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 \varphi vanishing at infinity, we set \varphi_\delta(\cdot)=\varphi(\cdot/\delta) and study the spectrum of the difference \varphi_\delta(H-\lambda)-\varphi_\delta(H_0-\lambda) for \delta\to0 . We prove that if \lambda is in the absolutely continuous spectrum of H_0 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_\varphi with the symbol \varphi .