We prove a family of dispersive estimates for the higher order Schrödinger equation \(iu_t=(-\Delta )^mu +Vu\) in n spatial dimensions, for \(m\in \mathbb {N}\) with \(m>1\) and \(2m<n<4m\) . Here V is a real-valued potential belonging to the closure of \(C_0\) functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that \(e^{-itH}P_{ac}(H)\) satisfies a \(|t|^{-\frac{n}{2m}}\) bound mapping \(L^1\) to \(L^\infty \) by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator \((-\Delta )^m +V\) and a family of dispersive estimates for operators of the form \(|H|^{\beta -\frac{n}{2m}}e^{-itH}P_{ac}(H)\) for \(0<\beta \le \frac{n}{2}\) . The results apply in both even and odd dimensions in the allowed range.
