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

In this paper, we study the fundamental solution of the higher order Schr\"odinger equation \begin{equation*} \mathrm{i}\partial_t u(x,t) = \big((-\Delta)^m + V(x)\big)u(x,t), \quad t \in \mathbb{R}, \ x \in \mathbb{R}^n, \end{equation*} for any odd dimension $n$ and integer $m \geq 1$ satisfying $n < 4m$, where $V$ is a real-valued bounded potential with suitable decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectral subspace of $H = (-\Delta)^m + V$, and assume $H$ has no positive embedded eigenvalues. Our main result says that the evolution operator $e^{-\mathrm{i}tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the pointwise estimate \begin{equation*} |K(t,x,y)| \leq C (1 + |t|)^{-h} (1 + |t|^{-\frac{n}{2m}}) \left(1 + |t|^{-\frac{1}{2m}}|x - y|\right)^{-\frac{n(m-1)}{2m-1}}, \quad t \neq 0, \ x,y \in \mathbb{R}^n, \end{equation*} where the exponent $h$ depends on $m$, $n$, and the zero energy resonance structure of $H$. We also prove analogous estimates for smoothing operators of the form $H^{\frac{\alpha}{2m}}e^{-\mathrm{i}tH}P_{ac}(H)$. The key innovation of this paper is a unified approach to deriving asymptotic expansions of the perturbed resolvents around zero, which comprehensively addresses all possible resonance types.