The $L^p$-boundedness of wave operators for higher order Schrödinger operator with zero singularities in low odd dimensions

TL;DR

This work establishes sharp $L^p$-boundedness results for wave operators $W_\pm(H;(-\Delta)^m)$ with $H=(-\Delta)^m+V$ in odd dimensions $1\le n\le 4m-1$, accounting for all zero-energy resonance types and eigenvalues. The authors develop a resonance-driven framework based on precise low-energy resolvent expansions of $(M^\pm(\lambda))^{-1}$ and a decomposition into low/high energy parts, exploiting projection tools $Q_j$ and oscillatory integral techniques. They derive explicit $p$-ranges depending on the resonance index $\mathbf{k}$ and a critical $k_c$, including full ranges for regular/low-kind resonances and sharp finite intervals for higher-kind resonances or eigenvalues, and prove corresponding unboundedness results to show sharpness. The approach bridges resolvent asymptotics, weighted $L^2$ decay (Kato–Jensen type estimates), and $L^p$-boundedness via careful kernel analysis, providing a robust foundation for dispersive estimates and potential nonlinear applications in higher-order Schrödinger settings. The results extend and unify known $m=1$ and $m=2$ cases and illuminate how zero-energy structures shape dispersive behavior in low odd dimensions for higher-order operators.

Abstract

This paper investigates the $L^p$-bounds of wave operators for higher-order Schrödinger operators $H = (-Δ)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the sharp $L^p$-boundedness of the wave operators $W_\pm(H; (-Δ)^m)$ in the presence of all types of zero-resonance singularities, for all odd dimensions $1 \le n \le 4m - 1$. Specifically, for odd $n$ with $1 \le n \le 4m - 1$, there exist $m_n$ types of zero resonances for $H$, along with a critical type $k_c$ (both depending on $n$ and $m$). If zero is a regular point of $H$ or a $\mathbf{k}$-th kind resonance with $1 \le \mathbf{k} \le k_c$, the wave operators $W_\pm(H; (-Δ)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \infty$. If zero is a $\mathbf{k}$-th kind resonance with $k_c < \mathbf{k} \le m_n$, we show that the range of $p$-boundedness for $W_\pm(H; (-Δ)^m)$ narrows to $1 < p < p_{\mathbf{k}}$, where $$p_{\mathbf{k}} = \frac{n}{n - 2m + \mathbf{k} + k_c - 1}.$$ Additionally, if zero is an eigenvalue of $H$ (i.e., $\mathbf{k} = m_n + 1$), then $W_\pm(H; (-Δ)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \frac{2n}{n - 1}$. Furthermore, it is shown that the wave operators $W_\pm(H; (-Δ)^m)$ are unbounded on $L^p(\mathbb{R}^n)$ for all $p_{\mathbf{k}} < p \le \infty$ if $k_c < \mathbf{k} \le m_n$, and for all $\frac{2n}{n - 1} < p \le \infty$ if zero is an eigenvalue of $H$ with a non-zero solution $φ$ to $Hφ= 0$ in $\bigcap_{s < -\frac{1}{2}} L^{2}_{s}(\mathbb{R}^n) \setminus L^2(\mathbb{R}^n)$(referred to as a $p$-wave resonance). The key idea of the proof is to reduce the $L^p$-unboundedness to establishing the optimality of time-decay estimates for $e^{itH}P_{ac}(H)$ in weighted $L^2$ spaces.

Theorems & Definitions (73)

• Definition 1.1
• Remark 1.2
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Corollary 1.10
• Remark 1.11