Jacobi operators and completely integrable nonlinear lattices

Gerald Teschl0821819402, 9780821819401

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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Table of contents :
Preface……Page 9
Part 1. Jacobi Operators……Page 15
1.1. General properties……Page 17
1.2. Jacobi operators……Page 27
1.3. A simple example……Page 33
1.4. General second order difference expressions……Page 35
1.5. The infinite harmonic crystal in one dimension……Page 36
2.1. Weyl m-functions……Page 41
2.2. Properties of solutions……Page 43
2.3. Positive solutions……Page 48
2.4. Weyl circles……Page 51
2.5. Canonical forms of Jacobi operators and the moment problem……Page 54
2.6. Some remarks on unbounded operators……Page 61
2.7. Inverse spectral theory……Page 68
3.1. Locating the spectrum and spectral multiplicities……Page 73
3.2. Locating the essential spectrum……Page 76
3.3. Locating the absolutely continuous spectrum……Page 79
3.4. A bound on the number of eigenvalues……Page 86
4.1. Prüfer variables and Sturm’s separation theorem……Page 89
4.2. Classical oscillation theory……Page 94
4.3. Renormalized oscillation theory……Page 97
5.1. Random Jacobi operators……Page 101
5.2. The Lyapunov exponent and the density of states……Page 105
5.3. Almost periodic Jacobi operators……Page 114
6.1. Asymptotic expansions……Page 119
6.2. General trace formulas and xi functions……Page 123
7.1. Floquet theory……Page 129
7.2. Connections with the spectra of finite Jacobi operators……Page 133
7.3. Polynomial identities……Page 137
7.4. Two examples: period one and two……Page 138
7.5. Perturbations of periodic operators……Page 140
8.1. Spectral analysis and trace formulas……Page 147
8.2. Isospectral operators……Page 154
8.3. The finite-gap case……Page 156
8.4. Further spectral interpretation……Page 164
9.1. Riemann surfaces……Page 167
9.2. Solutions in terms of theta functions……Page 169
9.3. The elliptic case, genus one……Page 177
9.4. Some illustrations of the Riemann-Roch theorem……Page 179
10.1. Transformation operators……Page 181
10.2. The scattering matrix……Page 185
10.3. The Gel’fand-Levitan-Marchenko equations……Page 189
10.4. Inverse scattering theory……Page 194
11.1. Commuting first order difference expressions……Page 201
11.2. The single commutation method……Page 203
11.3. Iteration of the single commutation method……Page 207
11.4. Application of the single commutation method……Page 210
11.5. A formal second commutation……Page 212
11.6. The double commutation method……Page 214
11.7. Iteration of the double commutation method……Page 220
11.8. The Dirichlet deformation method……Page 223
Notes on literature……Page 230
Part 2. Completely Integrable Nonlinear Lattices……Page 235
12.1. The Toda lattice……Page 237
12.2. Lax pairs, the Toda hierarchy, and hyperelliptic curves……Page 240
12.3. Stationary solutions……Page 247
12.4. Time evolution of associated quantities……Page 250
13.1. Finite-gap solutions of the Toda hierarchy……Page 255
13.2. Quasi-periodic finite-gap solutions and the time-dependent Baker-Akhiezer function……Page 261
13.3. A simple example — continued……Page 264
13.4. Inverse scattering transform……Page 265
13.5. Some additions in case of the Toda lattice……Page 268
13.6. The elliptic case — continued……Page 270
14.1. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy……Page 271
14.2. Kac and van Moerbeke’s original equations……Page 277
14.3. Spectral theory for supersymmetric Dirac-type difference operators……Page 278
14.4. Associated solutions……Page 279
14.5. N-soliton solutions on arbitrary background……Page 281
Notes on literature……Page 286
A.1. Basic notation……Page 289
A.2. Abelian differentials……Page 291
A.3. Divisors and the Riemann-Roch theorem……Page 293
A.4. Jacobian variety and Abel’s map……Page 297
A.5. Riemann’s theta function……Page 300
A.6. The zeros of the Riemann theta function……Page 302
A.7. Hyperelliptic Riemann surfaces……Page 305
Appendix B. Herglotz functions……Page 313
C.1. The package DiffEqs and first order difference equations……Page 323
C.2. The package JacDEqs and Jacobi difference equations……Page 327
C.3. Simple properties of Jacobi difference equations……Page 329
C.4. Orthogonal Polynomials……Page 333
C.5. Recursions……Page 335
C.6. Commutation methods……Page 336
C.7. Toda lattice……Page 342
Bibliography……Page 345
Glossary of notations……Page 349
Index……Page 361