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

Abstract

This paper investigates the $L^p$-bounds of wave operators for higher-order Schrödinger operators $H=(-\hat{I}”{)}^m+V$ 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 $W_{\pm }(H;(-\hat{I}”{)}^m)$ 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 $W_{\pm }(H;(-\hat{I}”{)}^m)$ 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 $W_{\pm }(H;(-\hat{I}”{)}^m)$ 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 $W_{\pm }(H;(-\hat{I}”{)}^m)$ are bounded on $L^p({\mathbb{R}}^n)$ for all $1<p<\frac{2n}{n-1}$. Furthermore, it is shown that the wave operators $W_{\pm }(H;(-\hat{I}”{)}^m)$ 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 $\ddot{I}•$ to $H\ddot{I}•=0$ in ${\bigcap }_{s<-\frac{1}{2}}{L}_s^2({\mathbb{R}}^n)\setminus L^2({\mathbb{R}}^n)$(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.

57 Pages