Spectral theory of piecewise continuous functions of self-adjoint operators

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.