The $\ell^p$-boundedness of wave operators for the fourth order Schrödinger operators on the lattice $\mathbb{Z}$
Sisi Huang, Xiaohua Yao
TL;DR
This work establishes sharp $\ell^p$-boundedness results for the wave operators $W_\pm(H,\Delta^2)$ with $H=\Delta^2+V$ on the lattice $\mathbb{Z}$, revealing that $W_\pm\in\mathbb{B}(\ell^p)$ for all $1<p<\infty$ under rapid decay of $V$ and absence of positive eigenvalues in $(0,16)$. The analysis hinges on precise limiting absorption principles and detailed asymptotics of the resolvent near the two lattice thresholds $0$ and $16$, including all zero-energy resonance types, combined with discrete singular integral techniques. Endpoint unboundedness is shown in resonant endpoint scenarios, establishing sharpness of the $\ell^p$-bounds. As a key application, the paper derives sharp $\ell^p-\ell^{p'}$ decay rates for solutions to the discrete beam equation, offering a lattice-analog of continuous results and confirming the robustness of the wave-operator approach in discrete higher-order dispersive systems.
Abstract
This paper investigates the $\ell^p$ boundedness of wave operators $W_\pm(H,Δ^2)$ associated with discrete fourth-order Schrödinger operators $H = Δ^2 + V$ on the lattice $\mathbb{Z}$, where $$(Δφ)(n)=φ(n+1)+φ(n-1)-2φ(n),\quad n\in\mathbb{Z},$$ and $V(n)$ is a real-valued potential on $\mathbb{Z}$. Under suitable decay assumptions on $V$ (depending on the types of zero resonance of $H$), we show that the wave operators $W_{\pm}(H, Δ^2)$ are bounded on $\ell^p(\mathbb{Z})$ for all $1 < p < \infty$: $$ \|W_{\pm}(H, Δ^2) f\|_{\ell^p(\mathbb{Z})} \lesssim \|f\|_{\ell^p(\mathbb{Z})}. $$ In particular, if both thresholds $0$ and $16$ are regular points of $H$, we prove that $W_{\pm}(H, Δ^2)$ are neither bounded on the endpoint space $\ell^1(\mathbb{Z})$ nor on $\ell^\infty(\mathbb{Z})$. We remark that the proof of these bounds relies fundamentally on the asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, and on the theory of {\it discrete singular integrals} on the lattice. As applications, we derive the following sharp $\ell^p-\ell^{p'}$ decay estimates for solutions to the discrete beam equation with a parameter $a\in \mathbb{R}$ on the lattice $\mathbb{Z}$: $$ \|{\rm cos}(t\sqrt {H+a^2})P_{ac}(H)\|_{\ell^p\rightarrow\ell^{p'}}+\left\|\frac{{\rm sin}(t\sqrt {H+a^2})}{t\sqrt {H+a^2}}P_{ac}(H)\right\|_{\ell^p\rightarrow\ell^{p'}}\lesssim|t|^{-\frac{1}{3}(\frac{1}{p}-\frac{1}{p'})},\quad t\neq0, $$ where $1<p\le 2$, ${p'}$ is the conjugated index of $p$ and $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$.