Schatten class conditions for functions of Schrödinger operators

We consider the difference f(H₁)−f(H₀), where H₀ = −Δ and H₁ = −Δ+V are the free and the perturbed Schrödinger operators in L²(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 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.