Jacobi operators and completely integrable nonlinear lattices / Gerald TeschlType de document : MonographieCollection : Mathematical surveys and monographs, 72Langue : anglais.Pays: Etats Unis.Éditeur : Providence : American Mathematical Society, 2000Description : 1 vol. (XVII-351 p.) : ill. ; 26 cmISBN: 0821819402.ISSN: 0885-4653.Bibliographie : Bibliogr. p. 333-342. Index.Sujet MSC : 39A70, Difference equations, Difference operators
35C05, Representations of solutions to partial differential equations, Solutions to PDEs in closed form
37K60, Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems, Lattice dynamics; integrable lattice equations
39A12, Difference equations, Discrete version of topics in analysis
47B36, Operator theory - Special classes of linear operators, Jacobi (tridiagonal) operators and generalizationsEn-ligne : Zentralblatt | MathScinet | AMS Item type: Monographie
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ISSN 0076-5376 figurant sur le document correspond à Mathematical surveys (1943-1981). A partir de 1984 le titre est Mathematical surveys and monographs avec ISSN 0885-4653
Bibliogr. p. 333-342. Index
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. (source : AMS)