  Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard
assumptions of the smooth version of scattering theory hold true. We give an
explicit description of the absolutely continuous spectrum of the operator
$\mathcal{D}_\theta=\theta(H)-\theta(H_0)$ for piecewise continuous functions
$\theta$. This description involves the scattering matrix for the pair $H_0$,
$H$, evaluated at the discontinuities of $\theta$. We also prove that the
singular continuous spectrum of $\mathcal{D}_\theta$ is empty and that the
eigenvalues of this operator have finite multiplicities and may accumulate only
to the "thresholds" of the absolutely continuous spectrum of
$\mathcal{D}_\theta$. Our approach relies on the construction of "model"
operators for each jump of the function $\theta$. These model operators are
defined as certain symmetrised Hankel operators which admit explicit spectral
analysis. We develop the multichannel scattering theory for the set of model
operators and the operator $\theta(H)-\theta(H_0)$. As a by-product of our
approach, we also construct the scattering theory for general symmetrised
Hankel operators with piecewise continuous symbols.
