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

TL;DR

This work analyzes decay properties for discrete bi-Schrödinger operators on the lattice $\mathbb{Z}$, showing that the free fourth-order operator $e^{-it\Delta^2}$ exhibits the sharp dispersive decay $|t|^{-1/4}$ and that, under a rapidly decaying potential $V$, the perturbed evolution $e^{-itH}P_{ac}(H)$ preserves the same rate provided threshold regularity at $0$ and $16$ holds. The authors develop a framework combining the limiting absorption principle (via Mourre theory), detailed asymptotic expansions of the perturbed resolvent $R_V^{\pm}(\mu^4)$ near the thresholds, and Van der Corput-type bounds for oscillatory integrals to control all kernel components. Central to the approach are expansions of $(M^{\pm}(\mu))^{-1}$, where $M^{\pm}(\mu)=U+vR_0^{\pm}(\mu^4)v$, and the careful treatment of degeneracy at $\mu=0$ and the non-degenerate endpoint at $\mu=2$, alongside a spectral decomposition into absolutely continuous and discrete parts. The results yield sharp $\ell^1$-$\ell^{\infty}$ dispersive estimates for the discrete bi-Schrödinger operator and corresponding beam-equation bounds, with potential implications for lattice quantum dynamics and discrete higher-order dispersive models.

Abstract

It is known that the discrete Laplace operator $Δ$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\left\|e^{itΔ}\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ 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 $Δ^2$ 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-Schrödinger operators of the form $H=Δ^2+V$ 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: $$\|{\rm cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\left\|\frac{{\rm sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\right\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$

Figures (1)

• Figure 1: Figure 1: The map $\theta(\omega)$ from $\mathbb{C}\setminus[0,4]$ to $\mathcal{D}$.

Theorems & Definitions (67)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof