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

TL;DR

This paper establishes sharp $L^p$-boundedness results for wave operators associated with higher-order Schrödinger operators in odd dimensions, considering various zero-resonance and eigenvalue scenarios, and identifies conditions for boundedness and unboundedness.

Contribution

It provides a comprehensive analysis of $L^p$-boundedness for wave operators with zero-resonance singularities in higher-order Schrödinger operators, extending previous results to all odd dimensions and various resonance types.

Findings

Wave operators are bounded on $L^p$ for $1<p<rac{2n}{n-1}$ when zero is an eigenvalue.

Boundedness on $L^p$ depends on the resonance type and narrows to specific $p$ ranges.

Wave operators are unbounded for $p$ outside the established bounded ranges under certain resonance conditions.

Abstract

This paper investigates the $L^p$-bounds of wave operators for higher-order Schr\"odinger operators $H = (-\Delta)^m + V$ 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 $W_\pm(H; (-\Delta)^m)$ 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 $W_\pm(H; (-\Delta)^m)$ 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 $W_\pm(H; (-\Delta)^m)$ 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 $W_\pm(H; (-\Delta)^m)$ are bounded on $L^p(\mathbb{R}^n)$ for all $1 < p < \frac{2n}{n - 1}$. Furthermore, it is shown that the wave operators $W_\pm(H; (-\Delta)^m)$ 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 $\phi$ to $H\phi = 0$ 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.