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Jacobi operators and completely integrable nonlinear lattices. (English) Zbl 1056.39029
Mathematical Surveys and Monographs 72. Providence, RI: American Mathematical Society (ISBN 0-8218-1940-2/hbk). xvii, 351 p. (2000).
Summary: 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. The author 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.

MSC:
39A70 Difference operators
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
35Q58 Other completely integrable PDE (MSC2000)
37K60 Lattice dynamics; integrable lattice equations
39A12 Discrete version of topics in analysis
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
82B23 Exactly solvable models; Bethe ansatz
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