Article

**J. Differential Equations 128, 252-299 (1996)**[DOI: 10.1006/jdeq.1996.0095]

## Commutation Methods for Jacobi Operators

### Fritz Gesztesy and Gerald Teschl

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.
** MSC91:** Primary 47B39, 34B20; Secondary 35Q51, 35Q58

**Keywords:** *Commutation methods, Jacobi operators, eigenvalues, solitons*

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