Additions for Jacobi Operators and the Toda Hierarchy of Lattice Equations Item Type text; Electronic Dissertation Authors Murphy, Dylan Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 30/09/2021 03:36:23 Link to Item http://hdl.handle.net/10150/636509 ADDITIONS FOR JACOBI OPERATORS AND THE TODA HIERARCHY OF LATTICE EQUATIONS by Dylan Murphy Copyright c Dylan Murphy 2019 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 1 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 Schr¨odingeroperators 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 Schr¨odingeroperators . 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 Schr¨odingeroperators 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 eigen- functions 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 Schr¨odingeroperator, 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 Schr¨odingeroperators, 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 ff(n)gn2Z by Hf(n) = bnf(n + 1) + anf(n) + bn−1f(n − 1) (1.1) 1 where fangn2Z, fbngn2Z 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 Schr¨odingeroperators. 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 commu- tation 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.