In the scattering theory framework, we consider a pair of operators , . For a continuous function vanishing at infinity, we set and study the spectrum of the difference for . We prove that if is in the absolutely continuous spectrum of and , then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix for the pair , and (ii) the singular values of the Hankel operator with the symbol .
Cite this article
Alexander Pushnitski, Scattering matrix and functions of self-adjoint operators. J. Spectr. Theory 1 (2011), no. 2, pp. 221–236DOI 10.4171/JST/10