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Jacobi Operators and Completely Integrable Nonlinear Lattices
Gerald Teschl, Institut für Mathematik, Universität Wien, Vienna, Austria
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Mathematical Surveys and Monographs
1999; 351 pp; hardcover
Volume: 72
ISBN-10: 0-8218-1940-2
ISBN-13: 978-0-8218-1940-1
List Price: US$79 Member Price: US$63.20
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 Kac-van Moerbeke hierarchy.

Beginning with second order difference equations, the author develops discrete Weyl-Titchmarsh-Kodaira 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 Kac-van 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.

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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

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
• Quasi-periodic Jacobi operators and Riemann theta functions
• Scattering theory
• Spectral deformations-Commutation methods
Completely integrable nonlinear lattices
• The Toda system
• The initial value problem for the Toda system
• The Kac-van Moerbeke system
• Notes on literature
• Compact Riemann surfaces-A review
• Herglotz functions
• Jacobi difference equations with Mathematica®
• Bibliography
• Glossary of notations
• Index