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In this paper, for any odd $n$ and any integer $m\geq1$ with $n<4m$, we studythe 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), \quadt\in \mathbb{R},\,\,x\in \mathbb{R}^n, \end{equation*} where $V$ is areal-valued potential with certain decay. Let $P_{ac}(H)$ denote the projectiononto the absolutely continuous spectrum space of $H=(-\Delta)^m+V$, and assumethat $H$ 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|)^{-h}(1+|t|^{-\frac{n}{2m}})\left(1+|t|^{-\frac{1}{2 m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quadt\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 zero energy resonances of $H$. A similarresult for smoothing operators like$H^\frac{\alpha}{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. One of the mainnovelties lies in a new and unified approach to obtain the asymptoticexpansions of perturbed resolvents under various types of zero energyresonances.

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