Scattering Matrix and Functions of Self-adjoint Operators

In the scattering theory framework, we consider a pair of operators H0, H. For a continuous function φ vanishing at infinity, we set φδ(·) = φ(·/δ) and study the spectrum of the difference φδ(H − λ)− φδ(H0 − λ) for δ → 0. We prove that if λ is in the absolutely continuous spectrum of H0 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(λ) for the pair H0, H and (ii) the singular values of the Hankel operator Hφ with the symbol φ.