Commutation Methods for Jacobi Operators

We offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded background Jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. In addition we compute the (matrix-valued) Weyl m-functions of the commuted operator in terms of the background Weyl m-functions. Finally we show how to iterate the above methods and give explicit formulas for various quantities (such as eigenfunctions and spectra) of the iterated operators in terms of the corresponding background quantities and scattering matrix. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and the N -soliton solutions of the Toda and Kac-van Moerbeke lattice equations with respect to arbitrary background solutions.