Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in low odd dimensions

Abstract

In this paper, we study the fundamental solution of the higher order Schrödinger equation \begin{equation*} \mathrm{i}\partial_t u(x,t) = \big((-Δ)^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 $m\ge 1$ 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.

Upon many reviews' valuable insights and comments, we decided to completely rewrite this paper and give a much clearer exposition of our approach, especially in regard to the study of resonances at the zero energy