Dispersive estimates for higher order Schr\"odinger operators with scaling-critical potentials

Abstract

We prove a family of dispersive estimates for the higher order Schr\"odinger equation $iu_t=(-\Delta)^mu +Vu$ 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 $0<\beta\leq \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range.