High frequency dispersive estimates for the Schrödinger equation in high dimensions

We prove optimal dispersive estimates at high frequency for the Schrödinger group for a class of real-valued potentials V(x)=O(〈x〉^−δ), δ>n−1, and V∈C^k(Rⁿ), k>k_n, where n≥4 and (n−3)/2≤k_n<n/2. We also give a sufficient condition in terms of L¹→L^∞ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.