Additions for Jacobi Operators and the Toda Hierarchy of Lattice Equations

Item Type text; Electronic Dissertation

Authors Murphy, Dylan

Publisher The University of Arizona.

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Link to Item http://hdl.handle.net/10150/636509 ADDITIONS FOR JACOBI OPERATORS AND THE TODA HIERARCHY OF LATTICE EQUATIONS

Dylan Murphy

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

2019

Acknowledgements

I would like to express my thanks to my advisor, Nick Ercolani, for his advice, encouragement, and patience throughout my studies; and also to the members of my dissertation committee, Sergey Cherkis, Hermann Flaschka, Dave Glickenstein, and Shankar Venkataramani.

Thanks are also due to the National Science Foundation, which partially supported this work under grant DMS-1615921.

I would like to thank my colleagues in the graduate program whose friendship and advice were especially valuable, including (but not limited to) Colin Dawson, Andrew Leach, Kyle Pounder, Megan McCormick Stone, and Patrick Waters.

Finally, this work would not have been possible without the support of my family and friends, including my parents in Wyoming, my brother in California, and my dear friends here in Tucson; I am thankful to all of them.

3 Contents

Abstract 6

1 Introduction 7 1.1 Main results ...... 7 1.1.1 Operator factorization and commutation methods ...... 8 1.1.2 Inversion of additions ...... 9 1.1.3 Inﬁnitesimal additions and the Toda hierarchy ...... 9 1.1.4 Poisson structures and Hamiltonian mechanics ...... 10 1.2 Future directions ...... 10 1.3 Structure of the document ...... 10

2 Background: Darboux transformations and additions for Schrödingeroperators on R 12 2.1 Operator factorizations and the Darboux transformation ...... 12 2.1.1 Operator factorization ...... 13 2.2 The Korteweg-de Vries equation ...... 14 2.2.1 The KdV equation ...... 14 2.2.2 Scattering theory and the KdV equation ...... 14 2.2.3 KdV solutions and Darboux transformations ...... 15 2.2.4 Example: the soliton solution ...... 16 2.2.5 Additions ...... 17 2.3 Infinitesimal additions ...... 17 2.3.1 Some spectral theory of Schrödingeroperators ...... 17 2.3.2 Constructing the infinitesimal addition ...... 18 2.4 Hamiltonian mechanics and the KdV hierarchy ...... 19 2.4.1 Definitions ...... 19 2.4.2 Generation of the KdV hierarchy by infinitesimal addition ...... 20 2.4.3 The transmission coefficient as an integral of motion ...... 21

3 Jacobi operators 22 3.1 Tridiagonal operators: discrete Schr¨odingeroperators and Jacobi operators . . . . . 22 3.1.1 Basic objects and notation ...... 22 3.1.2 Discrete Schr¨odingeroperators ...... 23 3.1.3 Jacobi operators ...... 23 3.2 Spectral theory of Jacobi operators ...... 24 3.2.1 The resolvent operator ...... 24 3.3 Scattering theory ...... 25 3.3.1 Scattering theory for Jacobi operators ...... 25 3.4 Factorization of Jacobi operators and Darboux transformations ...... 27 3.4.1 The single commutation method ...... 27

4 3.4.2 Another form of discrete Darboux transformation ...... 28 3.5 The Toda lattice equations ...... 31 3.5.1 A model for interacting particles ...... 31 3.5.2 Flaschka variables and the Lax representation ...... 32 3.5.3 Integrability of the Toda lattice equations ...... 32 3.5.4 The Toda hierarchy ...... 33

4 Additions for Jacobi operators 35 4.1 An example ...... 35 4.1.1 The free Jacobi operator and the soliton solution ...... 35 4.1.2 Additions ...... 36 4.2 Inversion of additions ...... 36 4.2.1 The inverse addition and infinitesimal addition ...... 37 4.3 The infinitesimal addition ...... 39 4.3.1 Trace formulas for the Green’s functions ...... 41 4.3.2 Infinitesimal addition and the Toda lattice equations ...... 42 4.4 Symplectic geometry and Poisson structures ...... 43 4.4.1 Poisson structures for Jacobi operators ...... 43 4.4.2 The gradient of T (k)...... 46

5 Discussion and future directions 50 5.1 Inﬁnite-period limits and theta functions ...... 50 5.2 Spectral shift theory ...... 51 5.3 Dressing transformations ...... 51 5.4 Other systems ...... 51 5.4.1 Orthogonal polynomials on the real line...... 51 5.4.2 Orthogonal polynomials on the unit circle ...... 52

A Numerical results for infinite-period limits of Floquet theory 54 A.1 Floquet theory ...... 54 A.1.1 Floquet discriminant for OPRL ...... 55 A.1.2 Floquet discriminant for OPUC ...... 57 A.2 Infinite-period limits ...... 58 A.2.1 Infinite-period limits for discrete systems ...... 59 A.3 Spike domains ...... 60 A.4 Description of numerical methods ...... 61

B Bibliography 66

5 Abstract

We develop a class of Darboux transformations called additions for Jacobi operators. We show that by conjugating by a reflection, an addition may be inverted by another addition with the same spectral parameter. This leads to the development of an “infinitesimal addition”, which allows the transformation to be interpreted as a vector field on a space of Jacobi operators rather than a discrete transformation. We show that in an appropriate limit, this vector field generates the flows of the Toda hierarchy of lattice equations, in analogy to the known fact that an infinitesimal addition on Schrödingeroperators can generate the Korteweg-de Vries hierarchy of PDEs. Furthermore, in the case of scattering-type operators, the same vector field appears as a gradient of the transmission coefficient, indicating that the values of the transmission coefficent form a commuting family of functionals with respect to the Poisson bracket corresponding to the Toda hierarchy.

6 Chapter 1

Introduction

A Darboux transformation is a type of transformation mapping a Sturm-Liouville operator to another operator. These transformations can, in general, be expressed in terms of operator factorizations, an interesting area of study in its own right. Generically, these transformations adjoin a point of discrete spectrum to the operator. Since their introduction in the nineteenth century, Darboux transformations and their relatives have been applied to a variety of problems, including quantum mechanics (where a version of the Darboux transformation appears as raising/lowering operators and provides an easy construction of the eigenfunctions of systems such as the harmonic oscillator or hydrogen atom) and more recently nonlinear integrable systems. In this dissertation, we examine a special subset of Darboux transformations, named additions by McKean [19]. Additions are distinguished by the property that, unlike the generic transformation, they do not alter the spectrum of the operator. This hints at a connection between additions and the spectrum-preserving flows of integrable systems. For the Korteweg-de Vries equation, whose flow preserves the spectrum of an associated Schrödingeroperator, this connection is explored in [19]. In particular, by computing an “infinitesimal” addition in order to produce a vector field (rather than a discrete dynamics) on the space of Schrödingeroperators, one may generate the flows of the KdV hierarchy.

1.1 Main results

The main results of this dissertation deal with Darboux transformations and related methods, as applied to a discrete analogue of a Sturm-Liouville operator known as a Jacobi operator. A Jacobi operator is an operator H acting on sequences {f(n)}n∈Z by

Hf(n) = bnf(n + 1) + anf(n) + bn−1f(n − 1) (1.1)

1 where {an}n∈Z, {bn}n∈Z are sequences of real numbers, with bn 6= 0. Such an operator may be represented as a bi-infinite matrix which is symmetric and tridiagonal. The situation in the case of Jacobi operators is alternately more and less complicated than in Schrödingeroperators. The discrete nature of these operators is an obstacle when spatial derivatives are concerned, since there is more than one first difference operator, and these operators are not skew-symmetric. However, other calculations are made easier by the presence of delta functions as honest members of `2(Z), where more circuitous arguments would be needed to work in L2(R). Despite the differences between these two settings, the major results are quite similar in both character and form.

1 Unless otherwise speciﬁed, the sequences an and bn are presumed to be bounded.

7 1.1.1 Operator factorization and commutation methods In the context of these discrete problems, the Darboux transformation is often known as a commutation method, referring to the exchange of the two ﬁrst-order factors after the starting operator is factored. Commutation methods for Jacobi operators have been studied more recently than Dar- boux transformations for diﬀerential operators. They were introduced by Gesztezy, Holden, Simon, and Zhao [15], and further studied by Gesztezy and Teschl [16], among others.

Theorem 0 (Theorem 11.2 of [26]). Let H be a Jacobi operator with coefficient sequences an, bn and bn < 0. Take λ < Spec H and let u(λ, n) be a positive solution to the difference equation Hu(λ, n) = λu(λ, n). Then H − λ may be factored as a product of first order (bidiagonal) operators Af(n) = o(λ, n)f(n + 1) + e(n)f(n) A∗f(n) = o(λ, n − 1)f(n − 1) + e(n)f(n) for sequences s b u(λ, n + 1) e(λ, n) = − n u(λ, n) s b u(λ, n) o(λ, n) = − − n u(λ, n + 1)

Specifically, H − λ = A∗A; these factors may be permuted to obtain a new operator H˜ − λ = AA∗. The matrix elements of H˜ are pb b u(λ, n)u(λ, n + 2) ˜b = n n+1 (1.2) n u(λ, n + 1) u(λ, n + 1) u(λ, n) a˜ = b + (1.3) n n u(λ, n) u(λ, n + 1) Among Darboux transformations, we identify a privileged subclass of transformations known as additions, defined in section 2.2.5. A generic Darboux transformation adjoins a point of discrete spectrum to the operator; additions are distinguished by their not doing so, and leaving the spectrum of the operator unchanged. The distinction lies in the dependence of the Darboux transformation on the function u(λ, n). 2 When λ < Spec H, there exist positive solutions u+, u− such that u+ ∈ ` (Z≥0) and u− ∈ 2 ` (Z≤0). A generic transformation is constructed using a linear combination of u+, u−. The transformation we call an addition is constructed with u = u+ or u = u−. It follows that additions are parameterized by pairs consisting of a spectral parameter λ and a choice of sign +/−. Henceforth we denote such pairs by p = (λ, sgn p) and the corresponding transformation on operators by Ap. The main results of this dissertation are all derived from and expressed in terms of the single commutation method above. However, we will briefly mention another factorization method that was independently developed by this author for discrete Schrödingeroperators and extended to a form of Jacobi operator by Rahalmeira-Tsu [23]. We define the operators of left and right shift, S+f(n) = f(n+1),S−f(n) = f(n−1), and forward and backward difference ∂ = S+−I, ∂∗ = S−−I.

Proposition 1. Let H be a discrete Schr¨odingeroperator (i.e. a Jacobi operator with bn ≡ 1, and any an), and let {f(n)}n∈Z be a nonvanishing formal solution of Hf = λf for some value λ. Denote f(n) − f(n − 1) µ (n) = = −f −1∂∗f (1.4) l f(n) f(n + 1) − f(n) µ (n) = = f −1∂f (1.5) r f(n) Then:

8 1. ∗ + H − λ = (∂ + µl)(−∂ − S µr)

2. Deﬁne H˜ by ∗ + − + H˜ − λ = (−∂ − S µr)(∂ + S µlS ) ∗ Then H˜ is a discrete Schr¨odingeroperator, and H˜ − H is the diagonal operator −∂ µr + ∂µl.

1.1.2 Inversion of additions An important feature of additions is that the original operator may be recovered by the application of another addition. In the case of Schrödingeroperators, this takes a very intuitive form. An addition is inverted by the addition specified by the same value of the spectral parameter, but the opposite sign. This is essential in the construction of the infinitesimal addition (to be discussed later) and consequently the generation of the KdV hierarchy. A similar feature is found in the commutation methods for Jacobi operators. It is slightly more complicated, however, because of the lack of skew-symmetry in the difference operator. As a result, changing the sign in the addition does not produce the “opposite” transformation. However, inspecting the reflection coefficient (in the case of scattering-type operators) reveals a way to work around this obstacle by conjugating an addition by a reflection in the (discrete) spatial variable. This leads us to our first main result. Theorem 1. Let H be a Jacobi operator. Fix λ < inf Spec H and let p = (λ, sgn p) where sgn p is either + or −. Let σ be the reflection n 7→ −n. Then,

σApσApH = H (1.6) ApσApσH = H (1.7)

This gives a recipe for inverting an addition by another addition, albeit with an intermediate transformation.

1.1.3 Infinitesimal additions and the Toda hierarchy Once the inverse addition has been computed, the infinitesimal addition is a straightforward cal- 0 culation. Starting with a pair p = (λ, sgn p), we compute σApσAp , where p0 = (λ + ∆λ, sgn p). Expanding this in terms of the small parameter ∆λ, we find

0 σApσAp H = H + XH∆λ + o(∆λ) where XH is expressed in terms of the Green’s function G(λ, n, m) of H, defined in (3.6) Regarding XH as a vector field on the space of Jacobi operators and exploiting standard trace formulas, we can expand this vector field in powers of λ, obtaining our second main result.

Theorem 2. XH may be expanded in decreasing powers of λ as λ → −∞:

∞ X −(j+1) XH = λ XjH, λ → −∞ j=1

The ﬂow generated by H˙ = XjH is the ﬂow of the jth system in the Toda hierarchy of lattice equations.

9 1.1.4 Poisson structures and Hamiltonian mechanics Our final result takes the form of a calculation of the dependence of the transmission coefficient for scattering-type Jacobi operators on perturbations to the original operator. An operator of scattering type is one satisfying hypothesis (3.8). Such operators have associated scattering data in the form of a transmission coefficent T (k) and reflection coefficients R±(k) (see (3.11)).

Theorem 3. Let H be a Jacobi operator of scattering type with coeﬃcients an, bn and transmission coeﬃcient T (k). Then

∂ log T (k) = −G(λ, n, n) (1.8) ∂an ∂ log T (k) = −2G(λ, n, n + 1) (1.9) ∂bn where G(λ, n, m) is the Green’s function of H. This is an analogue of a similar result for Schrödingeroperators; see proposition 6. As a corollary, the values of the transmission coefficient at different parameters k form a family of commuting constants of motion.

1.2 Future directions

We have identified several potential directions toward which to apply the results from this thesis. The most obvious route is to apply the same ideas to other systems. This thesis is itself an extension of an idea from the setting of the KdV hierarchy and Schrödingeroperators to the setting of the Toda hierarchy and Jacobi operators, so it is natural to proceed by extending it further afield. A likely candidate system is families of orthogonal polynomials on the unit circle. Existing results relate this system to Jacobi operators [25], and recent results computing Darboux transformations directly in this system suggest a good possibility for extending these methods there. From there, one could generalize to the discrete nonlinear Schrödingeror Ablowitz-Ladik hierarchies. There are also several options for refining and improving the present results. Applying Krein’s spectral shift theory has the potential to relax some of the assumptions used in deriving the results in this article. In particular, several results that rely on the assumption that an operator is of scattering type might be extendable to a more general case with the use of a less restrictive perturbation theory. Another avenue to extend the scope of the results presented here is to interpret additions in the context of dressing transformations. Similar work has been done for finite-dimensional Toda lattice equations [10], providing a more geometric interpretation of matrix factorization problems and offering a way to study singular solutions of the lattice equations. Finally, there are possibilities to apply these results to other problems. As an example, additions in the continuous case have been used as a tool in describing infinite-period limits of Floquet theory, which allow an algebrogeometric interpretation of the mechanisms of scattering theory for Schrödingeroperators. Preliminary results, numerical and analytical, suggest that a similar construction may prove effective in the case of Jacobi operators as well as orthogonal polynomials on the unit circle. [14]

1.3 Structure of the document

The structure of this document is as follows: chapter 2 contains an exposition of the application of Darboux transformations and additions to Schr¨odingeroperators on the real line, including the use of the inﬁnitesimal addition to generate the equations of the KdV hierarchy. Chapter 3 contains a

10 brief overview of the necessary components of the theory of Jacobi operators and the Toda hierarchy of equations. Chapter 4 contains the main results and proofs. Finally, in chapter 5 we discuss some related areas of research and possible future applications of these results. Appendix A contains a description of numerical simulations related to one of the applications described in chapter 5.

11 Chapter 2

Background: Darboux transformations and additions for Schr¨odingeroperators on R

This chapter describes the theory of Darboux transformations and additions for Schrödingeropera- tors on R. These transformations were introduced to study the theory of Sturm-Liouville eigenvalue problems. In short, the transformation allows one, given an eigenvalue problem and an eigenfunction as a starting point, to produce a new eigenvalue problem and a new eigenfunction with the same eigenvalue. In the case of Schrödingeroperators, the transformation can be expressed in terms of an operator factorization, and due to the close connection between the spectral theory of Schrödinger operators and the Korteweg-de Vries (KdV) equation, allows for the construction of new KdV solutions.

2.1 Operator factorizations and the Darboux transformation

We begin with a definition: a Schrödinger operator is an operator of the form Q = D2 + q(x) or 2 d Q = −D + q(x), where D = dx . Then, we consider a motivating question:1 given a Schrödingeroperator Q = D2 + q(x), when is there a first order operator T = D + µ(x) and another Schrödingeroperator Q˜ = D2 +q ˜(x) so that T intertwines with Q and Q˜, i.e.: TQ = QT˜ In this section we proceed with a purely formal construction of the answer to this problem; it may be made precise in certain function spaces. For our purposes we will assume that the functions we are working with are real analytic. We first inspect the kernel of the operator T . This may be described explicitly: if ψ ∈ ker T , then ψ0 + µψ = 0 ψ0 ψ must satisfy µ = − ψ = −D log ψ; since the kernel is one-dimensional, this characterizes it entirely. Moreover, if TQ = QT˜ then since QT˜ ψ = 0, it must be the case that Qψ = λψ, since Qψ ∈ ker T . So, we have a simple ˜ ψ0 answer to our original question: we may have T and Q provided T = D − ψ for some function ψ

1This approach to the Darboux transformation is based on unpublished notes by H. Flaschka.

12 satisfying Qψ = λψ for some number λ.2 This turns out to determine Q˜ entirely, so we need not impose any external conditions on Q˜.

2.1.1 Operator factorization Fix a value λ, and let ψ be a function satisfying Qψ = λψ. Formally, we may factor the operator Q − λ as Q − λ = (D − µ)(D + µ) := T 0T with µ = ψ0/ψ. To see this, consider the action of the right hand side on a test function f:

(D − µ)(D + µ)f = f 00 − µ0f − µ2f (2.1) ψ00ψ − (ψ0)2 ψ0 2 = f 00 − f − f (2.2) ψ2 ψ = f 00 − qf + λf (2.3) where in the last line we have used ψ00 = (q−λ)ψ. Then, to determine the operator Q˜ of the previous section, note that Q˜ = T QT −1 and since all operators commute with the scalar λ0,

Q˜ − λ = T (Q − λ)T −1 = TT 0 = (D + µ)(D − µ)

So, the eﬀect is to interchange the two factors. With another test function calculation we can verify that Q˜ is also a Schr¨odingeroperator.

(D − µ)(D + µ)f = f 00 + µ0f − µ2f (2.4) ψ00ψ − (ψ0)2 ψ0 2 = f 00 + f − f (2.5) ψ2 ψ ψ00 ψ0 = f 00 − f − 2 ∂ f (2.6) ψ x ψ ψ0 = f 00 − q − 2 ∂ + λf (2.7) x ψ where in the second-to-last line we have used ψ0 ψ00ψ − ψ02 ∂ = x ψ ψ2

Therefore, Q˜ = (D+µ)(D−µ)+λ is a Schr¨odinger operator with potentialq ˜ = q−2µ0 = q−2(log ψ)00. Moreover, the operator T = (D + µ) allows us to map solutions of Qψ = λψ to solutions of Qψ˜ = λψ. Speciﬁcally, let ψ1 satisfy Qψ1 = λ1ψ1 (λ1 not necessarily equal to λ). Then,

QT˜ ψ1 = T Qψ1 = T (λ1ψ1) = λ1(T ψ1) so that (T ψ1) satisﬁes the equation Qψ˜ = λ1ψ So the operator T transforms solutions of the original equation to solutions of the new equation. Henceforth we call the transformation Q 7→ Q − 2D2 log ψ a Darboux transformation.

2We have avoided the term ‘eigenfunction’ because we will not always make assumptions about the membership of 2 2 ψ in, e.g., L (R). Indeed, in the most important cases we will choose λ 6∈ Spec Q so that there are no L eigenfunctions.

13 2.2 The Korteweg-de Vries equation

Our principal motivation for studying Schrödingeroperators and their Darboux transformations is their close connection with the famous Korteweg-de Vries (KdV) equation. In this section we briefly review the KdV equation and a few features of its spectral and scattering theory. A detailed treatment of scattering theory for Schrödingeroperators and the connection to the KdV equation may be found in the article by Deift and Trubowitz [8].

2.2.1 The KdV equation The KdV equation is a nonlinear partial diﬀerential equation in one space and one time variable. There are several conventions regarding the dimensionless constants in the equation, but by one convention, the equation reads: ut + uxxx − 6uux = 0 This equation was derived by Korteweg and de-Vries3 in order to model water waves and describe what were then called “solitary waves” or “waves of translation”, to be described later in this section. The connection to Schr¨odingeroperators arises from the representation of the equation in the form of a so-called Lax pair. A Lax pair is a pair of linear operators, A and L, satisfying the Lax equation dL = [L, A] dt where the brackets denote the commutator. The Lax pair for the KdV equation is

2 L = −∂x + q(x, t) (2.8) 3 A = −4∂x + 6q∂x + 3∂x (2.9)

The Lax pair representation recasts the nonlinear KdV equation as the compatibility condition of a pair of linear equations:

Lψ = λψ (2.10)

∂tψ = Aψ (2.11)

The ﬁrst equation is the time-independent Schr¨odingerequation with potential q(x). The connection between the KdV equation and the spectral theory of this equation was originally discovered by Gardner, Greene, Kruskal, and Miura, and began an era of substantial progress in the study of nonlinear integrable systems.

2.2.2 Scattering theory and the KdV equation The essential idea behind scattering theory is to examine the Schr¨odingeroperator as if it were the analogous object from quantum mechanics. In quantum mechanics, the equation ∂ψ i = Qψ ∂t governs the time evolution of the wave functions that represent probability densities of particles. Seeking stationary solutions reduces the equation to the time-independent form

Qψ = λψ

3And also earlier by Boussinesq, who may be more entitled to the name of the equation (see [6]); however, when Kruskal et al. described the procedure of inverse scattering, they referred to it as the Korteweg-de Vries equation, and this nomenclature has stuck.

14 which is the ﬁrst of the equations in the Lax pair. In scattering theory, we study the spectral properties of a Schr¨odingeroperator under the restriction that the potential q(x) vanishes rapidly at ±∞.4 Then, near ±∞, the solutions of Qψ = λψ decompose into a linear combination of simple exponentials e±ikx (k2 = λ). In particular, we may seek two solutions f±(k, x), which we will refer to as scattering solutions, whose asymptotic behavior is summarized in the following table. x → −∞ x → +∞

−ikx −ikx ikx f− s11(k)e e + s21(k)e

ikx −ikx ikx f+ e + s12e s22e

Table 2.1: Asymptotic behavior of scattering solutions for Schrödingeroperators. In other words, these solutions look like radiation of constant wavelength and intensity incident from +∞ (in the case of f−) or −∞ (f+) along with the component of those waves that are reflected back or transmitted through the barrier represented by the potential q(x). In the above table, s11(k) = s22(k) is called the transmission coefficient, and s21(k) and s12(k) the right (resp. left) reflection coefficients. The significance of these coefficients is that under suitable assumptions, the operator Q is uniquely determined by the functions s11(k), s21(k).

Proposition 1. [12] Let Q1 and Q2 have the same transmission coefficient s11(k) and no bound states. Then, if Q1 and Q2 also have the same reflection coefficient s21(k), then Q1 = Q2. Moreover, the evolution of the KdV equation leaves the transmission coefficient invariant, and 3 the reflection coefficient s21(k) evolves according to the simple linear ODE ∂ts21(k) = 8ik s21(k), 3 so that s21(k, t) = s21(k, 0) exp(8ik t). [9] In other words, the map from potential to scattering data is a one-to-one change of coordinates that linearizes the KdV equation. This provides a recipe for solving the initial value problem for the KdV equation: compute the scattering data for the operator Q(t = 0), evolve the reflection coefficient forward in time, and recover the time-evolved operator Q(t) from the reflection coefficient. The latter step is called inverse scattering, and involves the solution of a certain integral equation called the Marcenko equation [8].

2.2.3 KdV solutions and Darboux transformations The renewed interest in Darboux transformations in the latter half of the previous century stems from their relationship to the KdV equation and other integrable systems. Most importantly, given a known solution q(x, t) of the KdV equation, if we apply a Darboux transformation to the corresponding operator Q(t) = −D2 + q(x, t) the resulting potentialq ˜(x, t) also satisﬁes the KdV equation. To see this, we return to the linear system (2.10).

2 Proposition 2. [18] Let L = −∂x + q(x, t) satisfy the Lax equation, and let ψ0(x, t) be a particular solution to the system (2.10):

Lψ = λ0ψ (2.12)

ψt = Aψ (2.13)

4 We will assume that q(x) ∈ S(R), the Schwartz space.

15 Then if ψ(x, t) is a solution to the system

Lψ = λψ (2.14)

ψt = Aψ (2.15)

˜ 0 the transformed function ψ = (∂x − ψ0/ψ0)ψ is a solution to the system

L˜ψ˜ = λψ˜ (2.16) ˜ ˜ ψt = A˜ψ (2.17) where L˜ = −D2 +q ˜(x, t) and A˜ = −4D3 + 6˜qD + 3˜q0. The existence of a solution to the transformed linear system (2.16) means that L,˜ A˜ satisfy ∂tL˜ = [L,˜ A˜], and therefore thatq ˜(x, t) satisﬁes the KdV equation. In the next section, we present an example in which we exploit this property to derive a solution of the KdV equation.

2.2.4 Example: the soliton solution An important feature of the KdV equation is that it supports soliton solutions; that is, localized waves that maintain their shape while traveling. Such a wave was famously observed in a canal by Scottish naval engineer John Scott Russell, who wrote a note describing the wave. This observation led to investigations by Boussinesq and later Korteweg and de Vries that resulted in the derivation of the KdV equation.5 It is a simple exercise to derive soliton solutions by seeking solutions of the form u(k, x − ωt); that is, solutions that depend only on x − ωt rather than on x and t independently (here ω is a wave velocity). Instead of approaching this derivation directly, though, we will apply the Darboux transformation, following the derivation in [18]. All we need to begin is a known solution of the KdV equation, and we can choose the simplest possible such solution: q(x, t) = 0. To carry out the transformation, we need a solution ψ(x, t) of (2.10) for a ﬁxed spectral parameter λ. Choose λ < 0, let k2 = −λ, and then the linear system reduces to

2 2 −∂xψ = k ψ (2.18) 3 ∂tψ = −4∂xψ (2.19)

We ﬁnd two linearly independent solutions

3 ψ±(x, t) = exp(±(kx − 4k t)

It is clear, though, that performing a Darboux transformation using ψ±(x, t) will yield no new po- 2 tential, because ∂x log ψ± = 0. Instead, we must choose a linear combination of these two solutions, and for this example we will choose

ψ(x, t) = cosh(kx − 4k3t) yielding the new potential

2 2 −2 3 q˜(x, t) = −2∂x log ψ(x, t) = −2k cosh (kx − 4k t) which describes a soliton moving with constant velocity 4k2.

5An interesting account of the history of this equation may be found in [6]

16 2.2.5 Additions In computing the previous example, we stumbled across an important feature of the Darboux transformation. Upon choosing λ < Spec Q, the equation Qu = λu has positive solutions u+, u− such 2 2 R R that u− ∈ L ((−∞, 0]) and u+ ∈ L ([0, ∞)) with (−∞,0] u+(x)dx = [0,∞) u−(x)dx = ∞. [19] These were given in the previous example by exp(±(kx − 4k3t)), which for any fixed t are simply exponential functions in x. The function f chosen to construct the Darboux transformation may, in general, be an arbitrary linear combination of the two functions u±(k, x). However, we observe qualitatively different behavior when f(k, x) is equal to one of u±(k, x). In the previous example, we chose the linear 1 combination f(k, x) = 2 (u+(k, x) + u−(k, x)). However, if we take f(k, x) = u±(k, x), the difference 2 is clear: ∂x log u±(k, x) ≡ 0, and so the resulting operator is the same as the starting operator 2 Q0 = −D . It is not the case, in general, that choosing f(k, x) = u±(k, x) in the Darboux transformation results in the identity transformation. What is true is that when f(k, x) is linear combination of 2 u+(k, x) and u−(k, x) with nonzero coefficients for each, the point λ = k is adjoined to the spectrum of the new operator. When f(k, x) = u±(k, x), the spectrum of Q˜ is always the same as the spectrum of Q. [19] Definition 1. Let Q be a Schrödingeroperator, λ < Spec Q, and denote by p the pair (λ, sgn p) where sgn p = ±. Then the addition Ap is the transformation

p 2 A Q = Q − 2D log usgn p(λ, x) (2.20)

Iterated additions may be described in terms of only the functions u+, u− for the original operator.

Lemma 1. [18] Let pi = (λi, sgn pi), i ∈ {1, . . . , n}. Then

p p 2 A1 ...AnQ = Q − 2D log[up1 , . . . , upn ] (2.21) where the brackets [, ] denote the Wronskian determinant

 u u0 u00 ...  p1 p1 p1 0 00 up u u ... [up , . . . , up ] = det  2 p2 p2  1 n  . . . .  . . . ..

Corollary 1. Additions commute with one another.

2.3 Inﬁnitesimal additions

An important feature of additions is that they may be inverted by another addition6. Because of this, we may consider an infinitesimal addition; intuitively speaking, this combines an addition of p + δp with an addition of −p to isolate the infinitesimal δp. This gives us an object that may be interpreted as a vector field with flows, rather than a transformation of finite size that generates a discrete dynamics on the space of Schrödingeroperators. The main result is that these flows are the same as the flows of the KdV hierarchy.

2.3.1 Some spectral theory of Schrödingeroperators Before constructing an infinitesimal addition, it is useful to consider a small amount of spectral theory for Schrödinger operators. The principal object of our interest is the resolvent operator

6The reproduction of this result in a discrete setting is a main result of chapter 4.

17 −1 2 RQ(λ) = (Q − λ) . The resolvent may be regarded as an integral operator on L (R) with a symmetric kernel Gxy(λ), called the Green’s function of the operator Q: Z −1 (Q − λ) f(x) = Gxy(λ)f(y)dy R The Green’s function may be expressed in terms of the scattering solutions we have already described: f+(λ, x)f−(λ, y) f+(λ, x)f−(λ, y) Gxy(λ) = = (x > y) (2.22) [f−, f+] 2iks11(k)

2.3.2 Constructing the inﬁnitesimal addition In order to construct the inﬁnitesimal addition, we require a means for “inverting” an addition. Fortunately, this is quite intuitive. Given p = (λ, sgn p), we denote −p = (λ, − sgn p); that is, a pair with the same spectral parameter λ but opposite sign. It is a trivial consequence of lemma 1 that −p generates the addition inverse to p. Proposition 3. For any operator Q and λ < σ(Q),

A−pApQ = Q

Proof. By lemma 1, −p p 00 A A Q = Q − 2(log[u−(λ), u+(λ)]) but since f−(λ) and f+(λ) are solutions of the same equation Qf = λf with the same value of λ, 00 their Wronskian is constant, so (log[u−(λ), u+(λ)]) = 0. Now that we can map Q → Q˜ → Q by additions, we may equally map Q → Q˜ → Q + ∆Q by using an inﬁnitesimally diﬀerent value of λ to compute one of the transformations. The result is expressible in terms of the Green’s function of the original operator Q.

Proposition 4. The inﬁnitesimal addition may be expressed as:

0 A−pAp Q = Q − XQ∆λ + O(∆λ) where XQ = 2DGxx(λ) (2.23) Proof. By lemma 1, the diﬀerence in the potential is

2 ∆Q = 2D log[f−(λ), f+(λ + ∆λ)]

Now,

˙ ! [f−(λ), f+(λ)] D log[f−(λ), f+(λ + ∆λ)] = D log 1 + ∆λ + o(∆λ) [f−(λ), f+(λ)] [f (λ), f˙ (λ)] = D − + ∆λ + o(∆λ) [f−(λ), f+(λ)] where the dot represents diﬀerentiation with respect to λ. Proceeding,

[f (λ), f˙ (λ)] f (λ)f˙00 (λ) − f (λ)00f˙ (λ) D − + = − + − + [f−(λ), f+(λ)] [f−(λ), f+(λ)]

18 ˙ Adding and subtracting qf+f− in the numerator, we get

f (λ)f˙00 (λ) − f (λ)00f˙ (λ) f (λ)Qf˙ (λ) − f˙ (λ)Qf (λ) − + − + = − + + − [f−(λ), f+(λ)] [f−(λ), f+(λ)] f˙ (λ)Qf (λ) − f (λ)(˙Qf (λ)) = + − − + [f−(λ), f+(λ)] λf˙ (λ)f (λ) − λf (λ)f˙ (λ)) − f (λ)f (λ) = + − − + − + [f−(λ), f+(λ)] f (λ, x)f (λ, x) = − + [f+(λ), f−(λ)]

= Gxx(λ) where in the last line we have used (2.22). The previous expression for the inﬁnitesimal addition is a nonlocal expression, in the sense that it depends on the Green’s function – the kernel of an integral operator. Local expressions can be obtained in the limit as λ → −∞. Proposition 5. [19] As λ → −∞,

∞ X −n+1/2 XQ = XnQλ n=0

The vector ﬁelds Q˙ = XnQ are the local isospectral ﬂows

qt = qx

qt = −qxxx + 3qqx ...

These ﬂows are spectrum-preserving because they inherit the spectrum-preserving properties of the additions.

2.4 Hamiltonian mechanics and the KdV hierarchy

Although we originally introduced the KdV equation directly in the form of a PDE, it may also be defined as a Hamiltonian system. For a particular Schrodinger operator Q with potential in S(R), we define the invariant manifold of Q to be the set of operators with potential in S(R) whose transmission coefficient is the same as Q.

2.4.1 Deﬁnitions In this section, we will be following the development of McKean [20], including his somewhat non- standard terminology, so we present a deﬁnition here. Let H be a functional on Schwartz space ∂H S(R); then, the gradient of H is a function ∂q(x) satisfying

Z ∂H H[q + εp] = H[q] + ε p(x)dx + o(ε) ∂q(x) R for any perturbation p ∈ S(R), provided such a function exists.

19 In order to describe the KdV flows as Hamiltonian systems, we define a Poisson bracket Z ∂f ∂g {f, g} = D dx (2.24) ∂q ∂q R By observing that " # Z ∂C 0 ∂2A ∂B 0 ∂2B ∂A0 [[A, B],C] = − dx ∂q ∂q2 ∂q ∂q2 ∂q R we see that this bracket obeys the Jacobi identity. The Hamiltonian flow associated with this Poisson bracket is, for a Hamiltonian H(q),

∂H 0 q˙ = ∂q

Then, the KdV equation is such a Hamiltonian ﬂow with the Hamiltonian Z 1 1 H = q3 + (q0)2dx 2 2 4 R so that ∂H 3 1 2 = q2 − q00 ∂q 2 2 ∂H 0 1 2 = 3qq0 − q000 ∂q 2

The Hamiltonian above is indexed H2 because it is the second in an infinite hierachy of Hamilto- nians, all of which generate isospectral flows on the space of Schrödingeroperators and all of which commute with respect to the Poisson bracket.7 This hierarchy is called the KdV hierarchy. We can obtain the Hamiltonians and vector fields for all systems of the KdV hierarchy by a recursive procedure. Specifically, define the operator R = qD + Dq − (1/2)D3, and denote the mth KdV Hamiltonian and vector field by Hm and Xm respectively. Then Theorem 1 (Theorem 2 of [21]). ∂H X q = R m−1 m ∂q 0 ∂Hm Since Xmq = ∂q , we may recursively obtain Hm,Xm by beginning with a known Xi, ∂Hi integrating it with respect to x to obtain ∂q , and applying R to obtain Xi+1. The homogeneous ∂H0 KdV hierarchy is obtained by beginning with H0 = q, ∂q = 1,X0 = 0 and running this recursive process, taking all subsequent integration constants to be 0.

2.4.2 Generation of the KdV hierarchy by infinitesimal addition The vector fields of the KdV hierarchy are contained in the expansion of the infinitesimal addition. The following derivation is due to McKean [19]. 0 ∗ Let K be the skew-symmetric operator [(3/2)Gxx − DGxx]G, and K its Hermitian conjugate. The skewness of this operator implies the vanishing of

(Q − λ)(K + K∗)(Q − λ)

7 R 1 2 The ﬁrst is H1 = 2 q dx, which generates the translation ﬂow qt = qx.

20 but this expression is equivalent to

3 [qD + Dq − (1/2)D − 2λD]Gxx so its vanishing implies 3 0 [qD + Dq − (1/2)D ]Gxx = 2λGxx

We recognize on the left hand side the operator R of the previous section. If we expand Gxx and 0 Gxx in decreasing powers of λ,

∞ ∞ X −j 0 X 0 −j Gxx = gj(x)λ ,Gxx = gj(x)λ j=0 j=0 then, matching powers of λ, we have 0 Rgj(x) = 2gj−1 So, the expansion coefficients of the infinitesimal addition generate the same hierarchy of flows as the Hamiltonians of the KdV hierarchy.

2.4.3 The transmission coefficient as an integral of motion The connection between the infinitesimal addition that we derived before and the symplectic structure of the KdV equation comes in the realization of the values of the transmission coefficient (or, more precisely, its logarithm log s11(k)) as a family of functionals in involution. This is accomplished by computing the gradient of s11(k) with respect to the potential q(x). Proposition 6. [Section 4.2 of [20]]

∂ log s (k) 11 = G (k2) q(x) xx

As a consequence, the Hamiltonian ﬂow generated by the values of −2 log s11(k) is

{q, −2 log s11(k)} = D(−2 log s11(k)) (2.25) 2 = −2DGxx(k ) (2.26) which is the same as the flow of the infinitesimal addition (2.23). Since additions commute, infinitesimal additions inherit this commutativity. As a result, the values of s11(k) also commute with respect to the Poisson bracket.

21 Chapter 3

Jacobi operators

3.1 Tridiagonal operators: discrete Schr¨odingeroperators and Jacobi operators

Our objects of study are symmetric tridiagonal operators on `2(I), where I is a discrete index space (Z if not otherwise mentioned). These operators may be generally regarded as discretizations of second-order differential operators, and share a number of the same properties. As we will see, many of the important results carry over in un- or minimally modified form. There are a few details, however, that are useful to note up front, as they encapsulate the major differences between the discrete and continuous theories.

3.1.1 Basic objects and notation In this section we introduce some of the elementary notation that we will use. Most of the objects we work with are described in terms of sequences; sometimes we will denote these sequences by subscripts (e.g. an, bn) and sometimes using function notation (e.g. f(n), g(n)). Most often, the former will be reserved for the coefficients (matrix elements) of an operator, and the latter for functions. The analogue of a derivative in this setting is a difference, and one of the first new things we notice is that there are two types of difference we will need:

(∂f)(n) = f(n + 1) − f(n) (3.1) (∂∗f)(n) = f(n − 1) − f(n) (3.2)

These forward and backward differences are formally adjoint, and one of the first distinctions between the `2 setting and the L2 setting of the previous chapter is this – that the adjoint of ∂ is not simply −∂. Further, the conventional second difference operator (∆f)(n) = f(n − 1) − 2f(n) + f(n + 1) may be written ∂ + ∂∗. Another consequence of discreteness when examining differences is the modified Leibniz rule for products:

(∂fg)(n) = f(n)(∂g)(n) + g(n + 1)(∂f)(n) (3.3) (∂∗fg)(n) = f(n)(∂∗g)(n) + g(n − 1)(∂∗f)(n) (3.4)

The last important diﬀerence between the discrete setting and the continuous is concerned with δ functions. The Dirac δ-function is of course not a true member of L2(R), and there are several situations where manipulations must be done under the integral sign to avoid running afoul of this

22 fact. However, the δ-function in the discrete setting is the Kronecker delta δn(m), which is 1 if n = m and 0 otherwise. This is, of course, a true function in `2(Z), which will simplify some calculations later.

3.1.2 Discrete Schrödingeroperators 2 A discrete Schrödingeroperator is an operator H on ` (I), where typically I = Z, Z≥0, or Z≤0, with the property that the matrix representation of H is tridiagonal with off-diagonal entries all equal to 1. This definition is motivated by considering the tridiagonal operator whose diagonal entries are equal to -2 and whose off-diagonal entries are equal to 1. Given a function f(n) defined on Z, the centered second difference operator is (∆f)(n) = f(n − 1) − 2f(n) + f(n + 1) Therefore, the matrix representation of the second difference is  . .  .. ..    ..   . −2 1     1 −2 1     ..   1 −2 .   . .  .. ..

Moreover, if the values f(n) are samples of a function f˜ defined on R, sampled at equally −2 ˜00 spaced points xn with spacing (δx), then (δx) (∆f)(n) converges to f (x) as the spacing δx → 0. Therefore, the tridiagonal operator ∆ above may be regarded as a discretization of the second derivative operator D2, and a generic Schrödingeroperator D2 +q(x) may be discretized in the form of a tridiagonal operator whose off-diagonal entries are equal to 1, and whose diagonal entries are given by a sequence q(n).

3.1.3 Jacobi operators Jacobi operators are a generalization of the class of discrete Schrödingeroperators, and share many of their properties. Instead of requiring that the off-diagonal elements are equal to 1, Jacobi operators satisfy the relaxed constraint that the off-diagonal elements are strictly positive (alternatively, strictly negative; the choice will be stated when necessary). A Jacobi operator, then, is parameterized by two sequences an and bn, subject to the constraint that bn 6= 0. It is common to require one of bn > 0 and bn < 0. The more convenient choice is dependent on the context. As we will see below, in the Toda lattice equations bn represent (a transformation of) the separation between adjacent particles, and so bn > 0 is the physically natural choice. However, bn < 0 yields a more direct analogue of the operator Q = −D2 + q(x) of chapter 2, and assures the positivity of certain solutions to the Jacobi difference equation (3.5) (below). 1 The sequence an gives the diagonal elements of the matrix; bn the off-diagonal elements. A Jacobi operator is more like a discretization of a Sturm-Liouville operator of the form d d p(x) + q(x) dx dx where the sequence bn corresponds to p(x), and an corresponds to q(x). A Jacobi operator H with coefficients an, bn is associated to a difference equation (H − λ)f = 0, or in coordinates

bn−1f(n − 1) + (an − λ)f(n) + bnf(n + 1) = 0 (3.5) 1Note: some authors follow a convention that exchanges the role of the symbols a and b.

23 which we will refer to as the Jacobi diﬀerence equation for that operator, with spectral parameter (eigenvalue) λ.

3.2 Spectral theory of Jacobi operators

In this section, we introduce those objects from the spectral theory of Jacobi operators that we will need to carry out constructions later.

3.2.1 The resolvent operator As in the case of Schrödingeroperators, the principal object we will need is the resolvent operator (H − λ)−1. The resolvent operator is defined for all λ ⊂ C \ Spec H. The matrix elements of the resolvent are defined as the Green’s function

−1 G(λ, n, m) = hδn, (H − λ) δmi (3.6)

An immediate use of the Green’s function is to construct solutions u±(λ, n) to the Jacobi difference equation Hu = λu that are square summable as n → ±∞ respectively. Set −1 u(λ, ·) = (H − λ) δ0(·) u(λ, ·) satisfies the Jacobi difference equation (H − λ)u = 0 except at n = 0, and moreover is square- summable at both +∞, −∞ (since away from Spec H, the resolvent is a bounded operator). So, we trade square-summability at one end for consistency with the difference equation at n = 0. Another useful fact is that if λ lies to the left of Spec H; that is, λ ∈ R, λ < inf Spec H, and bn < 0, then these solutions may be taken to be everywhere positive. This will be convenient (although not strictly necessary) in constructing the single commutation (a.k.a. Darboux transformation) in the next chapter. To see this, note that under these assumptions (H − λ) is a positive definite operator, 2 2 and so is its restriction (Hn,+ − λ) to ` (Z≥n). As a result

−1 hδn+1, (Hn,+ − λ) δn+1i > 0

Since −1 u+(λ, n + 1) hδn+1, (Hn,+ − λ) δn+1i = −bnu+(λ, n) this mean that u+(n), u+(n + 1) have the same sign, and this may be taken to be positive. It is also useful to note that u (λ, n)u (λ, m) G(λ, n, m) = − + (3.7) W (u−(λ), u+(λ)) for n ≤ m. If n ≥ m, instead u (λ, m)u (λ, n) G(λ, n, m) = − + W (u−(λ), u+(λ))

This exchange of u+ and u− ensures that G(λ, n, m) is symmetric in n and m. Now, we will look at how these objects appear in one of the simplest possible examples. Example 1 (The free Jacobi operator). [26] The simplest example of a Jacobi operator is the free Jacobi operator, which we take to have 3 an ≡ 0, bn ≡ 1/2 .

2This restriction is simply the semi-inﬁnite operator obtained by removing rows and columns indexed ≤ n. 3 Sometimes bn ≡ 1 is referred to as the free Jacobi operator, e.g. in [24]; however, the normalization bn ≡ 1/2 is often more convenient.

24 √ √ k+k−1 1. Solutions. For λ ∈ C \ [−1, 1], let k = λ + 1 + λ 1 − λ, so that λ = 2 . For |k| < 1, ±n 2 the functions u−(k, n), u+(k, n) are given by u±(k, n) = k ; it is clear then that u+ ∈ ` (Z≥0), u− ∈ 2 ± ` (Z≤0). (If |k| > 1, the roles of k are exchanged, of course, but the two functions are the same). 2 2. Spectrum. To ﬁnd the spectrum of H0, we consider the Fourier transform mapping ` (Z) → 2 P L (−π, π) by u(n) 7→ u(n)exp(inx). This maps H0 to the operator of multiplication by cos(x), n∈Z and cos(x) − λ is invertible only for λ outside [−1, 1]. Then, by unitarity of the Fourier transform, Spec H = [−1, 1]. 3. Green’s function. Using the established forms of u±(k, n), we can apply the formula (3.7) to obtain k|n−m| G(λ, n, m) = k − k−1 3.3 Scattering theory

Jacobi operators have a scattering theory that is very similar in character to that of Schr¨odinger operators.

3.3.1 Scattering theory for Jacobi operators The scattering theory for Jacobi operators was originally considered by Geronimo and Case [4] in the context of orthogonal polynomials on the real line. In scattering theory, we consider Jacobi operators that, intuitively, represent a localized barrier or potential well. That is to say, as n → ±∞, the coeﬃcients an, bn approach those of the free Jacobi operator an ≡ 0, bn ≡ 1/2 “fast enough.” Speciﬁcally, we say that a Jacobi operator is of scattering type if the sequences an, bn satisfy

1 1 n(1 − 2bn) ∈ ` (Z), nan ∈ ` (Z) (3.8) This is enough to treat a scattering type operator as a perturbation of the free Jacobi operator and apply a standard lemma to demonstrate the existence of Jost solutions.

Lemma 2 (Existence of Jost solutions). [26] Let an, bn satisfy (3.8). Then there exist Jost solutions f±(k, n) satisfying k + k−1 Hf (k, n) = f (k, n) ± 2 ± for 0 ≤ k ≤ 1 with asymptotic behavior

∓n lim f±(k, n)k = 1. n→±∞

−1 1−k2 We may calculate W (u±(k), u±(k )) = ± 2k (Lemma 3 below), so the functions u±(k, n) and −1 u±(k , n) are linearly independent, meaning that we may express u± in terms of u∓ as

−1 u±(k, n) = α(k)u∓(k , n) + β±(k)u∓(k, n) where

W (u∓(k), u±(k)) 2k α(k) = −1 = 2 W (u+(k), u−(k)) (3.9) W (u∓(k), u∓(k )) 1 − k −1 W (u∓(k), u±(k )) 2k −1 β±(k) = −1 = ± 2 W (u∓(k), u±(k )) (3.10) W (u±(k), u±(k )) 1 − k

25 −1 −1 We call the quantity α(k) := T (k) the transmission coefficient, and α(k) β±(k) := R±(k) the reflection coefficients. To understand why, we multiply the previous equation by α(k)−1 to define the modified Jost solutions

−1 f+(k, n) = u−(k , n) + R−(k)u−(k, n) = T (k)u+(k, n) (3.11) −1 f−(k, n) = u+(k , n) + R+(k)u+(k, n) = T (k)u−(k, n) (3.12)

The asymptotic behavior of these functions as n → ±∞ is summarized in the following table:

n → −∞ n → +∞

−n −n n f− T (k)k k + R+(k)k

n −n n f+ k + R−(k)k T (k)k

Table 3.1: Asymptotic behavior of scattering solutions for Jacobi operators.

This clarifies the use of the terms “transmission coefficient” and “reflection coefficient” for the functions T (k), R±(k). The function f+ represents a steady-state solution consisting of a “plane wave”4 kn incident from −∞ along with reflected and transmitted components at ∓∞ respectively. −n The function f− is similar, but the plane wave is k , incident from +∞.

Lemma 3. Assume |k| = 1, k2 6= 1. Then 1. 1 − k2 W (f (k), f (k−1)) = ± (3.13) ± ± 2k 2. 1 − k2 W (f (k), f (k)) = T (k) (3.14) + − 2k Proof. 1. Explicitly,

−1 −1 −1 Wn(f+(k), f+(k ) = bn[f+(k, n)f+(k , n + 1) − f+(k, n + 1)f+(k , n)]

By the asymptotic properties of f+ and since bn → 1/2 as n → ∞,

−1 2 −1 k − k 1 − k lim Wn(f+(k), f+(k ) = = n→∞ 2 2k but, since the Wronskian is constant, it must be equal to its limiting value, so

k−1 − k 1 − k2 W (f (k), f (k−1) = = n + + 2 2k

The proof for f−(k) is similar.

2. Expanding the Wronskian using the deﬁnitions of f+(k, n) and f−(k, n),

−1 −1 −1 W (f+(k), f−(k)) =W (u−(k ), u+(k )) + R+(k)W (u−(k ), u+(k)) (3.15) −1 R−(k)W (u−(k), u+(k )) + R−(k)R+(k)W (u−(k), u+(k)) (3.16)

4when |k| = 1

26 Substituting the deﬁnitions of R±(k) in and canceling terms yields

1 −1 −1 W (f+(k), f−(k)) = ×[W (u−(k , u+(k))W (u−(k), u+(k )) W (u+(k), u−(k)) −1 −1 −W (u−(k ), u+(k ))W (u−(k), u+(k))] Applying the Wronskian identity

W (f1, g1)W (f2, g2) − W (f1, g2)W (f2, g1) = W (f1, f2)W (g1, g2) yields

1 −1 −1 W (f+(k), f−(k)) = [W (u−(k ), u−(k))W (u+(k ), f+(k))] W (u+(k), u−(k)) after which applying 3.13 yields the claimed result.

3.4 Factorization of Jacobi operators and Darboux transformations

As we have seen, Jacobi operators share many properties with the Schr¨odingerand Sturm-Liouville operators that they discretize. Among these properties is their amenability to Darboux transformations, which are in this context referred to as commutation methods. These methods were ﬁrst introduced by Gesztesy, Holden, Simon, and Zhao and later extended by Gesztezy and Teschl [16]. They developed two forms of this transformation: the single commutation, which inserts an eigenvalue to the left of Spec H (or, by considering −H, to the right), and the double commutation, which may insert an eigenvalue into an arbitrary spectral gap. We will be concerned principally with additions, the subclass of transformations that do not insert eigenvalues, and so we will restrict our attention to the simpler single commutation.

3.4.1 The single commutation method The method of single commutation is an analogue of the classical Darboux transformation for Jacobi operators. Similarly to the Darboux transformation, it is based on the factorization of a second- order operator into two first-order parts, which are exchanged to produce the new operator. The distinctions between the discrete and continuous versions of this approach can be traced primarily to the fact that the differences that are the analogue of derivatives are constructed from finite rather than infinitesimal shifts. Let H be a Jacobi operator. In this section, we assume that the operator H satisfies bn < 0. This condition, with λ < inf Spec H, ensures the existence of solutions u±(k, n) square-integrable on a half line which are everywhere positive. This machinery can be developed for operators with the u±(k,n+1) weaker restriction of bn 6= 0, in which case the u±(k, n) change sign so that the quantity −bnu±(k,n) k+k−1 remains positive. Fix a spectral parameter λ ∈ R with λ < inf Spec H, λ = 2 . Let σ ∈ [0, 1] and define u(k, n) = σu−(k, n) + (1 − σ)u+(k, n). Denote by e(k, n), o(k, n) the sequences s b u(k, n + 1) e(k, n) = − n (3.17) u(k, n) s b u(k, n) o(k, n) = − − n (3.18) u(k, n + 1)

27 and let A, A∗ be the ﬁrst-order (i.e. bidiagonal) operators Af(n) = o(k, n)f(n + 1) + e(n)f(n) (3.19)

A∗f(n) = o(k, n − 1)f(n − 1) + e(n)f(n) (3.20)

By a direct calculation, we may observe that (H − λ) = A∗A, so that (H − λ) is factored into two ﬁrst order operators. Then, the transformed operator is obtained by exchanging these factors to obtain H˜ − λ = AA∗. We record the facts that we will need in the following theorem:

Theorem 2 (cf. Th. 11.2 in [26]). Let H be a Jacobi operator and let u(k, n) = σu−(k, n) + (1 − σ)u+(k, n) be used to construct a single commutation. 1. The transformed operator H˜ is a Jacobi operator with sequences s b b u(k, n)u(k, n + 2) ˜b = − n n+1 (3.21) n u(n + 1)2 u(k, n) u(k, n + 1) a˜ = λ − b + (3.22) n n u(k, n + 1) u(k, n) b u(k, n) = b + ∂∗ n (3.23) n u(k, n + 1)

2. The operator Wn(u(k, ·), f) Af(n) = p −bnu(k, n)u(k, n + 1) intertwines H and H˜ ; that is, HAf˜ = AHf

It will also be useful to record the effect of addition on the scattering data. Henceforth, Ap(p = (λ, sgn p)) denotes the transformation obtained by performing a single commutation with parameter λ and u(λ, n) = usgn p(λ, n). −1 p 0 p 0 Proposition 7. [Lemma 11.9 from [26]] Let p = ((k+k )/2, sgn p). Denote by A T (k ),A R±(k ) the transmission and reflection coefficients of the transformed operator ApH. Then ApT (k0) = T (k0) (3.24) ksgn p − k0∓1 R T (k0) = R (k0) (3.25) ± ksgn p − k0±1 ±

3.4.2 Another form of discrete Darboux transformation Another approach to operator factorization, developed independently by this author and extended by Rahlmeira-Tsu [23], bears an attractive similarity to the classical Darboux transformation. Although we will primarily use the single commutation method in the subsequent sections, we record the discrete Darboux transformation here as a comparison. Proposition 8. Let H be a discrete Schrödingeroperator, and f(n) a (formal) solution to the equation Hf = λf for a fixed value λ < inf Spec H, with f(n) 6= 0. Define sequences µl and µr by f(n) − f(n − 1) (µ )(n) = = −f −1∂∗f (3.26) l f(n) f(n + 1) − f(n) (µ )(n) = = f −1∂f (3.27) r f(n)

28 (Here f −1 is componentwise inverse.) Then

∗ + 1. H − λ = (∂ + µl)(−∂ − S µr)

∗ + − + 2. Deﬁne H˜ by setting H˜ − λ = (−∂ − S µr)(∂ + S µlS ). Then H˜ is a discrete Schr¨odinger ∗ operator with potential u˜ = u + (−∂ )µr + ∂µl. Proof. 1. Writing the factors in matrix form, we see

 . .  .. ..      −1 + (µl)i 1  (dr + µl) =   (3.28)    −1 + (µl)i+1 1   . .  .. ..  .  ..    ..   . 1  +   (dl − S µr) =  −1 − (µr)i 1  (3.29)    ..   −1 − (µr)i+1 .   .  ..

where in both matrices, the ﬁrst element written out is the (i, i) entry. From this, it is clear that the product is a tridiagonal matrix whose superdiagonal entries are 1.

(∂ + µl) is an upper bidiagonal operator whose superdiagonal elements are 1, and whose ith ∗ + diagonal element (i.e. (i, i) entry) is −1 + (µl)i. Similarly, (−∂ + S µr) is a lower bidiagonal operator whose diagonal elements are 1, and whose (i, i + 1) entry is −1 − (µr)i. It is clear ∗ + from this that the superdiagonal elements of (∂ + µl)(−∂ + S µr) are 1, being the product of the 1 entries from each operator. To verify that the diagonal and subdiagonal entries are correct, it is enough to check the following equations:

−2 + (µl)i − (µr)i = −2 + ui − λ (diagonal) (3.30)

(−1 + (µl)i+1)(−1 − (µr)i) = 1 (subdiagonal) (3.31)

hold for each i ∈ Z.

Diagonal equation: substituting in the deﬁnitions of µl and µr, the left hand side of the ﬁrst equation is

fi − fi−1 fi+1 − fi −2 + (µl)i − (µr)i = −2 + − (3.32) fi fi f + f = − i+1 i−1 (3.33) fi The equation Hf = λf, expressed elementwise, is

fi+1 + (−2 + ui)fi + fi−1 = λfi

from which we see that fi+1 + fi−1 = 2 − ui + λ fi

29 and so −2 + (µl)i − (µr)i = −2 + ui − λ.

Subdiagonal equation: again substituting the deﬁnitions of µl and µr, the left hand side is f − f f − f f − f f − f (f − f )2 −1 + i+1 i −1 − i+1 i = 1 − i+1 i + i+1 i + i+1 i (3.34) fi+1 fi fi+1 fi fi+1fi f (f − f ) − f (f − f ) (f − f )2 = 1 − i+1 i+1 i i i+1 i + i+1 i fi+1fi fi+1fi (3.35) (f − f )2 (f − f )2 = 1 − i+1 i + i+1 i (3.36) fi+1fi fi+1fi = 1 (3.37)

as needed. 2. Again it is clear that the product is a tridiagonal matrix whose superdiagonal entries are 1. Furthermore, the subdiagonal entry at index i is

(−1 + (µl)i+1)(−1 + (µr)i)

which by the previous section is 1. Therefore, H˜ − λ is a discrete Schr¨odingeroperator; hence, so is H˜ . To compute the potential, note that

(H˜ − λ) − (H − λ) =u ˜ − u

and by the factorization formulas

((H˜ − λ) − (H − λ))i = (−2 + (µl)i+1) − (µr)i−1) − (−2 + (µl)i − (µr)i) (3.38)

= (µl)i+1 − (µl)i + (µr)i − (µr)i−1 (3.39) ∗ = ∂µl − ∂ µr (3.40)

The final formula is particularly attractive, being a clear analogue to the second logarithmic derivative that appears in the addition for Schrödinger operators. As mentioned before, this may be extended to general Jacobi operators as well. It is convenient to transform the operator to an asymmetric form first. If H is any Jacobi operator, construct the −1 diagonal matrix D whose entries dn are any sequence satisfying dn+1dn = bn. (It is clear that an overall constant factor is irrelevant.) Then, DHD−1 has the form

 . .  .. ..    ..   . an 1   2   b an+1 1  (3.41)  n   .. ..   bn+12 . .   .  ..

Although this matrix lacks the symmetry of a typical Jacobi matrix, it is more convenient for some calculations: it relieves us of the need to be concerned about the sign of the bn coeﬃcients, 2 and the Toda equations (described below) depend on bn rather than on bn. This may also be interpreted in terms of orthogonal polynomials. The symmetric Jacobi operator, as we have seen, encodes a recurrence relation generating a sequence of polynomials orthonormal with respect to a

30 certain probability measure on R. This form of the operator instead encodes the recurrence relation generating the sequence of monic polynomials orthogonal with respect to the same measure. In this form, a general Jacobi operator may be factored in a manner similar to the way described above.

2 Proposition 9. [23] Let T be a Jacobi operator in monic form with coefficients (an), (bn). Let f(n) a solution to the equation T f = λf for a fixed value λ, with f(n) 6= 0. Define sequences µl and µr by f(n) − f(n + 1) (µ )(n) = (3.42) l f(n) f(n + 1) − b2 f(n) (µ )(n) = n+1 (3.43) r f(n + 1) Then + T − λ = (dl + S µr)(dr − µl)

3.5 The Toda lattice equations 3.5.1 A model for interacting particles The Toda lattice equations are a system of equations modeling a (finite or infinite) sequence of interacting particles. This may represent, for instance, atoms in a linear crystal whose motion is governed by a nonlinear nearest-neighbor interaction. Denoting the positions of the nth particle by q(n, t) and its momentum5 by p(n, t), define the following Hamiltonian: [27]

X p(n, t)2 H(p, q) = + V (q(n + 1, t) − q(n, t)) 2 where V is a potential energy function that determines the form of the interaction between particles. 1 2 If V (r) = 2 kr for some k > 0, then the resulting system is simply an array of masses coupled by linear springs, familiar to any beginning student of mechanics. However, in order to explain behavior seen in numerical simulations of interacting particle chains with weakly nonlinear nearest-neighbor interactions by Fermi, Pasta, and Ulam [13] (and Tsingou [5]) in the 1950s, it was necessary to seek a form of these equations that may support soliton solutions. Toda’s contribution was the choice of nonlinear interaction V (r) = e−r + r − 1, which he arrived at by considering addition formulas for elliptic integrals. [27] Hamilton’s equations of motion now read: ∂H(p, q) p˙(n, t) = − (3.44) ∂q(n, t) = e−(q(n,t)−q(n−1,t)) − e−(q(n+1,t)−q(n,t)) (3.45) ∂H(p, q) q˙(n, t) = − (3.46) ∂q(n, t) = p(n, t) (3.47)

Example 2 (Soliton solutions of the Toda lattice). The simplest interesting solutions to the Toda lattice equations are the soliton and multisoliton solutions, which are similar in character to those for the KdV equation. These solutions are characterized by a localized “bump” in the particle density that travels to the left in forward time. As in the case of the KdV solitons, the speed of travel is

5We take all particles to have unit mass

31 determined by the amplitude of the soliton, and two solitons that collide will interact nonlinearly before emerging unchanged in shape. The form of a single soliton solution, in the original variables q(n, t), is

1 + γ exp(−2κn − 2 sinh(κ)t) q(n, t) = q − log 0 1 + γ exp(−2κ(n − 1) − 2 sinh(κ)t) where γ and κ are parameters greater than zero. Similar to the KdV soliton, the Toda soliton has speed related to its size: the wave has width proportional to 1/κ and propagates with speed ± sinh(κ)/κ.

3.5.2 Flaschka variables and the Lax representation The connection to Jacobi operators is accomplished through a change of variables introduced by Flaschka. Deﬁne the variables a(n, t), b(n, t) as 1 a(n, t) = − p(n, t) (3.48) 2 1 b(n, t) = e−(q(n+1,t)−q(n,t))/2 (3.49) 2 Under this change of variables, the equations of motion read:

a˙(n, t) = 2(b(n, t)2 − b(n − 1, t)2) (3.50) b˙(n, t) = a(n, t)(b(n + 1, t) − b(n, t)) (3.51)

Since b(n, t) > 0, these variables parameterize a family H(t) of Jacobi operators.

3.5.3 Integrability of the Toda lattice equations Expressing the Toda lattice in these variable allows the equations to be written in terms of a Lax pair. The operators in the Lax pair take the matrix form

 . .  .. ..    ..   . a(n, t) b(n, t)    H =  b(n, t) a(n − 1, t) b(n + 1, t)  (3.52)    ..   b(n + 1, t) .   . .  .. ..  . .  .. ..    ..   . 0 b(n, t)    P =  −b(n, t) 0 b(n + 1, t)  (3.53)    ..   −b(n + 1, t) .   . .  .. ..

Deﬁning shift operators S± by S±f(n) = f(n ± 1), we may write these more compactly as H(t) = a(t) + b(t)S+ + (S−b(t))S− and P = b(t)S+ − (S−b(t))S−. Then, the Lax equation

d H(t) = P (t)H(t) − H(t)P (t) dt

32 reduces to the Toda lattice equations. As in the KdV case, the integrability comes from the fact that the Lax equation implies the existence of inﬁnitely many constants of motion. In this case, they take the form of traces. If H(t) is a solution to the Toda lattice equation, then

k k Ik = tr(H(t) − H0 ) is a constant of motion (where H0 is the constant solution a(n, t) ≡ 0, b(n, t) ≡ 1/2; this term ensures that the trace converges for k > 1).

3.5.4 The Toda hierarchy The Toda lattice equations, as defined above, are just the first system in a hierarchy of commuting isospectral flows. These systems may be derived by seeking a sequence of operators P2n+2(t) of order 2n + 2 such that the corresponding sequence of Lax equations dH = [P ,H] dt 2n+2 generate flows on the space of Jacobi operators; in particular, we must have that the commutators [P2n+2,H] are themselves symmetric tridiagonal operators. In short, the result of this calculation is that

j X r P2j+2(t) = cn−rP˜2r+2(t) + dnH r=0

j + j − where P˜2r+2 = (H ) −(H ) [26]. The choice of c0 is an overall scaling factor that may be removed by the transformation t 7→ t/c0, so we choose c0 = 1; the further choice of ci = di = 0, 1 ≤ i ≤ r yields the homogeneous Toda hierarchy.6 The Toda equations may be written in terms of the Green’s function of the operator H. It is convenient later to introduce abbreviations for some Green’s function elements: g(λ, n) = G(λ, n, n) (3.54)

h(λ, n) = 2bnG(λ, n, n + 1) − 1 (3.55) (3.56) These may be expanded in decreasing powers of λ as7

∞ X −(j+1) g(λ, n) = gj(n)λ (3.57) j=0 ∞ X −(j+1) h(λ, n) = −1 − hj(n)λ (3.58) j=0 where

j gj(n) = hδn,H δni (3.59) j hj(n) = 2bnhδn+1,H δni (3.60) The next proposition, a special case of Theorem 12.2 from [26], follows from a recurrence relation connecting the coeﬃcients gj(n), hj(n) at diﬀerent values of n; this relation will be revisited in chapter 4.

6 The constants ci play the same role as the integration constants in proposition 1. 7cf. Theorem 6.2 in [26]

33 Proposition 10. The jth system in the Toda hierarchy may be written as

a˙ n(t) = hj(n, t) − hj(n − 1, t) (3.61) ˙ bn(t) = bn(t)(gj(n + 1, t) − gj(n, t)) (3.62)

34 Chapter 4

Additions for Jacobi operators

4.1 An example

Before presenting the main results, it is illuminating to consider a straightforward example of the application of the single commutation method that we will explore in greater detail.

4.1.1 The free Jacobi operator and the soliton solution

We begin with an application of the single commutation method to the free Jacobi operator H0 from example 1. In the original variables q(n, t), p(n, t), this corresponds to the equilibrium solution with p(n, t) ≡ const., q(n + 1, t) − q(n, t) ≡ const.; i.e. all particles are equally spaced and moving with equal constant velocity. Recall from chapter 3 that the u+, u− solutions of the Jacobi diﬀerence equation for the operator ±n k+k−1 H0 are given by u±(k, n) = k , where 2 = λ. However, since we are looking for a solution of the Toda equations, we must also introduce time dependence. This may be obtained from the second Lax equation, du du(k, n, t) 1 1 = P u ⇒ = u(k, n + 1, t) − u(k, n − 1, t) dt dt 2 2 ±n with initial condition u±(k, n, 0) = k . This yields

±n ±(k−k−1)t u±(k, n, t) = k e

So, we choose σ ∈ (0, 1) and let u(k, n, t) = σu−(k, n, t) + (1 − σ)u−(k, n, t). Then, applying the single commutation, we obtain

1 pu(k, n, t)u(k, n + 2, t) b (k, t) = n 2 u(k, n + 1, t) u(k, n, t) a (k, t) = ∂∗ n u(k, n + 1, t) We need only use the ﬁrst equation to obtain a formula for the position of the nth particle q(n, t). Since 1 b (t) = exp(−(q(n + 1, t) − q(n, t))) n 2 to express this in the original variables we rewrite

− log 2bn(t) = q(n + 1, t) − q(n, t)

35 and so n X q(n, t) = q0 − log 2bn(t) i=0

Inspecting the form of bn(t), we see that the sum collapses and, after absorbing edge terms into q0,

u(k, n, t) q(n, t) = q − log 0 u(k, n − 1, t)

Writing k = exp(κ)

u(k, n, t) = σu−(k, n, t) + (1 − σ)u+(k, n, t) σ u−(k, n, t) = 1 + (1 − σ)u+(k, n, t)) 1 − σ u+(k, n, t) σ = 1 + exp(−2κn − 2 sinh(κ)t) (1 − σ)u (k, n, t)) 1 − σ +

σ Then, writing 1−σ = γ, we get u(k, n, t) 1 + exp(−2κn − 2 sinh(κ)t) log = k + log u(k, n − 1, t) 1 + exp(−2κ(n − 1) − 2 sinh(κ)t) and the term k may also be absorbed into q0 to obtain the conventional form of the one-soliton solution 1 + γ exp(−2κn − 2 sinh(κ)t) q(n, t) = q − log 0 1 + γ exp(−2κ(n − 1) − 2 sinh(κ)t)

4.1.2 Additions As we saw in the previous remark, commutations with σ ∈ {0, 1} have a diﬀerent character than those with σ ∈ (0, 1). The adjunction of a soliton to a free (or non-free) background comes with the adjunction of a point of discrete spectrum, so, since a transformation with σ ∈ {0, 1} does not change the spectrum of the operator, it does not add a soliton. As before, since these isospectral transformations occupy a priveleged place in the class of Darboux transformations, we will refer to them specially – and as before, we will call them additions.

4.2 Inversion of additions

Since additions do not alter the spectrum of the original Jacobi operator, it is possible that they may be “undone” by a transformation of the same type. However, the procedure for doing so is slightly more complex than in the case of Schrödingeroperators on R. In particular, it does not suffice to simply choose the same value of the spectral parameter but the opposite sign. To see why, it is instructive to examine specifically the case of scattering type operators and look at the reflection coefficient. In the setting of scattering theory for Schrödingeroperators, the additions act on the reflection coefficient by mapping sign p p k − k0 s (k0) A7→ s (k0) 12 k + k0 21 Since the reflection coefficient uniquely determines the operator for a given transmission coefficient p −p s11(k), this makes clear the fact that the A is inverted by another addition A , where −p =

36 (k2, − sgn p). This shows that choosing the opposite function in deﬁning the addition results in the inverse addition.1 However, a brief calculation shows that an attempt to replicate this calculation unmodiﬁed in the setting of Jacobi operators does not quite work. Recall from chapter 3 that:

p ksgn p − k0∓1 R (k0) A7→ R (k) ± ksgn p − k0±1 ± If one assumes that the inverse transformation also acts on the reflection coefficient by multiplication, then the multiplicative factor must be ksgn p − k0±1 ksgn p − k0∓1 0 which cannot be obtained by any choice of k or ±. Instead, the prefactor necessary to restore R±(k ) 0 is precisely the prefactor applied to R∓(k ) by the same transformation. This suggests the approach of conjugating by a transformation which exchanges the left and right reflection coefficients: a spatial reflection.

4.2.1 The inverse addition and inﬁnitesimal addition Now, we will show by explicit computation of matrix elements that the addition is inverted by conjugation by an appropriate spatial reﬂection.

Remark 1. Some care must be taken with these reflections. In particular, mapping n → −n independently in each of the three sequences an, bn, u±(n) does not preserve the Jacobi difference equation Ju = λu. This is because of a non-symmetric “offset” in the indices of an, bn. Instead, the operator J should be transformed by transposing across the anti-diagonal through the (0, 0) entry; the effect is to send an 7→ a−n, bn 7→ b−n−1. A trivial calculation shows the Jacobi difference equation is then preserved. We will denote this reflection by σ.

Proposition 11. Let J be a Jacobi operator with diagonal sequence an and oﬀ-diagonal sequence p bn. Assume that bn < 0, and let λ < inf(Spec J). Let p = (λ, ±), and let A denote the additon by p. Then J = σApσApJ.

Proof. For simplicity, we will assume p = (λ, −). Let u−(n) be a positive solution to Ju = λu which is square-summable near −∞. We proceed with four transformations. 1. ApJ is a Jacobi operator with matrix elements

0 p bn−1u−(n − 1) bnu−(n) an := A an = an − + u−(n) u−(n + 1) p 0 p bnbn+1u−(n)u−(n + 2) bn := A bn = − u−(n + 1)

To continue with the second addition, we must also push u+(λ, ·) forward to a solution of the p p p transformed Jacobi diﬀerence equation A JA u+ = λA u+.

0 p Wn(u−(λ), u+(λ)) un(λ, n) := A u+(λ, n) = p −bnu−(λ, n)u−(λ, n + 1) 1This holds true for operators not of scattering type as well, but the scattering coeﬃcients provide convenient motivation.

37 2. Applying the reﬂection:

00 p p b−n−1u−(−n − 1) b−nu−(−n) an := σA an = A a−n = a−n − + u−(−n) u−(−n + 1) p 00 p p b−n−1b−nu−(−n − 1)u−(−n + 1) bn := σA bn = A b−n−1 = − u−(−n) and 00 p p W−n(u−(λ), u+(λ)) u+(λ, n)σA u+(λ, n) = A u+(−n) = p −b−nu−(λ, −n)u−(λ, −n + 1)

p p p 3. Now, we apply the second addition. Notice that σA u+(n) is a positive solution of A JA u+ = p p p λA u+ square-summable near −∞, so it is the correct function to compute A (σA J). First, 000 an :

00 00 00 00 000 p p 00 bn−1u−(n − 1) bnu−(n) an := A (σA an) = an − 00 + 00 u−(n) u−(n + 1) b−n−1u−(−n − 1) b−nu−(−n) = an − + u−(−n) u−(−n + 1) " p p # − b−nb−n+1u−(−n)u−(−n + 2) W−n+1(u−(λ), u+(λ)) −b−nu−(−n)u−(−n + 1) − p u−(−n + 1) −b−n+1u−(−n + 1)u−(−n + 2) W−n(u−(λ), u+(λ)) " p p # − b−n−1b−nu−(−n − 1)u−(−n + 1) W−n(u−(λ), u+(λ)) −b−n−1u−(−n − 1)u−(−n) + p u−(−n) −b−nu−(−n)u−(−n + 1) W−n−1(u−(λ), u+(λ))

b−n−1u−(−n − 1) b−nu−(−n) = a−n − + u−(−n) u−(−n + 1) W (u (λ), u (λ)) b nu (−n) W (u (λ), u (λ)) b u (−n − 1) − −n+1 − + − − + −n − + −n−1 − W−n(u−(λ), u+(λ)) u−(−n + 1) W−n−1(u−(λ), u+(λ)) u−(−n)

000 Next, bn :

p 00 00 00 00 000 bnbn+1u−(n)u−(n + 2) bn = − u00(n+1) 1 1 ! 2 ! 2 pb b u (−n − 1)u (−n + 1) pb b u (−n − 2)u (−n) = − − −n−1 −n − − − −n−2 −n−1 − − u−(−n) u−(−n − 1)

1 1 ! 2 ! 2 W−n(u−(λ), u+(λ)) W−n−2(u−(λ), u+(λ)) × p p −b−nu−(λ, −n)u−(λ, −n + 1) −b−n−2u−(λ, −n − 2)u−(λ, −n − 2) ! p−b u (λ, −n − 1)u (λ, −n) × −n−1 − − W−n−1(u−(λ), u+(λ)) 1/2 1/2 W−n−2(u−(λ), u+(λ)) W−n(u−(λ), u+(λ)) ×b−n−1 W−n−1(u−(λ), u+(λ))

38 4. Applying the reflection again, we get p p bnu−(n) Wn+1(u−(λ), u+(λ)) (σA σA )an = an + 1 − u−(n + 1) Wn(u−(λ), u+(λ)) b u (n − 1) W (u (λ), u (λ)) − n−1 − 1 − n − + u−(n) Wn−1(u−(λ), u+(λ)) 1/2 1/2 p p Wn−1(u−(λ), u+(λ)) Wn+1(u−(λ), u+(λ)) (σA σA )bn = bn Wn(u−(λ), u+(λ)) We have left the Wronskian factors in the formulas above for clarity in a later calculation, but since u−(λ) and u+(λ) solve the Jacobi difference equation for the same value of λ, the Wronskian is independent of n and so the Wronskian factors cancel. As a result, the above simplifies to p p (σA σA )an = an p p (σA σA )bn = bn

Remark 2. Above, we computed (σApσ)Ap since it is slightly more convenient, primarily because one does not need to compute Wronskians in the reﬂected space. One may also compute Ap(σApσ); the computation is similar in character to the above, only somewhat more tedious. Omitting the bulk of the computation, the second-to-last step gives: p p bn−1u−(n) Wn−2(u−(λ), u+(λ)) A (σA σ)an = an + 1 − u−(n − 1) Wn−1(u−(λ), u+(λ)) b u (n + 1) W (u (λ), u (λ)) − n − 1 − n−1 − + u−(n) Wn(u−(λ), u+(λ)) 1/2 1/2 p p Wn−1(u−(λ), u+(λ)) Wn+1(u−(λ), u+(λ)) A (σA σ)bn = bn Wn(u−(λ), u+(λ)) After cancelling the Wronskian factors, the result is once again

p p (A σA σ)an = an p p (A σA σ)bn = bn, showing that σApσ is both a left and a right inverse to Ap.

4.3 The inﬁnitesimal addition

As in the KdV case, once we have obtained the inverse transformation to an addition, we can construct an infinitesimal addition. This infinitesimal addition is again expressible in terms of the Green’s function of the original operator. However, while the KdV infinitesimal addition depends only on the values of the Green’s function along the diagonal, the infinitesimal addition for Jacobi operators also includes the values along the first off-diagonal. Of course, this is unsurprising: the only part of a Schrödingeroperator which is modified by an addition is the potential, which is “diagonal” in the sense of being an operator that acts by pointwise multiplication. On the other hand, a Jacobi operator’s “potential” exists on the diagonal as well as on the off-diagonal. The infinitesimal addition is computed by applying an addition followed by an inverse addition with an infinitesimally different value of λ. The calculation proceeds similarly to the calculation in the previous proposition, except that the Wronskians are no longer independent of n and so do not drop out of the formulas. First, we state a useful expansion of the ratios of Wronskians that appeared above.

39 Lemma 4. W (u (λ), u (λ + ∆λ)) u (λ, n + 1)u (λ, n + 1) n+1 − + = 1 + − + ∆λ + o(∆λ) Wn(u−(λ), u+(λ + ∆λ)) W (u−(λ), u+(λ)) = 1 + G(λ, n + 1, n + 1)∆λ + o(∆λ) where G(λ, n, m) = u−(λ,n)u+(λ,m) is the Green’s function. W (u−(λ),u+(λ))

Proof. First, by expanding u+(λ + ∆λ), we have W (u (λ), u (λ + ∆λ)) W (u (λ), u˙ (λ)) n − + = 1 + n − + ∆λ + o(∆λ) W (u−(λ), u+(λ)) W (u−(λ), u+(λ)) where the dot denotes diﬀerentiation by λ. Henceforth, for compactness, we denote the constant Wronskian by W := W (u−(λ), u+(λ)). Then,

−1 Wn+1(u−(λ), u+(λ + ∆λ)) 1 + W Wn+1(u−(λ), u˙ +(λ))∆λ + o(∆λ) = −1 Wn(u−(λ), u+(λ + ∆λ)) 1 + W Wn(u−(λ), u˙ +(λ))∆λ + o(∆λ)

1 Using the expansion 1+ε = 1 − ε + ...,

Wn+1(u−(λ), u+(λ + ∆λ)) −1 = 1 + W (Wn+1(u−(λ), u˙ +(λ)) − Wn(u−(λ), u˙ +(λ))) ∆λ + o(∆λ) Wn(u−(λ), u+(λ + ∆λ)) So, we need to calculate

Wn+1(u−, u˙ +) − Wn(u−, u˙ +)

= bn+1u−(n + 1)u ˙ +(n + 2) + an+1u−(n + 1)u ˙ +(n + 1) + bnu−(n + 1)u ˙ +(n)

− bn+1u−(n + 2)u ˙ +(n + 1) − an+1u−(n + 1)u ˙ +(n + 1) − bnu−(n)u ˙ +(n + 1)

= (Ju˙ +)(n + 1)u−(n + 1) − u˙ +(n + 1)(Ju−)(n + 1)

= (Ju+)˙(n + 1)u−(n + 1) − u˙ +(n + 1)(Ju−)(n + 1)

= u−(n + 1)u+(n + 1) where in the last line we have used Ju = λu. Therefore, W (u (λ), u (λ + ∆λ)) u (n + 1)u (n + 1) n+1 − + = − + Wn(u−(λ), u+(λ + ∆λ)) W (u−(λ), u+(λ)) as desired. Corollary 2. 1. W (u (λ), u (λ + ∆λ)) n − + = 1 − G(λ, n + 1, n + 1)∆λ + o(∆λ) Wn+1(u−(λ), u+(λ + ∆λ))

2. W (u (λ), u (λ + ∆λ)) n+1 + − = 1 − G(λ, n + 1, n + 1)∆λ + o(∆λ) Wn(u+(λ), u−(λ + ∆λ))

1 Proof. (1) uses 1+ε+o(ε) = 1−ε+o(ε). For (2), note that the calculation proceeds identically, except that the constant Wronskian is Wn(u+(λ), u−(λ)) = −Wn(u−(λ), u+(λ)). Applying this lemma to the next-to-last step in proposition 11, we obtain the following ﬁrst-order expansions.

40 0 Proposition 12. Let J = ((an), (bn)), p = (λ, +), p = (λ + ∆λ, +). Then

p0 −p A A an = an + (bnG(λ, n, n + 1) − bn−1G(λ, n − 1, n))∆λ + o(∆λ) p0 −p 2 2 A A bn = bn (1 + (G(λ, n, n) − G(λ, n + 1, n + 1))∆λ + o(∆λ)) Proof. From proposition 11, we have

0 p0 −p bn−1u+(n) Wn−2(u+, u−) A A an = an + 1 − 0 u+(n − 1) Wn−1(u+, u−) 0 bnu+(n + 1) Wn−1(u+, u−) − 1 − 0 u+(n) Wn(u+, u−) 0 0 p0 −p 2 2 Wn−1(u+, u−)Wn+1(u+, u−) A A bn = bn 0 2 Wn(u+, u−) Applying the formulas from corollary 2, we see immediately that

p0 −p 2 2 A A bn = bn(1 + (G(λ, n, n) − G(λ, n + 1, n + 1)) ∆λ + o(∆λ)) and that p0 −p bn−1u+(n)G(λ, n − 1, n − 1) bnu+(n + 1)G(λ, n, n) A A an = an + − ∆λ + o(∆λ) u+(n − 1) u+(n)

Since G(λ, n, m) = u−(n)u+(m) , we have u+(n+1)G(λ,n,n) = G(λ, n, n + 1), which yields the claimed W (u−,u+) u+(n)

p0 −p A A an = an + (bnG(λ, n, n + 1) − bn−1G(λ, n − 1, n))∆λ + o(∆λ)

For the above to make sense, it is necessary to have some control on G(λ, n, n) as n → ±∞; 2 otherwise, for ﬁxed ∆λ, the formula for bn may become negative for suﬃciently large n. This is easy to see in the scattering case, where we may write:

f (λ, n)f (λ, n) G(λ, n, n) = + − W (f+, f−) The denominator is a constant, so we need only consider the numerator. Using the asymptotic behavior from table 2,

−2n f+(λ, n)f−(λ, n) ∼ T (k)(1 + R+(k)k )(n → −∞) (4.1) 2n f+(λ, n)f−(λ, n) ∼ T (k)(1 + R−(k)k )(n → ∞) (4.2)

In deﬁning k such that (k + k−1)/2 = λ, we may freely exchange k and k−1 as needed to ensure |k| ≤ 1, so that the above is O(1) as n → ±∞.

4.3.1 Trace formulas for the Green’s functions In order to connect the inﬁnitesimal addition to the Toda equations, we need a standard trace formula. The following expansions are a consequence of the fact that for a function δ ∈ `2(Z) with kδk = 1, the function g(λ) = hδ, (H − λ)−1δi is Herglotz, using the Neumann series expansion for the resolvent.

41 Proposition 13. [26] The Green’s function along the diagonal and oﬀ-diagonal have the following Laurent expansions:

∞ X gj(n) G(λ, n, n) = − , g = 1 λj+1 0 j=0 ∞ X hj(n) G(λ, n, n + 1) = − , h = 0 λj+1 0 j=0 where the coeﬃcients are

j gj(n) = hδn,H δni j hj(n) = hδn+1,H δni

The ﬁrst several coeﬃcients are

2 2 2 g0(n) = 1, g1(n) = an, g2 = bn + bn−1 + an,...

h0(n) = 0, h1(n) = bn, h2(n) = bn(an+1 + an),...

4.3.2 Inﬁnitesimal addition and the Toda lattice equations

By substituting the coeﬃcients g1(n), h1(n) into the expansion derived in proposition 12, we arrive immediately at the following.

0 2 Proposition 14. Let J = ((an), (bn)), p = (λ, +), p = (λ + ∆λ, +), and let Xan, Xb denote the 0 n order ∆λ term in the expansion of Ap A−pJ. Then 1 1 Xa = (b2 − b2 ) + O n n n−1 λ2 λ3 1 1 Xb2 = b2 (a − a ) + O n n n+1 n λ2 λ3 Specifically, the highest order term in this expansion generates the conventional Toda lattice flow. Remark 3. In the continuous analogue, there was a λ−1 term corresponding to the isospectral translation flow qt = qx. Such a term does not appear for the Toda lattice. This is probably best interpreted as a consequence of the fact that there is no infinitesimal translation in discrete space. Based on proposition 10, this in fact generates the entire Toda hierarchy. Recall that the rth system in the (homogeneous) Toda hierarchy may be written as:

a˙ n(t) = hj(n, t) − hj(n − 1, t) (4.3) ˙ bn(t) = bn(t)(gj(n + 1, t) − gj(n, t)) (4.4) hence the expression of the infinitesimal addition in terms of differences of Green’s functions, upon expansion near λ = ∞, contains the rth system in the homogeneous Toda hierarchy as its rth coefficient. Collecting the above propositions results in theorem 1.

42 4.4 Symplectic geometry and Poisson structures

As a final note in the discussion of additions for Jacobi operators, we will describe a few features of the symplectic geometry of isospectral sets in the space of such operators. This includes the definition of a Poisson structure that is compatible with the Hamiltonians of the Toda hierarchy, as well as computing the gradient of the transmission coefficient as an analogy to proposition 6.2 As we will see, the Green’s function which makes up the infinitesimal addition appears as the gradient of the transmission coefficient with respect to perturbations of the potential. This is similar to the KdV case, but has slightly more depth as a result of the fact that there are effectively two “potentials”: one on the diagonal, represented by an, and one on the off-diagonal, represented by bn. This means that there are two gradients to calculate, and it also means that values of the Green’s function along the off-diagonal play a role as well. However, we will see that the two gradients do not simply decouple into diagonal and off-diagonal Green’s function elements.

4.4.1 Poisson structures for Jacobi operators

Jacobi operators – and in general, difference operators of finite order – induce on `2(Z) a Poisson structure similar to that described in section 2.4. Such structures are described in Mumford and van Moerbeke [22] for the special case of periodic difference operators, where the sequences of coefficients along each diagonal are periodic with common period N. This simplifies the analysis by allowing one to inspect only finite matrices, and also allows the linearization of isospectral flows on the Jacobian variety of a certain algebraic curve.3 However, the ideas contained in that article may be extended to a more general case in a simple manner which also closely mirrors what is seen in the case of the KdV equation. We begin by defining a Poisson bracket on linear functionals on J , the space of infinite Jacobi operators. Extending [22] ∂F ∂G ∂a ∂a {F,G} = ∂F ,J ∂G (4.5) ∂b ∂b In the above equation, ∂F , etc. represent the sequences ∂F , etc., so we are abusing notation ∂a ∂an somewhat in stacking these infinite sequences as a column vector, but the meaning should be clear. Similar to the KdV Poisson bracket, the operator J is now a difference operator acting on the two ∂G ∂G sequences ∂a , ∂b . Abusing notation again, J is the “matrix” 0 B J = −BT 0 where B is the first-order (bidiagonal) operator (Bf)(n) = bnf(n) − bn−1f(n − 1); or, 0 −∂∗b J = b∂ 0 This means the bracket is X ∂F ∂G X ∂F ∗ ∂G {F,G} = bn∂ (n) − ∂ b (n) ∂bn ∂a ∂an ∂b n∈Z n∈Z

We note that in the case of a discrete Schrödingeroperator (bn ≡ 1) J reduces to 0 −∂∗ J = ∂ 0 so B is a simple first difference operator in analogy with (2.24).

2This part of the discussion, of course, is restricted to the case of operators of scattering type. 3This is the central idea of Floquet theory, mentioned in Chapter 5 and Appendix A.

43 Lemma 5. {, } as defined above satisfies the Jacobi identity. Proof. Following [22], we begin with a reduction: when expanding the derivative of a Poisson bracket with respect to the coefficients of H, the terms containing second derivatives will all cancel. So, we need only focus on those terms which contain only first derivatives. Note that a bracket {A, B} may be written explicitly as X ∂A ∂B X ∂A ∗ ∂B {A, B} = bn ∂ (n) − ∂ b (n) ∂bn ∂a ∂an ∂b n∈Z n∈Z Then, ∂ {A, B} = (2nd derivative terms) + 0 ∂an and ∂ ∂A ∂B ∂A ∂B {A, B} = (2nd derivative terms) + ∂ (n) − ∂ (n) ∂bn ∂bn ∂a ∂an ∂b Introducing a third functional C, this implies that

X ∂A ∂B {{A, B},C} = (2nd derivative terms) + ∂ (n) ∂bn ∂a n∈Z ∂A ∂B ∂C − ∂ (n) bn∂ (n) ∂an ∂b ∂a Permuting A, B, and C cyclically and summing, we see that all terms cancel and the Jacobi identity holds.

The Hamiltonians for the Toda hierarchy of equations are written in terms of traces: 1 F = tr Hj+1 − Hj+1 j j + 1 0 where H is the Jacobi operator with sequences an(t), bn(t), and H0 is the free Jacobi operator (required here to ensure convergence of the trace). In order to compute Poisson brackets, we need j+1 the derivatives of such traces with respect to the variables an, bn. The term H0 , included to ensure convergence of the trace, will not contribute; the necessary derivatives are computed in the following lemma. Lemma 6.

∂ j+1 j tr H = (j + 1)(H )n,n ∂an ∂ j+1 j tr H = 2(j + 1)(H )n,n+1 ∂bn Proof. We use a combinatorial representation of the matrix entries of Hj+1, counting paths in the following graph, whose edges are weighted with the matrix entries an and bn of H. The weight associated with a path π, denoted w(π), is the product of the weights of the edges traversed in the path. (The b edges are not directed, but the loops labeled with an may only be traversed in one direction to avoid double-counting.)

44 an−1 an an+1

...... bn−2 bn−1 bn bn+1 n − 1 n n + 1

j+1 The key observations are that (H )n,n is equal to the sum of path weights taken over paths of j+1 length j + 1 that begin and end at the vertex n, and that (H )n,n+1 is the sum of path weights taken over paths that begin at n and end at n + 1. For an: each path of length j + 1 that begins and ends at vertex n and traverses a loop may be obtained by inserting a loop traversal into a path π of length j that begins and ends at n. Each such insertion contributes a term equal to w(π) to the value of ∂ tr Hj+1, and there are j + 1 places to ∂an insert a loop traversal in each path. Therefore,

∂ X tr Hj+1 = (j + 1) × w(π) = (j + 1)(Hj) ∂a n,n n j-paths n → n

Note that the operation of inserting a loop in a path may not always insert the loop an (i.e. if the path travels to a diﬀerent vertex before inserting the loop). However, this is not a problem due to the isotropic nature of this graph (the graph looks the same from every vertex) and the fact that we are computing a trace (and thus considering all diagonal elements at once). We may think of paths in terms of their relative structures, as a sequence of steps up, down, or along a loop. The contribution will come from inserting the loop an into a path beginning and ending at a diﬀerent vertex. To clarify this, consider the example of j = 2. The n, n entry of H3 is

3 3 2 2 Hn,n = an + bn(an+1 + 2an) + bn−1(an−1 + 2an) obtained by counting the seven paths of length 3 from vertex n to n. Clearly

∂ 3 2 2 2 Hn,n = 3an + 2bn + 2bn−1 ∂an

2 2 j 2 2 2 which is missing bn + bn−1 from the claimed (j + 1)(H )n,n = 3(an + bn + bn−1). The missing 3 3 2 2 contributions come from Hn+1,n+1 and Hn−1,n−1, which contain terms bnan and bn−1an respectively; each of these terms corresponds to the insertion of an in paths that traverse the edges bn and bn−1 twice – but now starting from vertices n + 1 and n − 1 respectively. So, considering all insertions of an includes considering paths beginning and ending at other vertices; all contribute to the trace. For bn: by the same reasoning, we note that each path of length j + 1 beginning and ending at vertex n may be obtained by inserting a traversal of the edge bn into a path of length j that either begins at n and ends at n + 1, or begins at n + 1 and ends at n. By the symmetry of Hj, such paths j produce the same matrix element Hn,n+1; counting both is the source of the factor of 2. Before the ﬁnal proposition we quote a form of the Lax equations established in [26].

j + j − Lemma 7. Let P2j = (H ) − (H ) . Then

j j [P2j,H]i,i = 2 biHi,i+1 − bi−1Hi,i−1

45 Proposition 15. The flows of the homogeneous Toda hierarchy are Hamiltonian flows with respect to {, }. That is, the flow of the jth system may be written in the form (where c stands for a or b)

c˙n = {Fj, cn}

1 j+1 j+1 where Fj = j+1 tr(H − H0 ). Proof. Recall that the ﬂows of the homogeneous Toda hierarchy are deﬁned by the Lax equations d H = [P ,H] dt 2j

j + j − where P2j = (H ) − (H ) . Fix i ∈ Z; we compute {Fj, ai} and {Fj, bi}. ∂Fj j ∂Fj j By the lemma, = (H )i,i and = 2(H )i,i+1. Then, we may compute ∂ai ∂bi

* j ! + H(a) {Fj, ai} = j , Jδai H(b)

where δai is the Kronecker δ vector whose component at ai is 1, all others 0. Then, since

 0   .   .    Jδai =  −bi−1     bi   .  . we get j j {Fj, ai} = 2bi(H )i+1,i − 2bi−1(H )i,i−1 j + j − which is the same as the (i, i) entry of [(H ) − (H ) ,H], which by the Lax equation is ·ai. Similarly,  0   .   .    Jδbi =  −bi     bi   .  . so j j {Fj, bi} = bi Hi+1,i+1 − Hi,i which is the same as the (i, i + 1) entry of [(Hj)+ − (Hj)−,H]. The final piece in our exploration of these ideas is to realize the values of the transmission coefficient as a family of functionals in involution, by computing their gradients with respect to the coefficients an, bn.

4.4.2 The gradient of T (k)

In the case of Schrödingeroperators on R, establishing that the values of the transmission coefficient s11(k) commute with respect to the Poisson structure was accomplished by computing the gradient of the transmission coefficient directly.

46 A similar result holds in the setting of Jacobi operators, and can be proved a little bit more easily. The complication is that there are now two gradients to compute: one with respect to the an coefficients and one with respect to the bn coefficients. As may be expected because the sequence an is the analogue of the potential q(x) in a Schrödinger operator, the result for the an coefficients is an exact analogue of the result in the Schrödingersetting. 2 The bn coefficients do not have a direct analogue in the Schrodinger operator −∂x +q(x), but instead are similar to the function p(x) in a Sturm-Liouville operator ∂xp(x)∂x + q(x). As a consequence, the form of the result is slightly different for these coefficients, and the gradient with respect to bn depends on several values of the Green’s function (i.e. matrix entries of the resolvent).

Theorem 3. Let H be a Jacobi operator of scattering type with transmission coefficient T (k) and reflection coefficients R±(k). 1. ∂ log T (k) = −G(λ, n, n) ∂an 2. ∂ log T (k) = −2G(λ, n, n + 1) ∂bn ] Proof. We begin with (1). Let H denote the operator obtained by replacing an with an + θ, θ ∈ R. ∂T (k) We will compute ∂θ |θ=0. Let f+(k, m), f−(k, m) denote the modified Jost solutions of (H − λ)f = 0. Denote the modified ] ] ] Jost solutions of (H − λ)f = 0 by f+(k, m), f−(k, m). First, we describe the relationship between ] f± and f±. ] ] First, we may represent f− and f− in terms of the Jost solutions u−, u− as

]f−(k, m) = T (k)u−(k, m) and ] ] ] f−(k, m) = T (k)u−(k, m) However, for m < n, these satisfy the same difference equation and are both asymptotic to a constant −n times k . As a result, they must each be constant multiples of the same Jost solution u−(k, m); ] ] i.e. u− = u− and the difference between f− and f− appears only in the transmission coefficient. So, dividing the previous two equations yields

f ] (k, m) T ](k) − = (4.6) f−(k, m) T (k) Denoting the derivative with respect to θ at θ = 0 by ·, we also have

f˙ (k, m) T˙ (k) − = (4.7) f−(k, m) T (k) or T˙ (k) f˙ (k, m) = f (k, m) (4.8) − T (k) − ˙ Now we calculate f−(k, m) directly. Consider

(H − λ)(f ] − f ) lim − − (4.9) θ→0 θ

47 Since H − λ is a linear operator independent of θ, we may exchange it with the limit so that the above is ˙ (H − λ)f−(k, ·) (4.10) ] However, we may also calculate it directly: at n 6= m, f− and f− both solve (H − λ)f = 0, so the ] ] ] only nonzero entry of (H − λ)(f− − f−) is the nth entry. Since Hf− = Hf− − θf (k, n),

] ] (H − λ)(f− − f−)(n) = −θf (k, n) (4.11)

Inserting this into the limit we see ˙ 0 m 6= n ((H − λ)f−(k, ·))(m) = (4.12) −f−(k, n) m = n

By the deﬁnition of the Green’s function, then,

˙ X f−(k, m) = G(λ, m, `)(−f−(k, `)δk,`) = −G(λ, m, n)f−(k, n) `∈Z Evaluating this at m = n and equating this expression to the previous

T˙ (k) f˙ (k, n) = f (k, n) − T (k) − we obtain T˙ (k) = −G(λ, n, n) T (k) which is what was to be proved. For (2), we compute the derivative in the larger space of tridiagonal operators (not necessarily symmetric). Consider a perturbation of the form bn → bn + θ, applied only to the element bn appearing on the superdiagonal. In other words, H] is the non-symmetric operator in matrix form

 . .  .. ..    ..   . an−1 bn−1       bn−1 an bn + θ    (4.13)    bn an+1 bn+1     .. ..   bn+1 . .   .  ..

Although this is no longer a Jacobi operator, it induces the same diﬀerence equation at all but one row. Therefore, all of the asymptotic statements from the previous argument hold. In fact, the argument proceeds identically until we reach the analogue of (4.12), which is replaced by ˙ 0 m 6= n ((H − λ)f−(k, ·))(m) = (4.14) −f−(k, m + 1) m = n

Then, acting on the left by (H − λ)−1 yields

˙ X f−(k, m) = G(λ, m, `)(−f−(k, ` + 1)δk,`) = −G(λ, m, n)f−(k, n + 1) (4.15) `∈Z

48 and evaluating at m = n + 1 yields ˙ f−(k, n + 1) = −G(λ, n + 1, n)f−(k, n + 1) (4.16)

∂ log T (k) + As a result, we get + = −G(λ, n + 1, n) = −G(λ, n, n + 1) where bn denotes the element bn ∂bn ∂ log T (k) on the superdiagonal. An essentially identical argument yields the same − = −G(λ, n, n + 1) ∂bn ∂ log T (k) also. Then, varying both bn elements at once yields = −2G(λ, n, n + 1). ∂bn

Corollary 3. The values of the transmission coeﬃcient T (k) commute with respect to the Poisson bracket (4.5).

Proof. Computing the Hamiltonian vector ﬁeld associated to log T (k), we ﬁnd

* ∂ log T (k) !+ 2 0 ∂a {bn, log T (k)} = ,J ∂ log T (k) (4.17) 2bmδn ∂b 2 = 2bn(G(λ, n, n) − G(λ, n + 1, n + 1)) (4.18) and * !+ δ ∂ log T (k) {a , log T (k)} = n ,J ∂a (4.19) n 0 ∂ log T (k) ∂b ∗ = 2∂ (bnG(λ, n, n + 1)) (4.20)

= −2(bnG(λ, n, n + 1) − bn−1G(λ, n − 1, n)) (4.21)

In other words, the Hamiltonian vector field associated to log T (k) is, up to constant factors, the same as the infinitesimal addition. These vector fields then inherit the commutativity of addition.

49 Chapter 5

Discussion and future directions

To recap the results of the previous chapters: we have demonstrated that the isospectral Darboux transformations, aka additions, of Jacobi operators have inverse transformations and as a result support an infinitesimal limit, allowing such transformations to generate vector fields instead of discrete dynamics. Moreover, these vector fields are closely connected to the Hamiltonian vector fields of the Toda hierarchy of lattice equations, and in an appropriate expansion recover these vector fields as coefficients when the spectral parameter is sent to −∞. In this chapter, we discuss some related work on discrete integrable systems and orthogonal polynomials and propose some avenues for future extensions or applications of this work. The subjects include the adaptation of infinite-period limits for the KdV equation to the Toda hierarchy, the extension of discrete additions to other integrable lattice equations, and applications of spectral shift theory to relax assumptions in the constructions described in chapter 4.

5.1 Inﬁnite-period limits and theta functions

In chapters 2 and 3, we described scattering theory for Schrödinger and Jacobi operators, which is used to describe operators that represent a localized perturbation of a constant background, and to solve the KdV or Toda equations by linearizing them in the space of scattering data. Another approach to the spectral theory of such operators, Floquet theory, is used to describe operators whose potential (q(x) or an, bn) is periodic or quasiperiodic in the spatial variable (x or n). In Floquet theory, the change of coordinates instead leads to a set of coordinates describing a (compact or nearly so) algebraic curve and a divisor on that curve. The flow of the integrable systems is linearized on the Jacobian variety of the algebraic curve, where the image of the divisor under the Abel-Jacobi map travels along a straight line. Solutions to the original system are recovered in terms of the Riemann theta functions of the algebraic curve. In [12], Ercolani and McKean perform a limiting procedure to connect the algebrogeometric solutions of the KdV equation for periodic initial data to scattering data for rapidly decaying initial data. This leads to an interpretation of the scattering data in geometric terms, with the transmission coefficient defining a sort of (continuum-genus) algebraic curve, the reflection coefficient playing the role of a divisor on this curve, and a Fredholm determinant appearing in the inverse scattering transform playing the role of a theta function. The limiting procedure is carried out in part by investigating the Floquet discriminant, which defines the algebraic curve in the periodic case, as the period is taken to infinity. The long-period behavior of this discriminant is toward rapid oscillation within an envelope which is described by the transmission coefficient. For Jacobi operators, this behavior is reproduced in preliminary numerical simulations using expressions of the discriminant in terms of orthogonal polynomials. There are also

50 some results in this direction in [14], suggesting that this limiting procedure may be explored in detail for discrete systems.

5.2 Spectral shift theory

The greatest restriction of the results in this article is that many of them explicitly or implicitly use the assumption that the operator in question is of scattering type. The assumption of scattering type aided in motivation as well as in ensuring, by the rapid approach of the potential to the free Jacobi operator, that various sums and products converge. Krein’s spectral shift theory is a possible avenue to removing these restrictions. This theory is concerned with the spectral theory of perturbations of operators, and may oﬀer results similar to those we obtained in terms of scattering data in a more general setting. Indeed, scattering theory may itself be viewed through the lens of spectral shift theory by considering a scattering-type operator to be a perturbation of a zero background. Using this approach may allow us to relax the scattering assumptions on some of the existing results, and formulate an analogue of theorem 3 that does not require a transmission coeﬃcient.

5.3 Dressing transformations

Another perspective on additions is in the language of dressing transformations, initially introduced by Zakharov and Shabat in the study of the KdV equation. Dressing transformations for a ﬁnite-dimensional Toda hierarchy have been investigated by Ercolani, Flaschka, and Haine [10]. Approaching the additions of this dissertation from this perspective would involve taking a more algebrogeometric perspective. The approach taken to factorization in propositions 8 and 9 by this author and Rahalmeira-Tsu may be particularly amenable to this approach. One of the advantages of this perspective is the ability to naturally handle singularities. A major theme in [10] is the partitioning of the isospectral manifolds into cell structures according to the position and number of Toda matrix elements that become singular in ﬁnite time. This provides a geometric interpretation of algebraic aspects of matrix factorization, including the pivoting process in the LU decomposition.

5.4 Other systems

Since the objective of this thesis was to extend ideas of inﬁnitesimal addition from the KdV equation to the Toda hierarchy, it is natural to consider extending this further to other systems. A good candidate for this eﬀort is the system of orthogonal polynomials on the unit circle (OPUC). First, we review the connection between Jacobi operators and orthogonal polynomials on the real line (OPRL).

5.4.1 Orthogonal polynomials on the real line. A major historical (and contemporary) motivation for the study of Jacobi operators is in their connection to orthogonal polynomials, themselves used to approach moment problems in classical analysis. [1] A good exploration of this connection may be found in Deift’s book [7]; we reproduce the most essential facts here. A probability measure µ on the real line deﬁnes an associated family pn of polynomials orthonormal with respect to the L2 inner product: Z hpn, pmi = pn(x)pm(x)dx = δn,m (5.1) R

51 These polynomials may be constructed explicitly by applying the Gram-Schmidt procedure to the sequence of monomials 1, x, x2,.... Then, the polynomials pn are generated by the recurrence relation

xpn(x) = bn−1pn−1(x) + anpn(x) + bnpn+1(x) with initial values p−1(x) = 0, p0(x) = 1. This is the same as the Jacobi diﬀerence equation, and so may be expressed in terms of a Jacobi operator, albeit now a semi-inﬁnite operator acting on 2 ` (Z≥0). So, a measure gives rise to a sequence of polynomials and a Jacobi operator; in turn, a Jacobi operator has an associated spectral measure dµ, which is the measure of orthogonality for the polynomials generated by the recurrence relation above. The correspondence between measure and operator is 1-1 in the case that the measure is compactly supported, or, equivalently, that the operator is bounded. In the case of an unbounded operator or a non-compactly supported measure, a more subtle condition decides whether the correspondence is 1-1.

5.4.2 Orthogonal polynomials on the unit circle Here, we consider a probability measure defined on the unit circle and study the family of polynomials {φn(z)} satisfying Z φi(z)φj(z)dµ(z) = δij ∂D These polynomials satisfy a recurrence relation most easily expressed in the form of a 2 × 2 matrix equation φn+1(z) −1/2 z −α¯ φn(z) ∗ = ρn ∗ (5.2) φn+1(z) −αz 1 φn(z) 2 (ρn = 1 − |αn| ). The complex numbers αn are known as Verblunsky coefficients. They replace the coefficients an and bn in Jacobi operators. Note that the overall number of parameters is the same, since we have one sequence of complex numbers replacing two sequences of real numbers. The Verblunsky coefficients are subject to the condition |αn| ≤ 1, with |αn| < 1 for measures of infinite support.1 Orthogonal polynomials on the unit circle have a spectral theory that is very similar in spirit to that of OPRL and Jacobi matrices, including versions of Floquet and scattering theory. The spectral theory is in many ways somewhat simpler, owing to several symmetries that appear in the complex setting (in particular,z ¯ = z−1 on the unit circle). Additionally, OPUC are related to an integrable difference-differential system, the defocusing discrete nonlinear Schrödingerequation, in the same way that OPRL are related to the Toda lattice equations. Generalizations obtained by relaxing restrictions on the Verblunsky coefficient and allowing the spectral measure to be supported on arcs elsewhere in C instead of only on the unit circle allow this to be generalized to the focusing discrete NLS equation and, more generally still, the Ablowitz-Ladik hierarchy of systems. [11] Extending the techniques developed here for Jacobi operators to the OPUC/discrete NLS/Ablowitz- Ladik setting could take two forms. One potential approach would be to use a natural map between probability measures on the real line and probability measures on the unit circle that are symmetric with respect to complex conjugation. This map is generated by the two-to-one map S1 → [−2, 2] given by θ 7→ 2 cos θ, which establishes a one-to-one correspondence between certain restricted classes of measures on S1 and on R. Previous work on this map has formulas called Geronimus relations relating the Verblunsky coefficients αn to the Jacobi coefficients an, bn. [25] In principle, one could use this to transfer results from the setting of Jacobi operators to OPUC. The advantage of this approach lies in its overall simplicity. However, it does present some problems.

1 This condition is speciﬁc to OPUC; some integrable generalizations have fewer restrictions on αn.

52 The most obvious disadvantage is the limitation to a restricted class of measures on both sides of the mapping. Namely, the measure on S1 must be symmetric with respect to complex conjugation, and the measure on R must be supported on [−2, 2]. The first condition is quite simple, and is easily expressed in a condition on the Verblunsky coefficients: namely, αn ∈ R for all n. The second condition is much more subtle, and may only be expressed in terms of a recurrence relation (therefore, not in terms of satisfying as single closed-form condition). This is also reflected in the form of the Geronimus relations. The relations mapping αn to an and bn are local, in the sense that each an and bn depends only on at most three Verblunsky coefficients, whereas the relations in the other direction are nonlocal: the Verblunsky coefficient α2n depends on all Jacobi coefficients indexed less than or equal to n. This may present a challenge in transferring calculations performed on Jacobi operators to the recurrence matrices of OPUC. A second approach is to consider the problem of operator factorization directly, as we did for Jacobi operators. In the theory of OPUC, the Jacobi operator is replaced by an operator known as a CMV matrix 2. The CMV matrix is unitary (instead of symmetric) and has a five-diagonal structure (instead of tridiagonal). This is amenable to factorization and commutation methods just as a Jacobi operator is, and some progress has been made by CMV and Marcellánin the study of Darboux transformations for CMV matrices [2]. However, their work has also highlighted additional difficulties and subtleties that appear in this setting, particularly in the construction of the inverse transformation. As a result, this offers a promising but challenging avenue for future research. Remark 4. The previously mentioned avenue of research on infinite-period limits is also applicable to the OPUC recurrence. In fact, preliminary results on such limits are slightly stronger than those for Jacobi operators/OPRL. This is owing primarily to the fact that the spectrum of periodic OPUC lies on the unit circle, and no control over its infimum or supremum is necessary to describe the spectrum. In contrast, while the a.c. spectrum of a bounded Jacobi operator is contained in a compact interval in R, the endpoints of this interval are not fixed. The numerical results in the Appendix shed some light on this difference.

2for Cantero, Moral, and Vel´azquez[3]

53 Appendix A

Numerical results for inﬁnite-period limits of Floquet theory

In section 5.1, we mentioned a limiting procedure used to connect the algebrogeometric description of periodic Schr¨odingeroperators (and KdV solutions) to the scattering theory applied to rapidly decaying potentials. In this appendix we show some numerical results suggesting that the same procedure can be applied to discrete systems such as Jacobi operators and orthogonal polynomials on the unit circle.

A.1 Floquet theory

The basic object in Floquet theory is the discriminant, which characterizes the values of λ for which the solutions to the eigenvalue problem defined by a Schrödinger(or Jacobi, etc.) operator are stable. Let Q be a Schrödingeroperator whose potential q(x) satisfies q(x) = q(x + P ); then, we seek solutions of Qf = λf satisfying f(x + P ) = mf(x) (A.1) for some multiplier m. This requires that m be an eigenvalue of the matrix y1(x, λ) y2(x, λ) A(x, λ) = 0 0 (A.2) y1(x, λ) y2(x, λ)

0 evaluated at x = P , where y1 and y2 are the fundamental solutions deﬁned by y1(0) = 1, y1(0) = 0 0 and y2(0) = 0, y2(0) = 1. The determinant det A is the Wronskian of the solutions y2(x, λ), y2(x, λ), so is constant, and evaluating at x = 0 shows that this constant is 1. So, the eigenvalues may be characterized entirely in terms of the trace of A. Hence, we deﬁne the the Floquet discriminant 1 1 ∆(λ) = tr A = (y (P, λ) + y0 (P, λ)) (A.3) 2 2 1 2 This determines the possible values of m:

m2 − 2∆(λ)m + 1 = 0 (A.4)

If |m|= 6 1, then solutions to the eigenvalue problem are unstable: they grow exponentially as x → ∞ or −∞. However, if |m| = 1, then solutions are stable.

54 Figure A.1: Floquet discriminant for q(x) = sin(x) + sin(2x) + sin(5x) + sin(7x)

2 2 The values of ∆(λ) divide R+ into bands where ∆ < 1 and |m| = 1 and gaps where ∆ > 1 and |m|= 6 1. The union of the bands is called the Bloch (or Bloch-Floquet) spectrum of the operator; the boundaries of the gaps are the periodic (resp. antiperiodic) spectrum where m = 1 (resp. m = −1). Generically, ∆(λ) is an oscillatory (but not periodic) function; critical points of this function are always local minima or maxima, and critical values always lie outside (−1, 1). Because of this, a critical point is always associated with a gap, but it is possible for the gap to be closed, if value of ∆(λ) there is 1 or −1. An example of a Floquet discriminant appears in figure A.1. In the algebrogeometric theory, the equation (A.4) defines an algebraic curve in C2; this curve is generically not compact, unless all but finitely many gaps are closed. We investigate two discrete systems for the application of Floquet theory: orthogonal polynomials on the real line (OPRL), which are related to Jacobi operators, and orthogonal polynomials on the unit circle (OPUC), which are related to a different recurrence relation which we will describe below.

A.1.1 Floquet discriminant for OPRL 2 Let H be a Jacobi operator on ` (Z≥0). To deﬁne the Floquet discriminant for OPRL, we rewrite the diﬀerence equation Hf − xf = 0 to the equivalent sequence of 2 × 2 equations: f(n + 1) f(n) = A(n) f(n) f(n − 1) where the matrix A(n) is 1 a − x b A(n) = n n−1 bn bn 0

55 The transfer matrix is T (n) = A(n − 1)A(n − 2) ...A(0) and can be written as

p (x) q (x) T (n) = n n (A.5) pn−1(x) qn−1(x) where pi(x) is a polynomial of degree i and qi(x) is a polynomial of degree i − 1. They are the result of solving the eigenvalue equation with initial conditions (1, 0)T and (0, 1)T respectively; the polynomials pn(x) are the polynomials orthogonal with respect to the spectral measure of the operator H, and the polynomials qn(x) are called the second kind polynomials. Then, in analogy to what we have described before, if the sequences an, bn are N-periodic, the Floquet discriminant can be deﬁned by ∆(x) = tr T (N) = pN (x) + qN−1(x). There is another description of the Floquet discriminant that makes its relationship to quasiperiodic solutions to the eigenvalue equation somewhat more transparent. Suppose that H is N-periodic, and consider the matrix

 −1  a1 b1 h bN  b1 a2 b2     .. ..   b2 . .  Hh =    ..   .     bN−1  hbN bN−1 aN where the parameter h is a multiplier analogous to m in (A.1) The sequences f(n) satisfying both Hf = xf and f(n + N) = hf(n) are those sequences whose restriction to 1,...,N are eigenvectors of Hh with eigenvalue x. In other words, we can assess whether such sequences exist by considering the characteristic polynomial of Hh:

−1 −1 F (x, h, h ) = det(Hh − xI) = B(h + h ) − P (x) (A.6) where B = Q b(i) and P (x) is a polynomial of degree N. The polynomial B−1P (x) is exactly the Floquet discriminant ∆(x) as deﬁned above. Therefore, the discriminant ∆(x) is connected to the existence of stable solutions to the eigenvalue equation in the usual way: whenever a sequence satisﬁes both Hf = xf and f(n + N) = hf(n), h and f satisfy

h + h−1 − ∆(x) = 0 and so h2 − ∆(x) + 1 = 0 Stable solutions–those with |h| = 1–exist whenever ∆2(x)−4 < 0; that is, whenever ∆2(x) ∈ (−2, 2). The set of x for which this is true forms the Bloch-Floquet spectrum for the periodic Jacobi operator. An example discriminant is shown in ﬁgure A.2. The qualitative form of the spectrum is now made clear: since ∆2(x) is a polynomial of degree N, the spectrum consists of at most N bands, all of which are contained in compact interval since for suﬃciently large |x|, |∆(x)| > 2. In analogy to the previous cases, it can be shown that the critical values of ∆(x) must lie outside the interval (−2, 2), so that ∆(x) is strictly increasing or decreasing on bands except in the circumstance where a gap closes to a point–that is, where ∆(x) = ±2 has a double root.

56 Figure A.2: Floquet discriminant for Jacobi operator with 10 randomly generated a coeﬃcients (bn ≡ 1).

A.1.2 Floquet discriminant for OPUC As in the OPRL case, we will formulate the discriminant in terms of transfer matrices. Recall the recurrence relation (5.2) and write it as a matrix equation φn+1(z) −1/2 z −α¯ φn(z) ∗ = ρn ∗ (A.7) φn+1(z) −αz 1 φn(z) | {z } An(z)

In terms of An(z; α), the jth transfer matrix is the product of the ﬁrst j A matrices, so that φn(z) 1 ∗ = Tj(z; α) (A.8) φn(z) 1

We will also use the family of second kind polynomials ψn(z) which are produced by applying the same transfer matrices to a diﬀerent initial condition: ψn(z) 1 ∗ = Tj(z; α) −ψn(z) −1

To define a Floquet discriminant, suppose the sequence of Verblunsky coefficients {αj} is periodic 1 with even period p; that is, αj+p = αj for each j. The Floquet discriminant for this sequence of coefficients is −p/2 ∆(z; α) = z tr Tp(z; α) We consider the discriminant as a function of z = eiθ ∈ S1 ⊂ C, or simply as a periodic function of θ ∈ R. An example discriminant, with p = 8, is shown in figure A.3. 1The assumption of an even period is a convenience, avoiding any complications from factors of zp/2. A description of how to proceed without this assumption may be found in [25].

57 Figure A.3: Floquet discriminant for OPUC with randomly generated period 8 sequence of Verblun- sky coeﬃcients

A.2 Inﬁnite-period limits

A detailed description of the inﬁnite-period limit procedure for Schr¨odingeroperators may be found in [12]. The idea is to begin with a compactly supported potential q(x) (for simplicity, we assume no discrete spectrum) and introduce the periodized operator

2 X QP = −∂x + q(x + nP ) n∈Z The Floquet discriminant of this operator may be described in terms of the scattering data of the original potential q(x), because the scattering relations allow the fundamental solutions to be evaluated exactly at x = P .2 Speciﬁcally, in the vicinity of x = 0, 1 y (x, k2) = [eikx + e−ikx] 1 2 1 y (x, k2) = [eikx − e−ikx] 2 2ik From this we can compute 1 1 2∆(k) = eikP + e−ikP s11(k) s11(k) In particular, 1 2∆(k) = cos(kP + α(k)) |s11(k)|

2The assumption of compact support may be relaxed; then this evaluation will not be exact, so the procedure requires more careful analysis.

58 Figure A.4: OPUC Floquet discriminant with 8 random α coeﬃcients, extended by zeros to period 88

So, the qualitative behavior of the discriminant as P → ∞ is rapid oscillation within an envelope determined by the transmission coeﬃcient s11(k).

A.2.1 Infinite-period limits for discrete systems The idea is quite similar for the discrete systems. We begin with a periodic sequence of coefficients and insert zeros3 between each period to extend the length, leaving the original finitely supported sequence otherwise unmodified. The result is similar as well: the behavior of the discriminant in the long-period limit is rapid oscillation within an envelope that does not depend on this period. The figures A.4 and A.5 show such envelopes (starting from the same sequences as in figs. A.3 and A.2). In the case of OPUC, we may also describe the envelope in terms of the orthogonal polynomials.

φ2n(z)+ψ2n(z) Proposition 16. Let s(z; α) = 2zn . Then for z ∈ ∂D ∆(z; α(p)) = s(z)zp + s(z)z−p = |s(z)| cos(qθ + α(z)) (A.9) where s(z) = |s(z)|eiα(z), z = eiθ.

Proof. A general transfer matrix Tj(z; α) has the form

Tj(z) = by inspection of its action on the vectors (1, 1)T and (1, −1)T . Therefore, the Floquet discriminant for a sequence of period 2n + 2p is 1 ∆(z; α) = φ (z) + ψ (z) + φ∗ (z) + ψ∗ (z) 2zn+p 2n+2p 2n+2p 2n+2p 2n+2p 3or 1s, in the case of the Jacobi b coeﬃcients

59 Figure A.5: OPRL Floquet discriminant with 10 random a coeﬃcients, extended by zeros to period 60

Specializing to the case of α(p), we observe that if α = 0, the recurrence matrix is z 0 A = 0 1

∗ ∗ 2p so that φj+1(z) = zφj(z), φj+1(z) = φj (z) (and similarly for ψ). As a result, φ2n+2p(z) = z φ2n(z) ∗ −p and φ2n+2p(z) = φ2n(z) (similarly for ψ). Distributing the factor of z across, we arrive at φ (z) + ψ (z) φ∗ (z) + ψ∗ (z) ∆(z; α) = 2n 2n zp + 2n 2n z−p 2zn 2zn

Finally, we observe that for z ∈ ∂D and any polynomial p(z) of degree 2n p∗(z) p(z) = zn zn which completes the proof.

A.3 Spike domains

In performing the inﬁnite-period limiting procedure, it is useful to transform the discriminant by the so-called spike map of Marcenko and Ostrovskii [17]. This maps the spectral parameter k into C as follows: id∆(λ) ω = = d(arccos(∆)) (A.10) p∆(λ)2 − 1

2 1 Z k 1 k 7→ ω = arccos(∆(k)) (A.11) P 0 P

60 Figure A.6: Sketch of the spike domain.

Under this map, spectral bands are mapped to the real line and normalized to width 2π/P , and spectral gaps are mapped to vertical “spikes” with varying height (see ﬁgure A.6 for a sketch [12]). The spike heights encode the same information as the envelope of the discriminant, but the normalization of spike positions makes the limiting analysis more straightforward. As with the un-transformed discriminant, extending the period toward ∞ leads the spike map to approach an envelope traced by the spike ends. Figures A.8 and A.10 show this for OPUC and OPRL discriminants.

A.4 Description of numerical methods

The numerical simulations that contributed to this research were implemented primarily in Julia, although some early computations used Python and C. The code used for this simulation can be found at the author’s GitHub website, https://www.github.com/mpdylan, under the following repositories: • Floquet.jl, which includes routines for computing quantities related to Floquet theory (e.g. Floquet discriminant) as well as the scattering-theoretic quantities (e.g. transmission coefficient) used in the continuum limit. • OPUC.jl, which computes orthogonal polynomials on the unit circle (and real line, contrary to the name) as well as several associated objects, such as approximations to the spectral measures. In Julia, polynomial computations were typically performed using the Polynomials package. This package implements a data type for polynomials with arbitrary coefficient type, and includes support for polynomial arithmetic as well as operations such as root-finding and evaluation. Despite the availability of this package, however, there are several applications for which more elementary computations were performed. Evaluation for high-degree orthogonal polynomials is typically more accurate if performed directly using the recurrence relation, because the coefficients of the polynomial tend to suffer from loss of significance faster than values explicitly calculated by the recurrence relation. Additionally, the use of automatic differentiation (described in section A.4 below) requires operations simple enough for the automatic differentiation package to compute the derivative by the chain rule, but passing through non-number data types such as the Polynomial type makes this impossible.

61 (a) OPUC Floquet discriminant with 10 random (b) Spike domain for the same discriminant coeﬃcients

Figure A.8: Discriminant and spike domain for OPUC Floquet discriminant, with extensions by zeros

62 (a) OPRL Floquet discriminant with 10 random (b) Spike domain for the same discriminant a coeﬃcients

Figure A.10: Discriminant and spike domain for OPRL Floquet discriminant, with extensions by zeros

63 The ApproxFun package, which performs function approximation by representing functions by their series expansion in one of several bases (e.g. Chebyshev polynomials, Fourier series, etc.), was used for performing the computations which produced the spike domain plots. Rather than working with the Floquet discriminant for OPUC directly, we compute a Chebyshev polynomial expansion of it (as a function of θ ∈ [0, 2π]) and work with this approximant. This allows for fast and efficient operations such as differentiation and root-finding, without having to rely on numerical methods such as finite differences. Typically these Chebyshev approximations are accurate to within machine precision and therefore may be regarded as being as ‘exact’ as any other numerical computations. We finish with a few details of how the methods had to be different for each of the three domains investigated.

Orthogonal polynomials on the unit circle Computing Floquet-theoretic quantities for orthogonal polynomials on the unit circle is the most straightforward, because the Floquet discriminant is a polynomial and its domain S1 is compact. As a result, the Chebyshev approximation is fast and accurate; no signiﬁcant computational diﬃculties were found in computing these discriminants or the spike map.

Orthogonal polynomials on the real line The real-line case presented a handful of technical challenges which did not appear in the OPUC case, primarily stemming from the fact that the spectral measure for orthogonal polynomials on the real line (or discrete Schrödingeroperators) is supported inside an interval, which has endpoints, rather than the unit circle, which does not. The difficulty is that the endpoints of the interval containing the spectrum are not known a priori, and if the period is long (as it is when approximating the continuum limit), the Floquet discriminant grows quite rapidly outside this interval. As a result, attempting to approximate the discriminant by, e.g., Chebyshev polynomials, is problematic because if the domain of approximation extends well outside of the interval containing the spectrum, the approximation scheme will typically privilege accuracy outside this interval (where errors are high in magnitude) over accuracy inside this interval (where the interesting features of the discriminant appear). There are two approaches we attempted to mitigate this problem. First, since the primary quantity needed in the computation of the spike map is the set of critical points of the Floquet discriminant, we can take an alternative approach to finding these. The Julia package ForwardDiff implements forward-mode automatic differentiation. Automatic differentiation is an approach to differentiation that, philosophically speaking, lies somewhere between the numerical method of finite differences and a completely symbolic approach. The core idea is that a computer function can often be broken down into a composition of simple arithmetic operations and elementary functions whose derivatives are known. Provided that each step is an operation whose derivative is known and implemented in the language, the computer can analyze the sequence of operations and by applying the chain rule produce a function which computes the derivative. Second, we can attempt to estimate the endpoints of the interval on which we wish to approximate the discriminant. This is an effective approach, since (A.6) indicates how to compute the periodic and antiperiodic spectrum for a Jacobi matrix as eigenvalues of a matrix. The eigenvalue problem can be solved accurately and efficiently, and then the maximum and minimum points in the combined set of periodic/antiperiodic spectrum give upper and lower bounds for the Bloch spectrum of the operator. Fattening the interval by a small ε (which may be tuned by hand) then provides a good domain of approximation.

Schrödingeroperators on R Since our primary interest is in the discrete systems, our numerical investigation of the Schrödinger setting is limited to the production of a few illustrative examples (e.g. figure A.1). This was conve-

64 nient, because operators on the line are perhaps the most computationally problematic domain on which to numerically simulate Floquet-theoretic quantities and their scattering limits. The reason for this is that the discriminant is defined in terms of solutions to an ordinary differential equation. While this is not an obstacle to the computation of the discriminant itself–the ODE can be solved quickly and accurately for a reasonably well-behaved potential–it does disable many of the tricks described in the previous sections. Automatic differentiation is impossible, since the computer is unable to explicitly determine the dependence of the solution to the ODE on the parameter λ. Furthermore, although there is no in-principle obstacle to the approximation of the discriminant by Chebyshev series as was used for OPUC, the ApproxFun package was not able to produce approximations for the discriminant–when attempting to construct the approximation to the discriminant (or any of several test functions defined in terms of the solution to an ODE, even a very simple one) the software simply hung until the operation was canceled.

65 Appendix B

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