Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials

We prove a family of dispersive estimates for the higher order Schrödinger equation $i{u}_t=(-\mathrm{\Delta }{)}^{m}u+Vu$ 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 $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{n}{2m}}$ bound mapping $L^1$ to $L^{\mathrm{\infty }}$ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator $(-\mathrm{\Delta }{)}^m+V$ and a family of dispersive estimates for operators of the form $\mathrm{\mid }H{\mathrm{\mid }}^{\beta -\frac{n}{2m}}{e}^{-itH}{P}_{ac}(H)$ for $0<\beta \le \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range.