and

Completely Integrable Nonlinear Lattices

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.

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.

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.

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.

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

- 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.