Scattering matrix and functions of self-adjoint operators

In the scattering theory framework, we consider a pair of operators H-0, H. For a continuous function phi vanishing at infinity, we set phi delta(center dot) = phi(center dot/delta) and study the spectrum of the difference phi delta(H - lambda) - phi delta(H-0 - lambda) for delta -> 0. We prove that if lambda is in the absolutely continuous spectrum of H-O and H, 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 S(lambda) for the pair H-0, H and (ii) the singular values of the Hankel operator H-phi with the symbol phi.