Dispersive estimates for the Schrödinger equation in dimensions four and five

We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrödinger group e^it(−Δ+V) for a class of real-valued potentials V∈C^k(Rⁿ), V(x)=O(〈x〉^−δ), where n=4, 5, k>(n−3)/2, δ>3 if n=4 and δ>5 if n=5.