Commutation methods for Jacobi operators
type: Item
instance of: scholarly article
title: Commutation methods for Jacobi operators
zbMATH Open document ID: 0854.34079
published in: Journal of Differential Equations
publication date: 1996-08-28T00:00:00Z
review text: The authors observe that a complete spectral characterization of both the single and double commutation methods in the difference operator context is lacking in the literature and there seem to be no treatment of general backgrounds. In the present paper, the authors attempt to fill these gaps and provide a complete spectral characterization of the single commutation method and develop the corresponding results for the double commutation method. The paper starts with an informal discussion of commutation methods. It is followed by a detailed investigation of the single commutation method and its iteration. Next, it provides a complete characterization of the double commutation method for Jacobi operators. The authors prove unitary equivalence of commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with original background operators. Further the connection between the Weyl-Titchmarsh theory and the double commutation method is discussed. Finally, the authors demonstrate how to iterate the double commutation method and give explicit formulas for various quantities (such as eigenfunctions and spectra) of the iterated operators in terms of the background quantities and scattering matrix. As applications, the paper includes an explicit realization of the isospectral torus for algebra-geometric finite-gap Jacobi operators and the N-soliton solution of the Toda and Kac-van Moerbeke lattice equations with respect to arbitrary background solutions. The paper is supplemented with appendices giving the formulas for Jacobi operators and some results about Weyl-Titchmarsh theory for Jacobi operators which are quoted in the text.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 34B20
zbMATH DE Number: 895811
zbMATH Keywords: complete spectral characterization
zbMATH Keywords: single and double commutation methods
zbMATH Keywords: Jacobi operators
zbMATH Keywords: Weyl-Titchmarsh theory
zbMATH Keywords: Kac-van Moerbeke lattice equations
author: Fritz Gesztesy
author: Gerald Teschl
MaRDI profile type: Publication
OpenAlex ID: W2032984681
Wikidata QID: Q57343978
DOI: 10.1006/JDEQ.1996.0095
description: scientific article; zbMATH DE number 895811