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

Abstract

This paper investigates the ${\ell }^p$ boundedness of wave operators $W_{\pm }(H,{\Delta }^2)$ associated with discrete fourth-order Schrödinger operators $H={\Delta }^2+V$ on the lattice $\mathbb{Z}$, where $$(\Delta \varphi )(n)=\varphi (n+1)+\varphi (n-1)-2\varphi (n),n\in \mathbb{Z},$$ 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 $W_{\pm }(H,{\Delta }^2)$ are bounded on ${\ell }^p(\mathbb{Z})$ for all $1<p<\infty$: $$\Vert W_{\pm }(H,{\Delta }^2)f{\Vert }_{{\ell }^p(\mathbb{Z})}\lesssim \Vert f{\Vert }_{{\ell }^p(\mathbb{Z})}.$$ In particular, if both thresholds $0$ and $16$ are regular points of $H$, we prove that $W_{\pm }(H,{\Delta }^2)$ 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 ${\ell }^p-{\ell }^{p^{\prime }}$ decay estimates for solutions to the discrete beam equation with a parameter $a\in \mathbb{R}$ on the lattice $\mathbb{Z}$: $$\Vert \cos (t\sqrt{H+a^2})P_{ac}(H){\Vert }_{{\ell }^p\to {\ell }^{p^{\prime }}}+{\Vert \frac{\sin (t\sqrt{H+a^2})}{t\sqrt{H+a^2}}{P}_{ac}(H)\Vert }_{{\ell }^p\to {\ell }^{p^{\prime }}}\lesssim |t{|}^{-\frac{1}{3}(\frac{1}{p}-\frac{1}{p^{\prime }})},t\ne 0,$$ where $1<p\le 2$, $p^{\prime }$ is the conjugated index of $p$ and $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$.

60 pages