Title: Pointwise estimates for the fundamental solution of higher order Schrödinger equation in odd dimensions

Abstract: In this paper, for any odd $n$ and any integer $m\geq1$, 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 potential with certain decay, smoothness, and spectral properties. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-\Delta)^m+V$. 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|)^{-h}(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 the constants $C, h>0$, and $h$ can be specified by $m, n$ and the spectral property of $H$.