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In this paper, for any odd $n$ and any integer $m>1$, we study thefundamental solution of the higher order Schrd\"{o}dinger equation $$\mathrm{i}\partial_tu(x, t)=((-\Delta)^m+V(x))u(x, t), \quad t\in\mathbb{R},\,\,x\in \mathbb{R}^n, $$ where $V$ is a real-valued potential withcertain decay, smoothness, and spectral properties. Let $P_{ac}(H)$ denote theprojection 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 $$  |K(t, x,y)|\le C (1+|t|)^{-h}(1+|t|^{-\frac{n}{2 m}})\left(1+|t|^{-\frac{1}{2m}}|x-y|\right)^{-\frac{n(m-1)}{2 m-1}},\quad t\neq0,\,x,y\in\mathbb{R}^n, $$where the constants $C, h>0$, and $h$ can be specified by $m, n$ and thespectral property of $H$.

Pointwise estimates for the fundamental solution of higher order Schr\"{o}dinger equations in odd dimensions