Jacobi operators

Jacobi Operators
and
Completely Integrable Nonlinear Lattices

Gerald Teschl

The name of the game

Jacobi operators appear in a variety of applications. They can be viewed as the discrete analogue of Sturm-Liouville operators and their investigation has many similarities with Sturm-Liouville theory. Spectral and inverse spectral theory for Jacobi operators play a fundamental role in the investigation of completely integrable nonlinear lattices, in particular the Toda lattice and its modified counterpart, the Kac-van Moerbeke lattice.

Why I have written this book

Whereas numerous books about Sturm-Liouville operators have been written, only few on Jacobi operators exist. In particular, there is currently no monograph available which covers all basic topics (like spectral and inverse spectral theory, scattering theory, oscillation theory and positive solutions, (quasi-)periodic operators, spectral deformations, etc.) typically found in textbooks on Sturm-Liouville operators. In the case of the Toda lattice a textbook by M. Toda [6] exists, but none of the recent advances in the theory of nonlinear lattices are covered there.

Audience and prerequisites

As audience I had researchers in mind. This book can be used to get acquainted with selected topics as well as to look up specific results. Nevertheless, no previous knowledge on difference equations is assumed and all results are derived in a self-contained manner. Hence the present book is accessible to graduate students as well. Previous experience with Sturm-Liouville operators might be helpful but is not necessary. Still, a solid working knowledge from other branches of mathematics is needed. In particular, I have assumed that the reader is familiar with the theory of (linear) self-adjoint operators in Hilbert spaces which can be found in (e.g.) [5] or [7]. This theory is heavily used in the first part. In addition, the reader might have to review material from complex analysis (see Appendix A and B) and differential equations on Banach manifolds (second part only) if (s)he feels (s)he doesn't have the necessary background. However, this knowledge is mainly needed for understanding proofs rather than the results themselves.

The style of this book

The style of this monograph is strongly influenced by my personal bias. I have striven to present an intuitive approach to each subject and to include the simplest possible proof for every result. Most proofs are rather sketchy in character, so that the main idea becomes clear instead of being drowned by technicalities. Nevertheless, I have always tried to include enough information for the reader to fill in the remaining details (her)himself if desired. To help researchers, using this monograph as a reference, to quickly spot the result they are looking for, most information is found in display style formulas.

The entire treatment is supposed to be mathematically rigorous. I have tried to prove {\em every} statement I make and, in particular, these little obvious things, which turn out less obvious once one tries to prove them. In this respect I had Marchenko's monograph on Sturm-Liouville operators [4] and the one by Weidmann [7] on functional analysis in mind.

Literature

The first two chapters are of an introductory nature and collect some well-known "folklore", the successive more advanced chapters are a synthesis of results from research papers. In most cases I have rearranged the material, streamlined proofs, and added further facts which are not published elsewhere. All results appear without special attribution to who first obtained them but there is a section entitled "Notes on literature" in each part which contains references to the literature plus hints for additional reading. The bibliography is selective and far from being complete. It contains mainly references I (am aware of and which I) have actually used in the process of writing this book.

Terminology and notation

For the most part, the terminology used agrees with generally accepted usage. Whenever possible, I have tried to preserve original notation. Unfortunately I had to break with this policy at various points, since I have given higher priority to a consistent (and self-explaining) notation throughout the entire monograph. A glossary of notation can be found towards the end.

For convenience of the reader, I have split the material into two parts; one on Jacobi operators and one on completely integrable lattices. In particular, the second part is to a wide extent independent of the first one and anybody interested only in completely integrable lattices can move directly to the second part (after browsing Chapter 1 to get acquainted with the notation).

Part I

Chapter 1 gives an introduction to the theory of second order difference equations and bounded Jacobi operators. All basic notations and properties are presented here. In addition, this chapter provides several easy but extremely helpful gadgets. We investigate the case of constant coefficients and, as a motivation for the reader, the infinite harmonic crystal in one dimension is discussed to some extent.

Chapter 2 establishes the pillars of spectral and inverse spectral theory for Jacobi operators. Here we develop what is known as discrete Weyl-Titchmarsh-Kodaira theory. Basic things like eigenfunction expansions, connections with the moment problem, and important properties of solutions of the Jacobi equation are shown in this chapter.

Chapter 3 considers qualitative theory of spectra. It is shown how the essential, absolutely continuous, and point spectrum of specific Jacobi operators can be located in some cases. The connection between existence of alpha-subordinate solutions and alpha-continuity of spectral measures is discussed. In addition, we investigate under which conditions the number of discrete eigenvalues is finite.

Chapter 4 covers discrete Sturm-Liouville theory. Both classical oscillation and renormalized oscillation theory are developed.

Chapter 5 gives an introduction to the theory of random Jacobi operators. Since there are monographs ([1]) devoted entirely to this topic only basic results on the spectra and some applications to almost periodic operators are presented.

Chapter 6 deals with trace formulas and asymptotic expansions which play a fundamental role in inverse spectral theory. In some sense this can be viewed as an application of Krein's spectral shift theory to Jacobi operators. In particular, the tools developed here will lead to a powerful reconstruction procedure from spectral data for reflectionless (e.g., periodic) operators in Chapter 8.

Chapter 7 considers the special class of operators with periodic coefficients. This class is of particular interest in the one-dimensional crystal model and several profound results are obtained using Floquet theory. In addition, the case of impurities in one-dimensional crystals (i.e., perturbation of periodic operators) is studied.

Chapter 8 again considers a special class of Jacobi operators, namely reflectionless ones, which exhibit an algebraic structure similar to periodic operators. Moreover, this class will show up again in Chapter 12 as the stationary solutions of the Toda equations.

Chapter 9 shows how reflectionless operators with no eigenvalues (which turn out to be associated with quasi-periodic coefficients) can be expressed in terms of Riemann theta functions. These results will be used in Chapter 13 to compute explicit formulas for solutions of the Toda equations in terms of Riemann theta functions.

Chapter 10 provides a comprehensive treatment of (inverse) scattering theory for Jacobi operators with constant background. All important objects like reflection/transmission coefficients, Jost solutions and Gel'fand-Levitan-Marchenko equations are considered. Again this applies to impurities in one-dimensional crystals. Furthermore, this chapter forms the main ingredient of the inverse scattering transform for the Toda equations.

Chapter 11 tries to deform the spectra of Jacobi operators in certain ways. We compute isospectral transformations and transformations which insert a finite number of eigenvalues. The standard transformations like single, double, or Dirichlet commutation methods are developed. These transformations can be used as powerful tools in inverse spectral theory and they allow us to compute new solutions from old solutions of the Toda and Kac-van Moerbeke equations in Chapter 14.

Part II

Chapter 12 is the first chapter on integrable lattices and introduces the Toda system as hierarchy of evolution equations associated with the Jacobi operator via the standard Lax approach. Moreover, the basic (global) existence and uniqueness theorem for solutions of the initial value problem is proven. Finally, the stationary hierarchy is investigated and the Burchnall-Chaundy polynomial computed.

Chapter 13 studies various aspects of the initial value problem. Explicit formulas in case of reflectionless (e.g., (quasi-)periodic) initial conditions are given in terms of polynomials and Riemann theta functions. Moreover, the inverse scattering transform is established.

The final Chapter 14 introduces the Kac van-Moerbeke hierarchy as modified counterpart of the Toda hierarchy. Again the Lax approach is used to establish the basic (global) existence and uniqueness theorem for solutions of the initial value problem. Finally, its connection with the Toda hierarchy via a Miura-type transformation is studied and used to compute N-soliton solutions on arbitrary background.

Appendix

The first appendix reviews the theory of Riemann surfaces as needed in this monograph. While most of us will know Riemann surfaces from a basic course on complex analysis or algebraic geometry, this will be mainly from an abstract viewpoint like in [2] or [3], respectively. Here we will need a more "computational" approach and I hope that the reader can extract this knowledge from this Appendix A.

The second appendix compiles some relevant results from the theory of Herglotz functions. Since not everybody is familiar with them, they are included for easy reference.

The final appendix shows how a program for symbolic computation, Mathematica, can be used to do some of the computations encountered during the main bulk (see Jacobi equations with Mathematica). While I don't believe that programs for symbolic computations are an indispensable tool for doing research on Jacobi operators (or completely integrable lattices), they are at least useful for checking formulas.

Acknowledgments

This book has greatly profited from collaborations and discussions with W. Bulla, F. Gesztesy, H. Holden, M. Krishna, and B. Simon. In addition, many people generously devoted considerable time and effort to reading earlier versions of the manuscript and making many corrections. In particular, I wish to thank D. Damanik, H. Hanßmann, A. von der Heyden, R. Killip, T. Sørensen, S. Timischl, K. Unterkofler, and H. Widom. Next, I am happy to express my gratitude to P. Deift, J. Geronimo, and E. Lieb for helpful suggestions and advise. I also like to thank the staff at the American Mathematical Society for the fast and professional production of this book.

Partly supported by the Austrian Science Fund under Grant No. P12864-MAT.

Finally, no book is free of errors. So if you find one, or if you have comments or suggestions, please let me know. I will make all corrections and complements available at the errata page.

Gerald Teschl
Vienna, Austria
May, 1999

References

R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
O. Forster, Lectures on Riemann Surfaces, Springer, New York, 1991.
R. Hartshorne, Algebraic Geometry, Springer, Berlin, 1977.
V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, rev. and enl. edition, Academic Press, San Diego, 1980.
M. Toda, Theory of Nonlinear Lattices, 2nd enl. edition, Springer, Berlin, 1989.
J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.