  We consider the difference $f(H_1)-f(H_0)$, where $H_0=-\Delta$ and
$H_1=-\Delta+V$ are the free and the perturbed Schr\"odinger operators in
$L^2(\mathbb R^d)$, and $V$ is a real-valued short range potential. We give a
sharp sufficient condition for this difference to belong to a given Schatten
class $\mathbf S_p$, depending on the rate of decay of the potential and on the
smoothness of $f$ (stated in terms of the membership in a Besov class). In
particular, for $p>1$ we allow for some unbounded functions $f$.
