Decay estimates for discrete bi-Schr\"{o}dinger operators on the lattice $\mathbb{Z}$

Abstract

It is known that the discrete Laplace operator $\Delta$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\left\|e^{it\Delta}\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $|t|^{-\frac{1}{2}}$ decay in the continuous case on $\mathbb{R}$. However in this paper, we have showed that the discrete bi-Laplacian $\Delta^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\frac{1}{4}}$ as its continuous counterpart. In view of these free decay estimates, this paper further investigates the discrete bi-Schr\"{o}dinger operators of the form $H=\Delta^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V(n)$ is a real valued potential of $\mathbb{Z}$. Under suitable decay conditions on $V$ and assuming that both 0 and 16 are regular spectral points of $H$, we establish the following sharp $\ell^1-\ell^{\infty}$ dispersive estimates: $$\left\|e^{-itH}P_{ac}(H)\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the following decay estimates for beam equation are also derived: $$\|{\rm cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\left\|\frac{{\rm sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$