  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 $\phi$.
