The $\ell^p$-boundedness of wave operators for the fourth order Schrödinger operators on the lattice $\mathbb{Z}$

Abstract: This paper investigates the $\ell<sup>p$ boundedness of wave operators $W_\pm(H,Δ<sup>2)$ associated with discrete fourth-order Schrödinger operators $H = Δ² + V$ 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 $W_{\pm}(H, Δ<sup>2)$ are bounded on $\ell<sup>p(\mathbb{Z})$ for all $1 &lt; p &lt; \infty$: $$|W_{\pm}(H, Δ²⁾ f|<em>{\ell^{p(\mathbb{Z})}} \lesssim |f|</em>{\ell^{p(\mathbb{Z})}.}$$ In particular, if both thresholds $0$ and $16$ are regular points of $H$, we prove that $W_{\pm}(H, Δ<sup>2)$ are neither bounded on the endpoint space $\ell<sup>1(\mathbb{Z})$ nor on $\ell<sup>\infty(\mathbb{Z})$. 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 $\ell<sup>p-\ell<sup>{p&#39;}$ decay estimates for solutions to the discrete beam equation with a parameter $a\in \mathbb{R}$ on the lattice $\mathbb{Z}$: $$ |{\rm cos}(t\sqrt {H+a^{2})P_{ac}(H)|_{\ell^{p\rightarrow\ell^{{p&#39;}}+\left|\frac{{\rm}}} sin}(t\sqrt {H+a^2})}{t\sqrt {H+a^{2}}P_{ac}(H)\right|_{\ell^{p\rightarrow\ell^{{p&#39;}}\lesssim|t|^{{-\frac{1}{3}(\frac{1}{p}-\frac{1}{p&#39;})},\quad}}}} t\neq0, $$ where $1&lt;p\le 2$, ${p&#39;}$ is the conjugated index of $p$ and $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$.