Commutation methods for Jacobi operators
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.
scientific article; zbMATH DE number 895811