High frequency dispersive estimates for the Schrodinger equation in high dimensions

We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^{-\delta })$, $\delta >n-1$, and $V\in C^k({\mathbf{R}}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)\mathrm{/}2\le k_n<n\mathrm{/}2$. We also give a sufficient condition in terms of $L^1\to L^{\mathrm{\infty }}$ bounds for the formal iterations of Duhamel’s formula, which might be satisfied for potentials of less regularity.