Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions I: low dimensional case

Abstract: 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 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|)^{-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 zero energy resonances of $H$. A similar result for smoothing operators like $H^\frac{\alpha}{2m}e^{-\mathrm{i}tH}P_{ac}(H)$ is also given. One of the main novelties lies in a new and unified approach to obtain the asymptotic expansions of perturbed resolvents under various types of zero energy resonances.