see Lemma 3.2 below. Afterwards, we introduce a quantity that is better suited for the iteration scheme which is at the heart of proving Theorem 3.3,seeGμ(k) in (3.37) below. Lastly, we will state and prove Theorem 3.3. We begin with the following lemma on second order derivatives of the unperturbed system.

Lemma 3.2 Fix μ>0 and let x ∈ M. There exists a number C = C(μ,x)>0 and a function h, depending on μ and x, for which given any n, k, ∈ Z, the estimate ∂2 ∂2 ≤ −μ|n−| −μ|n−k| 2μv|t| max ˜ an(t), ˜ bn(t) Ce e e h(t) (3.17) ∂z∂bk ∂z∂bk ∈ holds for all t R. Here an(t) and bn(t) are the solutions of (2.4) with initial condition ∂ ∂ ∂ x ∈ M, ˜ = − , z ∈{a,b}, and the number v is as in Theorem 2.1. The function ∂bk ∂bk+1 ∂bk h grows at most exponentially.

Proof Fix x ∈ M. As in Theorem 2.1, it is clear that the function Fn(t) is well-deﬁned for ∈ ∈ = ˜ each n Z and t R. In addition, both (2.18)and(2.19) still hold with the choice z bk . Taking a second derivative, we ﬁnd that ∂2 t ∂ ∂ ∂2 F (t) = D (s) F + (s) + D (s) F + (s) ds, (3.18) ˜ n ∂z n,e ˜ n e n,e ˜ n e ∂z∂bk |e|≤1 0 ∂bk ∂z∂bk

∂2 for each z ∈{a,b},since ˜ Fn(0) = 0. ∂z∂bk The bound ∂2 8 |t| ∂ ≤ √ −μ|n−| μvs ˜ Fn(t) e e De ˜ Fn+e(s) ds ∂z∂bk 17 |e|≤1 0 ∂bk |t| 2 ∂ + L(0) D F + (s) ds (3.19) 2 e ˜ n e |e|≤1 0 ∂z∂bk with (eμ + 1)δ (e) δ (e) + δ (e) = 0 0 1 De μ (3.20) 4(δ0(e) + e δ−1(e)) 0 and De as in (2.23), follows using Theorem 2.1 and the argument therein. A further application of Theorem 2.1 implies that ∂ 8 ≤ √ μ −μ|n+e−k| μvs 1 ˜ Fn+e(s) e e e (3.21) ∂bk 17 1 and therefore the ﬁrst term on the right hand side of (3.19) can be estimated by |t| μ + 64 μ −μ|n−| −μ|n−k| 2μvs 2(e 1) e e e e ds 2μ . (3.22) 17 0 4(e + 1) Lieb–Robinson Bounds for the Toda Lattice 455

Here we have used that 32 μ + − | + − | 1 − | − | 2(e 1) e μ n e k D ≤ e μ n k . (3.23) e 1 4(e2μ + 1) |e|≤1 Let us introduce the notation μ + 64 μ 2(e 1) Cμ = e and y = . (3.24) 17 4(e2μ + 1)

We have proven that ∂2 |t| ≤ −μ|n−| −μ|n−k| 2μvs · ˜ Fn(t) Cμe e e ds y ∂z∂bk 0 |t| 2 ∂ + L(0) D F + (s) ds. (3.25) 2 e ˜ n e 0 |e|≤1 ∂z∂bk Iteration now yields

∞ 2 |t| t1 tj ∂ j 2μvt + F (t) ≤ C L(0) ··· e j 1 dt + ···dt ˜ n μ 2 j 1 1 ∂z∂bk j=0 0 0 0 − | + +···+ − | − | + +···+ − | × ··· μ n e1 ej μ n e1 ej k ··· e e De1 De2 Dej y |e1|≤1 |ej |≤1 (3.26) and the convergence is guaranteed as before. A short calculation shows that | | ∞ t t1 tj | | m 2μvt + 1 (2μv t ) ··· e j 1 dt + ···dt = (3.27) j 1 1 (2μv)j+1 m! 0 0 0 m=j+1 whereas the bound e−μ|z+e| ≤ eμ|e|e−μ|z| immediately implies that

− | + + +···+ − | | | | | − | − | e μ n e1 e2 ej ≤ eμ e1 ···eμ ej e μ n . (3.28)

This proves that

∞ ∞ 2 j | | m ∂ Cμ −μ|n−| −μ|n−k| L(0) 2 (2μv t ) ˜ j Fn(t) ≤ e e (D) y (3.29) ˜ 2μv 2μv m! ∂z∂bk j=0 m=j+1 where 2μ + ˜ 2μ|e| 2 e 1 D = e De = . (3.30) 4(e2μ + 1) 0 |e|≤1 ˜ The eigenvalues of D are λ± = 1 ± 1 + 4(e2μ + 1)2 and a convenient choice of eigenvectors are λ± v± = . (3.31) 4(e2μ + 1) 456 U. Islambekov et al.

In terms of these, it is clear that 33 μ μ 2(e + 1) − λ− λ+ − 2(e + 1) y = y1v+ + y2v− with y1 = and y2 = . (3.32) λ+ − λ− λ+ − λ−

Inserting this into (3.29), we ﬁnd that 2 ∂ ˜ Fn(t) ∂z∂bk ∞ ∞ j ∞ | | m Cμ −μ|n−| −μ|n−k| L(0) 2 (2μv t ) j j ≤ e e y λ+v+ + y λ−v− 2μv 2μv m! 1 2 ∞ j=0 m=j+1 ∞ j ∞ m L(0) λ+ (2μv|t|) ≤ Ce−μ|n−|e−μ|n−k| 2 . (3.33) 2μv m! j=0 m=j+1

It is easy to see that for any A,B ∈ R,

∞ ∞ ∞ − (B|t|)m (B|t|)m m 1 Aj = Aj m! m! j=0 m=j+1 m=1 j=0 1 (A−1)B|t| − B|t| = − (e 1)e ,A 1, = A 1 (3.34) B|t|eB|t|,A= 1.

This proves Lemma 3.2. One may choose Cμ C = |y | v+ ∞ +|y | v− ∞ (3.35) 2μv 1 2

= L(0) 2λ+ − and by setting β 2μv 1, we see that 1 (e2μvβ|t| − 1) if β = 0, h(t) = β (3.36) 2μv|t| if β = 0.

−1 Note that β depends only on μ, limt→0 h(t) = 0, and moreover, if β<0, then h(t) ≤|β| . r|t| For any μ,however,h(t) ≤ he ,whereh = h(μ, x) and r = r(μ,x) ≥ 0.

Our proof of Theorem 3.3 will again use an iteration scheme. Due to interpolation, it will be necessary to sum certain terms over all integers. To this end, we make the following observation. For each μ>0, the function

e−μ|k| G (k) = (3.37) μ (1 +|k|)2 satisﬁes the estimate Gμ(j − l)Gμ(l − k) ≤ γGμ(j − k) (3.38) ∈ l Z Lieb–Robinson Bounds for the Toda Lattice 457 −2 2 with γ = 4 ∈ (1 +|k|) . In fact, setting f(k)= (1 +|k|) , it is clear that f(j + k) ≤ k Z 2(f (j) + f(k)), and thus 34

−1 −1 −1 μ(|j−k|−|j−l|−|l−k|) Gμ(j − k) Gμ(j − l)Gμ(l − k) = f(j − k)f(j − l) f(l− k) e ≤ 2 f(j − l)−1 + f(l− k)−1 . (3.39)

The claim follows after summing over l. Clearly the choice of power 2 in the denominator −1−δ of Gμ is merely for convenience; a factor (1 +|k|) for any δ>0 would sufﬁce. Note, however, that there is no bound of the type in (3.38) for the exponential function with no inverse polynomial weight. As a ﬁnal comment, it is clear that for any μ>0andε>0, the bound −(μ+ε)|x| e ≤ CεGμ |x| (3.40)

2 −ε|x| holds for all x ∈ . The number C = sup ∈ (1 +|x|) e < ∞. R ε x R We can now state the second result of this section.

∈ 2 ∈ ∞ ∈ Theorem 3.3 Fix W C (R) with W ,W L (R), take μ>0, and let x Mb. For any ε>0, there are positive numbers C = C(ε), D = D(ε,μ,x,W), and δ = δ(ε,μ,x,W)such that ∂ w ∂ w (μ+ε)v|t| δ|t| max a (t), b (t) ≤ CGμ |n − m| e 1 + D e − 1 (3.41) ∂z n ∂z n ∈ ∈ = + ∈{ } holds for all t R and n, m Z. Here v v(x,μ ε) is as in Theorem 2.1 and z am,bm .

In words, this result shows that for each μ>0 the perturbed Lieb–Robinson velocity w v (x,μ), corresponding to x ∈ Mb, satisﬁes, for each ε>0, ε δ vw(x,μ)≤ 1 + v(x,μ+ ε) + (3.42) μ μ and, as is shown in the proof below, the dependence of δ on W can be made explicit = + + δ δ1 W ∞ δ2 W ∞ δ3 (3.43) for some δi = δi (ε, μ, x) and i = 1, 2, 3.

∈ ∈ Proof Fix W ,x Mb,andt R. We begin by interpolating between the Toda and perturbed dynamics. As in Section 2.1,denoteby

w w αt (A) = A ◦ Φt and α (A) = A ◦ Φ for any A ∈ , (3.44) t t A the Toda and perturbed dynamics respectively. For any A ∈ , the observable-valued equa- A tion t d w − = w αt (A) αt (A) αs αt−s (A) ds (3.45) 0 ds is clear. Due to the form of the perturbation, 458 U. Islambekov et al. d w w w w α α − (A) = α α − (A), H − α α − {A,H} ds s t s s t s s t s 35 = w αs αt−s (A), Wk , (3.46) ∈ k Z and moreover, one finds that =−1 · ∂ αt−s (A), Wk Wk ˜ αt−s (A). (3.47) 2 ∂bk Combining these expressions, we have shown that 1 t ∂ w = − w · αt (A) αt (A) αs Wk ˜ αt−s (A) ds (3.48) 2 ∈ 0 ∂bk k Z as a formal expansion. For specific choices of observables, the expression above can be estimated. ∈ Fix n Z and denote by An and Bn the observables introduced in Example 2.2. Consider the observable-valued αw(A ) α (A ) ˜ w = t n ˜ = t n Fn (t) w and Fn(t) , (3.49) αt (Bn) αt (Bn) whichforanygivenx∈ M satisfies aw(t) a (t) ˜ w = n = w ˜ = n = Fn (t) (x) w Fn (t) and Fn(t) (x) Fn(t), (3.50) bn (t) bn(t) in terms of our previous notation. Our starting point is the equation t w ∂ 1 αs ( ˜ αt−s (An)) ˜ w = ˜ − w · ∂bk Fn (t) Fn(t) αs Wk w ∂ ds. (3.51) α ( α − (B )) 2 ∈ 0 s ˜ t s n k Z ∂bk Since W is sufficiently smooth, the formula ∂ ∂ t ∂ F˜ w(t) = F˜ (t) − D (s; n) F˜ w(s) ds ∂z n ∂z n k ∂z k k∈ 0 Z 1 t ∂ − w · ; ˜ w αs Wk Dk,(s n) F (s) ds (3.52) 2 ∈ 0 ∂z k, Z where w · −1 · w ∂ αs (Wk Ak ) αs ( ˜ αt−s (An)) 0 D (s; n) = ∂bk (3.53) k w · −1 · w ∂ αs (Wk Ak ) αs ( ˜ αt−s (Bn)) 0 ∂bk and ⎛ ⎞ w ∂2 w ∂2 αs ( ˜ αt−s (An)) αs ( ˜ αt−s (An)) D (s; n) = ⎝ ∂a∂bk ∂b∂bk ⎠ (3.54) k, w ∂2 w ∂2 αs ( ˜ αt−s (Bn)) αs ( ˜ αt−s (Bn)) ∂a∂bk ∂b∂bk readily follows. Lieb–Robinson Bounds for the Toda Lattice 459

For any x ∈ M, the estimate 36 ∂ ∂ |t| ∂ ˜ w ≤ ˜ + ; ˜ w Fn (t) (x) Fn(t) (x) Dk(s n) Fk (s) (x) ds ∂z ∞ ∂z ∞ ∈ 0 ∂z ∞ k Z |t| W ∞ ∂ + ; ˜ w Dk,(s n) F (s) (x) ds (3.55) 2 ∈ 0 ∂z ∞ k, Z is clear. Using Theorem 2.1, the ﬁrst term can be bounded by ∂ ˜ ∂ Fn(t) (x) = Fn(t) ∂z ∞ ∂z ∞ 8 ≤ √ e−(μ+ε)(|n−m|−v|t|) 17 8 (μ+ε)v|t| ≤ √ CεGμ |n − m| e (3.56) 17 wherewehavesetv = v(x,μ+ ε) and used (3.40). For each x ∈ Mb, the matrix appearing in the second term satisﬁes 8 ; ≤ √ μ+ε −(μ+ε)(|n−k|−v(s)(|t|−s)) Dk(s n) (x) ∞ C2 W ∞ e e 17 8 ≤ √ μ+ε | − | (μ+ε)v(s)(|t|−s) C2 W ∞ e CεGμ n k e (3.57) 17 where we have denoted by √ = w + = + w μ+ε+1 + + −1 v(s) v Φs (x), μ ε (1 17 ) L Φs (x) 2 e (μ ε) (3.58) w ∈ the unperturbed Lieb–Robinson corresponding to the initial condition Φs (x).Sincex Mb, w ≤ w + w ≤ L Φs (x) 2 2 a (s) ∞ b (s) ∞ 3C1, (3.59) and so the quantity in (3.58) can be estimated independent of s. Clearly,

v = v(0) ≤ sup v(s) := v∗. (3.60) ∈ s R The matrix in the third term can be dominated using Lemma 3.2. In fact, the bound ; ≤ −(μ+ε)|n−| −(μ+ε)|n−k| 2(μ+ε)v(s)(|t|−s) | |− Dk,(s n) (x) ∞ 2Ce e e h t s (3.61) follows immediately. Some comments are in order. First, the prefactor C from Lemma 3.2, appearing above, seems to depend on s through the velocity v(s);see(3.35). If, however, one repeats the argument of Lemma 3.2 and replaces the v(s) in (3.27) with v∗ above, then the new prefactor is independent of s. Next, as discussed at the end of Lemma 3.2,the function h grows at most exponentially. In fact, in this particular application, it is clear that

∗ h |t|−s ≤ he2(μ+ε)γ v (|t|−s) (3.62) 460 U. Islambekov et al. with numbers h = h(x,μ+ ε) and γ = γ(x,μ+ ε), each independent of s. This proves that

∗ 37 ; ≤ ˜ | − | | − | 2(γ +1)(μ+ε)v (|t|−s) Dk,(s n) (x) ∞ CGμ n Gμ n k e . (3.63) Putting everything together and suppressing the x-dependence, we have found that ∂ w ≤ | − | (μ+ε)v|t| Fn (t) B1Gμ n m e ∂z ∞ |t| ∗ ∂ + | − | (μ+ε)v (|t|−s) w B2 Gμ n k e Fk (s) ds ∈ 0 ∂z ∞ k Z + B3 Gμ |n − | Gμ |n − k| ∈ k, Z |t| ∗ ∂ × 2(γ +1)(μ+ε)v (|t|−s) w e F (s) ds 0 ∂z ∞ (μ+ε)v|t| ≤ B1Gμ |n − m| e |t| ∗ ∂ + | − | 2(γ +1)(μ+ε)v (|t|−s) w B Gμ n k e Fk (s) ds. (3.64) ∈ 0 ∂z ∞ k Z Upon iteration, we ﬁnd that ∂ w ≤ | − | a|t| Fn (t) B1Gμ n m e ∂z ∞ ∞ j + B1 B ··· Gμ |n − k1| ···Gμ |kj − m| = ∈ ∈ j 1 k1 Z kj Z |t| s1 sj−1 b(|t|−s1) b(s1−s2) b(sj−1−sj ) asj × ··· e e ···e e dsj ···ds1 0 0 0 wherewehaveset

a = (μ + ε)v and b = 2(γ + 1)(μ + ε)v∗. (3.65)

From (3.38), it is clear that j ··· Gμ |n − k1| ···Gμ |kj − m| ≤ γ Gμ |n − m| (3.66) ∈ ∈ k1 Z kj Z and the iterated integral can also be calculated: |t| s1 sj−1 b(|t|−s1) b(s1−s2) b(sj−1−sj ) asj ··· e e ···e e dsj ···ds1 0 0 0 |t| s1 sj−1 b|t| (a−b)sj = e ··· e dsj ···ds1 0 0 0 ∞ eb|t| ((a − b)|t|)k = . (3.67) (a − b)j k! k=j Lieb–Robinson Bounds for the Toda Lattice 461

This shows that ∞ ∞ 38 ∂ Bγ j ((a − b)|t|)k w ≤ | − | a|t| + b|t| Fn (t) B1Gμ n m e e . (3.68) ∂z ∞ a − b k! j=1 k=j

∗ Since vμ+ε ≤ v , it is clear that a − b<0. In this case, ∞ ∞ Bγ j ((a − b)|t|)k Bγ = eBγ|t| − e(a−b)|t| (3.69) a − b k! Bγ + b − a j=1 k=j follows from (3.34)andso ∂ Bγ w ≤ | − | a|t| + (Bγ +b−a)|t| − Fn (t) B1Gμ n m e 1 e 1 . (3.70) ∂z ∞ Bγ + b − a

This proves (3.41) and completes the proof.

3.3 Comments on These Results

It is clear that results analogous to Theorems 3.1 and 3.3 hold for more general, ﬁnite range potentials W . We stated the results as above for simplicity of presentation. It is also clear that analogues of Theorem 2.3, for more general observables, hold as direct corollaries of Theorems 3.1 and 3.3. Since the statements are clear, we do not rewrite them for the sake of brevity.

4 Bounds for the Hierarchy

In this section, we will demonstrate that the results from Section 2 also apply to the Toda hierarchy. We begin, in Section 4.1, by introducing the Toda hierarchy using the recursive approach from [8]. We will use essentially the same notation as [40] and refer the interested reader to this text for further information. In Section 4.2, we state the Lieb–Robinson bound valid for the hierarchy, see Theorem 4.4. To prove this result, we first establish the crucial solution estimate, an analogue of Theorem 2.1, which shows that a solution of the hierarchy at site n has weak dependence on the initial condition at site m if |n − m|1. The bound is explicit in terms of a quantity vr which, in particular, depends on the initial condition through the infinity norm of a specific matrix, see (4.11). To make our bound even more concrete, we estimate the norm of this matrix in Lemma 4.3 below. Theorem 4.4 follows as in Section 2. We note that our results also immediately apply to the Kac–van Moerbeke hierarchy. This hierarchy can be viewed as a special case of the Toda hierarchy which is obtained by setting bn ≡ 0 in the even order Toda equations. As is discussed, e.g. in [27], this gives precisely the equations of the Kac–van Moerbeke hierarchy.

4.1 The Toda Hierarchy ∈ = ∪{ } = We introduce the Toda Hierarchy as follows. Fix r N0 N 0 .Takec0 1 and choose ∈ ≤ ≤ ∈ constants cj R for 1 j r. For each x M, define sequences g(x,r) and h(x,r) com- ponentwise by setting r = ˜(j+1) ˜(j) = j gn(x,r) cr−j gn (x) with gn (x) δn,L(x) δn , (4.1) j=0 462 U. Islambekov et al. and r 39 = ˜(j+1) ˜(j) = j hn(x,r) cr−j hn (x) with hn (x) 2an δn+1,L(x) δn , (4.2) j=0 with L(x) as in (2.8). We now define a system of equations for the components of an un- known sequence x(t, r) ={(an(t, r), bn(t, r))}: a˙n(t, r) = an(t, r) gn+1(t, r) − gn(t, r) (4.3) and ˙ bn(t, r) = hn(t, r) − hn−1(t, r) (4.4) with initial condition x(0) ={(an(0,r),bn(0,r))}=x ∈ M. As in the previous section, we have, for example, denoted by gn(t, r) = gn(x(t, r), r) to simplify notation. The system (4.3) and (4.4) is known to have global solutions (see e.g. [40, Theorem 12.6]) for initial condi- ∈ (r) tions in M. Varying r N0 describes the Toda hierarchy. Let us denote by Φt the flow cor- (r) (r) responding to the Toda hierarchy, i.e. Φt : M → M satisfies Φt (x) ={(an(t, r), bn(t, r))}, (r) = the solution of (4.3)and(4.4) above, with Φ0 (x) x. The simplest example corresponds to r = 0. In this case, the system becomes a˙n(t, 0) = an(t, 0) bn+1(t, 0) − bn(t, 0) and (4.5) ˙ 2 2 bn(t, 0) = 2 an(t, 0) − an−1(t, 0) , which is, of course, the Toda system (2.4).

4.2 The Lieb–Robinson Bound

The above choice of sequences g and h guarantee that the Lax-formalism of the Toda Lat- tice, see (2.8)–(2.12), still holds for the hierarchy. In fact, set r = ˜ (j+1) ˜ (j) = j − j P(x,r) cr−j P (x) with P (x) L(x) + L(x) −, (4.6) j=0 where [A]± denote the upper and lower triangular parts of an operator with respect to the standard basis δm(n) = δm,n (with δm,n the usual Kronecker delta). It is known, see e.g. [40], that the Toda hierarchy is equivalent to the Lax equation d L(t) = P(t,r),L(t) , (4.7) dt

(r) (r) where we have set L(t) = L(Φt (x)) and P(t,r)= P(Φt (x), r) again to ease notation. It is easy to see that the operator P(t,r), which is of order 2r + 2, is skew-adjoint and differentiable. Like before then, there exists a unique unitary propagator U (r)(t, s) for P(t,r).It follows from the Lax equation that

L(t) = U (r)(t, s)L(s)U (r)(t, s)−1 (4.8) implying again an a-priori estimate ≤ = max a(t,r) ∞, b(t,r) ∞ L(t) 2 L(0) 2. (4.9) For later use we record the following structure. Lieb–Robinson Bounds for the Toda Lattice 463

Lemma 4.1 ([23]) For each x ∈ M and any integer j ≥ 1, the sequences g˜(j)(x) and h˜(j)(x), as deﬁned in (4.1) and (4.2), have components that are homogeneous. In fact, they are sums 40 of monomials of the components of x with degree j and j + 1, respectively, which have the form: ⎧ ⎪ k−1 2 + + − − + ⎪( =0 an+)bn+k R(n k 1,n k 1) ⎪ ⎨ k 2 k−1 + ( = a − )(bn−k + 2 = bn−), j = 2k + 1, g˜(j)(x) = 1 n 0 n ⎪ k−2 2 2 + 2 + k−2 ⎪( =0 an+)(an+k−1 bn+k−1 2bn+k−1 =0 bn+) ⎩⎪ + + − − + + k 2 = R(n k 2,n k 1) =1 an−,j2k, and ⎧ ⎪ k−1 2 2 + 2 + k−1 ⎪2( =0 an+)(an+k bn+k 2bn+k =0 bn+) ⎪ ⎨ k 2 + R(n + k − 1,n− k + 1) + 2 = a − ,j= 2k + 1, h˜(j)(x) = 0 n n ⎪ k−1 2 + + − − + ⎪2( =0 an+)bn+k R(n k 1,n k 2) ⎩⎪ + k−1 2 + + k−2 = 2 =0 an−(bn+1 bn−k+1 2 =0 bn−), j 2k, for j>1. Here R(n,m) denotes terms which involve only a and b with m ≤ ≤ n and we set R(n,m) = 0 if n

The result below is an analogue of Theorem 2.1 for the Toda hierarchy.

∈ ∈ ∈ Theorem 4.2 Fix x M, μ>0 and r N0. For any n, m Z, the bound ∂ ∂ r −μ(|n−m|/( +1)−vr |t|) max an(t, r), bn(t, r) ≤ e 2 , (4.10) ∂z ∂z ∈ ∈{ } holds for all t R, where z am,bm , 1 v = v (x,μ)= D(r) L(0) eμ+1 + , (4.11) r r ∞ 2 μ and D(r) depends on x, r, and the numbers c1,...,cr .

≥ ∈ : Proof Without loss of generality assume t 0. For each n Z, deﬁne the function Fn R → R2 by an(t, r) Fn(t, r) = . (4.12) bn(t, r) Since r will be ﬁxed for the remainder of the argument, we will drop it from our notation. Using the equations of motion (4.3)and(4.4), it is clear that t an(s)(gn+1(s) − gn(s)) Fn(t) = Fn(0) + ds 0 hn(s) − hn−1(s) r t ˜(j+1) −˜(j+1) an(s)(gn+1 (s) gn (s)) = F (0) + c − ds. (4.13) n r j ˜(j+1) − ˜(j+1) j=0 0 hn (s) hn−1 (s) 464 U. Islambekov et al.

≤ ≤ ˜(j+1) −˜(j+1) Observe that by Lemma 4.1, for each 0 j r, the quantities an(s)(gn+1 (s) gn (s)) ˜(j+1) − ˜(j+1) + 41 and hn (s) hn−1 (s) are homogeneous polynomials (of degree j 2) in the variables j + + | |≤ + an e(s) and bn e(s) for e 2 1. Differentiating with respect to z ∈{am,bm} we get ∂ ∂ r t ∂ = + (j) Fn(t) Fn(0) cr−j Dn,e(s) Fn+e(s) ds, (4.14) ∂z ∂z 0 ∂z j=0 | |≤ j + e 2 1

(j) where the entries of Dn,e(s) are homogeneous polynomials of degree j + 1. In particular, the following estimate, analogous to the bound in (2.22), holds (j) ≤ j+1 (j) Dn,e(s) L(0) 2 De , (4.15)

(j) an explicit formula for De appears in (4.27) below. As a result, it is clear that r t ∂ ∂ j+1 (j) ∂ ≤ + | − | + Fn(t) Fn(0) cr j L(0) 2 De Fn e(s)ds. (4.16) ∂z ∂z 0 ∂z j=0 | |≤ j + e 2 1

Upon iteration of the above inequality (in the case that z = am), we ﬁnd that ∞ r r k + + ∂ t j1 1 jk 1 ≤ | − | ··· | − | Fn(t) cr j1 L(0) 2 cr jk L(0) 2 ∂am k! k=0 j1=0 jk =0 (j1) (jk) 1 × ··· δm+e +···+e (n)D ···D 1 k e1 ek 0 | |≤ j1 + | |≤ jk + e1 2 1 ek 2 1 ∞ k r ( L(0) 2t) j1 ≤ |c − | L(0) ··· k! r j1 2 =| − | r + j =0 k n m /( 2 1) 1 r jk (j ) (j ) 1 × |c − | L(0) D 1 ···D k r jk 2 0 jk =0 ∞ k ( L(0) 2t) 1 ≤ D(r)k (4.17) k! 0 =| − | r + k n m /( 2 1) wherewehaveset

r (j) (j) j (j) = = | − | D De and D(r) cr j L(0) 2D . (4.18) | |≤ j + j=0 e 2 1

A similar estimate holds for the case z = bm. Lieb–Robinson Bounds for the Toda Lattice 465

Taking the inﬁnity norm one obtains that for any μ>0 42 ∞ k ∂ ( L(0) 2t) k Fn(t) ≤ D(r) ∂z k! ∞ ∞ =| − | r + k n m /( 2 1)

−μ(|n−m|/( r +1)−v |t|) ≤ e 2 r , (4.19)

μ+1 1 = ∞ + wherewehavesetvr D(r) L(0) 2(e μ ).

We now provide a rough, but explicit, estimate on the velocity corresponding to the Toda hierarchy.

∈ ∈ Lemma 4.3 Fix x M, μ>0 and r N0. The velocity corresponding to the Toda hierarchy, see (4.10) and (4.11) satisﬁes r μ+1 1 j+1 j v ≤ 8 e + |c − | L(0) (j + 2)3 . (4.20) r μ r j 2 j=0

Proof It is clear from (4.11) that we need only estimate the quantity D(r) ∞, with D(r) as defined in (4.18). To see this, first recall that for each 1 ≤ j ≤ r, the quantities ˜(j+1) −˜(j+1) ˜(j+1) − ˜(j+1) an(s)(gn+1 (s) gn (s)) and hn (s) hn−1 (s) are homogeneous polynomials (of de- j + + + | |≤ + gree j 2) in the variables an e(s) and bn e(s) for e 2 1. We need an estimate on the number of terms in each of these polynomials. Both sequences g˜(j+1) and h˜(j+1),see(4.1)and(4.2), have components defined in terms ± 2 ± = of the operator L(x) given by (2.8). In terms of the shifts S on (Z), i.e. (S f)n fn±1, one can write

L(x) = aS+ + a−S− + b. (4.21)

Here a and b are regarded a multiplication operators and a± is the multiplication operator ± given by (a f)n = an±1fn. An upper bound on the desired number of terms can be obtained by taking both a and b to be constant sequences, i.e., calculating j+1 j+ j + 1 k η(j+1) = δ , S+ + S− + I 1δ = (4.22) n n k k/2 k=0,keven and j+1 + (j+1) + − j+1 j 1 k ξ = δ + , S + S + I δ = . (4.23) n 1 n k (k + 1)/2 k=0,kodd

The bounds η(j+1) ≤ 2ξ (j+1) and ξ (j+1) ≤ 3j readily follow. Next, using this estimate, we can expand

+ 2η(j 1) j+2 ˜(j+1) −˜(j+1) = an(s) gn+1 (s) gn (s) dn+ek, (4.24) k=1 =1 466 U. Islambekov et al. and + 43 2ξ(j 1) j+2 ˜(j+1) ˜(j+1) h (s) − h − (s) = 2 d + (4.25) n n 1 n ek, k=1 =1 where dn+e ∈{±an+e(s), ±bn+e(s)}. Here we have inserted a 2 because of the deﬁnition of ˜(j+1) (j) h ,see(4.2). This form enables us to determine the entries of the matrix Dn,e(s). In fact, differentiation of the left hand side of (4.24), with respect to z ∈{am,bm}, yields

η(j+1) j+2 j+2 η(j+1) j+2 j+2 2 ∂ ∂ 2 d + · d + = d + δ (e ) d + (4.26) ∂z n ek,o n ek, ∂z n e e k,o n ek, k=1 o=1 =o | |≤ j + k=1 o=1 =o e 2 1 and an analogous formula holds for the partial derivative of the left hand side of (4.25). (j) These expressions determine the matrix entries of Dn,e(s). The bound in (4.15) now follows with ⎛ ⎞ (j+1) + (j+1) + 2η j 2 δ (e ) 2η j 2 δ (e ) (j) = ⎝ k=1 o=1 e k,o k=1 o=1 e k,o ⎠ D + + . (4.27) e 2ξ (j 1) j+2 2ξ (j 1) j+2 2 k=1 o=1 δe(ek,o) 2 k=1 o=1 δe(ek,ξ ) Summing on e yields, (j) = (j) = + (j+1) D ∞ De 8(j 2)ξ (4.28) ∞ | |≤ j + e 2 1 wherewehaveusedη(j+1) ≤ 2ξ (j+1). This shows that

r j j ≤ | − | + D(r) ∞ 8 cr j L(0) 2(j 2)3 , (4.29) j=0 and we are done.

(r) We end this section with an analogue of Theorem 2.3 for the hierarchy. Let αt denote the dynamics corresponding to the Toda hierarchy, i.e., α(r)(A) = A ◦ Φ(r) for all A ∈ . t t A ∈ ∈ Theorem 4.4 Let r N0,x M, and μ>0. There exist numbers Cr and vr for which given any observables A,B ∈ (1), the estimate A (r) αt (A), B (x) ∂A ∂A ∂B ∂B −μ(|n−m|/( r +1)−v |t|) ≤ C + + e 2 r ∈ ∂am x ∂bm x ∂an x ∂bn x n,m Z (4.30)

∈ = + μ/( r +1) holds for all t R. Here 4Cr a ∞(1 e 2 ) and vr is as in Theorem 4.2.

Proof The estimate (4.30) follows as in the proof of Theorem 2.3 using the results of Theo- rem 4.2 as input. Lieb–Robinson Bounds for the Toda Lattice 467

5 Results for the Perturbed Hierarchy 44 The purpose of this section is to demonstrate that the methods from Section 3 also apply to the Toda hierarchy. Many of the proofs follow closely the previous arguments, and so we only sketch the details. To introduce the relevant class of perturbations, we ﬁrst recall the Hamiltonian formula- ∈ tion of the hierarchy, see e.g. Section 1.7 of [15]. For each r N0, consider the following formal Hamiltonian

r = 4 ˜(j+2) − Hr (x) cr−j gk (x) λr+2 , r + 2 ∈ = kZ j 0 1 r (5.1) 2r r/2 ,reven, λr = 0,rodd,

˜(j+2) with gk as deﬁned in (4.1). It can be shown that Hr generates the Toda hierarchy in the sense that for any A ∈ 0, A d α(r)(A) = α(r) {A,H } = α(r)(A), H (5.2) dt t t r t r

(r) where αt is the dynamics associated with the Toda hierarchy as introduced before Theo- rem 4.4. In fact, it is shown in e.g. [15, Thm. 1.71] that

∂Hr ∂Hr (x) = 4gn(x,r) and An (x) = 4hn(x,r) (5.3) ∂an ∂bn and therefore the evolution equations d (r) (r) 1 (r) ∂Hr ∂Hr αt (An) = αt {An,Hr } = αt An − (5.4) dt 4 ∂bn+1 ∂bn and similarly, d (r) (r) 1 (r) ∂Hr ∂Hr αt (Bn) = αt {Bn,Hr } = αt An − An−1 (5.5) dt 4 ∂an an−1 follow; compare with (4.3)and(4.4). Note that for r = 0 = 2 + 2 − H0(x) 2bk 4ak 1 (5.6) ∈ k Z is different from (2.5). However, the equations of motion (2.4)arethesameasabove. ∈ : →[ ∞ We can now introduce perturbations as in Section 3.Fixr N0 and let W R 0, ) satisfy W ∈ C2(R). Consider the formal Hamiltonian w = + Hr Hr Wn (5.7) ∈ n Z = 2 where Hr is as in (5.1) above and Wn is the observable with Wn(x) W(ln(4an)). w The equations of motion corresponding to Hr are ˙ w = w − an (t, r) an (t, r) gn+1(t, r) gn(t, r) (5.8) 468 U. Islambekov et al. and 45 ˙w = − + bn (t, r) hn(t, r) hn−1(t, r) Rn(t) (5.9) where 1 w 2 w 2 R (t) = W ln 4a (t, r) − W ln 4a − (t, r) . (5.10) n 2 n n 1 Again, local existence and uniqueness of solutions of (5.8)and(5.9), corresponding to initial conditions x ∈ M0, follows from standard results, [1, Thm. 4.1.5]. As before, let w w = us denote by Φt,r the perturbed ﬂow of the Toda hierarchy, i.e., the function Φt,r(x) { w w } = (an (t, r), bn (t, r)) . Our arguments apply to the set of initial conditions Mb,r Mb,r (W) with bounded trajectories, i.e., for which there exist numbers C1,C2 < ∞ with 1 w ≤ ≤ sup Φt,r(x) M C1 and sup sup C2. (5.11) ∈ ∈ ∈ | w | t R t R n Z an (t, r)

For initial conditions x ∈ Mb,r , there are results similar to the two main estimates from Section 3. We ﬁrst state an analogue of Theorem 3.1.

∈ ∈ 2 ∈ ∞ ∈ Theorem 5.1 Fix r N0, W C (R) with W L (R), and x Mb,r . For each μ>0, w = w ∈ there exists a number vr vr (μ, x) for which given any n, m Z, the estimate ∂ w ∂ w −μ(|n−m|/( r +1)−vw|t|) max a (t, r), b (t, r) ≤ e 2 r , (5.12) ∂z n ∂z n ∈ ∈{ } holds for all t R. Here z am,bm .

Proof We will follow closely the proof of Theorem 3.1 using Theorem 4.2 as input. Take t ≥ 0 and, as before, introduce aw(t, r) w = n Fn (t, r) w . (5.13) bn (t, r)

w We will suppress the dependence on r.SinceW is sufﬁciently smooth, Fn is differentiable with respect to z ∈{am,bm} and the bound

+ ∂ ∂ r 1 t ∂ w ≤ w + (j) w Fn (t) Fn (0) αj De Fn+e(s) ds (5.14) ∂z ∂z 0 ∂z j=0 |e|≤βj follows as in Theorem 4.2. Here we have denoted by =| | j+1 ≤ ≤ = αj cr−j C1 for 0 j r with αr+1 C2 W ∞ (5.15) and

βj =j/2+1for0≤ j ≤ r with βr+1 = 1. (5.16) (j) Moreover, for 0 ≤ j ≤ r,wehavetakenDe as in the proof of Theorem 4.2 and set 00 + 00 D(r+1) = for e =−1, 0andD(r 1) = . (5.17) e 10 1 00 Lieb–Robinson Bounds for the Toda Lattice 469

Taking z = am and iterating yields 46 ∞ k ∂ w t w k 1 F (t) ≤ D (5.18) ∂z n k! r 0 =| − | r + k n m /( 2 1) with r+1 w = (j) (j) = (j) Dr αj D and D De . (5.19) j=0 |e|≤βj

A similar estimate holds for the case z = bm. As in the proof of Theorem 4.2, the bound (5.12) now follows with 1 vw = Dw eμ+1 + . (5.20) r r ∞ μ

Note also that r w ≤ | | j (j) + Dr ∞ C1 cr−j C1 D ∞ 2C2 W ∞, (5.21) j=0 and so the estimate from Lemma 4.3 applies here as well.

The interpolation argument proven in Section 3 generalizes to the hierarchy as well. To see this, we ﬁrst prove an analogue of Lemma 3.2.

∈ ∈ = Lemma 5.2 Fix r N0, μ>0, and let x M0. There exist a number C C(r,μ,x)>0 and a function h, depending on r, μ, and x, for which given any n, k, ∈ Z, the estimate 2 2 ∂ ∂ max ˜ an(t, r), ˜ bn(t, r) ∂z∂bk ∂z∂bk −μ(|n−|/( r +1)) −μ(|n−k|/( r +1)) 2μv |t| ≤ Ce 2 e 2 e r h(t) (5.22) ∈ holds for all t R. Here an(t, r) and bn(t, r) are the solutions of (2.4) with initial condition ∂ ∂ ∂ x ∈ M, ˜ = − , z ∈{a,b}, and the number vr is as in Theorem 2.1. The function ∂bk ∂bk+1 ∂bk h grows at most exponentially.

Proof Again, since this proof follows closely the arguments of Lemma 3.2,wewillonly sketch the details. Keeping with the previous notation, it is clear that 2 r t ∂ ∂ (j) ∂ F (t) = c − D (s) F + (s) ds. (5.23) ˜ n r j ∂z n,e ˜ n e ∂z∂bk j=0 |e|≤j/2+1 0 ∂bk

Differentiation with respect to z produces two terms. We begin by estimating the ﬁrst term, ∂ (j) | |≤ + i.e., the one that contains ∂zDn,e(s).UsingTheorem4.2, it is clear that if e j/2 1, then ∂ − | − | r + − | | 1 ≤ μ μ( n k /( 2 1) vr s ) ˜ Fn+e(s) e e . (5.24) ∂bk 1 470 U. Islambekov et al.

For convenience, let us denote by 47 −μ(|x|/( r +1)) E(x) = e 2 . (5.25)

Using (5.24), one can show that |t| ∂ ∂ |t| (j) 2μvr s 1 D (s) Fn+e(s) ds ≤ γj (μ)E(n − k)E(n − ) e ds ∂z n,e ˜ 1 |e|≤j/2+1 0 ∂bk 0 (5.26) where = 2μ j + + γj (μ) 8e L(0) 2(j 1)(j 2). (5.27) This proves that | | ∂2 t 1 ≤ − − 2μvr s ˜ Fn(t) Cμ(r)E(n k)E(n ) e ds ∂z∂bk 0 1 r |t| 2 j (j) ∂ + L(0) |c − | L(0) D F + (s) ds 2 r j 2 e ˜ n e j=0 |e|≤j/2+1 0 ∂z∂bk (5.28) where r Cμ(r) = |cr−j |γj (μ), (5.29) j=0

(j) and De is as in (4.15). Iteration, as in the proof of Lemma 3.2, yields

∞ 2 m ∂ Cμ(r) L(0) 2 2μ Fn(t) ≤ E(n − k)E(n − ) e ˜ 2μv 2μv ∂z∂bk r m=0 r ∞ p (2μvr |t|) 1 × D(r)˜ m , (5.30) p! 1 p=m+1 where r ˜ j (j) = | − | D(r) cr j L(0) 2 De . (5.31) j=0 |e|≤j/2+1 Taking ∞-norms, and using (3.34), we see that ∂2 ≤ − − 2μvr |t| ˜ Fn(t) CE(n k)E(n )e h(t) (5.32) ∂z∂bk ∞ where 2μ ˜ C (r) e L(0) D(r) ∞ C = μ and β = 2 − 1 (5.33) 2μvr 2μvr h is as in (3.36). This proves (5.22) and we are done. Lieb–Robinson Bounds for the Toda Lattice 471

Given Lemma 5.2 above, the analogue of Theorem 3.3 follows with only minor modiﬁ- cations. We state it below. 48

∈ ∈ 2 ∈ ∞ ∈ Theorem 5.3 Fix r N0, W C (R) with W ,W L (R), and let x Mb,r . For each μ>0 and any ε>0, there are positive numbers C = C(r,ε), D = D(r,ε,μ,x,W), and δ = δ(r,ε,μ,x,W)such that ∂ ∂ w w (μ+ε)vr |t| δ|t| max a (t, r), b (t, r) ≤ CGμ |n − m| e 1 + D e − 1 (5.34) ∂z n ∂z n r ∈ = + = r + holds for all t R. Here vr vr (μ ε, x) is as in Theorem 4.2, μr μ/( 2 1), and z ∈{am,bm}.

Proof Interpolating as before, it is clear that for any x ∈ M, the bound ∂ ∂ |t| ∂ ˜ w ≤ ˜ + ; ˜ w Fn (t) (x) Fn(t) (x) Dk(s n) Fk (s) (x) ds ∂z ∞ ∂z ∞ ∈ 0 ∂z ∞ k Z |t| W ∞ ∂ + ; ˜ w Dk,(s n) F (s) (x) ds (5.35) 2 ∈ 0 ∂z ∞ k, Z follows as in the proof of (3.55); here all quantities depend now also on r, but we have suppressed this in our notation. With Theorem 4.2, it is clear that ∂ ˜ ∂ Fn(t) (x) = Fn(t) ∂z ∞ ∂z ∞

−(μ+ε)(|n−m|/( r +1)−v |t|) ≤ e 2 r ≤ | − | (μ+ε)vr |t| Cε,r Gμr n m e , (5.36) wherewehaveset

− r + μ = + 2 ε( x/( 2 1) ) = = + Cε,r sup(1 x) e ,μr r , and vr vr (x,μ ε). (5.37) ≥ + x 0 2 1

Similarly, for each x ∈ Mb,r , the matrix appearing in the second term satisﬁes + − + | − | r + − | |− ; ≤ μ ε (μ ε)( n k /( 2 1) vr (s)( t s)) Dk(s n) (x) ∞ C2 W ∞e e + + ∗ | |− ≤ μ ε | − | (μ ε)vr ( t s) C2 W ∞e Cε,r Gμr n k e (5.38) with = w + ∗ = vr (s) vr Φs,r(x), μ ε and vr sup vr (s). (5.39) ∈ s R Lastly, the bound + + ∗ | |− ; ≤ ˜ | − | | − | 2(γ 1)(μ ε)vr ( t s) Dk,(s n) (x) ∞ CGμr n Gμr n k e (5.40) follows as in the proof of (3.63); with (possibly) different values of C˜ and γ . Iteration yields (5.34) as in the proof of Theorem 3.3. 472 U. Islambekov et al.

6 Locality Bounds for More General Initial Conditions 49 In this section, we return to the class of perturbations considered in Section 3, see also Section 5. The goal here is to prove a locality estimate for more general initial conditions. Let us recall the basic set-up. ∞ Fix W : R →[0, ∞) satisfying W ∈ C2(R) with W ,W ∈ L (R). The formal Hamil- tonian is given by w H = H + Wn, (6.1) ∈ n Z where H is the Toda Hamiltonian as in (2.5), and for each n, the perturbation Wn is taken to = 2 be the observable Wn(x) W(ln(4an)). The corresponding equations of motion are ˙ w = w w − w an (t) an (t) bn+1(t) bn (t) , (6.2) ˙w = w 2 − w 2 + bn (t) 2 an (t) an−1(t) Rn(t) where 1 w 2 w 2 R (t) = W ln 4a (t) − W ln 4a − (t) . (6.3) n 2 n n 1 As we discussed before, local existence and uniqueness of solutions of (6.2) corresponding to initial conditions x ∈ M0 follows again from Theorem 4.1.5 in [1]. For x ∈ M0, let us again w ={ w w } = w denote by Φt (x) (an (t), bn (t)) the perturbed Toda flow. If we set, L(t) L(Φt (x)) = w and P(t) P(Φt (x)) with L and P as in (2.8)and(2.9), it is easy to check that d L(t) = P(t),L(t) + R(t), (6.4) dt where the multiplication operator R(t) : 2(Z) → 2(Z) is defined by = R(t)f n Rn(t)fn. (6.5) From this definition, it is clear that ≤ R(t) 2 W ∞, (6.6) and hence the operator norm of R(t) is bounded uniformly in both t and the initial condition x ∈ M0. One can check that P(t) still corresponds to a family of unitary propagators, which we denote by U(t,s). A short calculation shows then that L(t)˜ = U(t,s)∗L(t)U(t, s) satisfies

d L(t)˜ = U(t,s)∗R(t)U(t,s). (6.7) dt In this case, |t| = ˜ ≤ ˜ + ≤ + | | L(t) 2 L(t) 2 L(0) 2 R(s) 2 ds L(0) 2 W ∞ t . (6.8) 0 Since the bound w w ≤ max sup an (t) , sup bn (t) L(t) 2 (6.9) n n Lieb–Robinson Bounds for the Toda Lattice 473 holds, (6.8) produces a linear bound on the growth of solutions. From this, the existence of global solutions follows from Proposition 4.1.22 in [1]. 50 A result analogous to Theorem 2.1 follows.

={ } ∈ ∈ ∗ = | | Theorem 6.1 Let x (an,bn) n Z M0 with a infn an > 0. Then for any μ>0 and n, m ∈ Z the estimate ∂ ∂ 2 w w 2 w ≤ −μ(|n−m|−v (t)) max ln an (t) , bn (t) max 1, e (6.10) ∂z ∂z a∗

∈ ∈{ } w holds for all t R. Here z am,bm and v (t) is an explicit cubic polynomial with vw(0) = 0.

∈ ∈ Proof Fix x M0 with a∗ > 0. Global existence on M guarantees that for each n Z,the w : → 2 function Fn R R given by ln(aw(t)2) w = n Fn (t) w (6.11) bn (t) is well-deﬁned for all t ∈ R. In fact, since a∗ > 0, the equations of motion ensure that t w = w w − w an (t) an (0) exp bn+1(s) bn (s) ds (6.12) 0 preventing a singularity in the logarithm. It is then clear that (6.2) implies t w w b + (s) − b (s) F w(t) = F w(0) + 2 n 1 n ds. (6.13) n n w 2 − w 2 + 1 0 an (s) an−1(s) 2 Rn(s)

For any z ∈{am,bm}, a relation similar to (2.18) holds, i.e. ∂ ∂ t ∂ F w(t) = F w(0) + 2 Dw (s) F w (s) ds, (6.14) ∂z n ∂z n n,e ∂z n+e |e|≤1 0 where 0 δ (e) − δ (e) w 1 0 D (s) = . n,e w 2 W ∞ w 2 W ∞ + − + − (an (s) 4 )δ0(e) (an−1(s) 4 )δ 1(e) 0 (6.15) The bounds in (6.8)and(6.9) show that w 2 ≤ + | | 2 an (s) L(0) 2 W ∞ s , (6.16) and so we ﬁnd that |t| ∂ w ∂ w w ∂ w F (t) ≤ F (0) + g(s)D F + (s) ds (6.17) ∂z n ∂z n e ∂z n e |e|≤1 0 474 U. Islambekov et al. holds for any t ∈ R. Here we have denoted by 51 W ∞ g(s) = 1 + L(0) + W |s| 2 + and 2 ∞ 4 0 δ (e) + δ (e) (6.18) w = 1 0 De δ0(e) + δ−1(e) 0

Iteration (with z = am) yields ∞ | | ∂ t t1 tk−1 w ≤ ··· ··· ··· Fn (t) g(t1) g(tk) (6.19) ∂z 0 0 0 k=|n−m| |e1|≤1 |ek |≤1 w w ∂ w × D ···D F + +···+ (0) dtk ···dt1 (6.20) e1 ek ∂z n e1 ek ∞ k 2 h(t) k 1 ≤ Dw , (6.21) |a | k! 0 m k=|n−m| = |t| where h(t) 2 0 g(s)ds and 1 01 Dw = Dw = . (6.22) 2 e 10 |e|≤1

In the case that z = bm, it is clear that ∞ k ∂ w h(t) w k 0 F (t) ≤ D . (6.23) ∂z n k! 1 k=|n−m|

Taking the inﬁnity norm, we get for any μ>0 ∞ k ∂ w 2 h(t) F (t) ≤ max 1, ∂z |a | k! ∞ m k=|n−m| 2 w ≤ max 1, e−μ(|n−m|−v (t)), (6.24) a∗

w μ+1 −1 where v (t) = h(t)(e + μ ).

Again, everything extends to the hierarchy. Since the statements are clear, we do not rewrite them for the sake of brevity.

Acknowledgements The authors would like to thank Bruno Nachtergaele for suggesting the problem discussed in this article and for several productive discussions. Also, the authors would like to acknowledge the hospitality of the Erwin Schrödinger Institute in Vienna, Austria where parts of this paper were discussed. Lieb–Robinson Bounds for the Toda Lattice 475

Appendix Existence of Bounded Solutions for Hamiltonian Systems 52 In this appendix we want to look at a general Hamiltonian system with nearest neighbor interaction: 2 pn H(x) = + V(qn+1 − qn) , (A.1) ∈ 2 n Z 2 ={ } ∈ ∈ ≥ = = where x (pn,qn) n Z.LetV C (R) with V(x) 0andV(0) V (0) 0 such that 0 is a fixed point of the system. Since we want to obtain bounded solutions we will assume that our interaction potential is confining in the sense that V(x)→+∞as x →±∞.Since 0 0 0 0 the uniform motion qn(t) = q + p t (with q ,p some real constants) of the system is unbounded we switch to relative coordinates rn = qn+1 −qn in which the equation of motions read r˙n = pn+1 − pn, p˙n = V (rn) − V (rn−1). (A.2) We will consider this system in the Hilbert space = 2(Z) × 2(Z). X Theorem A.1 Suppose V ∈ C2(R) such that 0 is a unique global minimum with V(0) = V (0) = 0, V (0)>0 and V(x)→+∞as |x|→∞. Then the system (A.2) has a unique global solution in for which the energy X 2 pn H(p,r)= + V(rn) (A.3) ∈ 2 n Z is finite and conserved. This solution is C1 with respect to the initial condition. Moreover,0is a stable fixed point and all solutions satisfy (p(t), r(t)) 2 ≤ C as well as (p(t), r(t)) ∞ ≤ C, where the constant C depends only on the initial condition.

Proof First of all note that by our assumption on V we can find constants cR and CR for 2 2 every R>0 such that |V(x)|≤CRx , |V (x)|≤CR|x| and |V(x)|≥cRx for |x|≤R. ∈ 2 ≤ ≤ In particular, for r (Z) with r 2 R we have V (r) 2 CR r 2 and it follows → 1 2 → that the map rn V (rn) is C on (Z). Since the same is true for the shift operator xn 1 xn−1, our vector field is C and local existence and uniqueness follow from standard results [1, Thm. 4.1.5]. This also implies that the flow is C1 with respect to the initial condition [1, Lem. 4.1.9]. Moreover, |H(p,r)|≤CR (p, r) 2 implies that H is finite on and a straightforward calculation shows that it is conserved by the flow. X Moreover, H(p,r)≥ c1 ε for (p, r) 2 = ε and ε ≤ 1. Hence [1, Thm. 4.3.11] shows that 0 is a stable fixed point. Finally, if V(x)→+∞there is a constant ME such that |x|≤ME whenever |V(x)√ |≤E. Hence setting E = H(p(0), r(0)) we have H (p(t), r(t)) = E implying√ p(t) 2 ≤ 2E and −1 −1 r(t) ∞ ≤ M . But this implies r(t) ≤ c V(r) ≤ c E. Hence our vector field E 2 ME 2 ME remains bounded along integral curves on finite t intervals and hence all solutions are global in time by [1, Prop. 4.1.22]. = + t Note that, using qn(t) qn(0) 0 pn(s)ds, for our original variables we get ≤ + | | ≤ + | | q(t) 2 q(0) 2 C t , q(t) ∞ q(0) ∞ C t . (A.4) Moreover, clearly the Toda potential V(r)= e−r + r − 1 satisfies the above assumptions. 476 U. Islambekov et al.

Theorem A.2 Let x = (p, r) ∈ and μ>0. There exists a number v = v(μ,x) for which X given any n, m ∈ Z, the bound 53 ∂ ∂ −μ(|n−m|−v|t|) max pn(t), rn(t) ≤ Ce , (A.5) ∂z ∂z ∈ ∈{ } holds for all t R and each z pm,rm , where 1/2 C = C(x) = max sup V rn(t) , 1 . (A.6) ∈ × (t,n) R Z In fact, one may take 1 v = 2C eμ+1 + . (A.7) μ

Proof Fix x ∈ . Our previous theorem guarantees that for each x ∈ M and n ∈ Z,the : X→ 2 function Fn R R given by rn(t) Fn(t; x) = (A.8) pn(t) is well-deﬁned and differentiable with respect to each z ∈{pm,rm}. When convenient, we will suppress the dependence of Fn on x. Using the equations of motion, i.e. (A.2), it is clear that t p + (s) − p (s) = + n 1 n Fn(t) Fn(0) ds. (A.9) 0 V (rn(s)) − V (rn−1(s)) Differentiating with respect to z we obtain ∂ ∂ t ∂ F (t) = F (0) + D (s) F + (s) ds, (A.10) ∂z n ∂z n n,e ∂z n e |e|≤1 0 with 0 δ (e) − δ (e) = 1 0 Dn,e(s) . (A.11) V (rn)δ0(e) − V (rn−1)δ−1(e) 0 Hence |t| ∂ ∂ ∂ Fn(t) ≤ Fn(0) + De Fn+e(s) ds, (A.12) ∂z ∂z ∂z |e|≤1 0 where 0 δ (e) + δ (e) = 1 0 De 2 . (A.13) C (δ0(e) + δ−1(e)) 0

Let us now consider the case that z = rm, i.e., ∂ 1 Fn(0) = δm(n). (A.14) ∂rm 0 In this case, iteration yields Lieb–Robinson Bounds for the Toda Lattice 477 ∞ k ∂ |t| 1 F (t) ≤ ··· δ + +···+ (n)D ···D n m e1 ek e1 ek 54 ∂rm k! 0 k=0 |e1|≤1 |ek|≤1 ∞ |2Ct|k 1 ≤ Dk , (A.15) k! 0 k=|n−m| wherewehaveset 1 0 C−1 D = De = . (A.16) 2C C 0 |e|≤1

∂ Again, convergence is guaranteed since ∂zFn(t) is continuous and thus bounded on compact time intervals. Using D2 = 1, one obtains ∞ k ∂ |2Ct| Fn(t) ≤ C (A.17) ∂r ∞ k! m k=|n−m|

The rest follows as in Theorem 2.1.

Note that in Flaschka variables (2.3) the equations of motion read a˙n(t) = an(t) bn+1(t) − bn(t) and 1 (A.18) b˙ (t) =− V − ln 4a2 − V − ln 4a2 , n 2 n n−1 and (p, r) will be bounded if and only if (a, a−1,b) are bounded. The ﬁxed point in these = 1 ∈ − = − new coordinates is (a0,b0) ( 2 , 0) and (p, r) if and only if (a, b) (a0,b0) (a 1 ∈ X 2 ,b) . Finally,X by t an(t) = an(0) exp bn+1(s) − bn(s) ds (A.19) 0 the sign of an is preserved under the ﬂow.

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Appendix B – An Estimate for Infinite Range Perturbations

In this Appendix, we consider inﬁnite range perturbations of the Toda system and prove a result similar to Theorem 3.1. We will follow the same notation for consis- tency.

Z 2 Z For each k ∈ Z, let Wk : R → [0, ∞) be C (R ) function satisfying the following decay condition: for any µ > 0, there exists α1 > 0 such that for all n, k, j ∈ Z the following bound holds: 2 2 ∂ Wk ∂ Wk := sup ({xi}i∈Z) ≤ α1Gµ(|n − k|)Gµ(|k − j|), ∂xj∂xn ∞ ∂xj∂xn ∞ {xi}i∈Z∈` (Z,R\{0}) (2.0.1) where Gµ is as in Section 3.2. We introduce the perturbed Hamiltonian as follows:

w X 2 H (x) = H(x) + Wk({ln(4aj )}j∈Z), (2.0.2) k∈Z where H is the Toda Hamiltonian and x = {(an, bn)}n∈Z is an initial condition in ∞ ∞ M0 = ` (Z, R \{0}) × ` (Z, R). The corresponding equations of motion take the form

w w w w a˙ n (t) = an (t) bn+1(t) − bn (t) (2.0.3) ˙w w 2 w 2 bn (t) = 2 an (t) − an−1(t) + Rn(t), where 1 X ∂Wk ∂Wk w 2 Rn(t) = − {ln(4ai (t) )}i∈Z . (2.0.4) 2 ∂xn ∂xn−1 k∈Z Existence and uniqueness of local solutions of (2.0.3) associated with an initial condition x ∈ M0 again follows from standard results (see e.g. Theorem 4.1.5 in [1]). As in Section 3, consider the set of initial conditions x ∈ Mb for which the bounds

w 1 sup kΦt (x)k∞ ≤ C1 and sup sup w ≤ C2 (2.0.5) t∈R t∈R n∈Z |an (t)| 58

w hold for some finite constants C1 and C2. Here Φt is the perturbed Toda flow. The theorem below is an analogue of Theorem 3.1 for the infinite range perturbations of the Toda system.

Theorem 2.0.1. Let Wk, k ∈ Z, be as above. Then for any µ > 0 and x ∈ Mb there exists a constant vw = vw(µ, x) such that for any given n, m ∈ Z, the estimate ∂ w ∂ w vw|t| max a (t) , b (t) ≤ Gµ(|n − m|)e (2.0.6) ∂z n ∂z n

w holds for all t ∈ R and each z ∈ {am, bm}. Moreover, v has an explicit form and can be taken as

w µ v = γ (1 + 4e ) (4C1 + γα1C2), (2.0.7) where γ is as in Section 3.2 (see (3.38)).

w w Proof. Let {(an (t), bn (t))}n∈Z be global solutions of the system (2.0.3) associated with an initial condition x ∈ Mb. Let

w w an (t) Fn (t) = w (2.0.8) bn (t) By (2.0.3), it is clear that

Z t w w w w w an (s) bn+1(s) − bn (s) Fn (t) = Fn (0) + w 2 w 2 ds. (2.0.9) 0 2 an (s) − an−1(s) + Rn(s)

Diﬀerentiating both sides with respect to z ∈ {am, bm} and observing that 2 2 ∂ 1 X X ∂ Wk ∂ Wk 2 ∂ w Rn(s) ≤ + w aj (s) ∂z 2 ∂xj∂xn ∂xj∂xn−1 aj (s) ∂z k∈Z j∈Z ∞ ∞

X ∂ w ≤ α1γC2 (Gµ(|n − j|) + Gµ(|n − 1 − j|)) a (s) ∂z j j∈Z

µ X ∂ w ≤ α1γ (1 + 4e ) C2 Gµ(|n − j|) a (s) , ∂z j j∈Z we ﬁnd that |t| ∂ ∂ X Z ∂ F w(t) ≤ F w(0) + Dw F w(s) ds, (2.0.10) n n n,j ∞ j ∂z ∞ ∂z ∞ 0 ∂z ∞ j∈Z 59 where w 2C1δn(j) C1 (δn(j) + δn+1(j)) Dn,j = (2.0.11) 4C1 (δn(j) + δn−1(j)) + α2C2Gµ(|n − j|) 0

µ with α2 = α1γ (1 + 4e ). Using the fact that δn(j) ≤ Gµ(|n−j|) and Gµ(|n+e−j|) ≤ µ 4e Gµ(|n − j|) for |e| ≤ 1, we get

Z |t| ∂ w X ∂ w Fn (t) ≤ δn(m) + C3 Gµ(|n − j|) Fj (s) , (2.0.12) ∂z ∞ 0 ∂z ∞ j∈Z

µ where C3 = 4C1 (1 + 4e ) + α2C2. Iteration yields

∞ k ∂ w X X X (C3|t|) Fn (t) ≤ ··· Gµ(|n − j1|) ··· Gµ(|jk−1 − jk|)δjk (m) ∂z ∞ k! k=0 j1∈Z jk∈Z ∞ k X X X (C3|t|) ≤ ··· G (|n − j |) ··· G (|j − j |)G (|j − m|) µ 1 µ k−1 k µ k k! k=0 j1∈Z jk∈Z ∞ k X (γC3|t|) ≤ G (|n − m|) µ k! k=0 γC3|t| ≤ Gµ(|n − m|)e . 60

Appendix C – Supplemental material to Section 2.4

This short Appendix provides a visual comparison between the Lieb–Robinson velocity and the velocity of the system in the context of one-soliton solutions. Consider

the behavior of an(t) for n = −50,..., 50 at times t = 0, 1, 2, 3, 4, 5 depicted below.

sinh(κ) In this picture, the soliton (the bump) is traveling with velocity κ , where κ is κ chosen to equal 0.7. The line passing through the peaks of the soliton has slope sinh(κ)

(reciprocal of the velocity). Since the Lieb-Robinson velocity, v(xκ, µ0), is an upper bound on the actual velocity, the soliton stays within the region above the black lines having slope ±1/v(xκ, µ0).