Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$

TL;DR

This paper demonstrates that the discrete bi-Laplacian on the integer lattice exhibits the same sharp decay rate as its continuous counterpart, and extends these decay estimates to perturbed operators with potentials, including resonance cases.

Contribution

It establishes sharp decay estimates for the discrete bi-Laplacian and bi-Schrödinger operators on $ extbf{Z}$, including detailed analysis of resonances and resolvent expansions.

Findings

Discrete bi-Laplacian exhibits $|t|^{-1/4}$ decay similar to continuous case.

Decay estimates for operators with potentials depend on resonance types.

Resonance characterizations in weighted spaces are fully classified.

Abstract

It is known that the discrete Laplace operator $\Delta$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\big\|e^{it\Delta}\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $ O(|t|^{-\frac{1}{2}})$ decay in the continuous case on $\mathbb{R}$. However, this paper shows 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 the free decay estimate, we further investigate the discrete bi-Schr\"{o}dinger operators of the form $H=\Delta^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V$ is a class of real-valued decaying potentials on $\mathbb{Z}$. First, we establish the limiting absorption principle for $H$, and then derive the full asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, including resonance cases. In particular, we provide a complete characterizations of the different resonance types in $\ell^2$-weighted spaces. Based on these results above, we establish the following sharp $\ell^1-\ell^{\infty}$ decay estimates for all different resonances types of $H$ under suitable decay conditions on $V$: $$\big\|e^{-itH}P_{ac}(H)\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the decay estimates for the evolution flow of discrete beam equation are also derived: $$\|{\cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\Big\|\frac{{\sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\Big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$