Pointwise estimates for the fundamental solutions of higher order Schr\"{o}dinger equations in odd dimensions II: high dimensional case

In this paper, for any odd $n$ and any integer $m\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schr\"{o}dinger equation \begin{equation*} \mathrm{i}\partial_tu(x,t)=((-\Delta)^m+V(x))u(x,t),\quad t\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is a real-valued $C^{\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-\Delta)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K(t,x,y)$ satisfying \begin{equation*} |K(t, x,y)|\le C(1+|t|)^{-(\frac{n}{2m}-\sigma)}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, \end{equation*} where $\sigma=2$ if $0$ is an eigenvalue of $H$, and $\sigma=0$ otherwise. A similar result for smoothing operators $H^\frac{\alpha}{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. The regularity condition $V\in C^{\frac{n+1}{2}-2m}$ is optimal in the second order case, and it also seems optimal when $m>1$.