  We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$
and $H_1$ acting in a Hilbert space. We establish a new class of estimates for
the operator norm and the Schatten class norms of this difference. Our
estimates utilise ideas of scattering theory and involve conditions on $H_0$
and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of
functions $f$ (including some unbounded ones) than previously available results
do. As an important technical tool, we propose a new notion of Schatten class
valued smoothness and develop a new framework for double operator integrals.
