**Mathematical Surveys and Monographs**

Volume: 72;
1999;
351 pp;
Hardcover

MSC: Primary 39; 47; 58;

Print ISBN: 978-0-8218-1940-1

Product Code: SURV/72

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

**Electronic ISBN: 978-1-4704-1299-9
Product Code: SURV/72.E**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

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#### Supplemental Materials

# Jacobi Operators and Completely Integrable Nonlinear Lattices

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*Gerald Teschl*

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.

#### Readership

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

#### Reviews & Endorsements

[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

# Table of Contents

## Jacobi Operators and Completely Integrable Nonlinear Lattices

- Contents ix10 free
- Preface xiii14 free
- Part 1. Jacobi Operators 120 free
- Chapter 1. Jacobi operators 322
- Chapter 2. Foundations of spectral theory for Jacobi operators 2746
- Chapter 3. Qualitative theory of spectra 5776
- Chapter 4. Oscillation theory 7392
- Chapter 5. Random Jacobi operators 87106
- Chapter 6. Trace formulas 105124
- Chapter 7. Jacobi operators with periodic coefficients 115134
- Chapter 8. Reflectionless Jacobi operators 133152
- Chapter 9. Quasi-periodic Jacobi operators and Riemann theta functions 153172
- Chapter 10. Scattering theory 167186
- Chapter 11. Spectral deformations – Commutation methods 185204
- 11.1. Commuting first order difference expressions 185204
- 11.2. The single commutation method 187206
- 11.3. Iteration of the single commutation method 191210
- 11.4. Application of the single commutation method 194213
- 11.5. A formal second commutation 196215
- 11.6. The double commutation method 198217
- 11.7. Iteration of the double commutation method 204223
- 11.8. The Dirichlet deformation method 207226

- Notes on literature 215234

- Part 2. Completely Integrable Nonlinear Lattices 219238
- Chapter 12. The Toda system 221240
- Chapter 13. The initial value problem for the Toda system 239258
- 13.1. Finite-gap solutions of the Toda hierarchy 239258
- 13.2. Quasi-periodic finite-gap solutions and the time-dependent Baker-Akhiezer function 245264
- 13.3. A simple example – continued 248267
- 13.4. Inverse scattering transform 249268
- 13.5. Some additions in case of the Toda lattice 252271
- 13.6. The elliptic case – continued 254273

- Chapter 14. The Kac-van Moerbeke system 255274
- 14.1. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy 255274
- 14.2. Kac and van Moerbeke's original equations 261280
- 14.3. Spectral theory for supersymmetric Dirac-type difference operators 262281
- 14.4. Associated solutions 263282
- 14.5. N-soliton solutions on arbitrary background 265284

- Notes on literature 271290

- Appendix A. Compact Riemann surfaces – a review 273292
- Appendix B. Herglotz functions 297316
- Appendix C. Jacobi Difference Equations with Mathematica 307326
- C.1. The package DiffEqs and first order difference equations 307326
- C.2. The package JacDEqs and Jacobi difference equations 311330
- C.3. Simple properties of Jacobi difference equations 313332
- C.4. Orthogonal Polynomials 317336
- C.5. Recursions 319338
- C.6. Commutation methods 320339
- C.7. Toda lattice 326345
- C.8. Kac-van Moerbeke lattice 328347

- Bibliography 333352
- Glossary of notations 343362
- Index 349368