Mathematical Surveys and Monographs 1999; 351 pp; hardcover Volume: 72 ISBN10: 0821819402 ISBN13: 9780821819401 List Price: US$75 Member Price: US$60 Order Code: SURV/72
 This volume can serve as an introduction and a reference source on spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of those theories to the Toda and Kacvan Moerbeke hierarchy. Beginning with second order difference equations, the author develops discrete WeylTitchmarshKodaira theory, covering all classical aspects, such as Weyl \(m\)functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Teschl then investigates more advanced topics, such as locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi)periodic operators, scattering theory, and spectral deformations. Utilizing the Lax approach, he introduces the Toda hierarchy and its modified counterpart, the Kacvan Moerbeke hierarchy. Uniqueness and existence theorems for solutions, expressions for solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are derived. This text covers all basic topics of Jacobi operators and includes recent advances. It is suitable for use as a text at the advanced graduate level. ® Mathematica is a registered trademark of Wolfram Research Inc. Readership Graduate students and research mathematicians interested in finite differences and functional equations; theoretical physicists. Reviews "[The author] does an admirable job of bringing out the ideas of the subject without getting lost in the details. This is certainly an important reference for the researcher in integrable lattices."  Mathematical Reviews Table of Contents Jacobi operators  Jacobi operators
 Foundations of spectral theory for Jacobi operators
 Qualitative theory of spectra
 Oscillation theory
 Random Jacobi operators
 Trace formulas
 Jacobi operators with periodic coefficients
 Reflectionless Jacobi operators
 Quasiperiodic Jacobi operators and Riemann theta functions
 Scattering theory
 Spectral deformationsCommutation methods
Completely integrable nonlinear lattices  The Toda system
 The initial value problem for the Toda system
 The Kacvan Moerbeke system
 Notes on literature
 Compact Riemann surfacesA review
 Herglotz functions
 Jacobi difference equations with Mathematica®
 Bibliography
 Glossary of notations
 Index
