Spectral theory of piecewise continuous functions of self-adjoint operators

Let H₀, 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 (a.c.) spectrum of the operator D_θ=θ(H)-θ(H₀) for piecewise continuous functions θ. This description involves the scattering matrix for the pair H₀, H, evaluated at the discontinuities of θ. We also prove that the singular continuous spectrum of D_θ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the 'thresholds' of the a.c. spectrum of D_θ. Our approach relies on the construction of 'model' operators for each jump of the function θ. These model operators are defined as certain symmetrized Hankel operators (SHOs) which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator θ(H)-θ(H₀). As a by-product of our approach, we also construct the scattering theory for general SHOs with piecewise continuous symbols.