Preface xv 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 equa- tions and bounded Jacobi operators. All basic notations and properties are pre- sented 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. 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 locat- ed in some cases. The connection between existence of ce-subordinate solutions and a-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 (e.g., [40]) 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 renectionless (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 sev- eral 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 renec- tionless ones, which exhibit an algebraic structure similar to periodic operators. Moreover, this class will show up again in Chapter 10 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 theo- ry for Jacobi operators with constant background. All important objects like re- flection/transmission coefficients, Jost solutions and Gel'fand-Levitan-Marchenko

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