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Jacobi Operators and Completely Integrable Nonlinear Lattices

Gerald Teschl Institut für Mathematik, Universität Wien, Vienna, Austria
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Hardcover ISBN: 978-0-8218-1940-1
Product Code: SURV/72
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Electronic ISBN: 978-1-4704-1299-9 Product Code: SURV/72.E List Price:$79.00
MAA Member Price: $71.10 AMS Member Price:$63.20
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List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Click above image for expanded view Jacobi Operators and Completely Integrable Nonlinear Lattices Gerald Teschl Institut für Mathematik, Universität Wien, Vienna, Austria Available Formats:  Hardcover ISBN: 978-0-8218-1940-1 Product Code: SURV/72  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
 Electronic ISBN: 978-1-4704-1299-9 Product Code: SURV/72.E
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$126.00 MAA Member Price: $113.40 AMS Member Price:$100.80
• Book Details

Mathematical Surveys and Monographs
Volume: 721999; 351 pp
MSC: Primary 39; 47; 58;

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.

Graduate students and research mathematicians interested in finite differences and functional equations; theoretical physicists.

• Chapters
• 1. Jacobi operators
• 2. Foundations of spectral theory for Jacobi operators
• 3. Qualitative theory of spectra
• 4. Oscillation theory
• 5. Random Jacobi operators
• 6. Trace formulas
• 7. Jacobi operators with periodic coefficients
• 8. Reflectionless Jacobi operators
• 9. Quasi-periodic Jacobi operators and Riemann theta functions
• 10. Scattering theory
• 11. Spectral deformations – Commutation methods
• 12. The Toda system
• 13. The initial value problem for the Toda system
• 14. The Kac-van Moerbeke system

• 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
• Request Review Copy
• Get Permissions
Volume: 721999; 351 pp
MSC: Primary 39; 47; 58;

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.

Graduate students and research mathematicians interested in finite differences and functional equations; theoretical physicists.

• Chapters
• 1. Jacobi operators
• 2. Foundations of spectral theory for Jacobi operators
• 3. Qualitative theory of spectra
• 4. Oscillation theory
• 5. Random Jacobi operators
• 6. Trace formulas
• 7. Jacobi operators with periodic coefficients
• 8. Reflectionless Jacobi operators
• 9. Quasi-periodic Jacobi operators and Riemann theta functions
• 10. Scattering theory
• 11. Spectral deformations – Commutation methods
• 12. The Toda system
• 13. The initial value problem for the Toda system
• 14. The Kac-van Moerbeke system
• [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
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