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 $\phi_\delta(\cdot)=\phi(\cdot/\delta)$ and study the spectrum of the difference $\phi_\delta(H-\lambda)-\phi_\delta(H_0-\lambda)$ 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 $H_\phi$ with the symbol $\phi$.